Carleman estimates and unique continuation property for abstract elliptic equations

Abstract

The unique continuation theorems for elliptic differential-operator equations with variable coefficients in vector-valued L _p -space are investigated. The operator-valued multiplier theorems, maximal regularity properties and the Carleman estimates for the equations are employed to obtain these results. In applications the unique continuation theorems for quasielliptic partial differential equations and finite or infinite systems of elliptic equations are studied.

AMS: 34G10; 35B45; 35B60.

1 Introduction

The aim of this article, is to present a unique continuation result for solutions of a differential inequalities of the form:

$$\Vert P(x,D)u(x)\Vert {}_E\le \Vert V(x)u(x)\Vert {}_E,$$

(1)

where

$$P\left(x;D\right)u=\sum _{i,j=1}^n{a}_{ij}\frac{{\partial }^{2}u}{\partial x_i\partial x_j}+Au+\sum _{k=1}^n{A}_k\frac{\partial u}{\partial x_k},$$

here a _ij are real numbers, A = A (x), A _k = A _k (x) and V (x) are the possible linear operators in a Banach space E.

Jerison and Kenig started the theory of L _p Carleman estimates for Laplace operator with potential and proved unique continuation results for elliptic constant coefficient operators in [1]. This result shows that the condition V ∈ L_n/2,locis in the best possible nature. The uniform Sobolev inequalities and unique continuation results for second-order elliptic equations with constant coefficients studied in [2]. This was latter generalized to elliptic variable coefficient operators by Sogge in [3]. There were further improvement by Wolff [4] for elliptic operators with less regular coefficients and by Koch and Tataru [5] who considered the problem with gradients terms. A comprehensive introductions and historical references to Carleman estimates and unique continuation properties may be found, e.g., in [5]. Moreover, boundary value problems for differential-operator equations (DOEs) have been studied extensively by many researchers (see [6–18] and the references therein).

In this article, the unique continuation theorems for elliptic equations with variable operator coefficients in E-valued L _p spaces are studied. We will prove that if $n\left(\frac{1}{p}-\frac{1}{p^{\prime }}\right)\le 2,$ $\frac{1}{\mu }=\frac{1}{p}-\frac{1}{p^{\prime }}$, $\frac{1}{p}+\frac{1}{p^{|}}=1$, V ∈ L _μ (Rⁿ; L(E)), p, μ∈ (1, ∞) and $u\in W_p^2\left(R^n;E\left(A\right),E\right)$ satisfies (1), then u is identically zero if it vanishes in a nonempty open subset, where $W_p^2\left(R^n;E\left(A\right),E\right)$ is an E-valued Sobolev-Lions type space. We prove the Carleman estimates to obtain unique continuation. Specifically, we shall see that it suffices to show that if $w\left(x\right)=x_1+\frac{x_1^2}{2}$, then

$$\begin{array}{c}\Vert e^{tw}u\Vert {}_{L_{p^{|}}(R^n;E)}\le C\Vert e^{tw}L(\epsilon x,D)u\Vert {}_{L_p(R^n;E)},\frac{1}{p}+\frac{1}{p^{|}}=1,\\ {{\displaystyle \sum _{\left|\alpha \right|\le 1}t}}^{(1+\frac{1}{n}-|\alpha |)}\Vert e^{tw}{D}^{\alpha }u\Vert {}_{L_p(R^n;E)}+\Vert e^{tw}Au\Vert {}_{L_p(R^n;E)}\le \\ C\Vert e^{tw}L(\epsilon x,D)u\Vert {}_{L_p(R^n;E)}.\end{array}$$

In the Hilbert space L₂ (Rⁿ; H), we derive the following Carleman estimate

$${\displaystyle \sum _{\left|\alpha \right|\le 2}{t}^{\frac{3}{2}}{}^{-\left|\alpha \right|}}\Vert e^{tw}{D}^{\alpha }u\Vert {}_{L_2(R^n;H)}+\Vert e^{tw}Au\Vert {}_{L_2(R^n;H)}\le C\Vert e^{tw}{L}_{0}u\Vert {}_{L_2(R^n;H)}.$$

Any of these inequalities would follow from showing that the adjoint operator L _t (x; D) = e^twL (x; D) e^-tw satisfies the following relevant local Sobolev inequalities

$$\begin{array}{l}{\Vert u\Vert }_{L_{p^{|}}(R^n;E)}\le C{\Vert L_{t}u\Vert }_{L_p(R^n;E)},\frac{1}{p}+\frac{1}{p^{|}}=1,\\ {{\displaystyle \sum _{\left|\alpha \right|\le 1}t}}^{\left(1+\frac{1}{n}-\left|\alpha \right|\right)}{\Vert D^{\alpha }u\Vert }_{L_p(R^n;E)}{\Vert Au\Vert }_{L_p(R^n;E)}\le C{\Vert L_{t}u\Vert }_{L_p(R^n;E)},\end{array}$$

uniformly to t, where L_0t = e^twL₀e^-tw. In application, putting concrete Banach spaces instead of E and concrete operators instead of A, we obtain different results concerning to Carleman estimates and unique continuation.

2 Notations, definitions, and background

Let R and C denote the sets of real and complex numbers, respectively. Let

$$S_{\phi }=\left\{\xi \in C,\left|\text{arg}\xi \right|\le \phi \right\}\cup \left\{0\right\},\phi \in \left[0,\pi \right).$$

Let E and E₁ be two Banach spaces, and L (E, E₁) denotes the spaces of all bounded linear operators from E to E₁. For E₁ = E we denote L (E, E₁) by L (E). A linear operator A is said to be a φ-positive in a Banach space E with bound M > 0 if D (A) is dense on E and

$$\Vert {(A+\xi I)}^{-1}\Vert {}_{L(E)}\le M{(1+|\xi |)}^{-1}$$

with λ ∈ S _φ , φ ∈ (0, π], I is identity operator in E. We will sometimes use A + ξ or A _ξ instead of A + ξI for a scalar ξ and (A + ξI)^-1 denotes the inverse of the operator A + ξI or the resolvent of operator A. It is known [19, §1.15.1] that there exist fractional powers A^θ of a positive operator A and

$$E\left(A^{\theta }\right)=\left\{u\in D\left(A^{\theta }\right),{∥u∥}_{E\left(A^{\theta }\right)}={∥A^{\theta }u∥}_E+∥u∥<\infty ,-\infty <\theta <\infty \right\}.$$

We denote by L _p (Ω; E) the space of all strongly measurable E-valued functions on Ω with the norm

$${∥u∥}_{L_p}={∥u∥}_{L_p\left(\Omega ;E\right)}={\left({\int }_{\Omega }{∥u\left(x\right)∥}_E^{p}dx\right)}^{1/p},1\le p<\infty .$$

By L_p,q(Ω) and $W_{p,q}^l\left(\Omega \right)$ let us denoted, respectively, the (p, q)-integrable function space and Sobolev space with mixed norms, where 1 ≤ p, q < ∞, see [20].

Let E₀ and E be two Banach spaces and E₀ is continuously and densely embedded E.

Let l be a positive integer.

We introduce an E-valued function space $W_p^l\left(\Omega ;E_0,E\right)$ (sometimes we called it Sobolev-Lions type space) that consist of all functions u ∈ L _p (Ω; E₀) such that the generalized derivatives $D_k^{l}u=\frac{{\partial }^{l}u}{\partial x_k^l}\in L_p\left(\Omega ;E\right)$ are endowed with the

$${\Vert u\Vert }_{W_p^l(\Omega ;E_0,E)}={\Vert u\Vert }_{L_p(\Omega ;E_0)}+{\displaystyle \sum _{k=1}^n\Vert D_k^{l}u\Vert {}_{L_p(\Omega ;E)}}<\infty ,1\le p<\infty .$$

The Banach space E is called an UMD-space if the Hilbert operator $\left(Hf\right)\left(x\right)=\underset{\epsilon \to 0}{\text{lim}}\underset{|x-y|>\epsilon }{\int }\frac{f\left(y\right)}{x-y}$dy is bounded in L _p (R, E), p ∈ (1, ∞) (see e.g., [21, 22]). UMD spaces include, e.g., L _p , l _p spaces and Lorentz spaces L _pq , p, q ∈ (1, ∞).

Let E₁ and E₂ be two Banach spaces. Let S (Rⁿ; E) denotes a Schwartz class, i.e., the space of all E-valued rapidly decreasing smooth functions on Rⁿ. Let F and F^-1denote Fourier and inverse Fourier transformations, respectively. A function Ψ ∈ C^m (Rⁿ; L (E₁, E₂)) is called a multiplier from L _p (Rⁿ; E₁) to L _q (Rⁿ; E₂) for p, q ∈ (1, ∞) if the map u → Ku = F^-1 Ψ (ξ) Fu, u ∈ S (Rⁿ; E₁) is well defined and extends to a bounded linear operator

$$K:L_p\left(R^n;E_1\right)\to L_q\left(R^n;E_2\right).$$

We denote the set of all multipliers from L _p (Rⁿ; E₁) to L _q (Rⁿ; E₂) by $M_p^q\left(E_1,E_2\right)$. For E₁ = E₂ = E and q = p we denote $M_p^q\left(E_1,E_2\right)$ by M _p (E). The L _p -multipliers of the Fourier transformation, and some related references, can be found in [19, § 2.2.1-§ 2.2.4]. On the other hand, Fourier multipliers in vector-valued function spaces, have been studied, e.g., in [23–28].

A set K ⊂ L (E₁, E₂) is called R-bounded [22, 23] if there is a constant C such that for all T₁, T₂, . . . , T _m ∈ K and u_1,u₂, . . . , u _m ∈ E₁, m ∈ N

$$\underset{0}{\overset{1}{\int }}{∥\sum _{j=1}^m{r}_j\left(y\right)T_j{u}_j∥}_{E_2}dy\le C\underset{0}{\overset{1}{\int }}{∥\sum _{j=1}^m{r}_j\left(y\right)u_j∥}_{E_1}dy,$$

where {r _j } is a sequence of independent symmetric {-1, 1}-valued random variables on [0,1]. The smallest C for which the above estimate holds is called a R-bound of the collection K and denoted by R (K).

Let

$$\begin{array}{c}{U}_n=\left\{\beta =\left({\beta }_1,{\beta }_2,\dots ,{\beta }_n\right),{\beta }_i\in \left\{0,1\right\},i=1,2,\dots ,n\right\},\\ {\xi }^{\beta }={\xi }_1^{{\beta }_1}{\xi }_2^{{\beta }_2}\dots {\xi }_n^{{\beta }_n},\left|{\xi }^{\beta }\right|=\left|{\xi }_1{|}^{{\beta }_1}\right|{\xi }_2{|}^{{\beta }_2}\dots |{\xi }_n{|}^{{\beta }_n}.\end{array}$$

For any r = (r₁, r₂, . . . , r _n ), r _i ∈ [0, ∞) the function (iξ)^r, ξ ∈ Rⁿ will be defined such that

$${\left(i\xi \right)}^r=\left\{\begin{array}{c}\hfill {\left(i{\xi }_1\right)}^{r_1}\dots {\left(i{\xi }_n\right)}^{r_n},{\xi }_1,{\xi }_2,\dots ,{\xi }_n\ne 0,\hfill \\ \hfill 0,{\xi }_1,{\xi }_2,\dots ,{\xi }_n=0,\hfill \end{array}\right.$$

where

$${\left(it\right)}^{\nu }=|t{|}^{\nu }\text{exp}\left(\frac{i\pi }{2}\mathsf{\text{sign }}t\right),t\in \left(-\infty ,\infty \right),\nu \in \left[0,\infty \right).$$

Definition 2.1. The Banach space E is said to be a space satisfying a multiplier condition with respect to p, q ∈ (1, ∞) (with respect to p if q = p) when for Ψ ∈ C⁽ⁿ⁾(Rⁿ; L (E₁, E₂)) if the set

$$\left\{{\xi }^{\left|\beta \right|+\frac{1}{p}-\frac{1}{q}}{D}^{\beta }\Psi \left(\xi \right):\xi \in R^n\backslash 0,\beta \in U_n\right\}$$

is R-bounded, then $\Psi \in M_p^q\left(E_1,E_2\right)$.

Definition 2.2. The φ-positive operator A is said to be a R-positive in a Banach space E if there exists φ ∈ [0, π) such that the set

$$L_A=\left\{\xi {\left(A+\xi I\right)}^{-1}:\xi \in S_{\phi }\right\}$$

is R-bounded.

Remark 2.1. By virtue of [29] or [30] UMD spaces satisfy the multiplier condition with respect to p ∈ (1, ∞).

Note that, in Hilbert spaces every norm bounded set is R-bounded. Therefore, in Hilbert spaces all positive operators are R-positive. If A is a generator of a contraction semigroup on L _q , 1 ≤ q ≤ ∞ [31], A has the bounded imaginary powers with $\Vert (-A^{it})\Vert {}_{L(E)}\le C{e}^{\nu \left|t\right|}$, $\nu <\frac{\pi }{2}$ or if A is a generator of a semigroup with Gaussian bound in E ∈ UMD then those operators are R-positive (see e.g., [24]).

It is well known (see e.g., [32]) that any Hilbert space satisfies the multiplier condition with respect to p ∈ (1, ∞). By virtue of [33] Mikhlin conditions are not sufficient for operator-valued multiplier theorem. There are however, Banach spaces which are not Hilbert spaces but satisfy the multiplier condition (see Remark 2.1).

Let $H_k=\{{\Psi }_h\in M_p^q\left(E_1,E_2\right)$, $h=(h_1,h_2,\dots ,h_n)\in K\}$ be a collection of multipliers in $M_p^q\left(E_1,E_2\right)$. We say that H _k is a uniform collection of multipliers if there exists a constant M > 0, independent on h ∈ K, such that

$${∥F^{-1}{\Psi }_{h}Fu∥}_{L_q\left(R^n;E_2\right)}\le M{∥u∥}_{L_p\left(R^n;E_1\right)}$$

for all h ∈ K and u ∈ S (Rⁿ; E₁).

We set

$$C_b\left(\Omega ;E\right)=\left\{u\in C\left(\Omega ;E\right),\underset{\left|x\right|\to \infty }{\text{lim}}u\left(x\right)\mathsf{\text{ exists}}\right\}.$$

In view of [17, Theorem A₀], we have

Theorem 2.0. Let E₁ and E₂ be two UMD spaces and let

$$\Psi \in C^{\left(n\right)}\left(R^n\backslash 0;L\left(E_1,E_2\right)\right)\mathsf{\text{for }}p,q\in \left(1,\infty \right).$$

If

$$R\left\{{\xi }^{\left|\beta \right|+\frac{1}{p}-\frac{1}{q}}{D}_{\xi }^{\beta }{\Psi }_h\left(\xi \right):\xi \in R^n\backslash 0,\beta \in U_n\right\}\le K_{\beta }<\infty$$

uniformly with respect to h ∈ K then Ψ _h (ξ) is a uniformly collection of multipliers from L _p (Rⁿ; E₁) to L _q (Rⁿ; E₂).

Let

$$\chi =\frac{\left|\alpha \right|+n\left(\frac{1}{p}-\frac{1}{q}\right)}{l},\alpha =\left({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n\right).$$

Embedding theorems in Sobolev-Lions type spaces were studied in [13–18, 32, 34]. In a similar way as [17, Theorem 3] we have

Theorem 2.1. Suppose the following conditions hold:

• (1) E is a Banach space satisfying the multiplier condition with respect to p, q ∈ (1, ∞) and A is a R-positive operator on E;

• (2) l is a positive and α _k are nonnegative integer numbers such that 0 ≤ μ ≤ 1 - ϰ, t and h are positive parameters.

Then the embedding

$$D^{\alpha }{W}_p^l\left(R^n;E\left(A\right),E\right)\subset L_q\left(R^n;E\left(A^{1-\chi -\mu }\right)\right)$$

is continuous and there exists a positive constant C _µ such that for

$$u\in W_p^l\left(R^n;E\left(A\right),E\right)$$

the uniform estimate holds

$${∥D^{\alpha }u∥}_{L_q\left(R^n;E\left(A^{1-\chi -\mu }\right)\right)}\le C_{\mu }\left[h^{\mu }{∥u∥}_{W_p^l\left(R^n;E\left(A\right),E\right)}+h^{-\left(1-\mu \right)}{∥u∥}_{L_p\left(R^n;E\right)}\right].$$

Moreover, for $u\in W_p^l\left(R^n;E\left(A\right),E\right)$ the following uniform estimate holds

$${∥A^{1-\chi -\mu }u∥}_{L_p\left(R^n;E\right)}\le C_{\mu }\left[h^{\mu }{∥u∥}_{W_p^l\left(R^n;E\left(A\right),E\right)}+h^{-\left(1-\mu \right)}{∥u∥}_{L_p\left(R^n;E\right)}\right].$$

3 Carleman estimates for DOE

Consider at first the equation with constant coefficients

$$L_{0}u=\sum _{k=1}^n{D}_k^{2}u+Au=f\left(x\right),$$

(2)

where $D_k=\frac{\partial }{i{\partial }_k}$ and A is the possible unbounded operator in a Banach space E.

Let $w\left(x\right)=x_1+\frac{x_1^2}{2}$ and t is a positive parameter.

Remark 3.1. It is clear to see that

$$\begin{array}{ll}\hfill e^{tw}{L}_0\left[e^{-tw}u\right]& =L_{0t}\left(x,D\right)u=e^{tw}\left(\sum _{k=1}^n{D}_k^2\left(e^{-tw}u\right)+e^{-tw}Au\right)\\ =\sum _{k=1}^n{D}_k^{2}u+Au+2t{w}_1\frac{\partial u}{\partial x_1}+\left[-t^2{w}_1^2+t\right]u,\end{array}$$

(3)

where $w_1=\frac{\partial w}{\partial x_1}$. Let L_0t(x, ξ) is the principal operator symbol of L_0t(x, D) on the domain B₀, i.e.,

$$L_{0t}\left(x,\xi \right)={\xi }_1^2-2i{\xi }_1{w}_{1}t+A+|{\xi }^{|}{|}^2-t^2{w}_1^2=G_t\left(x,\xi \right)B_t\left(x,\xi \right),$$

where

$$\begin{array}{c}{G}_t\left(x,\xi \right)={\xi }_1-i\left[{\left(A+|{\xi }^{|}{|}^2\right)}^{\frac{1}{2}}+t{w}_1\right],\\ B_t\left(x,\xi \right)={\xi }_1+i\left[{\left(A+|{\xi }^{|}{|}^2\right)}^{\frac{1}{2}}-t{w}_1\right],|{\xi }^{|}{|}^2=\sum _{k=2}^n{\xi }_k^2.\end{array}$$

Our main aim is to show the following result:

Remark 3.2. Since Q(ξ) ∈ S (φ) for all φ ∈ [0, π) due to positivity of A, the operator function A + |ξ^||², ξ ∈ Rⁿ is uniformly positive in E. So there are fractional powers of A+|ξ^||² and the operator function ${\left(A+|{\xi }^{|}{|}^2\right)}^{\frac{1}{2}}$ is positive in E (see e.g., [19, §1. 15.1]).

First, we will prove the following result.

Theorem 3.1. Suppose A is a positive operator in a Hilbert space H. Then the following uniform Sobolev type estimate holds for the solution of Equation (3)

$${\displaystyle \sum _{\left|\alpha \right|\le 2}{t}^{\frac{3}{2}}{}^{-\left|\alpha \right|}}\Vert e^{tw}{D}^{\alpha }u\Vert {}_{L_2(R^n;H)}+\Vert e^{tw}Au\Vert {}_{L_2(R^n;H)}\le C\Vert e^{tw}{L}_{0}u\Vert {}_{L_2(R^n;H)}.$$

(4)

By virtue of Remark 3.1 it suffices to prove the following uniform coercive estimate

$${\displaystyle \sum _{\left|\alpha \right|\le 2}{t}^{\frac{3}{2}}{}^{-\left|\alpha \right|}}\Vert D^{\alpha }u\Vert {}_{L_2(R^n;H)}+\Vert Au\Vert {}_{L_2(R^n;H)}\le C\Vert L_{0t}u\Vert {}_{L_2(R^n;H)}$$

(5)

for $u\in W_2^2\left(R^n;H\left(A\right),E\right)$.

To prove the Theorem 3.1, we shall show that L_0t(x, D) has a right parametrix T, with the following properties.

Lemma 3.1. For t > 0 there are functions K = K _t and R = R _t so that

$$L_{0t}\left(x,D\right)K\left(x,y\right)=\delta \left(x-y\right)+R\left(x,y\right),x,y\in B_0,$$

(6)

where δ denotes the Dirac distribution. Moreover, if we let T = T _t be the operator with kernel K, i.e.,

$$Tf\left(x\right)=\underset{B_0}{\int }K\left(x,y\right)f\left(y\right)dy,f\in C_0^{\infty }\left(B_0;E\right),$$

and R is the operator with kernel R (x, y), then for large t > 0, the adjoint of these operators satisfy the following estimates

$${\displaystyle \sum _{\left|\alpha \right|\le 2}{t}^{2-\left|\alpha \right|}}\Vert D^{\alpha }{T}^{*}f\Vert {}_{L_2(B_0;H)}\le C\Vert f\Vert {}_{L_2(B_0;H)},\Vert A{T}^{*}f\Vert {}_{L_2(B_0;H)}\le C\Vert f\Vert {}_{L_2(B_0;H)},$$

(7)

$$t^{\frac{1}{2}}\Vert R^{*}f\Vert {}_{L_2(B_0;H)}\le C\Vert f\Vert {}_{L_2(B_0;H)},$$

(8)

$$t^{-\frac{1}{2}}\Vert D^{\nu }{R}^{*}f\Vert {}_{L_2(B_0;H)}\le C{\displaystyle \sum _{\left|\alpha \right|\le \left|\nu \right|-1}\Vert D^{\alpha }f\Vert {}_{L_2(B_0;H)}},1\le \left|\nu \right|\le 2.$$

(9)

Proof. By Remark 3.2 the operator function ${\left(A+|{\xi }^{|}{|}^2\right)}^{\frac{1}{2}}$ is positive in E for all ξ ∈ Rⁿ. Since tw₁ + iξ₁ ∈ S(φ), due to positivity of A, for $\phi \in \left[\frac{\pi }{2},\pi \right)$ the factor $G_t\left(x,\xi \right)=-i\left[{\left(A+|{\xi }^{|}{|}^2\right)}^{\frac{1}{2}}+w_{1}t+i{\xi }_1\right]$ has a bounded inverse $G_t^{-1}\left(x,\xi \right)$for all ξ ∈ Rⁿ, t > 0 and

$$\Vert G_t^{-1}(x,\xi )\Vert {}_{B(H)}\le C{(1+|t{w}_1+i{\xi }_1|)}^{-1}.$$

(10)

Therefore, we call G _t (x, ξ) the regular factor. Consider now the second factor

$$B_t\left(x,\xi \right)=i\left[{\left(A+|{\xi }^{|}{|}^2\right)}^{\frac{1}{2}}-\left(w_{1}t+i{\xi }_1\right)\right].$$

By virtue of operator calculus and fractional powers of positive operators (see e.g., [19, §1.15.1] or [35]) we get that - [tw₁ + iξ₁] ∉ S (φ) for ξ₁ = 0 and tw₁ = |ξ^||, i.e., the operator B _t (x, ξ) does not has an inverse, in the following set

$${\Delta }_t=\left\{\left(x,\xi \right)\in B_0\times R^n:{\xi }_1=0,\left|{\xi }^{|}\right|=t{w}_1\right\}.$$

So we will called B _t the singular factor and the set Δ _t call singular set for the operator function B _t . The operator $B_t^{-1}$ cannot be bounded in the set Δ _t . Nevertheless, the operator $B_t^{-1}$, and hence $L_{0t}^{-1}$, can be bounded when (x, ξ) is sufficiently far from Δ _t . For instance, if we define

$${\Gamma }_t=\left\{\left(x,\xi \right)\in B_0\times R^n:\left|{\xi }^{|}\right|\in \left[\frac{t}{4},4t\right],\left|{\xi }_1\right|\le \frac{t}{4}\right\},$$

by properties of positive operators we will get the same estimate of type (10) for the singular factor B _t . Hence, using this fact and the resolvent properties of positive operators we obtain the following estimate

$$\Vert L_{0t}^{-1}(x,\xi )\Vert {}_{B(E)}\le C{(1+|\xi {|}^2+t^2)}^{-1}\text{when }(x,\xi )\in {}^c{\Gamma }_t,$$

(11)

where the constant C is independent of x, ξ, t and ^cΓ _t denotes the complement of Γ _t .

Let $\beta \in C_0^{\infty }\left(R\right)$ such that, β(ξ) = 0 if $\left|\xi \right|\in \left[\frac{1}{4},4\right]$ and β (ξ) = 0 near the origin. We then define

$${\beta }_0\left(\xi \right)={\beta }_{0t}\left(\xi \right){\beta }_0\left(\xi \right)=1-\beta \left(\left|{\xi }^{|}\right|/t\right)\beta \left(1-{\xi }_1/t\right)$$

and notice that β₀ (ξ) = 0 on Γ _t . Hence, if we define

$$K_0\left(x,y\right)={\left(2\pi \right)}^{-n}\underset{R^n}{\int }{\beta }_0\left(\xi \right)e^{i\left(\left(x-y\right),\xi \right)}{L}_{0t}^{-1}\left(y,\xi \right)d\xi$$

(12)

and recall (11), then by [31] it follows from standard microlocal arguments that

$$L_{0t}\left(x,D\right)K_0\left(x,y\right)={\left(2\pi \right)}^{-n}\underset{R^n}{\int }{\beta }_0\left(\xi \right)e^{i\left(\left(x-y\right),\xi \right)}d\xi +R_{0t}\left(x,y\right),$$

where R_0tbelongs to a bounded subset of S^-1 which is independent of t. Since operator $R_{0t}^{*}$ also has the same property, it follows that for all $f\in C_0^{\infty }\left(B_0;H\right)$

$$\Vert D^{\nu }{R}_{0t}^{*}f\Vert {}_{L_2(B_0;H)}\le C{\displaystyle \sum _{\left|\alpha \right|\le \left|\nu \right|-1}\Vert D^{\alpha }f\Vert {}_{L_2(B_0;H)}},1\le \left|\nu \right|\le 2.$$

By reasoning as in [31] we get that tR_0tbelongs to a bounded subset of S⁰. So, we have the following estimate

$$t\Vert D^{\nu }{R}_{0t}^{*}f\Vert {}_{L_2(B_0;H)}\le C\Vert f\Vert {}_{L_2(B_0;H)}.$$

Moreover, the Remark 3.2, positivity properties of A and, (11) and (12) imply that, the operator functions $\sum _{\left|\alpha \right|\le 2}{\beta }_0\left(\xi \right)t^{2-\left|\alpha \right|}{\xi }^{\alpha }{L}_{0t}^{-1}\left(x,\xi \right)$ and ${\beta }_0\left(\xi \right)A{L}_{0t}^{-1}\left(x,\xi \right)$ are uniformly bounded. Then, if we let T₀ be the operator with kernel K₀ (x, y), by using the Minkowski integral inequality and Plancherel's theorem we obtain

$${\displaystyle \sum _{\left|\alpha \right|\le 2}{t}^{2-\left|\alpha \right|}}\Vert D^{\alpha }{T}_{0}f\Vert {}_{L_2(B_0;H)}\le C\Vert f\Vert {}_{L_2(B_0;H)},\Vert A{T}_{0}f\Vert {}_{L_2(B_0;H)}\le C\Vert f\Vert {}_{L_2(B_0;H)}.$$

For inverting L_0t(x, D) on the set Γ _t we will require the use of Fourier integrals with complex phase. Let β₁ (ξ) = 1 - β₀ (ξ). We will construct a Fourier integral operator T₁ with kernel

$$K_1\left(x,y\right)={\left(2\pi \right)}^{-n}\underset{R^n}{\int }{\beta }_1\left(\xi \right)e^{i\mathrm{\Phi }\left(x,y,\xi \right)}{L}_{0t}^{-1}\left(y,\xi \right)d\xi$$

(13)

so that the analogs of (16) and the estimates (7)-(9) are satisfied. Since $G_t^{-1}\left(x,\xi \right)$ is uniformly bounded on Γ _t , we should expect to construct the phase function Φ in (13) using the factor B _t (x, ξ). Specifically, we would like Φ to satisfy the following equation

$$B_t\left(x,{\mathrm{\Phi }}_x\right)=B_t\left(y,\xi \right),y\in B_0,\left(x,\xi \in {\Gamma }_t\right).$$

(14)

The Equation (14) leads to complex eikonal equation (i.e., a non-linear partial differential equation with complex coefficients).

$$\begin{array}{c}{\left(A+|{\mathrm{\Phi }}_{x^{|}}\left(x,y,\xi \right){|}^2\right)}^{\frac{1}{2}}-\left[w_1\left(x\right)t+i{\mathrm{\Phi }}_{x_1}\left(x,y,\xi \right)\right]=\\ {\left(A+|{\xi }^{|}{|}^2\right)}^{\frac{1}{2}}-\left(w_1\left(y\right)t+i{\xi }_1\right).\end{array}$$

(15)

Since w₁ (x) = 1 + x₁, w₁ (y) = 1 + y₁, we have

$$\mathrm{\Phi }=\left(x-y,\xi \right)+\frac{{\left(x_1-y_1\right)}^2{\xi }_1}{2\left(1+y_1\right)}+\frac{i{\left(x_1-y_1\right)}^2\left|{\xi }^{|}\right|}{2\left(1+y_1\right)}$$

(16)

is a solution of (15). To use this we get

$$L_{0t}\left(x,D\right)e^{i\mathrm{\Phi }\left(x,y,\xi \right)}=e^{i\mathrm{\Phi }}{L}_{0t}\left(x,{\mathrm{\Phi }}_x\right)+e^{i\mathrm{\Phi }}\frac{{\partial }^2\mathrm{\Phi }}{\partial x_1^2}.$$

Next, if we set

$$r\left(x,y,\xi \right)=G_t\left(y,\xi \right)-G_t\left(x,\xi \right)=-i\left[w_1\left(y\right)-w_1\left(x\right)\right]t$$

(17)

then it follows from L_0t(x, ξ) = G _t (x, ξ)B _t (x, ξ) and (14) that

$$L_{0t}\left(x,{\mathrm{\Phi }}_x\right)=L_{0t}\left(y,\xi \right)+B_t\left(y,\xi \right)r\left(x,y,\xi \right).$$

(18)

Consequently, (16)-(18) imply that

$$\begin{array}{c}{\left(2\pi \right)}^n{L}_{0t}\left(x,D\right)K_1\left(x,y\right)=\underset{R^n}{\int }{\beta }_1\left(\xi \right)e^{i\mathrm{\Phi }}d\xi +\underset{R^n}{\int }{\beta }_1\left(\xi \right)r\left(x,y,\xi \right)G_t^{-1}\left(y,\xi \right)e^{i\mathrm{\Phi }}d\xi \\ \underset{R^n}{\int }{\beta }_1\left(\xi \right)A{L}_{0t}^{-1}\left(y,\xi \right)e^{i\mathrm{\Phi }}d\xi +\underset{R^n}{\int }{\beta }_1\left(\xi \right)\frac{{\partial }^2\mathrm{\Phi }}{\partial x_1^2}{L}_{0t}^{-1}\left(y,\xi \right)e^{i\mathrm{\Phi }}d\xi .\end{array}$$

(19)

By reasoning as in [3] we obtain that the first and second summands in (19) belong to a bounded subset of S⁰. So, we see that the equality (5) must hold. Now we let K (x, y) = K₀ (x, y) + K₁ (x, y) and R (x, y) = R₀ (x, y) + R₁ (x, y), where

$$\begin{array}{c}{R}_1\left(x,y\right)=R_{10}\left(x,y\right)+R_{11}\left(x,y\right),R_{10}\left(x,y\right)=\underset{R^n}{\int }{\beta }_1\left(\xi \right)r\left(x,y,\xi \right)G_t^{-1}\left(y,\xi \right)e^{i\mathrm{\Phi }}d\xi ,\\ R_{11}\left(x,y\right)=\underset{R^n}{\int }{\beta }_1\left(\xi \right)\frac{{\partial }^2\mathrm{\Phi }}{\partial x_1^2}{L}_{0t}^{-1}\left(y,\xi \right)e^{i\mathrm{\Phi }}d\xi ,T_{0}f\left(x\right)=\\ \underset{B_0}{\int }{K}_0\left(x,y\right)f\left(y\right)dy,T_{1}f\left(x\right)=\underset{B_0}{\int }{K}_1\left(x,y\right)f\left(y\right)dy.\end{array}$$

Due to regularity of kernels, by using of Minkowski and Hölder inequalities we get the analog estimate as (7) and (9) for the operators T₀ and R₁₀. Thus, in order to finish the proof, it suffices to show that for f ∈L₂ (B₀; E) one has

$${\displaystyle \sum _{\left|\alpha \right|\le 2}{t}^{2-\left|\alpha \right|}}\Vert D^{\alpha }{T}_1^{*}f\Vert {}_{L_2(B_0;H)}+\Vert A{T}_1^{*}f\Vert {}_{L_2(B_0;H)}\le C\Vert f\Vert {}_{L_2(B_0;H)},$$

(20)

$$t^{\frac{1}{2}}\Vert R_{11}^{*}f\Vert {}_{L_2(B_0;H)}\le C\Vert f\Vert {}_{L_2(B_0;H)},$$

(21)

$$t^{-\frac{1}{2}}\left|\right|D^{\nu }{R}_{11}^{*}f\Vert {}_{L_2(B_0;H)}\le C{\displaystyle \sum _{\left|\alpha \right|\le \left|\nu \right|-1}\Vert D^{\alpha }f\Vert {}_{L_2(B_0;H)}},1\le \left|\nu \right|\le 2.$$

(22)

However, since R_1,1 ≈ tT₁, we need only to show the following

$$t^{3/2}\Vert T_1^{*}f\Vert {}_{L_2(B_0;H)}\le C\Vert f\Vert {}_{L_2(B_0;H)}.$$

(23)

By using the Minkowski inequalities we get

$$\Vert T_1^{*}f\Vert {}_{L_2(R^{n-1};E)}\le {\displaystyle \underset{-\frac{1}{4}}{\overset{\frac{1}{4}}{\int }}\Vert {\displaystyle \underset{B_0}{\int }{K}_1^{*}(x,y)f(y)d{y}^{|}}\Vert d{y}_1,}$$

where $K_1^{*}\left(x,y\right)={\bar{K}}_1\left(y,x\right)$. The estimates (13) and (16) imply that

$$K_1^{*}\left(x,y\right)={\left(2\pi \right)}^{-n}\underset{R^{n-1}}{\int }{e}^{i\left(x^{\prime }-y^{\prime }\right)}m\left(x_1,y_1,{\xi }^{|}\right)d{\xi }^{|},$$

where

$$m\left(x_1,y_1,{\xi }^{|}\right)=\underset{-\infty }{\overset{\infty }{\int }}{\beta }_1\left(\xi \right)e^{i\left[\left(x_1-y_1\right){\xi }_1+{\left(x_1-y_1\right)}^2\left(i\left|{\xi }^{|}\right|-{\xi }_1\right)/2\left(1+x_1\right)\right]}{L}_{0t}^{-1}\left(y,\xi \right)d{\xi }_1.$$

Consequently, it follows from Plancherel's theorem that

$$∥\underset{R^{n-1}}{\int }{K}_1^{*}\left(x,y\right)f\left(y\right)d{y}^{|}∥\le \underset{{\xi }^{|}}{\text{sup}}\left|m\left(x_1,y_1,{\xi }^{|}\right)\right|{\left(\underset{R^{n-1}}{\int }|f\left(y\right){|}^{2}d{y}^{|}\right)}^{\frac{1}{2}}.$$

(24)

Note that for every N we have

$$e^{i\left[{\left(x_1-y_1\right)}^2\left|{\xi }^{|}\right|/2\left(1+x_1\right)\right]}\le C_N{\left[1+t{\left(x_1-y_1\right)}^2\right]}^{-N}\mathsf{\text{on supp }}{\beta }_1.$$

Since A is a positive operator in E, we have

$$\Vert L_{0t}^{-1}(x,\xi )\Vert {}_{B(E)}\le 1+|-2i{\xi }_1{w}_{1}t+|\xi {|}^2-t^2{w}_1^2{|}^{-1}$$

when $-2i{\xi }_1{w}_{1}t+A+|\xi {|}^2-t^2{w}_1^2\in S\left(\phi \right)$. Then by using the above estimate it not easy to check that

$$\underset{-\infty }{\overset{\infty }{\int }}{\beta }_1\left(\xi \right)e^{i{\xi }_1\left[\left(x_1-y_1\right)-{\left(x_1-y_1\right)}^2/2\left(1+x_1\right)\right]}{L}_{0t}^{-1}\left(y,\xi \right)d{\xi }_1=O\left(t^{-1}\right),$$

i.e.,

$$\left|m\left(x_1,y_1,{\xi }^{|}\right)\right|\le C{t}^{-1}{\left[1+t{\left(x_1-y_1\right)}^2\right]}^{-1}.$$

Moreover, it is clear that

$$\underset{-\infty }{\overset{\infty }{\int }}{\left(1+t{x}_1\right)}^{-1}d{x}_1=O\left(t^{-\frac{1}{2}}\right).$$

Thus from (24) by using the above relations and Young's inequality we obtain the desired estimate

$$\begin{array}{ll}\hfill {∥T_1^{*}f∥}_{L_2\left(B_0;H\right)}& \le C{t}^{-1}\int \left|\int \left[1+t{\left(x_1-y_1\right)}^2\right]{∥f\left(y_1,\cdot \right)∥}_{L_2}d{y}_1\right|d{x}_1\\ \le C{t}^{-3/2}{∥f∥}_{L_2\left(R^n;H\right)}.\end{array}$$

Moreover, by using the estimate (10) and the resolvent properties of the positive operator A we have

$${∥A{T}_1^{*}f∥}_{L_2\left(B_0;H\right)}\le C{∥f∥}_{L_2\left(B_0;H\right)}.$$

The last two estimates then, imply the estimates (20)-(22).

Proof of Theorem 3.1: The estimates (7)-(9) imply the estimate (5), i.e., we obtain the assertion of the Theorem 3.1.

4 L _p -Carleman estimates and unique continuation for equation with variable coefficients

Consider the following DOE

$$L\left(x,D\right)u=\sum _{i,j=1}^n{a}_{ij}\left(x\right)D_{ij}^{2}u+Au=f\left(x\right),x\in R^n,$$

(25)

where $D_k=\frac{\partial }{i{\partial }_k}$ and A is the possible unbounded operator in a Banach space E and a _ij are

real-valued smooth functions in B _ε = {x ∈ Rⁿ, |x| < ε}.

Condition 4.1. There is a positive constant γ such that $\sum _{i,j=1}^n{a}_{ij}\left(x\right){\xi }_i{\xi }_j\ge \gamma |\xi {|}^2$ for all ξ ∈ Rⁿ, x ∈ $B_0=\left\{x\in R^n,\left|x\right|<\frac{1}{4}\right\}.$

The main result of the section is the following

Theorem 4.1. Let E be a Banach space satisfies the multiplier condition and A be a R-positive operator in E. Suppose the Condition 4.1 holds, n ≥ 3, p = $\frac{2n}{n+2}$and p' is the conjugate of p, w = $x_1+\frac{x_1^2}{2}$ and a _ij ∈ C^∞ (B _ε )_. Then for $u\in C_0^{\infty }$( B _ε ; E(A)) and $\in >0,\frac{1}{t}<\frac{1}{2}$ the following estimates are satisfied:

$${∥e^{tw}u∥}_{L_{p^{|}}\left(R^n;E\right)}\le C{∥e^{tw}L\left(\epsilon x,D\right)u∥}_{L_p\left(R^n;E\right)},\frac{1}{p}+\frac{1}{p^{|}}=1,$$

(26)

$$\sum _{\left|\alpha \right|\le 1}{t}^{\left(1+\frac{1}{n}-\left|\alpha \right|\right)}{∥e^{tw}{D}^{\alpha }u∥}_{L_p\left(R^n;E\right)}+{∥e^{tw}Au∥}_{L_p\left(R^n;E\right)}\le$$

(27)

$$C{∥e^{tw}L\left(\epsilon x,D\right)u∥}_{L_p\left(R^n;E\right)}.$$

Proof. As in the proof of Theorem 3.1, it is sufficient to prove the following estimates

$${∥v∥}_{L_{p^{|}}\left(R^n;E\right)}\le C{∥L_t\left(\epsilon x,D\right)v∥}_{L_p\left(R^n;E\right)},\frac{1}{p}+\frac{1}{p^{|}}=1,$$

(28)

$$\sum _{\left|\alpha \right|\le 1}{t}^{\left(1+\frac{1}{n}-\left|\alpha \right|\right)}{∥D^{\alpha }v∥}_{L_p\left(R^n;E\right)}+{∥Av∥}_{L_p\left(R^n;E\right)}\le C{∥L_t\left(\epsilon x,D\right)v∥}_{L_p\left(R^n;E\right)}$$

(29)

where,

$$L_t\left(\epsilon x,D\right)=e^{tw}L\left(\epsilon x,D\right)e^{-tw}=L\left(\epsilon x,D\right)+2t{w}_1\frac{\partial }{\partial x_1}-{\left(t{w}_1\right)}^2-t^2,w_1=\frac{\partial w}{\partial x_1}.$$

Consequently, since w₁ ≃ 1 on B _ε , it follows that, if we let Q _t (εx, D) be the differential operator whose adjoint equals

$$Q_t^{*}\left(\epsilon x,D\right)=w_1^{-2}L\left(\epsilon x,D\right)+2t{w}_1^{-1}\frac{\partial }{\partial x_1}-t^2,$$

then it suffices to prove the following

$$\begin{array}{c}{∥v∥}_{L_{p^{|}}\left(R^n;E\right)}\le C{∥Q_t\left(\epsilon x,D\right)v∥}_{L_p\left(R^n;E\right)},\frac{1}{p}+\frac{1}{p^{|}}=1,\\ \sum _{\left|\alpha \right|}{t}^{\left(1+\frac{1}{n}-\left|\alpha \right|\right)}\left|\right|D^{\alpha }v|{|}_{L_p\left(R^n;E\right)}+|\left|Av\right|{|}_{L_p\left(R^n;E\right)}\le C\left|\right|Q_t\left(\epsilon x,D\right)v|{|}_{L_p}\left(R^n;E\right),\\ v\in C_0^{\infty }\left(B_{\epsilon };E\left(A\right)\right).\end{array}$$

(30)

The desired estimates will follow if we could constrict a right operator-valued parametrix T, for Q _t * (εx, D) satisfying L _p estimates. these are contained in the following lemma.

Lemma 4.1. For t > 0 there are functions K = K _t and R = R _t , so that

$$Q_t^{*}\left(\epsilon x,D\right)K\left(x,y\right)=\delta \left(x-y\right)+R\left(x,y\right),x,y\in B_{\epsilon },$$

(31)

where δ denotes the Dirac distribution. Moreover, if we let T = T _t be the operator with kernel K (x, y) and R be the operator with kernel R (x, y), then if ε and $\frac{1}{t}$ are sufficiently small, the adjoint of these operators satisfy the following uniform estimates

$${∥T^{*}f∥}_{L_{p^1}\left(R^n;E\right)}\le C{∥f∥}_{L_p\left(R^n;E\right)},\frac{1}{p}+\frac{1}{p^{|}}=1,$$

(32)

$$\sum _{\left|\alpha \right|\le 1}{t}^{\left(1+\frac{1}{n}-\left|\alpha \right|\right)}{∥D^{\alpha }{T}^{*}f∥}_{L_p\left(R^n;E\right)}\le C{∥f∥}_{L_p\left(R^n;E\right)},$$

(33)

$${∥A{T}^{*}f∥}_{L_p\left(R^n;E\right)}\le C{∥f∥}_{L_p\left(R^n;E\right)},$$

$$t^{\frac{1}{n}}{∥R^{*}f∥}_{L_q\left(R^n;E\right)}\le C{∥f∥}_{L_q\left(R^n;E\right),}q=p,p^{|},$$

(34)

$$t^{-1+\frac{1}{n}}{∥\nabla R^{*}f∥}_{L_p\left(R^n;E\right)}\le C{∥f∥}_{L_p\left(R^n;E\right)},f\in C_0^{\infty }\left(B_{\epsilon };E\right).$$

(35)

Proof. The key step in the proof is to find a factorization of the operator-valued symbol $Q_t^{*}\left(\epsilon x,\xi \right)$ that will allow to microlocally invert $Q_t^{*}\left(\epsilon x,D\right)$ near the set where $Q_t^{*}\left(\epsilon x,\xi \right)$ vanishes. Note that, after making a suitable choice of coordinates, it is enough to show that if L (x, D) is of the form

$$L\left(x,D\right)=D_1^2+\sum _{i,j=2}^n{a}_{ij}{D}_i{D}_j,D_j=\frac{1}{i}\frac{\partial }{\partial x_j}$$

therefore, we can expressed $Q_t^{*}\left(\epsilon x,\xi \right)$ as

$$Q_t^{*}\left(\epsilon x,\xi \right)=B_t\left(\epsilon x,\xi \right)G_t\left(\epsilon x,\xi \right),$$

(36)

where

$$\begin{array}{c}{B}_t\left(x,\xi \right)=w_1^{-1}{\xi }_1+i\left[\left(A+w_1^{-1}a\left(\epsilon x,{\xi }^{|}\right)\right)-t\right],\\ G_t\left(x,\xi \right)=w_1^{-1}{\xi }_1-i\left[\left(A+w_1^{-1}a\left(\epsilon x,{\xi }^{|}\right)\right)+t\right],\end{array}$$

where

$$a\left(x,{\xi }^{|}\right)=\sum _{i,j=2}^n{a}_{ij}\left(x\right){\xi }_i{\xi }_j.$$

The ellipticity of Q(x, D) and the positivity of the operator A, implies that the factor G _t (x, ξ) never vanishes and as in the proof of Theorem 3.1 we get that

$${∥G_t^{-1}\left(\epsilon x,\xi \right)∥}_{B\left(H\right)}\le C{\left(1+|w_1^{-1}a\left(\epsilon x,{\xi }^{|}\right){|}^{\frac{1}{2}}+|t+w_1^{-1}{\xi }_1|\right)}^{-1},$$

(37)

$$x\in B_{\epsilon },\xi \in R^n,$$

i.e., the operator function G _t (εx, ξ) has uniformly bounded inverse for (x, ξ) ∈ B _ε ×Rⁿ. One can only investigate the factor B _t (εx, ξ). In fact, if we let

$${\Delta }_t=\left\{\left(x,\xi \right)\in B_{\epsilon }\times R^n:{\xi }_1=0,\left|{\xi }^{|}\right|=t{w}_{\mathsf{\text{1}}}\right\},$$

then the operator function B _t (x, ξ) is not invertible for (x, ξ) ∈ Δ _t . Nonetheless, B _t (εx, ξ) and $Q_t^{*}\left(\epsilon x,\xi \right)$ can be have a bounded inverse when (x, ξ) is sufficiently far away. For instance, if we define

$${\Gamma }_t=\left\{\left(x,\xi \right)\in B_{\epsilon }\times R^n:\left|{\xi }^{|}\right|\in \left[\frac{t}{4},4t\right],\left|{\xi }_1\right|\le \frac{t}{4}\right\},$$

by properties of positive operators we will get the same estimate of type (37) for the singular factor B _t . Hence, we using this fact and the resolvent properties of positive operators we obtain the following estimate

$${∥{\left(Q_t^{*}\right)}^{-1}\left(\epsilon x,\xi \right)∥}_{B\left(E\right)}\le C{\left(1+\left|{\xi }^{|}\right|+|t+w_1^{-1}{\xi }_1|\right)}^{-1}\mathsf{\text{when}}\left(x,\xi \right)\in {}^c{\Gamma }_t.$$

(38)

As in § 3, we can use (38) to microlocallity invert $Q_t^{*}\left(\epsilon x,D\right)$ away from Γ _t . To do this, we first fix $\beta \in C_0^{\infty }\left(R\right)$ as in § 3. We then define

$${\beta }_0={\beta }_{0t}=1-\beta \left(\left|{\xi }^{|}\right|/t\right)\beta \left(1-{\xi }_1/t\right).$$

It is clear that β₀ (ξ) = 0 on Γ _t . Consequently, if we define

$$K_0\left(x,y\right)={\left(2\pi \right)}^{-n}\underset{R^n}{\int }{\beta }_0\left(\xi \right)e^{i\left(\left(x-y\right),\xi \right)}{\left(Q_t^{*}\right)}^{-1}\left(\epsilon y,\xi \right)d\xi$$

(39)

and recall (37), then we can conclude that standard microlocal arguments give that

$$Q_t^{*}\left(\epsilon x,D\right)K_0\left(x,y\right)={\left(2\pi \right)}^{-n}\underset{R^n}{\int }{\beta }_0\left(\xi \right)e^{i\left(\left(x-y\right),\xi \right)}d\xi +R_0\left(x,y\right),$$

(40)

where R₀ belongs to a bounded subset of S^-1 that independent of t. Since the adjoint operator $R_0^{*}$ also is abstract pseudodifferential operator with this property, by reasoning as in [31, Theorem 6] it follows that

$${∥\nabla R_0^{*}f∥}_{L_p\left(R^n;E\right)}\le C{∥f∥}_{L_p\left(R^n;E\right)},f\in C_0^{\infty }\left(B_{\epsilon };E\right),$$

(41)

$$t{∥R_0^{*}f∥}_{L_q\left(R^n;E\right)}\le C{∥f∥}_{Lq\left(R^n;E\right)},f\in C_0^{\infty }\left(B_{\epsilon };E\right),$$

(42)

$$q=p,p^{\prime },\frac{1}{p}+\frac{1}{p^{\prime }}=1.$$

Moreover, the positivity properties of A and the estimate (38) imply that the operator functions $\sum _{\left|\alpha \right|\le 2}{\beta }_0\left(\xi \right)t^{2-\left|\alpha \right|}{\xi }^{\alpha }{\left(Q_t^{*}\right)}^{-1}\left(\epsilon x,\xi \right)\mathsf{\text{and}}{\beta }_0\left(\xi \right)A{\left(Q_t^{*}\right)}^{-1}\left(\epsilon x,\xi \right)$ are uniformly bounded. Next, let T₀ be the operator with kernel K₀. Then in a similar way as in [31] we obtain that

$$\sum _{\left|\alpha \right|\le 1}{t}^{\left(2-\left|\alpha \right|\right)}{∥D^{\alpha }{T}_0^{*}f∥}_{L_p\left(R^n;E\right)}\le C{∥f∥}_{L_p\left(R^n;E\right)},$$

(43)

$${∥A{T}_0^{*}f∥}_{L_p\left(R^n;E\right)}\le C{∥f∥}_{L_p\left(R^n;E\right)}$$

which also the first estimate is stronger than the corresponding inequality in Lemma 4.1. Finally, since T₀ ∈ S^-2 and $\frac{1}{p}-\frac{1}{p^{|}}=\frac{2}{n}$ it follows from imbedding theorem in abstract Sobolev spaces [17] that

$${∥T_0^{*}f∥}_{L_{p^{|}}\left(R^n;E\right)}\le C{∥f∥}_{L_p\left(R^n;E\right)},f\in C_0^{\infty }\left(B_{\epsilon };E\right).$$

(44)

Thus, we have shown that the microlocal inverse corresponding to ^cΓ _t , satisfies the desired estimates.

Let β₁ (ξ) = 1-β₀ (ξ). To invert $Q_t^{*}\left(\epsilon x,D\right)$ for (x, ξ) ∈ Γ _t , we have to construct a Fourier integral operator T₁, with kernel

$$K_1\left(x,y\right)={\left(2\pi \right)}^{-n}\underset{R^n}{\int }{\beta }_1\left(\xi \right)e^{i\mathrm{\Phi }\left(x,y,\xi \right)}{Q}_{0t}^{{*}^{-1}}\left(\epsilon y,\xi \right)d\xi ,$$

(45)

such that the analogs of (39) and (32)-(35) are satisfied. For this step the factorization (36) of the symbol $Q_t^{*}\left(\epsilon y,\xi \right)$ will be used. Since the factor G _t (εx, ξ) has a bounded inverse for (x, ξ) ∈ Γ _t , the previous discussions show that we should try to construct the phase function in (46) using the factor B _t (εx, ξ). We would like Φ (x, y, ξ) to solve the complex eikonal equation

$$B_t\left(\epsilon x,{\mathrm{\Phi }}_x\right)=B_t\left(\epsilon y,\xi \right),x,y\in B_{\epsilon },\xi \in \mathsf{\text{supp }}{\beta }_1,$$

(46)

Since B _t (εx, Φ _x ) - B _t (εy, ξ) is a scalar function (it does not depend of operator A ), by reasoning as in [3, Lemma 3.4] we get that

$$\mathrm{\Phi }\left(x,y,\xi \right)=\varphi \left(x^{\prime },y,{\xi }^{\prime }\right)+\psi \left(x,y,\xi \right),$$

where ϕ is real and defined as

$$\varphi \left(x^{\prime },y,{\xi }^{\prime }\right)=\left(x_1-y_1\right){\xi }_1+O\left({\left|x^{\prime }-y^{\prime }\right|}^2\left|{\xi }^{\prime }\right|\right),$$

while

$$\psi \left(x,y,\xi \right)=\left(x_1-y_1\right){\xi }_1+O\left({\left|x_1-y_1\right|}^2\left|{\xi }^{\prime }\right|\right)$$

and

$$\mathsf{\text{Im}}\psi \left(x,y,\xi \right)\ge c{\left(x_1-y_1\right)}^2\left|{\xi }^{\prime }\right|,c>0.$$

(47)

Then we obtain from the above that

$$Q_t^{*}\left(\epsilon x,D\right)e^{i\mathrm{\Phi }\left(x,y,\xi \right)}=e^{i\mathrm{\Phi }}{Q}_t^{*}\left(\epsilon x,{\mathrm{\Phi }}_x\right)+e^{i\mathrm{\Phi }}{w}_1^{-2}L\left(\epsilon x,D\right)\mathrm{\Phi }.$$

Next, if we set

$$\begin{array}{c}r\left(x,y,\xi \right)=G_t\left(\epsilon y,\xi \right)-G_t\left(\epsilon x,\xi \right)=w_1^{-1}\left(y\right)\left[{\xi }_1-ia\left(\epsilon y,{\xi }^{\prime }\right)\right]\\ -w_1^{-1}\left(x\right)\left[{\xi }_1-ia\left(\epsilon x,{\xi }^{\prime }\right)\right]\end{array}$$

(48)

then it follows from (36) and (48) that

$$e^{i\mathrm{\Phi }}{Q}_t^{*}\left(\epsilon x,{\mathrm{\Phi }}_x\right)=e^{i\mathrm{\Phi }}{Q}_t^{*}\left(\epsilon y,\xi \right)+e^{i\mathrm{\Phi }}{B}_t\left(\epsilon y,\xi \right)r\left(x,y,\xi \right)+O\left(t^{-N}\right)$$

(49)

for every N when β₁ (ξ) ≠ 0. Consequently, (49), (50) imply that

$${\left(2\pi \right)}^n{Q}_t^{*}\left(\epsilon x,D\right)K_1\left(x,y\right)=\int {\beta }_1\left(\xi \right)e^{i\mathrm{\Phi }}d\xi +\int {\beta }_1\left(\xi \right)r\left(x,y,\xi \right)G_t^{-1}\left(\epsilon y,\xi \right)e^{i\mathrm{\Phi }}d\xi$$

$$w_1^{-2}\int {\beta }_1\left(\xi \right)Q_t^{*-1}\left(\epsilon y,\xi \right)\left(L\left(\epsilon x,D\right)\mathrm{\Phi }\right)e^{i\mathrm{\Phi }}d\xi +O\left(t^{-N}\right).$$

(50)

By reasoning as in Theorem 3.1 we obtain from (51) that

$$Q_t^{*}\left(\epsilon x,D\right)K_1\left(x,y\right)={\left(2\pi \right)}^{-n}\int {\beta }_1\left(\xi \right)e^{i\left(x-y,\xi \right)}d\xi +R_{10}\left(x,y\right)+R_{11}\left(x,y\right),$$

where

$$R_{11}\left(x,y\right)={\left(2\pi \right)}^{-n}{w}_1^{-2}\int {\beta }_1\left(\xi \right)Q_t^{*-1}\left(\epsilon y,\xi \right)\left(L\left(\epsilon x,D\right)\mathrm{\Phi }\right)e^{i\mathrm{\Phi }}d\xi$$

(51)

while R₁₀ belongs to a bounded subset of S^-1 and tR₁₀ belongs to a bounded subset of S⁰. In view of this formula, we see that if we let K (x, y) = K₀ (x, y) + K₁ (x, y) and R (x, y) = R₀ (x, y)+R₁ (x, y), where R₁ = R₁₀ +R₁₁, then we obtain (31). Moreover, since R₁₀ satisfies the desired estimates, we see from Minkowski inequality that, in order to finish the proof of Lemma 4.1, it suffices to show that for $f\in C_0^{\infty }\left(B_{\epsilon };E\right)$

$${\Vert T_1^{*}f\Vert }_{L_{p^{|}}(R^n;E)}\le C{\Vert f\Vert }_{L_p(R^n;E)},$$

(52)

$${{\displaystyle \sum _{\left|\alpha \right|\le 1}t}}^{(1+\frac{1}{n}-|\alpha |)}\Vert D^{\alpha }{T}_1^{*}f\Vert {}_{L_p(R^n;E)}\le C\Vert f\Vert {}_{L_p(R^n;E)},$$

(53)

$$t^{\frac{1}{n}}\Vert R_{11}^{*}f\Vert {}_{L_q(R^n;E)}\le C\Vert f\Vert {}_{L_q(R^n;E)},q=p,p^{|},$$

(54)

$$t^{-1+\frac{1}{n}}\Vert \nabla R_{11}^{*}f\Vert {}_{L_p(R^n;E)}\le C\Vert f\Vert {}_{L_p(R^n;E)},$$

(55)

where $\frac{1}{p}+\frac{1}{p^{|}}=1$.

To prove the above estimates we need the following prepositions for oscillatory integral in E-valued L _p spaces which generalize the Carleson and Sjolin result [36].

Preposition 4.1. Let E be Banach spaces and $A\in C_0^{\infty }\left(R^n,L\left(E\right)\right)$. Moreover, suppose Φ ∈ C^∞ satisfies | ∇Φ| ≥ γ > 0 on supp A. Then for all λ > 1 the following holds

$${∥\int e^{i\lambda \mathrm{\Phi }\left(x\right)}A\left(x\right)dx∥}_{L\left(E\right)}\le C_N{\lambda }^{-N},N=1,2,\dots$$

where C _N -depends only on γ if Φ and A (x) belong to a bounded subset of C^∞ and C^∞ (Rⁿ, L (E)) and A is supported in a fixed compact set.

Proof. Given x₀ ∈ supp A. There is a direction ν ∈ S^n-1such that |(ν, ∇Φ)| ≥$\frac{\gamma }{2}$ on some ball centered at x₀. Thus, by compactness, we can choose a partition of unity ${\phi }_j\in C_0^{\infty }$ consisting of a finite number of terms and corresponding unit vectors ν _j such that $\sum _{j=1}^m{\phi }_j\left(x\right)=1$ on supp A and $\left|\left({\nu }_j,\nabla \mathrm{\Phi }\right)\right|\ge \frac{\gamma }{2}$ on supp φ _j . For A _j = φ _j A it suffices to prove that for each j

$${∥\int e^{i\lambda \mathrm{\Phi }\left(x\right)}{A}_j\left(x\right)dx∥}_{L\left(E\right)}\le C_N{\lambda }^{-N},N=1,2,\dots .$$

After possible changing coordinates we may assume that ν _j = (1, 0, . . . , 0) which means that $\left|\frac{\partial \mathrm{\Phi }}{\partial x_1}\right|\ge \frac{\gamma }{2}$ on supp φ _j . If let $L\left(x;D\right)={\left(\frac{\partial \mathrm{\Phi }}{\partial x_1}\right)}^{-1}\frac{1}{i\lambda }\frac{\partial }{\partial x_1}$, then $L\left(x;D\right)e^{i\lambda \mathrm{\Phi }\left(x\right)}=e^{i\lambda \mathrm{\Phi }\left(x\right)}$. Consequently, if $L^{*}=\frac{\partial }{\partial x_1}\left(\frac{1}{i\lambda }{\left(\frac{\partial \mathrm{\Phi }}{\partial x_1}\right)}^{-1}\right)$ is a adjoint, then

$$\int e^{i\lambda \mathrm{\Phi }\left(x\right)}A\left(x\right)dx=\int e^{i\lambda \mathrm{\Phi }\left(x\right)}{\left(L^{*}\right)}^N{A}_j\left(x\right)dx.$$

Since our assumption imply that (L*)^N A _j (x) = O (λ^-N), the result follows.

Preposition 4.2. Suppose Φ ∈ C^∞ is a phase function satisfying the non-degeneracy condition det $\left[\frac{{\partial }^2\mathrm{\Phi }}{\partial x_i\partial x_j}\right]\ne 0$on the support of

$$A\left(x,y\right)\in C_0^{\infty }\left(R^n\times R^n,L\left(E\right)\right).$$

Then for $T_{\lambda }f=\underset{R^n}{\int }{e}^{i\lambda \mathrm{\Phi }\left(x,y\right)}A\left(x,y\right)f\left(y\right)dx,\lambda >0$ the following estimates hold

$$\begin{array}{c}{∥T_{\lambda }f∥}_{L_p\left(R^n;E\right)}\le C{\lambda }^{-\frac{n-1}{p^{\prime }}}{∥f∥}_{L_p\left(R^n;E\right)},1\le p\le 2,\\ {∥T_{\lambda }f∥}_{L_p\left(R^n;E\right)}\le C{\lambda }^{-\frac{n}{p^{\prime }}}{∥f∥}_{L_p\left(R^n;E\right)},\frac{1}{p}+\frac{1}{p^{\prime }}=1.\end{array}$$

Proof. In view of [3, Remark 2.1] we have

$$\left|{\nabla }_x\left[\mathrm{\Phi }\left(x,y\right)-\mathrm{\Phi }\left(x,z\right)\right]\right|\simeq |y-z|$$

(56)

where |y - z| is small. By using a smooth partition of unity we can decompose A (x, y) into a finite number of pieces each of which has the property that (57) holds on its support. So, by (57) we can assume

$$\left|{\nabla }_x\left[\mathrm{\Phi }\left(x,y\right)-\mathrm{\Phi }\left(x,z\right)\right]\right|\ge C|y-z|$$

(57)

on supp A for same C > 0. To use this we notice that

$${∥T_{\lambda }f∥}_2^2=\int \int K_{\lambda }\left(y,z\right)f\left(y\right)\bar{f}\left(z\right)dydz,$$

where

$$K_{\lambda }\left(y,z\right)={\int }_{R^n}{e}^{i\lambda \left[\mathrm{\Phi }\left(x,y\right)-\mathrm{\Phi }\left(x,z\right)\right]}A\left(x,y\right)Ā\left(x,z\right)dx.$$

(58)

Hence, by virtue of Preposition 4.1 and by (58) we obtain that

$${∥K_{\lambda }\left(y,z\right)∥}_{L\left(E\right)}\le C_N\left(1+\left|\lambda \right||y-z{|}^{-N}\right),\mathsf{\text{ for all }}N.$$

Consequently, by Young's inequality, the operator with kernel K _λ acts

$$L_p\left(R^n;E\right)\mathsf{\text{ to }}{L}_p\left(R^n;E\right).$$

By (59) we get that

$${∥T_{\lambda }f∥}_{L_2\left(R^n;E\right)}\le C{\lambda }^{-n}{∥f∥}_{L_2\left(R^n;E\right)}.$$

Moreover, it is clear to see that

$${∥T_{\lambda }f∥}_{L{}_{\infty }\left(R^n;E\right)}\le C{\lambda }^{-n}{∥f∥}_{L_1\left(R^n;E\right)}.$$

Therefore, by applying Riesz interpolation theorem for vector-valued L _p spaces (see e.g., [19, § 1.18]) we get the assertion.

In a similar way as in [3, Preposition 3.6] we have.

Preposition 4.3. The kernel K₁ (x, y) can be written as

$$K_1\left(x,y\right)=\sum _{j=0,1}{A}_j\left(x,y\right)\frac{t^{n-2}{e}^{it{\phi }_j\left(x^{\prime },y\right)}}{\left|t\left(x^{\prime }-y^{\prime }\right){|}^{\left(n-2\right)/2}\right|t\left(x-y\right)|},$$

where, for every fixed N, the operator functions A _j satisfy

$$∥D^{\alpha }{A}_j\left(x,y\right)∥\le C_{\alpha }{\left(1+t{\left(x_1-y_1\right)}^2\right)}^{-N}|x^{\prime }-y^{\prime }{|}^{-\left|\alpha \right|},$$

and moreover, the phase functions φ _j are real and the property that when ε is small enough, 0 < δ ≤ ε and y₁ ∈ [-ε, ε] is fixed, the dilated functions

$$\left(x^{\prime },y^{\prime }\right)\to {\left(-1\right)}^j{\delta }^{-1}{\phi }_j\left(\delta x^{\prime },y_1,\delta y^{\prime }\right)$$

in the some fixed neighborhood of the function ${\phi }_0\left(x^{\prime },y^{\prime }\right)=|x^{\prime }-y^{\prime }|$ in the C^∞ topology. Then, the following estimates holds

$$\left|K_1\left(x,y\right)\right|\le C{t}^{n-2}{\left(1+t|x_1-y_1|\right)}^{-1}.$$

(59)

Proof. By representation of K₁ (x, y) and Φ (x, y, ξ) we have

$$K_1\left(x,y\right)\simeq t^{n-2}{\int }_{R^n}{\beta }_1\left(t\xi \right)e^{it\mathrm{\Phi }\left(x,y,\xi \right)}{Q}_{0t}^{*-1}\left(\epsilon y,\xi \right)d\xi .$$

Then, by using (36) in view of positivity of operator A, by reasoning as in [3, Preposition 3.6] we obtain the assertion.

Let us now show the end of proof of Lemma 4.1. Let $\eta \in C_0^{\infty }\left(R\right)$ be supported in $\left[\frac{1}{4},4\right]$ such that $\sum _{\nu \mathsf{\text{ = }}-\infty }^{\infty }\eta \left(2^{\nu }s\right)=1,s>0$ and set ${\eta }_0\left(s\right)=1-\sum _{\nu \mathsf{\text{ = }}-\infty }^0\eta \left(2^{\nu }s\right)$. Then we define kernels K_1,ν, ν = 0, 1, 2, . . . , as follows

$$K_{1,\nu }=\left\{\begin{array}{c}\eta \left(t{2}^{-\nu }|x^{\prime }-y^{\prime }|\right)K_1\left(x,y\right),\nu >0\\ {\eta }_0\left(t|x^{\prime }-y^{\prime }|\right)K_1\left(x,y\right),\nu =0.\end{array}\right.$$

Let T_1,νdenotes the operators associated to these kernels. Then, by positivity properties of the operator A and by Prepositions 4.2, 4.3 we obtain for $f\in C_0^{\infty }\left(B_{\epsilon };E\right)$ the following estimates

$${\Vert T_{1,\nu }^{*}f\Vert }_{L_{p^{\prime }}(R^n;E)}\le C{2}^{-2\nu /n}{\Vert f\Vert }_{L_p(R^n;E)},\frac{1}{p}+\frac{1}{p^1}=1,$$

(60)

$${\Vert T_{1,\nu }^{*}f\Vert }_{L_p(R^n;E)}\le C{(t{2}^{-\nu })}^{-1/p^{\prime }}{t}^{-\left(1+\frac{1}{n}\right)}{\Vert f\Vert }_{L_p(R^n;E)}.$$

(61)

By summing a geometric series one sees that these estimates imply (52) and (53) for case of α = 0.

Let us first to show (60). One can check that the estimate (59) implies that the L _r norm of $K_{10}^{*}$ is O (t^n-2t ^-n/r). But, if we let r = n/n - 2, it is follows from Young inequality and the fact that $\frac{1}{p}-\frac{1}{p^{\prime }}=\frac{2}{n}$ that

$${\Vert T_{1,0}^{*}f\Vert }_{L_{p^{|}}(R^n;E)}\le C{t}^{n-2}{t}^{-n/r}{\Vert f\Vert }_{L_p(R^n;E)}=C{\Vert f\Vert }_{L_p(R^n;E)}$$

as desired. To prove the result for ν > 0, set $B_{\epsilon }^{\prime }=\left\{x^{\prime }\in R^{n-1},\left|x^{\prime }\right|<\epsilon \right\}$ and let $K_{1\nu }^{*}$ be the kernel of the operator $T_{1,\nu }^{*}$. Then, if we fix x₁ and y₁, it follows that the $L_p\left(B_{\epsilon }^{\prime };E\right)\to L_p\left(B_{\epsilon }^{\prime };E\right)$ norm of the operator

$$T_{1,v}^{{}^{\prime }*}g\left(x^{\prime }\right)=\underset{{B^{\prime }}_{\epsilon }}{\int }{K}_{1\nu }^{*}\left(x,y\right)g\left(y^{\prime }\right)d{y}^{\prime }$$

equal ${{\left(2^{\nu }{t}^{-1}\right)}^{\left(n-1\right)}}^{\left(1-\frac{1}{p}+\frac{1}{p^{\prime }}\right)}$ times the norm of the dilated operator

$${\tilde{T}}_{1,\nu }^{*}g({}^{}$$