Open Access

Asymptotics of solutions of the heat equation in cones and dihedra under minimal assumptions on the boundary

Boundary Value Problems20122012:142

DOI: 10.1186/1687-2770-2012-142

Received: 25 June 2012

Accepted: 14 November 2012

Published: 3 December 2012

Abstract

In the first part of the paper, the authors obtain the asymptotics of Green’s function of the first boundary value problem for the heat equation in an m-dimensional cone K. The second part deals with the first boundary value problem for the heat equation in the domain K × R n m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq1_HTML.gif. Here the right-hand side f of the heat equation is assumed to be an element of a weighted L p , q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq2_HTML.gif-space. The authors describe the behavior of the solution near the ( n m ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq3_HTML.gif-dimensional edge of the domain.

Introduction

The paper is concerned with the first boundary value problem for the heat equation
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equ1_HTML.gif
(1)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equ2_HTML.gif
(2)
in the domain
D = { x = ( x , x ) : x K , x R n m } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equa_HTML.gif
where K = { x = ( x 1 , , x m ) : x / | x | Ω } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq4_HTML.gif is a cone in R m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq5_HTML.gif, 2 m n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq6_HTML.gif, Ω denotes a subdomain of the unit sphere, and M = { x = ( x , x ) : x = 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq7_HTML.gif is the ( n m ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq3_HTML.gif-dimensional edge of  D https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq8_HTML.gif. We are interested in the asymptotics of solutions in the class of the weighted Sobolev spaces W p , q ; β 2 , 1 ( D × R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq9_HTML.gif. Here the space W p , q ; β 2 l , l ( D × R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq10_HTML.gif is defined for an arbitrary integer l 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq11_HTML.gif and real p > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq12_HTML.gif, q > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq13_HTML.gif, β as the set of all function u ( x , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq14_HTML.gif on D × R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq15_HTML.gif with the finite norm
u W p , q ; β 2 l , l ( D × R ) = ( R ( D | α | + 2 k 2 l | x | p ( β 2 l + 2 k + | α | ) | t k x α u ( x , t ) | p d x ) q / p d t ) 1 / q . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equ3_HTML.gif
(3)

In the case l = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq16_HTML.gif, we write W p , q ; β 0 , 0 = L p , q ; β https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq17_HTML.gif. If, moreover, β = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq18_HTML.gif, then we write L p , q ; 0 = L p , q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq19_HTML.gif.

For the case of smooth boundary Ω (of class C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq20_HTML.gif), the asymptotics of solutions was obtained in our previous paper [1]. For the particular case p = q = 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq21_HTML.gif, m = n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq22_HTML.gif, we refer also to the paper [2] by Kozlov and Maz’ya, and for the case p = q 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq23_HTML.gif, m = n = 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq24_HTML.gif, to the paper [3] by de Coster and Nicaise. The goal of the present paper is to describe the asymptotics of solutions with a remainder in W p , q ; β 2 , 1 ( D × R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq9_HTML.gif under minimal smoothness assumptions on the boundary. Throughout the paper, we assume that Ω C 1 , 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq25_HTML.gif.

The paper consists of two parts. The first part (Section 1) deals with the asymptotics of the Green function for the heat equation in the cone K. We obtain the same decomposition
G ( x , y , t ) = λ j + < σ k = 0 m j t k c j ( y , t ) | x | λ j + + 2 k ϕ j ( ω x ) 4 k k ! ( σ j + k ) ( k ) + R σ ( x , y , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equb_HTML.gif

as in [4, 5] (for the definition of λ j + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq26_HTML.gif, ϕ j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq27_HTML.gif, m j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq28_HTML.gif, c j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq29_HTML.gif and σ ( k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq30_HTML.gif, see Section 1.1). However, the proof in [4, 5] does not work if Ω is only of the class C 1 , 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq31_HTML.gif. We give a new proof, which is completely different from that in [4, 5]. Our tools are estimates for solutions of the Dirichlet problem for the Laplace equation in a cone in weighted L p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq32_HTML.gif Sobolev spaces and asymptotic formulas for solutions of this problem which were obtained in the papers [6, 7] by Maz’ya and Plamenevskiĭ. Moreover, we use the estimates of the Green function in the recent paper [8] by Kozlov and Nazarov. In contrast to the case Ω C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq33_HTML.gif, the estimates for the second order x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq34_HTML.gif- and y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq35_HTML.gif-derivatives of the remainder R σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq36_HTML.gif contain an additional factor ( | x | 1 d ( x ) ) ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq37_HTML.gif with a negative exponent −ε. Here, d ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq38_HTML.gif is the distance from the boundary of ∂K.

In the second part of the paper (Section 2), we apply the results of Section 2 in order to obtain the asymptotics of solutions of the problem (1), (2) for f L p , q ; β ( D × R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq39_HTML.gif. We show that, under a certain condition on β, there exists a solution of the form
u ( x , t ) = λ j + < 2 β m / p k = 0 m j ( t Δ x ) k H j ( x , t ) 4 k k ! ( σ j + k ) ( k ) | x | λ j + + 2 k ϕ j ( ω x ) + w ( x , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equc_HTML.gif
with a remainder w W p , q ; β 2 , 1 ( D × R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq40_HTML.gif. Here, H j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq41_HTML.gif is an extension of the function
h j ( x , t ) = t D c j ( y , t τ ) Φ ( x , y , t τ ) f ( y , τ ) d y d τ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equd_HTML.gif

Φ denotes the fundamental solution of the heat equation in R n m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq42_HTML.gif. The proof of this result (Theorem 2.2) is essentially the same as in [1]. However, the proofs of some lemmas in [1] have to be modified under our weaker assumptions on Ω.

At the end of the paper, we show that the extensions of the functions h j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq43_HTML.gif can be defined as
H j ( x , t ) = ( E h j ) ( x , t ) = 0 R n m T ( τ ) R ( z ) h j ( x r z , t r 2 τ ) d z d τ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Eque_HTML.gif

where T and R are certain smooth functions on R + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq44_HTML.gif and R n m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq42_HTML.gif, respectively (see the beginning of Section 3 for their definition). This extends the result of [[1], Corollary 4.5] to the case p q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq45_HTML.gif.

1 The Green function of the heat equation in a cone

We start with the problem
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equ4_HTML.gif
(4)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equ5_HTML.gif
(5)
Let G ( x , y , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq46_HTML.gif be the Green function for the problem (4), (5). It is defined for every y K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq47_HTML.gif as the solution of the problem
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equf_HTML.gif

Furthermore, ( 1 ζ ) G ( , y , ) W 2 ; β 2 , 1 ( K × R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq48_HTML.gif if λ 1 < 2 β m / 2 < λ 1 + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq49_HTML.gif ( λ 1 ± https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq50_HTML.gif are defined below), and ζ is a function in C 0 ( K × R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq51_HTML.gif equal to one in a neighborhood of the point ( x , t ) = ( y , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq52_HTML.gif. Here W 2 , β 2 , 1 ( K × R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq53_HTML.gif is the space of all functions u = u ( x , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq54_HTML.gif on K × R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq55_HTML.gif such that | x | β 2 + 2 k + | α | t k x α u L 2 ( K × R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq56_HTML.gif for 2 k + | α | 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq57_HTML.gif. The goal of this section is to describe the behavior of the Green function for | x | < t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq58_HTML.gif.

1.1 Asymptotics of Green’s function

Let { Λ j } j = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq59_HTML.gif be the nondecreasing sequence of eigenvalues of the Beltrami operator −δ on Ω (with the Dirichlet boundary condition) counted with their multiplicities, and let { ϕ j } j = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq60_HTML.gif be an orthonormal (in L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq61_HTML.gif) sequence of eigenfunctions corresponding to the eigenvalues  Λ j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq62_HTML.gif. Furthermore, we define
λ j ± = 2 m 2 ± ( 1 m / 2 ) 2 + Λ j and σ j = λ j + 1 + m 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equg_HTML.gif

This means that λ j ± https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq63_HTML.gif are the solutions of the quadratic equation λ ( m 2 + λ ) = Λ j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq64_HTML.gif. Obviously, λ j + > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq65_HTML.gif and λ j < 2 m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq66_HTML.gif for j = 1 , 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq67_HTML.gif .

By [[8], Theorem 3],
| t k x α y γ G ( x , y , t ) | c t k ( m + | α | + | γ | ) / 2 ( | x | | x | + t ) λ 1 + | α | ε ( | y | | y | + t ) λ 1 + | γ | ε × ( d ( x ) | x | ) ε α ( d ( y ) | y | ) ε γ exp ( κ | x y | 2 t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equ6_HTML.gif
(6)

for | α | 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq68_HTML.gif, | γ | 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq69_HTML.gif. Here d ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq38_HTML.gif denotes the distance of the point x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq34_HTML.gif from the boundary ∂K. Furthermore, ε α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq70_HTML.gif is defined as zero for | α | 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq71_HTML.gif, while ε α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq70_HTML.gif is an arbitrarily small positive real number if | α | = 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq72_HTML.gif. Actually, the estimate (6) is proved in [8] only for k = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq73_HTML.gif, but for a more general class of operators, parabolic operators with discontinuous in time coefficients. If the coefficients in [8] do not depend on t, then one can use the same argument as in the proof of [[8], Theorem 3] when treating the derivatives along the edge of the domain D = K × R n m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq74_HTML.gif. This argument shows that the k th derivative with respect to t will bring only an additional factor t k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq75_HTML.gif to the right-hand side of (6).

The following lemma will be applied in the proof of Lemma 1.2. Here and in the sequel, we use the notation r = | x | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq76_HTML.gif and ω x = x / | x | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq77_HTML.gif.

Lemma 1.1 Let G ( x , y , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq46_HTML.gif be the Green function introduced above, and let G j ( r , ρ , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq78_HTML.gif denote the Green function of the initial-boundary value problem
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equh_HTML.gif
Then
Ω G ( x , y , t ) ϕ j ( ω x ) d ω x = | y | 1 m G j ( | x | , | y | , t ) ϕ j ( ω y ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equ7_HTML.gif
(7)
Proof The solution of the problem
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equ8_HTML.gif
(8)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equ9_HTML.gif
(9)
is given by the formula
u ( x , t ) = K G ( x , y , t ) ϕ ( y ) d y . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equi_HTML.gif
We define
U j ( r , t ) = Ω u ( x , t ) ϕ j ( ω x ) d ω x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equj_HTML.gif
Then it follows from (8) and (9) that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equk_HTML.gif
Furthermore,
U j ( r , 0 ) = Φ j ( r ) = def Ω ϕ ( x ) ϕ j ( ω x ) d ω x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equl_HTML.gif
Therefore,
U j ( r , t ) = 0 G j ( r , ρ , t ) Φ j ( ρ ) d ρ = 0 Ω G j ( r , ρ , t ) ϕ j ( ω y ) ϕ ( y ) d ω y d ρ = K G j ( r , | y | , t ) ϕ j ( ω y ) ϕ ( y ) | y | 1 m d y . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equm_HTML.gif
Comparing this with the formula
U j ( r , t ) = Ω u ( x , t ) ϕ j ( ω x ) d ω x = K Ω G ( x , y , t ) ϕ j ( ω x ) d ω x ϕ ( y ) d y , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equn_HTML.gif

we get (7). □

In the sequel, σ is an arbitrary real number satisfying the conditions
σ > λ 1 , σ λ j + for all  j . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equ10_HTML.gif
(10)
We define G σ ( x , y , t ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq79_HTML.gif for σ < λ 1 + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq80_HTML.gif, while
G σ ( x , y , t ) = λ j + < σ u j ( m j ) ( x , t ) c j ( y , t ) for  σ > λ 1 + , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equ11_HTML.gif
(11)
where
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equ12_HTML.gif
(12)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equ13_HTML.gif
(13)
and m j = [ σ λ j + 2 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq81_HTML.gif. Here, we used the notation
σ ( μ ) = σ ( σ 1 ) ( σ μ + 1 ) for  μ = 1 , 2 , and σ ( 0 ) = 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equo_HTML.gif
We define V p , β l ( K ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq82_HTML.gif as the weighted Sobolev space with the norm
u V p , β l ( K ) = ( K | α | l r p ( β l + | α | ) | x α u ( x ) | p d x ) 1 / p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equp_HTML.gif

for 1 < p < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq83_HTML.gif and integer l 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq11_HTML.gif.

Lemma 1.2 Suppose that σ is a real number such that σ > λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq84_HTML.gif and ( σ λ j + ) / 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq85_HTML.gif is not integer for λ j + σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq86_HTML.gif. Furthermore, let 1 < p < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq83_HTML.gif and β = 2 σ m / p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq87_HTML.gif. Then
G ( x , y , t ) = G σ ( x , y , t ) + R σ ( x , y , t ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equq_HTML.gif

where t k y γ R σ ( , y , t ) V p , β 2 ( K ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq88_HTML.gif for y K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq47_HTML.gif, t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq89_HTML.gif, | γ | 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq69_HTML.gif.

Proof We prove the lemma by induction in m 1 = [ ( σ λ 1 + ) / 2 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq90_HTML.gif.

First, let λ 1 < σ < λ 1 + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq91_HTML.gif. Then it follows from [[7], Corollary 4.1 and Theorem 4.2] (see also [[6], Theorem 3.2]) that t k y γ G ( , y , t ) V p , β 2 ( K ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq92_HTML.gif for all y K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq47_HTML.gif, t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq89_HTML.gif, | γ | 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq69_HTML.gif, where β = 2 σ m / p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq93_HTML.gif. Thus, the assertion of the lemma is true for σ < λ 1 + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq80_HTML.gif.

Suppose the assertion is proved for σ < λ 1 + + 2 l https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq94_HTML.gif. Now let λ 1 + + 2 l < σ < λ 1 + + 2 ( l + 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq95_HTML.gif. We set σ = σ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq96_HTML.gif if l > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq97_HTML.gif and σ = λ 1 + ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq98_HTML.gif if l = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq16_HTML.gif, where ε is a sufficiently small positive number. Then
[ σ λ j + 2 ] = [ σ λ j + 2 ] 1 = m j 1 for  λ j + < σ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equr_HTML.gif
By the induction hypothesis, we have
G ( x , y , t ) = G σ ( x , y , t ) + R σ ( x , y , t ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equs_HTML.gif
where G σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq99_HTML.gif is given by (11) (with σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq100_HTML.gif instead of σ and m j 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq101_HTML.gif instead of m j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq28_HTML.gif), t k y γ R σ ( , y , t ) V p , β 2 ( K ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq102_HTML.gif, β = 2 σ m / p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq103_HTML.gif. The coefficients c j ( y , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq104_HTML.gif in G σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq99_HTML.gif are given by (13) and satisfy the equation ( t Δ y ) c j ( y , t ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq105_HTML.gif. Therefore,
( t Δ y ) R σ ( x , y , t ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equt_HTML.gif
for x , y K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq106_HTML.gif, t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq89_HTML.gif. Obviously, G σ ( a x , a y , a 2 t ) = a m G σ ( x , y , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq107_HTML.gif for a > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq108_HTML.gif. Using the same equality for the Green function G ( x , y , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq46_HTML.gif, we obtain
R σ ( a x , a y , a 2 t ) = a m R σ ( x , y , t ) for  a > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equu_HTML.gif
Furthermore,
Δ x R σ ( x , y , t ) = Δ x G ( x , y , t ) Δ x G σ ( x , y , t ) = ( t Δ x ) G σ ( x , y , t ) + t R σ ( x , y , t ) = ( t Δ x ) λ j + < σ k = 0 m j 1 t k c j ( y , t ) 4 k k ! ( σ j + k ) ( k ) r λ j + + 2 k ϕ j ( ω x ) + t R σ ( x , y , t ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equv_HTML.gif
Using the formula
Δ x r λ j + + 2 k ϕ j ( ω ) = 4 k ( σ j + k ) r λ j + + 2 k 2 ϕ j ( ω x ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equw_HTML.gif
we get
Δ x R σ ( x , y , t ) = λ j + < σ t m j c j ( y , t ) r λ j + + 2 m j 2 ϕ j ( ω x ) 4 m j 1 ( m j 1 ) ! ( σ j + m j 1 ) ( m j 1 ) + t R σ ( x , y , t ) = Δ x Σ + t R σ ( x , y , t ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equ14_HTML.gif
(14)
where
Σ = λ j + < σ t m j c j ( y , t ) 4 m j m j ! ( σ j + m j ) ( m j ) r λ j + + 2 m j ϕ j ( ω x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equx_HTML.gif
( Σ = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq109_HTML.gif for l = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq16_HTML.gif). Let χ be a smooth function with compact support on [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq110_HTML.gif such that χ ( r ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq111_HTML.gif for r < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq112_HTML.gif. Using the notation r = | x | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq76_HTML.gif, the function χ can be also considered as a function in K. Since σ < λ j + + 2 m j < σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq113_HTML.gif for λ j + < σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq114_HTML.gif, we have χ t k y γ Σ ( , y , t ) V p , β 2 ( K ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq115_HTML.gif and ( 1 χ ) t k y γ Σ ( , y , t ) V p , β 2 ( K ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq116_HTML.gif for all y K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq47_HTML.gif, t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq89_HTML.gif. Consequently, t k y γ ( R σ ( , y , t ) χ Σ ( , y , t ) ) V p , β 2 ( K ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq117_HTML.gif and
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equy_HTML.gif
Applying [[7], Theorem 4.2], we obtain
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equ15_HTML.gif
(15)
where v k , γ ( , y , t ) V p , β 2 ( K ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq118_HTML.gif. The coefficients c μ , k , γ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq119_HTML.gif are given by the formula
c μ , k , γ ( y , t ) = K t k y γ ( t R σ ( x , y , t ) + Δ x ( 1 χ ) Σ ( x , y , t ) ) v μ ( x ) d x , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equ16_HTML.gif
(16)
where v μ ( x ) = 1 2 σ μ r λ μ ϕ μ ( ω x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq120_HTML.gif. The integral in (16) is well defined, since
t k y γ ( t R σ ( , y , t ) + Δ x ( 1 χ ) Σ ( , y , t ) ) V p , β 0 ( K ) V p , β 0 ( K ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equz_HTML.gif
and v μ V p , β 0 ( K ) + V p , β 0 ( K ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq121_HTML.gif, p = p / ( p 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq122_HTML.gif, for σ < λ μ + < σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq123_HTML.gif. The remainder v k , γ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq124_HTML.gif and the coefficients c μ , k , γ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq119_HTML.gif in (15) satisfy the estimate
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equ17_HTML.gif
(17)
Obviously, c μ , k , γ ( y , t ) = t k y γ c μ ( y , t ) = t k y γ c μ , 0 , 0 ( y , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq125_HTML.gif. This means that
R σ ( x , y , t ) χ ( r ) Σ ( x , y , t ) = σ < λ μ + < σ c μ ( y , t ) r λ μ + ϕ μ ( ω x ) + v ( x , y , t ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equaa_HTML.gif
where t k y γ v ( , y , t ) = v k , γ ( , y , t ) V p , β 2 ( K ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq126_HTML.gif. Consequently,
R σ ( x , y , t ) = Σ ( x , y , t ) + R σ ( x , y , t ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equ18_HTML.gif
(18)
where
Σ ( x , y , t ) = Σ ( x , y , t ) + σ < λ μ + < σ c μ ( y , t ) r λ μ + ϕ μ ( ω x ) = λ j + < σ t m j c j ( y , t ) r λ j + + 2 m j ϕ j ( ω x ) 4 m j m j ! ( σ j + m j ) ( m j ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equab_HTML.gif
and R σ ( x , y , t ) = v ( x , y , t ) + ( χ 1 ) Σ ( x , y , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq127_HTML.gif. Obviously, t k y γ R σ ( , y , t ) V p , β 2 ( K ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq128_HTML.gif for | γ | 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq69_HTML.gif. Using (18) and the equality
G σ ( x , y , t ) + Σ ( x , y , t ) = G σ ( x , y , t ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equac_HTML.gif
we conclude that
G ( x , y , t ) = G σ ( x , y , t ) + R σ ( x , y , t ) = G σ ( x , y , t ) + R σ ( x , y , t ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equad_HTML.gif
It remains to show that the coefficients
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equ19_HTML.gif
(19)
in (15) have the form (13) for σ < λ μ + < σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq129_HTML.gif. First, note that
( t Δ y ) c μ ( y , t ) = 0 for  y K , t > 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equae_HTML.gif

since ( t Δ y ) R σ ( x , y , t ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq130_HTML.gif and ( t Δ y ) Σ ( x , y , t ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq131_HTML.gif.

Obviously, the functions t G σ ( x , y , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq132_HTML.gif and
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equaf_HTML.gif
contain only functions ϕ j ( ω x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq133_HTML.gif with λ j + < σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq134_HTML.gif. Thus, the orthogonality of the functions ϕ j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq27_HTML.gif implies
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equ20_HTML.gif
(20)
for λ μ + > σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq135_HTML.gif. Applying Lemma 1.1, we conclude that c μ ( y , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq136_HTML.gif has the form
c μ ( y , t ) = ρ 1 m ϕ μ ( ω y ) f μ ( ρ , t ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equ21_HTML.gif
(21)
where ρ = | y | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq137_HTML.gif. Since R σ ( a x , a y , a 2 t ) = a m R σ ( x , y , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq138_HTML.gif and Σ ( a x , a y , a 2 t ) = a m Σ ( x , y , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq139_HTML.gif for all a > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq108_HTML.gif, it follows from (18) that
σ < λ μ + < σ ( a λ μ + c μ ( a y , a 2 t ) a m c μ ( y , t ) ) r λ μ + ϕ μ ( ω x ) = a m R σ ( x , y , t ) R σ ( a x , a y , a 2 t ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equag_HTML.gif
The function on the right-hand side belongs to V p , β 2 ( K ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq140_HTML.gif for all y K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq47_HTML.gif, t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq89_HTML.gif, a > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq108_HTML.gif, while the left-hand side belongs only to V p , β 2 ( K ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq140_HTML.gif if
c μ ( a y , a 2 t ) = a m λ μ + c μ ( y , t ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equah_HTML.gif
Combining the last equality with (21), we get the representation
c μ ( y , t ) = ρ m λ μ + ϕ μ ( ω y ) h μ ( ρ 2 4 t ) = ρ λ μ 2 ϕ μ ( ω y ) h μ ( ρ 2 4 t ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equai_HTML.gif
Inserting this into the equation ( t Δ y ) c μ ( y , t ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq141_HTML.gif, we obtain
r 2 h μ ( r ) + ( r σ μ 1 ) r h μ ( r ) + ( σ μ + 1 ) h μ ( r ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equaj_HTML.gif
The substitution h μ ( r ) = e r r σ μ + 1 u ( r ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq142_HTML.gif leads to the differential equation
r 2 u ( r ) + ( σ μ + 1 r ) r u ( r ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equak_HTML.gif
which has the solution
u ( r ) = d 1 + d 2 r 1 s σ μ 1 e s d s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equal_HTML.gif
with arbitrary constants d 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq143_HTML.gif and d 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq144_HTML.gif. Consequently,
c μ ( y , t ) = ρ λ μ 2 ϕ μ ( ω y ) ( ρ 2 4 t ) σ μ + 1 exp ( ρ 2 4 t ) ( d 1 + d 2 ρ 2 / ( 4 t ) 1 s σ μ 1 e s d s ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equ22_HTML.gif
(22)
Using (6) and (17), one gets the estimate
| t k c μ ( y , t ) | C k ( t ) ρ λ 1 + ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equam_HTML.gif
with certain functions C k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq145_HTML.gif for ρ = | y | < t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq146_HTML.gif. Thus, the constant d 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq144_HTML.gif in (22) must be zero. Integrating (19), we get
0 c μ ( y , t ) d t = v μ ( y ) = 1 2 σ μ ρ λ μ ϕ μ ( ω y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equan_HTML.gif
by means of (20). Hence,
d 1 ρ λ μ 2 ϕ μ ( ω y ) 0 ( ρ 2 4 t ) σ μ + 1 exp ( ρ 2 4 t ) d t = 1 2 σ μ ρ λ μ ϕ μ ( ω y ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equao_HTML.gif
The integral on the left-hand side is equal to 1 4 ρ 2 Γ ( σ μ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq147_HTML.gif. Thus, we get u ( r ) = d 1 = 2 / Γ ( σ μ + 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq148_HTML.gif and
h μ ( r ) = 2 Γ ( σ μ + 1 ) r σ μ + 1 e r . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equap_HTML.gif

This means that the formula (13) is valid for the coefficients c j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq29_HTML.gif if σ < λ j + < σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq149_HTML.gif. The proof of the lemma is complete. □

1.2 Point estimates for the remainder in the asymptotics of Green’s function

We are interested in point estimates for the remainder R σ ( x , y , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq150_HTML.gif in Lemma 1.2 in the case | x | < t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq58_HTML.gif. For this, we need the following lemma.

Lemma 1.3 Suppose that u L p , β ( K ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq151_HTML.gif and d u L p , β ( K ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq152_HTML.gif, where p > m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq153_HTML.gif. Then
sup x K d ( x ) m / p r ( x ) β | u ( x ) | c ( K r p β ( | d ( x ) u ( x ) | p + | u ( x ) | p ) d x ) 1 / p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equaq_HTML.gif

with a constant c independent of u.

Proof Let x 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq154_HTML.gif be a point int K, and let B 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq155_HTML.gif be a ball centered at x 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq154_HTML.gif with radius d 0 / 2 = d ( x 0 ) / 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq156_HTML.gif. We introduce the new coordinates y = d 0 1 x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq157_HTML.gif and set v ( y ) = u ( d 0 y ) = u ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq158_HTML.gif. Obviously, the point y 0 = d 0 1 x 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq159_HTML.gif has the distance 1 from ∂K. Hence,
| v ( y 0 ) | p c | y y 0 | < 1 / 2 ( | y v ( y ) | p + | v ( y ) | p ) d y . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equar_HTML.gif
This implies
| u ( x 0 ) | p c d 0 m B 0 ( | d 0 x u ( x ) | p + | u ( x ) | p ) d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equas_HTML.gif
Since d 0 / 2 < d ( x ) < 3 d 0 / 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq160_HTML.gif and r ( x 0 ) / 2 < r ( x ) < 3 r ( x 0 ) / 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq161_HTML.gif for x B 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq162_HTML.gif, we obtain
d 0 m r ( x 0 ) p β | u ( x 0 ) | p c B 0 r p β ( | d ( x ) x u ( x ) | p + | u ( x ) | p ) d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equat_HTML.gif

The result follows. □

Using the last two lemmas, we can prove the following theorem.

Theorem 1.1 Suppose that σ is a real number satisfying (10). Then
G ( x , y , t ) = G σ ( x , y , t ) + R σ ( x , y , t ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equau_HTML.gif
where
| t k x α y γ R σ ( x , y , t ) | c t k ( m + | α | + | γ | ) / 2 ( | x | t ) σ | α | ( | y | | y | + t ) λ 1 + | γ | ε × ( d ( x ) | x | ) ε α ( d ( y ) | y | ) ε γ exp ( κ | y | 2 t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equ23_HTML.gif
(23)

for | x | < t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq58_HTML.gif, | α | 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq68_HTML.gif, | γ | 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq69_HTML.gif. Here ε α = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq163_HTML.gif for | α | 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq71_HTML.gif, while ε α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq70_HTML.gif is an arbitrarily small positive real number if | α | = 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq72_HTML.gif.

Proof Since G σ = G σ + ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq164_HTML.gif for small positive ε, we may assume, without loss of generality, that ( σ λ j + ) / 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq85_HTML.gif is not integer for λ j + < σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq165_HTML.gif. We prove the theorem by induction in m 1 = [ ( σ λ 1 + ) / 2 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq90_HTML.gif.

If λ 1 < σ < λ 1 + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq91_HTML.gif, then the assertion of the theorem follows from [[8], Theorem 3]. Suppose that λ 1 + + 2 l < σ < λ 1 + + 2 ( l + 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq95_HTML.gif, l 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq166_HTML.gif, and that the theorem is proved for σ < λ 1 + + 2 l https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq94_HTML.gif. We set σ = σ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq96_HTML.gif if l > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq97_HTML.gif. In the case l = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq16_HTML.gif, let σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq100_HTML.gif be an arbitrary real number satisfying the inequalities λ 1 < σ < λ 1 + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq167_HTML.gif and σ σ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq168_HTML.gif. By the induction hypothesis, we have
G ( x , y , t ) = G σ ( x , y , t ) + R σ ( x , y , t ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equav_HTML.gif
where G σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq99_HTML.gif is given by (11) (with σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq100_HTML.gif instead of σ and m j 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq101_HTML.gif instead of m j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq28_HTML.gif). Since G σ = G σ + δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq169_HTML.gif for sufficiently small δ, it follows from the induction hypothesis that
| t k x α y γ R σ ( x , y , t ) | c t k ( m + | α | + | γ | ) / 2 ( | x | t ) σ + δ | α | ( | y | | y | + t ) λ 1 + | γ | ε × ( d ( x ) | x | ) ε α ( d ( y ) | y | ) ε γ exp ( κ | y | 2 t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equ24_HTML.gif
(24)
for | x | < 2 t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq170_HTML.gif, | α | 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq68_HTML.gif, | γ | 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq69_HTML.gif. As was shown in the proof of Lemma 1.2, the remainder R σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq171_HTML.gif admits the decomposition
R σ ( x , y , t ) = Σ ( x , y , t ) + R σ ( x , y , t ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equaw_HTML.gif
where
Σ ( x , y , t ) = λ j + < σ r λ j + + 2 m j ϕ j ( ω x ) t m j c j ( y , t ) 4 m j m j ! ( σ j + m j ) ( m j ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equax_HTML.gif
and t k y γ R σ ( , y , t ) V p , β 2 ( K ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq172_HTML.gif for t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq89_HTML.gif, y K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq47_HTML.gif, | γ | 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq69_HTML.gif. Here β = 2 σ m / p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq93_HTML.gif. Furthermore (cf. (14)),
Δ x R σ ( x , y , t ) = Δ x ( R σ ( x , y , t ) Σ ( x , y , t ) ) = Δ x ( R σ ( x , y , t ) Σ ( x , y , t ) ) = t R σ ( x , y , t ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equay_HTML.gif
Let χ be a smooth cut-off function on the interval [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq110_HTML.gif, χ = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq173_HTML.gif in [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq174_HTML.gif and χ = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq175_HTML.gif on ( 2 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq176_HTML.gif. We define χ 1 ( x , t ) = χ ( t 1 / 2 | x | ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq177_HTML.gif for x K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq178_HTML.gif, t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq89_HTML.gif. Then
Δ x ( χ 1 ( x , t ) y γ t k R σ ( x , y , t ) ) = f ( x , y , t ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equaz_HTML.gif
where
f = χ 1 y γ t k + 1 R σ + 2 x χ 1 x y γ t k ( R σ Σ ) + ( Δ x χ 1 ) y γ t k ( R σ Σ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equba_HTML.gif
Thus, by [[7], Theorem 4.1], there exists a constant c such that
χ 1 ( , t ) y γ t k R σ ( , y , t ) V p , β 2 ( K ) c f ( , y , t ) V p , β 0 ( K ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equ25_HTML.gif
(25)
for all y K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq47_HTML.gif, t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq89_HTML.gif, | γ | 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq69_HTML.gif. We estimate the norm of f. Using (24), we get
χ 1 t k + 1 y γ R σ ( , y , t ) V p , β 0 ( K ) c t k 1 ( m + | γ | + σ + δ ) / 2 ( | y | | y | + t ) λ 1 + | γ | ε exp ( κ | y | 2 t ) × ( d ( y ) | y | ) ε γ ( | x | < 2 t | x | p ( β + σ + δ ) d x ) 1 / p . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equbb_HTML.gif
Here, p ( β + σ + δ ) > m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq179_HTML.gif. Thus,
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equbc_HTML.gif
Since x χ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq180_HTML.gif vanishes outside the region t < | x | < 2 t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq181_HTML.gif and | x α χ 1 ( x , t ) | c t | α | / 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq182_HTML.gif, the estimate (24) also yields
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equbd_HTML.gif
Finally, it follows from the inequality
| y γ t k c μ ( y , t ) | c t k ( m + | γ | + λ μ + ) / 2 ( | y | t ) λ μ + | γ | exp ( | y | 2 6 t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Eqube_HTML.gif
that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equbf_HTML.gif
Consequently, by (25),
χ 1 ( , t ) y γ t k R σ ( , y , t ) V p , β 2 ( K ) c t k ( m + | γ | + σ ) / 2 ( | y | | y | + t ) λ 1 + | γ | ε × ( d ( y ) | y | ) ε γ exp ( κ | y | 2 t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equ26_HTML.gif
(26)
with a positive constant κ. Applying the estimate
| α | 1 | x | β 2 + | α | + m / p | x α χ 1 ( x , t ) y γ t k R σ ( x , y , t ) | c χ 1 y γ t k R σ ( , y , t ) V p , β 2 ( K ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equbg_HTML.gif

for p > m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq153_HTML.gif (cf. [[9], Lemma 1.2.3]), we obtain (23) for | α | 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq71_HTML.gif.

It remains to prove the estimate (23) for | α | = 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq72_HTML.gif. Let ρ ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq183_HTML.gif be the “regularized distance” of the point x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq34_HTML.gif to the boundary ∂K, i.e., ρ is a smooth function in K satisfying the inequalities
c 1 d ( x ) ρ ( x ) c 2 d ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equbh_HTML.gif
with positive constants c 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq184_HTML.gif and c 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq185_HTML.gif (cf. [[10], Chapter VI, § 2.1]). Moreover, ρ satisfies the inequality
| x α ρ ( x ) | c r ( x ) 1 | α | . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equ27_HTML.gif
(27)
We consider the function
v ( x , y , t ) = χ 1 ( x , t ) ρ ( x ) x j y γ t k R σ ( x , y , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equbi_HTML.gif
for 1 j m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq186_HTML.gif. It follows from the equation Δ x R σ = t R σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq187_HTML.gif that
Δ x v = f 1 + f 2 + f 3 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equbj_HTML.gif
where f 1 = χ 1 ρ x j y γ t k + 1 R σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq188_HTML.gif, f 2 = ( Δ x ( χ 1 ρ ) ) x j y γ t k R σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq189_HTML.gif and f 3 = 2 x ( χ 1 ρ ) x x j y γ t k R σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq190_HTML.gif. Using (24) and (27), we obtain
f 1 ( , y , t ) V p , β 0 ( K ) c t k ( m + | γ | + σ ) / 2 ( | y | | y | + t ) λ 1 + | γ | ε ( d ( y ) | y | ) ε γ exp ( κ | y | 2 t ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equbk_HTML.gif
Let χ 2 ( x , t ) = χ ( | x | / ( 2 t ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq191_HTML.gif. The inequalities | Δ x ( χ 1 ρ ) | c r 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq192_HTML.gif and | x ( χ 1 ρ ) | c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq193_HTML.gif yield
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equbl_HTML.gif
(see (26)). Consequently by [[7], Theorem 4.1], the function v = χ 1 ρ x j y γ t k R σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq194_HTML.gif satisfies the estimate
v ( , y , t ) V p , β 2 ( K ) c f 1 + f 2 + f 3 V p , β 0 ( K ) c t k ( m + | γ | + σ ) / 2 ( | y | | y | + t ) λ 1 + | γ | ε ( d ( y ) | y | ) ε γ exp ( κ | y | 2 t ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equbm_HTML.gif
Applying Lemma 1.3 to the function u ( x , y , t ) = χ 1 ( x , t ) x α y γ t k R σ ( x , y , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq195_HTML.gif with an arbitrary multi-index α with length | α | = 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq72_HTML.gif, we get
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equbn_HTML.gif

for | α | = 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq72_HTML.gif, | γ | 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq69_HTML.gif, p > m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq153_HTML.gif. Since p can be chosen arbitrarily large, the estimate (23) holds in the case | α | = 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq72_HTML.gif. The proof is complete. □

2 Asymptotics of solutions of the problem in D https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq8_HTML.gif

Now we consider the problem (1), (2) in the domain D https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq8_HTML.gif. Throughout this section, it is assumed that f L p , q ; β ( D × R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq39_HTML.gif, where p and β satisfy the inequalities
2 β m / p > λ 1 = 2 m λ 1 + and 2 β m / p λ j + for  j = 1 , 2 , , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equ28_HTML.gif
(28)
and q is an arbitrary real number >1. Let G ( x , y , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq46_HTML.gif be the Green function of the problem (4), (5). Furthermore, let
Φ ( x , y , t ) = ( 4 π t ) ( m n ) / 2 exp ( | x y | 2 4 t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equbo_HTML.gif
be the fundamental solution of the heat equation in R n m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq42_HTML.gif. Then
G ( x , y , t ) = G ( x , y , t ) Φ ( x , y , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equbp_HTML.gif
is the Green function of the problem (1), (2). We consider the solution
u ( x , t ) = t D G ( x , y , t τ ) f ( y , τ ) d y d τ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equ29_HTML.gif
(29)

of the problem (1), (2).

We again denote by G σ ( x , y , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq196_HTML.gif the function (11) introduced in Section 1. In the sequel, σ is an arbitrary real number such that
σ > 2 β m / p , λ j + [ 2 β m / p , σ ] for all  j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equ30_HTML.gif
(30)
and
m j = [ σ λ j + 2 ] = [ 2 β λ j + m / p 2 ] for  λ j + < 2 β m / p . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equ31_HTML.gif
(31)
Then G σ ( x , y , t ) = G 2 β m / p ( x , y , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq197_HTML.gif. Let χ be an infinitely differentiable function on R + = ( 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq198_HTML.gif equal to one on the interval ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq199_HTML.gif and vanishing on ( 2 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq176_HTML.gif. We define
χ 1 ( x , y ) = χ ( | x | | y | ) , χ 2 ( x , t , τ ) = χ ( | x | t τ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equbq_HTML.gif
Obviously,
u = Σ + v , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equbr_HTML.gif
where
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equ32_HTML.gif
(32)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equ33_HTML.gif
(33)
We also consider the decomposition
u = Σ + w , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equbs_HTML.gif
where
Σ = λ j + < 2 β m / p u j ( m j ) ( x , t Δ x ) H j ( x , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equ34_HTML.gif
(34)
and
H j ( x , t ) = t D χ 1 ( x , y ) χ 2 ( x , t , τ ) c j ( y , t τ ) Φ ( x , y , t τ ) f ( y , τ ) d y d τ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equ35_HTML.gif
(35)
is an extension of the function
h j ( x , t ) = t D c j ( y , t τ ) Φ ( x , y , t τ ) f ( y , τ ) d y d τ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equ36_HTML.gif
(36)

with c j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq29_HTML.gif defined by (13). Our goal is to show that both remainders v and w are elements of the space W p , q ; β 2 , 1 ( D × R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq9_HTML.gif. We start with the case p = q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq200_HTML.gif.

2.1 Estimates in weighted L p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq32_HTML.gif Sobolev spaces

Let W p , q ; β 2 l , l ( D × R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq10_HTML.gif be the weighted Sobolev space with the norm (3). Furthermore, let
W p ; β 2 l , l ( D × R ) = W p , p ; β 2 l , l ( D × R ) , L p ; β ( D × R ) = W p ; β 0 , 0 ( D × R ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equbt_HTML.gif
In this subsection, we assume that f L p ; β ( D × R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq201_HTML.gif, where p and β satisfy (28). First, we prove that Σ Σ W p ; β 2 , 1 ( D × R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq202_HTML.gif. This was shown in [[1], Corollary 2.3] for the case Ω C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq33_HTML.gif. In the case Ω C 1 , 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq203_HTML.gif, we must keep in mind that the second-order derivatives of the eigenfunctions ϕ j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq27_HTML.gif must not be bounded. Then we have the estimate
| x α ϕ j ( ω x ) | c | x | | α | ( d ( x ) | x | ) ε α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equ37_HTML.gif
(37)

for | α | 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq68_HTML.gif, where ε α = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq163_HTML.gif for | α | 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq71_HTML.gif and ε α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq70_HTML.gif is an arbitrarily small positive real number if | α | 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq71_HTML.gif. However, this requires only a small modification of the proof in [1].

Lemma 2.1 Suppose that f L p , β ( D × R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq204_HTML.gif. Then x α t k ( Σ Σ ) L p ; β 2 + | α | + 2 k ( D × R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq205_HTML.gif and
x α t k ( Σ Σ ) L p ; β 2 + | α | + 2 k ( D × R ) c f L p , β ( D × R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equbu_HTML.gif

for | α | 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq68_HTML.gif and all k.

Proof A simple calculation (see the proof of [[1], Corollary 1]) yields
Σ Σ = λ j + < σ t D χ 1 ( x , y ) ( [ u j ( m j ) ( x , t ) , χ 2 ] c j ( y , t τ ) ) × Φ ( x , y , t τ ) f ( y , τ ) d y d τ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equbv_HTML.gif
where [ u j ( m j ) ( x , t ) , χ 2 ] = u j ( m j ) ( x , t ) χ 2 χ 2 u j ( m j ) ( x , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq206_HTML.gif denotes the commutator of u j ( m j ) ( x , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq207_HTML.gif and χ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq208_HTML.gif. Obviously, the inequalities
| x | 2 | y | and t τ | x | 2 t τ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equbw_HTML.gif
are satisfied on the support of the kernel
K j ( x , y , t , τ ) = χ 1 ( x , y ) ( [ u j ( m j ) ( x , t ) , χ 2 ] c j ( y , t τ ) ) Φ ( x , y , t τ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equ38_HTML.gif
(38)
Since, moreover, the eigenfunctions ϕ j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq27_HTML.gif satisfy the inequality (37) for | α | 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq68_HTML.gif, we obtain
| x α t k K j ( x , y , t , τ ) | c ( t τ ) n / 2 ( d ( x ) | x | ) ε | x | | α | 2 k σ | y | σ exp ( | y | 2 + | x y | 2 8 ( t τ ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equbx_HTML.gif
for | α | 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq68_HTML.gif. Using Hölder’s inequality, we obtain
| x α t k ( Σ Σ ) ( x , t ) | c ( d ( x ) | x | ) ε | x | | α | 2 k σ A 1 / p B 1 / p , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equby_HTML.gif
where
A = t | x | 2 t | x | 2 / 4 D ( t τ ) n / 2 | y | p β | f ( y , τ ) | p exp ( | y | 2 + | x y | 2 8 ( t τ ) ) d y d τ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equbz_HTML.gif
and
B = t | x | 2 t | x | 2 / 4 D | y | > | x | / 2 ( t τ ) n / 2 | y | p ( σ β ) exp ( | y | 2 + | x y | 2 8 ( t τ ) ) d y d τ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equca_HTML.gif
The substitution y = z t τ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq209_HTML.gif, y = x + z t τ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq210_HTML.gif yields
B c t | x | 2 t | x | 2 / 4 ( t τ ) p ( σ β ) / 2 d τ | z | > 1 / 2 | z | p ( σ β ) exp ( | z | 2 8 ) d z × R n m exp ( | z | 2 8 ) d z , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equcb_HTML.gif
i.e., B c | x | p ( σ β ) + 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq211_HTML.gif. Consequently,
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equcc_HTML.gif
where
D ( y , τ ) = τ τ + | y | 2 D t τ < | x | < 2 t τ | x | 2 ( d ( x ) | x | ) p ε ( t τ ) n / 2 × exp ( | y | 2 + | x y | 2 8 ( t τ ) ) d x d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equcd_HTML.gif
Substituting x = z t τ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq212_HTML.gif and x = y + z t τ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq213_HTML.gif, we obtain
D ( y , τ ) = τ τ + | y | 2 ( t τ ) 1 exp ( | y | 2 8 ( t τ ) ) d t K 1 < | z | < 2 | z | 2 ( d ( z ) | z | ) p ε d z . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equce_HTML.gif

This means that D ( y , τ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq214_HTML.gif is a constant. This proves the lemma. □

Next, we estimate the first-order x-derivatives of the remainder v. For this, we employ the following lemma (cf. [[11], Lemma A.1]).

Lemma 2.2 Let K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq215_HTML.gif be the integral operator
( K f ) ( x , t ) = t R n K ( x , y , t , τ ) f ( y , τ ) d y d τ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equ39_HTML.gif
(39)
with a kernel K ( x , y , t , τ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq216_HTML.gif satisfying the estimate
| K | c ( t τ ) ( n + 2 r ) / 2 ( | x | | x | + t τ ) a + r ( | y | | y | + t τ ) b | x | μ r | y | μ exp ( κ | x y | 2 t τ ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equcf_HTML.gif

where κ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq217_HTML.gif, 0 < r 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq218_HTML.gif, a + b > m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq219_HTML.gif, m p a < μ < m m p + b https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq220_HTML.gif. Then K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq215_HTML.gif is bounded on L p ( R n × R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq221_HTML.gif.

In the proof of the following assertion, we use another decomposition of the remainder v as in [[1], Lemma 2.4]. This allows us to apply directly the estimate in Theorem 1.1.

Lemma 2.3 Let p and β satisfy the condition (28). Furthermore, let v be the function (33), where f L p ; β ( D × R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq222_HTML.gif, 1 < p < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq83_HTML.gif. Then x α v L p ; β 2 + | α | ( D × R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq223_HTML.gif for | α | 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq71_HTML.gif and
| α | 1 x α v L p ; β 2 + | α | ( D × R ) c f L p ; β ( D × R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equcg_HTML.gif

with a constant c independent of f. The same is true for the function w.

Proof Obviously,
v = j = 1 3 t D Ψ j ( x , y , t , τ ) f ( y , τ ) d y d τ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equch_HTML.gif
where
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equci_HTML.gif
and
Ψ 3 ( x , y , t , τ ) = ( 1 χ 1 ( x , y ) ) χ 2 ( x , t , τ ) G σ ( x , y , t τ ) Φ ( x , y , t τ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equcj_HTML.gif
We show that the integral operators with the kernels
K j ( α ) ( x , y , t , τ ) = | x | β 2 + | α | | y | β x α Ψ j ( x , y , t , τ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equck_HTML.gif
are bounded in L p ( D × R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq224_HTML.gif for j = 1 , 2 , 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq225_HTML.gif and | α | 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq226_HTML.gif. Using Theorem 1.1, we get
| K 1 ( α ) ( x , y , t , τ ) | c | x | β 2 + | α | | y | β ( t τ ) ( n + | α | ) / 2 ( | x | t τ ) σ | α | ( | y | | y | + t τ ) λ 1 + ε × exp ( κ | x y | 2 t τ ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equcl_HTML.gif

where ε is an arbitrarily small positive number. Applying Lemma 2.2 with r = 2 | α | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq227_HTML.gif, μ = β https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq228_HTML.gif, a = σ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq229_HTML.gif, b = λ 1 + ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq230_HTML.gif, we conclude that the integral operator with the kernel K 1 ( α ) ( x , y , t , τ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq231_HTML.gif is bounded in L p ( D × R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq224_HTML.gif for | α | 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq71_HTML.gif.

Since | x | | x | + t τ 2 | x | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq232_HTML.gif on the support of K 2 ( α ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq233_HTML.gif, the estimate (6) implies
| K 2 ( α ) ( x , y , t , τ ) | c | x | β 2 + | α | | y | β ( t τ ) ( n + | α | ) / 2 ( | x | | x | + t τ ) a ( | y | | y | + t τ ) λ 1 + ε × exp ( κ | x y | 2 t τ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equcm_HTML.gif

with arbitrary real a. Thus, by Lemma 2.2, the integral operator with the kernel K 2 ( x , y , t , τ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq234_HTML.gif is bounded in L p ( D × R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq224_HTML.gif for | α | 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq71_HTML.gif.

We consider the kernel K 3 ( α ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq235_HTML.gif. Since G σ ( x , y , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq196_HTML.gif has the form
G σ ( x , y , t ) = λ j + < σ k = 0 m j c j , k | x | λ j + + 2 k | y | λ j + ϕ j ( ω x ) ϕ j ( ω y ) t k t λ j + m / 2 exp ( | y | 2 4 t ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equcn_HTML.gif
we get the representation
K 3 ( α ) ( x , y , t , τ ) = λ j + < σ k = 0 m j K j , k ( x , y , t , τ ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equco_HTML.gif
where
| K j , k ( x , y , t , τ ) | c | x | β 2 + | α | | y | β | x | λ j + + 2 k | α | | y | λ j + ( t τ ) k λ j + n / 2 exp ( κ | x y | 2 t τ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equcp_HTML.gif
Here we used the fact that | y | | x | 2 t τ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq236_HTML.gif on the support of the function ( 1 χ 1 ) χ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq237_HTML.gif. The inequalities | y | | x | 2 t τ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq236_HTML.gif and λ j + + 2 k σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq238_HTML.gif imply
| K j , k ( x , y , t , τ ) | c | x | β 2 + | α | | y | β ( t τ ) ( n + | α | ) / 2 ( | x | t τ ) σ | α | ( | y | t τ ) 2 λ 1 + σ × exp ( κ | x y | 2 t τ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equcq_HTML.gif
It is no restriction to assume that σ < 2 λ 1 + + m β m / p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq239_HTML.gif in addition to (30) and (31). Therefore, we can apply Lemma 2.2 with r = 2 | α | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq227_HTML.gif, a = σ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq229_HTML.gif and b = 2 λ 1 + σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq240_HTML.gif to the integral operator with the kernel K j , k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq241_HTML.gif. It follows that the integral operator with the kernel K 3 ( α ) ( x , y , t , τ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq242_HTML.gif is bounded in L p ( D × R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq224_HTML.gif for | α | 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq71_HTML.gif. Consequently, the integral operator with the kernel
K ( α ) ( x , y , t , τ ) = j = 1 3 K j ( α ) ( x , y , t , τ ) = | x | β 2 + | α | | y | β j = 1 3 x α Ψ j ( x , y , t , τ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equcr_HTML.gif

is bounded in L p ( D × R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq224_HTML.gif for | α | 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq71_HTML.gif. This proves the lemma. □

Furthermore, the assertions of [[1], Lemmas 2.5, 2.6, Theorem 2.7] are also valid if Ω is only of the class C 1 , 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq31_HTML.gif. The proof under this weaker assumption on Ω does not require any modifications of the method in [1]. We give here only the formulation of [[1], Theorem 2.7].

Theorem 2.1 Let f L p ; β ( D × R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq243_HTML.gif, where p and β satisfy the condition (28). Then there exists a solution of the problem (1), (2) which has the form
u = λ j + < 2 β m / p u j ( m j ) ( x , t Δ x ) H j ( x , t ) + w , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equcs_HTML.gif
where w W p ; β 2 , 1 ( D × R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq244_HTML.gif and u j ( k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq245_HTML.gif, m j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq28_HTML.gif, H j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq41_HTML.gif are given by (12), (31) and (35), respectively. The functions H j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq41_HTML.gif depend only on | x | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq246_HTML.gif, x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq247_HTML.gif and t and satisfy the estimates
t k x γ H j L p ; β + λ j + + 2 k + | γ | 2 ( D × R ) c k , γ f L p ; β ( D × R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equ40_HTML.gif
(40)
for 2 k + | γ | > 2 β λ j + m / p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq248_HTML.gif and
t k x α x γ H j L p ; β + λ j + + 2 k + | α | + | γ | 2 ( D × R ) c k , α , γ f L p ; β ( D × R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equ41_HTML.gif
(41)

for all k, α, γ, | α | 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq249_HTML.gif.

2.2 Weighted L p , q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq2_HTML.gif estimates for the remainder

We assume now that f L p , q ; β ( D × R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq250_HTML.gif and consider the decomposition
u = Σ + w https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equct_HTML.gif

of the solution (29), where Σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq251_HTML.gif is defined by (34). Our goal is to show that w W p , q ; β 2 , 1 ( D × R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq252_HTML.gif if p and β satisfy the condition (28). For the proof, we will use the next lemma which follows directly from [[12], Theorem 3.8].

Lemma 2.4 Suppose that K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq215_HTML.gif is a linear operator on L p ( R n × R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq253_HTML.gif satisfying the following conditions:
  1. (i)

    K h L p ( R n × R ) c 1 h L p ( R n × R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq254_HTML.gif for all h L p ( R n × R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq255_HTML.gif,

     
  2. (ii)

    | t t 0 | > 2 δ ( K h ) ( , t ) L p ( R n ) d t c 2 R h ( , t ) L p ( R n ) d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq256_HTML.gif for all δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq257_HTML.gif and for all functions h with support in the layer | t t 0 | < δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq258_HTML.gif such that R h ( x , t ) d t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq259_HTML.gif .

     
Then the inequality
K h L p , q ( R n × R ) c h L p , q ( R n × R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equcu_HTML.gif

holds for arbitrary q, 1 < q < p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq260_HTML.gif. Here the constant c depends only on c 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq184_HTML.gif, c 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq185_HTML.gif, p and q.

The condition (ii) of the last lemma can be verified in some cases by means of the following lemma (cf. [[8], Lemma 10]).

Lemma 2.5 Suppose that the kernel of the integral operator (39) satisfies the estimate
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equcv_HTML.gif
for t > t 0 + 2 δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq261_HTML.gif, | τ t 0 | δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq262_HTML.gif, where κ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq217_HTML.gif, 0 r 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq263_HTML.gif, a + b > m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq219_HTML.gif, m p a < μ < m m p + b https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq220_HTML.gif, 0 ε 1 < 1 / p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq264_HTML.gif, 0 ε 2 < 1 1 / p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq265_HTML.gif. Then
t 0 + 2 δ ( K h ) ( , t ) L p ( D ) d t c h L p , 1 ( D × R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equcw_HTML.gif

for all h L p , 1 ( D × R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq266_HTML.gif with support in the layer | t t 0 | δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq267_HTML.gif. Here, the constant c is independent of t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq268_HTML.gif and δ.

It is more easy to estimate the remainder v = u Σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq269_HTML.gif, where Σ is defined by (32). For this reason, we estimate the difference Σ Σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq270_HTML.gif first.

Lemma 2.6 Let Σ and Σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq251_HTML.gif be the functions (32) and (34), respectively. If f L p , q ; β ( D × R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq39_HTML.gif, then t k x α ( Σ Σ ) L p , q ; β 2 + 2 k + | α | ( D × R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq271_HTML.gif and
t k x α ( Σ Σ ) L p , q ; β 2 + 2 k + | α | ( D × R ) c k , α f L p ; β ( D × R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equcx_HTML.gif

for all k and α, | α | 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq68_HTML.gif. Here, the constants c k , α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq272_HTML.gif are independent of f. In particular, Σ Σ W p , q ; β 2 , 1 ( D × R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq273_HTML.gif.

Proof We have
Σ Σ = λ j + < σ t D K j ( x , y , t , τ ) f ( y , τ ) d y d τ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equcy_HTML.gif
where K j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq274_HTML.gif is given by (38). Let K j , k , α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq275_HTML.gif be the integral operator with the kernel
K j , k , α ( x , y , t , τ ) = | x | β 2 + 2 k + | α | | y | β x α t k K j ( x , y , t , τ ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_Equcz_HTML.gif
where | α | 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-142/MediaObjects/13661_2012_Article_247_IEq68_HTML.gif. As was shown in the proof of Lemma 2.1, this operator is bounded in L p ( D × R )