RETRACTED ARTICLE: A new application of boundary integral behaviors of harmonic functions to the least harmonic majorant

Abstract

Our main aim in this paper is to obtain a new type of boundary integral behaviors of harmonic functions in a smooth cone. As an application, the least harmonic majorant of a nonnegative subharmonic function is also given.

1 Introduction

Let $B(P,R)$ denote the open ball with center at P and radius R in $\mathbf{R}^{n}$, where $\mathbf{R}^{n}$ is the n-dimensional Euclidean space, $P\in\mathbf{R}^{n}$ and $R>0$. Let $B(P)$ denote the neighborhood of P and $S_{R}=B(O,R)$ for simplicity. The unit sphere and the upper half unit sphere in $\mathbf{R}^{n}$ are denoted by $\mathbf{S}_{1}$ and $\mathbf{S}_{1}^{+}$, respectively. For simplicity, a point $(1,\Theta)$ on $\mathbf{S}_{1}$ and the set $\{\Theta; (1,\Theta)\in\Gamma\}$ for a set Γ, $\Gamma\subset\mathbf{S}_{1}$, are often identified with Θ and Γ, respectively. Let $\Lambda\times\Gamma$ denote the set $\{(r,\Theta)\in \mathbf{R}^{n}; r\in\Lambda,(1,\Theta)\in\Gamma\}$, where $\Lambda\subset\mathbf{R}_{+}$ and $\Gamma\subset\mathbf{S}_{1}$. We denote the set $\mathbf{R}_{+}\times\mathbf{S}_{1}^{+}=\{(X,x_{n})\in\mathbf{R}^{n}; x_{n}>0\}$ by $\mathbf{T}_{n}$, which is called the half space.

We shall also write $h_{1}\approx h_{2}$ for two positive functions $h_{1}$ and $h_{2}$ if and only if there exists a positive constant a such that $a^{-1}h_{1}\leq h_{2}\leq ah_{1}$. We denote $\max\{u(r,\Theta),0\}$ and $\max\{-u(r,\Theta),0\}$ by $u^{+}(r,\Theta)$ and $u^{-}(r,\Theta)$, respectively.

The set $\mathbf{R}_{+}\times\Gamma$ in $\mathbf{R}^{n}$ is called a cone. We denote it by $\mathfrak{C}_{n}(\Gamma)$, where $\Gamma\subset\mathbf{S}_{1}$. The sets $I\times\Gamma$ and $I\times \partial{\Gamma}$ with an interval on R are denoted by $\mathfrak {C}_{n}(\Gamma;I)$ and $\mathfrak{S}_{n}(\Gamma;I)$, respectively. We denote $\mathfrak {C}_{n}(\Gamma)\cap S_{R}$ and $\mathfrak{S}_{n}(\Gamma; (0,+\infty))$ by $\mathfrak {S}_{n}(\Gamma; R)$ and $\mathfrak{S}_{n}(\Gamma)$, respectively.

Furthermore, we denote by dσ (resp. $dS_{R}$) the $(n-1)$-dimensional volume elements induced by the Euclidean metric on $\partial{\mathfrak{C}_{n}(\Gamma)}$ (resp. $S_{R}$) and by dw the elements of the Euclidean volume in $\mathbf{R}^{n}$.

It is known (see, e.g., [1], p.41) that

$$ \begin{gathered} \Delta^{*}\varphi(\Theta)+\lambda \varphi(\Theta)=0 \quad \textrm{in } \Gamma, \\ \varphi(\Theta)=0 \quad \textrm{on } \partial{\Gamma}, \end{gathered} $$

(1.1)

where $\Delta^{*}$ is the Laplace-Beltrami operator. We denote the least positive eigenvalue of this boundary value problem (1.1) by λ and the normalized positive eigenfunction corresponding to λ by $\varphi(\Theta)$, $\int_{\Gamma}\varphi^{2}(\Theta)\,dS_{1}=1$.

We remark that the function $r^{\aleph^{\pm}}\varphi(\Theta)$ is harmonic in $\mathfrak{C}_{n}(\Gamma)$, belongs to the class $C^{2}(\mathfrak{C}_{n}(\Gamma )\backslash\{O\})$ and vanishes on $\mathfrak{S}_{n}(\Gamma)$, where

$$2\aleph^{\pm}=-n+2\pm\sqrt{(n-2)^{2}+4\lambda}. $$

For simplicity we shall write χ instead of $\aleph^{+}-\aleph^{-}$.

For simplicity we shall assume that the boundary of the domain Γ is twice continuously differentiable, $\varphi\in C^{2}(\overline{\Gamma})$ and $\frac{\partial\varphi}{\partial n}>0$ on ∂Γ. Then (see [2], pp.7-8)

$$ \operatorname{dist}(\Theta,\partial{\Gamma})\approx\varphi(\Theta), $$

(1.2)

where $\Theta\in\Gamma$.

Let $\delta(P)=\operatorname{dist}(P,\partial{\mathfrak{C}_{n}(\Gamma)})$, we have

$$ \varphi(\Theta)\approx\delta(P) $$

(1.3)

for any $P=(1,\Theta)\in\Gamma$ (see [3, 4]).

Let $u(r,\Theta)$ be a function on $\mathfrak{C}_{n}(\Gamma)$. For any given $r\in\mathbf{R}_{+}$, the integral

$$\int_{\Gamma}u(r,\Theta)\varphi(\Theta)\,d S_{1} $$

is denoted by $\mathcal{N}_{u}(r)$ when it exists. The finite or infinite limit

$$\lim_{r\rightarrow\infty}r^{-\aleph^{+}}\mathcal{N}_{u}(r) $$

is denoted by $\mathscr{U}_{u}$ when it exists.

Remark 1

A function $g(t)$ on $(0,\infty)$ is $\mathbb{A}_{d_{1},d_{2}}$-convex if and only if $g(t)t^{d_{2}}$ is a convex function of $t^{d}$ ($d=d_{1}+d_{2}$) on $(0,\infty)$ or, equivalently, if and only if $g(t)t^{-d_{1}}$ is a convex function of $t^{-d}$ on $(0,\infty)$.

Remark 2

$\mathcal{N}_{u}(r)$ is $\mathbb{A}_{\aleph^{+},\gamma-1}$-convex on $(0,\infty)$, where u is a subharmonic function on $\mathfrak{C}_{n}(\Gamma)$ such that

$$ \limsup_{P\in\mathfrak{C}_{n}(\Gamma),P\rightarrow Q }u(P)\leq0 $$

(1.4)

for any $Q\in \partial{\mathfrak{C}_{n}(\Gamma)}$ (see [5]).

The function

$$\mathbb{P}_{\mathfrak{C}_{n}(\Gamma)}(P,Q)=\frac{\partial\mathbb {G}_{\mathfrak{C}_{n}(\Gamma)}(P,Q)}{\partial n_{Q}} $$

is called the ordinary Poisson kernel, where $\mathbb{G}_{\mathfrak{C}_{n}(\Gamma)}$ is the Green function.

The Poisson integral of g relative to $\mathfrak{C}_{n}(\Gamma)$ is defined by

$$\mathbb{PI}_{\mathfrak{C}_{n}(\Gamma)} [g](P)=\frac{1}{c_{n}} \int_{\mathfrak{S}_{n}(\Gamma)}\mathbb{P}_{\mathfrak {C}_{n}(\Gamma)}(P,Q)g(Q)\,d\sigma, $$

where g is a continuous function on $\partial{\mathfrak{C}_{n}(\Gamma)}$ and $\frac{\partial}{\partial n_{Q}}$ denotes the differentiation at Q along the inward normal into $\mathfrak{C}_{n}(\Gamma)$.

We set functions f satisfying

$$ \int_{\mathfrak{S}_{n}(\Gamma)}\frac{ \vert f(t,\Phi) \vert ^{p}}{1+t^{\gamma}}\,d\sigma< \infty, $$

(1.5)

where $p>0$ and

$$\gamma>\frac{-\aleph^{+}-n+2}{p}+n-1. $$

Further, we denote by $\mathcal{A}_{\Gamma}$ the class of all measurable functions $g(t,\Phi)$ ($Q=(t,\Phi)=(Y, y_{n})\in \mathfrak{C}_{n}(\Gamma)$) satisfying the following inequality:

$$ \int_{\mathfrak{C}_{n}(\Gamma)}\frac{ \vert g(t,\Phi) \vert ^{p}\varphi}{1+t^{\gamma+1}}\,dw< \infty, $$

(1.6)

and the class $\mathcal{B}_{\Gamma}$ consists of all measurable functions $h(t,\Phi)$ ($(t,\Phi)=(Y, y_{n})\in\mathfrak{S}_{n}(\Gamma)$) satisfying

$$ \int_{\mathfrak{S}_{n}(\Gamma)}\frac{ \vert h(t,\Phi) \vert ^{p}}{1+t^{\gamma-1}}\frac{\partial\varphi}{\partial n}\,d\sigma< \infty. $$

(1.7)

We will also consider the class of all continuous functions $u(t,\Phi)$ ($(t,\Phi)\in\overline{\mathfrak{C}_{n}(\Gamma)}$) harmonic in $\mathfrak{C}_{n}(\Gamma)$ with $u^{+}(t,\Phi)\in \mathcal{A}_{\Gamma}$ ($(t,\Phi)\in\mathfrak{C}_{n}(\Gamma)$), and $u^{+}(t,\Phi)\in\mathcal{B}_{\Gamma}$ ($(t,\Phi)\in\mathfrak {S}_{n}(\Gamma)$) is denoted by $\mathcal{C}_{\Gamma}$.

Remark 3

If we denote $\Gamma=S_{1}^{+}$ in (1.6) and (1.7), we have

$$\int_{T_{n}}y_{n} \bigl\vert f(Y,y_{n}) \bigr\vert \bigl(1+t^{n+2}\bigr)^{-1}\,dQ< \infty \quad \textrm{and}\quad \int_{\partial{T_{n}}} \bigl\vert g(Y,0) \bigr\vert \bigl(1+t^{n}\bigr)^{-1}\,dY< \infty. $$

Recently Zhao and Yamada (see [6]) obtained the following result.

Theorem A

Letgbe a measurable function on $\partial{T_{n}}$such that

$$\int_{\partial{T_{n}}}\frac{ \vert g(Q) \vert }{1+ \vert Q \vert ^{n}} \,dQ< \infty. $$

Then the harmonic function $\mathbb{PI}_{T_{n}}[g]$satisfies $\mathbb{PI}_{T_{n}}[g](P)=o(r \sec^{n-1}\theta_{1})$as $r\rightarrow\infty$in $T_{n}$.

Recently Wang and Qiao (see [7]) generalized Theorem A to the conical case.

Theorem B

Letgbe a continuous function on $\partial{\mathfrak{C}_{n}(\Gamma)}$satisfying (1.5) with $p=1$and $\gamma=-\aleph^{-}+1$. Then

$$\mathscr{U}_{\mathbb{PI}_{\mathfrak{C}_{n}(\Gamma)}[g]}=\mathscr {U}_{\mathbb{PI}_{\mathfrak{C}_{n}(\Gamma)}[ \vert g \vert ]}=0. $$

2 Results

Our main aim in this paper is to give the least harmonic majorant of a nonnegative subharmonic function on $\mathfrak{C}_{n}(\Gamma)$. For related results, we refer the reader to the papers [8, 9].

Theorem 1

Ifuis a subharmonic function on a domain containing $\overline{\mathfrak{C}_{n}(\Gamma)}$, $u\geq0$on $\mathfrak{C}_{n}(\Gamma)$and $u'=u\vert\partial{\mathfrak{C}_{n}(\Gamma )}$ (the restriction ofuto $\partial{\mathfrak{C}_{n}(\Gamma)}$) satisfies (1.5), then the limit $\mathscr{U}_{u}$ ($0\leq\mathscr{U}_{u}\leq+\infty$) exists. Further, if $\mathscr{U}_{u}<+\infty$, then

$$ u(P)\leq h_{u}(P)=\mathbb{PI}_{\mathfrak{C}_{n}(\Gamma)} \bigl[u'\bigr](P)+M\mathscr {U}_{u}r^{\aleph^{+}} \varphi(\Theta)\quad \bigl(P=(r,\Theta)\in\mathfrak {C}_{n}(\Gamma) \bigr), $$

(2.1)

where $h_{u}(P)$is the least harmonic majorant ofuon $\mathfrak{C}_{n}(\Gamma)$.

3 Main lemmas

Lemma 1

Letube a function subharmonic on $\mathfrak{C}_{n}(\Gamma)$satisfying (1.4) for any $Q\in\partial {\mathfrak{C}_{n}(\Gamma)}$. Then the limit $\mathscr{U}_{u}$ ($-\infty<\mathscr{U}_{u}\leq+\infty$) exists.

Proof

It suffices to prove that the limit $\lim_{r\rightarrow0}r^{\gamma-1}\mathcal{N}_{u}(r)$ exists, then apply it to the function

$$u''(r,\Theta)=r^{2-n}(u\circ K) (r,\Theta), $$

where $K: (r,\Theta)\rightarrow(r^{-1},\Theta)$ is the Kelvin transform (see [10], pp.36-37). Consider the auxiliary function

$$I(s)=s^{\frac{\aleph^{+}}{\chi}}\mathcal{N}_{u}\bigl(s^{-\frac{1}{\chi}}\bigr) $$

on $(a^{-\chi},+\infty)$. Then, from Remarks 1 and 2, $I(s)$ is a convex function on $(a^{-\chi},+\infty)$. Hence

$$\zeta=\lim_{s\rightarrow\infty}s^{-1}I(s)=\lim_{r\rightarrow0}r^{\gamma -1} \mathcal{N}_{u}(r)\quad (-\infty< \zeta\leq+\infty) $$

exists. □

Lemma 2

Letube a nonnegative subharmonic function on $\mathfrak{C}_{n}(\Gamma)$satisfying (1.4) for any $Q\in \partial{\mathfrak{C}_{n}(\Gamma)}$and

$$ \mathscr{U}_{u^{+}}< +\infty. $$

(3.1)

Then

$$u(r,\Theta)\leq M \mathscr{U}_{u^{+}}r^{\aleph^{+}}\varphi(\Theta) $$

for any $(r,\Theta)\in\mathfrak{C}_{n}(\Gamma)$, whereMis a positive constant.

Proof

Take any $(r,\Theta)\in\mathfrak{C}_{n}(\Gamma)$ and any pair of numbers $R_{1}$, $R_{2}$ ($0<2R_{1}<r<\frac{1}{2}R_{2}<+\infty$). We define a boundary function on $\partial{\mathfrak{C}_{n}(\Gamma;(R_{1},R_{2}))}$ by

$$\nu(r,\Theta)= \textstyle\begin{cases} u(R_{i},\Theta) & \mbox{on } \{R_{i}\}\times\Gamma\ (i=1,2), \\ 0 &\mbox{on } [R_{1},R_{2}]\times\partial{\Gamma}. \end{cases} $$

This is an upper semi-continuous function which is bounded above. If we denote Perron-Wiener-Brelot solution of the Dirichlet problem on $\mathfrak{C}_{n}(\Gamma;(R_{1},R_{2}))$ with ν by $H_{\nu}((r,\Theta);\mathfrak{C}_{n}(\Gamma;(R_{1},R_{2})))$, then we have

$$\begin{aligned} u(r,\Theta) \leq& H_{\nu}\bigl((r,\Theta); \mathfrak{C}_{n}\bigl(\Gamma;(R_{1},R_{2})\bigr) \bigr) \\ \leq& \frac{1}{c_{n}} \int_{\Gamma}u^{+}(R_{1},\Theta) \frac{\partial \mathbb{G}_{\mathfrak{C}_{n}(\Gamma;(R_{1},R_{2}))}((R_{1},\Phi),(r,\Theta ))}{\partial R}R_{1}^{n-1}\,dS_{1} \\ &{}-\frac{1}{c_{n}} \int_{\Gamma}u^{+}(R_{2},\Theta) \frac{\partial \mathbb{G}_{\mathfrak{C}_{n}(\Gamma;(R_{1},R_{2}))}((R_{2},\Phi),(r,\Theta ))}{\partial R}R_{2}^{n-1}\,dS_{1}, \end{aligned}$$

which gives that

$$ u(r,\Theta)\leq M R_{1}^{\gamma-1} \mathcal{N}_{u^{+}}(R_{1})r^{\aleph ^{-}}\varphi(\Theta)+M R_{2}^{-\aleph^{+}}\mathcal {N}_{u^{+}}(R_{2})r^{\aleph^{+}} \varphi(\Theta). $$

(3.2)

As $R_{1}\rightarrow0$ and $R_{2}\rightarrow+\infty$ in (3.2), we complete the proof by (3.1). □

Lemma 3

Letgbe a locally integrable function on $\partial{\mathfrak{C}_{n}(\Gamma)}$satisfying (1.5) andube a subharmonic function on $\mathfrak{C}_{n}(\Gamma)$satisfying

$$ \limsup_{P\in\mathfrak{C}_{n}(\Gamma), P\rightarrow Q}\bigl\{ u(P)-\mathbb{PI}_{\mathfrak{C}_{n}(\Gamma)}[g](P) \bigr\} \leq0 $$

(3.3)

and

$$ \limsup_{P\in\mathfrak{C}_{n}(\Gamma), P\rightarrow Q}\bigl\{ u^{+}(P)- \mathbb{PI}_{\mathfrak{C}_{n}(\Gamma)}\bigl[ \vert g \vert \bigr](P)\bigr\} \leq0 $$

(3.4)

for any $Q\in\partial{\mathfrak{C}_{n}(\Gamma)}$. Then the limits $\mathscr{U}_{u}$and $\mathscr{U}_{u^{+}}$ ($-\infty<\mathscr{U}_{u}\leq+\infty$, $0\leq\mathscr{U}_{u^{+}}\leq+\infty$) exist, and if (3.1) is satisfied, then

$$ u(P)\leq \mathbb{PI}_{\mathfrak{C}_{n}(\Gamma)}[g](P)+M\mathscr{U}_{u^{+}}r^{\aleph ^{+}} \varphi(\Theta), $$

(3.5)

whereMis a positive constant and $P=(r,\Theta)\in\mathfrak {C}_{n}(\Gamma)$.

Proof

Consider two subharmonic functions

$$U(P)=u(P)-\mathbb{PI}_{\mathfrak{C}_{n}(\Gamma )}[g](P)\quad \mbox{and}\quad U'(P)=u^{+}(P)-\mathbb{PI}_{\mathfrak{C}_{n}(\Gamma )}\bigl[ \vert g \vert \bigr](P) $$

on $\mathfrak{C}_{n}(\Gamma)$. From (3.3) and (3.4) we have

$$\limsup_{P\in\mathfrak{C}_{n}(\Gamma), P\rightarrow Q}U(P)\leq0\quad \mbox{and}\quad \limsup _{P\in\mathfrak{C}_{n}(\Gamma), P\rightarrow Q}U'(P)\leq0 $$

for any $Q\in\partial{\mathfrak{C}_{n}(\Gamma)}$. Hence it follows from Lemma 1 that the limits $\mathscr{U}_{U}$ and $\mathscr{U}_{U'}$ ($-\infty<\mathscr{U}_{U}\leq+\infty$, $0\leq\mathscr{U}_{U'}\leq+\infty$) exist. Since

$$\mathcal{N}_{U}(r)=\mathcal{N}_{u}(r)- \mathcal{N}_{\mathbb{PI}_{\mathfrak {C}_{n}(\Gamma)}[g]}(r)\quad \mbox{and}\quad \mathcal{N}_{U'}(r)= \mathcal {N}_{u^{+}}(r)-\mathcal{N}_{\mathbb{PI}_{\mathfrak{C}_{n}(\Gamma)}[ \vert g \vert ]}(r), $$

Theorem B (Theorem 1 will be proved in the next section) gives the existences of the limits $\mathscr{U}_{u}$, $\mathscr{U}_{u^{+}}$,

$$ \mathscr{U}_{U}=\mathscr{U}_{u} \quad \mbox{and}\quad \mathscr{U}_{U'}=\mathscr{U}_{u^{+}}. $$

(3.6)

Since $0\leq U^{+}(P)\leq u^{+}(P)+(\mathbb{PI}_{\mathfrak{C}_{n}(\Gamma )}[g])^{-}(P)$ on $\mathfrak{C}_{n}(\Gamma)$, it also follows from Theorem B and (3.1) that

$$\mathscr{U}_{U^{+}}\leq\mathscr{U}_{u^{+}}< \infty. $$

Hence, by applying Lemma 2 to $U(P)$, we obtain the conclusion from (3.6). □

Lemma 4

Letgbe a nonnegative lower semi-continuous function on $\partial{\mathfrak{C}_{n}(\Gamma)}$satisfying (1.5) andube a nonnegative subharmonic function on $\mathfrak{C}_{n}(\Gamma )$such that

$$ \limsup_{P\in\mathfrak{C}_{n}(\Gamma), P\rightarrow Q}u(P)\leq g(Q) $$

(3.7)

for any $Q\in\partial{\mathfrak{C}_{n}(\Gamma)}$. Then the limit $\mathscr{U}_{u}$ ($0\leq\mathscr{U}_{u}\leq+\infty$) exists, and if $\mathscr{U}_{u}<+\infty$, then

$$u(P)\leq \mathbb{PI}_{\mathfrak{C}_{n}(\Gamma)}[g](P)+M\mathscr{U}_{u}r^{\aleph ^{+}} \varphi(\Theta) $$

for any $P=(r,\Theta)\in\mathfrak{C}_{n}(\Gamma)$.

Proof

Since −g is an upper semi-continuous function $\partial{\mathfrak{C}_{n}(\Gamma)}$, it follows from [11], p.3, that

$$ \liminf_{P\in\mathfrak{C}_{n}(\Gamma), P\rightarrow Q}\mathbb{PI}_{\mathfrak{C}_{n}(\Gamma)}[g](P) \geq g(Q) $$

(3.8)

for any $Q\in \partial{\mathfrak{C}_{n}(\Gamma)}$. We see from (3.7) and (3.8) that

$$\limsup_{P\in\mathfrak{C}_{n}(\Gamma), P\rightarrow Q}\bigl\{ u(P)-\mathbb{PI}_{\mathfrak{C}_{n}(\Gamma)}[g](P)\bigr\} \leq0 $$

for any $Q\in \partial{\mathfrak{C}_{n}(\Gamma)}$, which gives (3.3). Since g and u are nonnegative, (3.4) also holds. Thus we obtain the conclusion from Lemma 3. □

Lemma 5

Letube subharmonic on a domain containing $\overline{\mathfrak{C}_{n}(\Gamma)}$such that $u'=u\vert\partial{\mathfrak{C}_{n}(\Gamma)}$satisfies (1.5) and $u\geq0$on $\mathfrak{C}_{n}(\Gamma)$. Then $\mathbb{PI}_{\mathfrak{C}_{n}(\Gamma )}[u'](P)\leq h(P)$on $\mathfrak{C}_{n}(\Gamma)$, where $h(P)$is any harmonic majorant ofuon $\mathfrak{C}_{n}(\Gamma)$.

Proof

Take any $P'=(r',\Theta')\in \mathfrak{C}_{n}(\Gamma)$. Let ϵ be any positive number. In the same way as in the proof of Lemma 2, we can choose R such that

$$ \frac{1}{c_{n}} \int_{\mathfrak{S}_{n}(\Gamma;(R,\infty))}\mathbb {P}_{\mathfrak{C}_{n}(\Gamma)}\bigl(P',Q \bigr)u'(Q)\,d\sigma< \frac{\epsilon}{2}. $$

(3.9)

Further, take an integer j ($j>R$) such that

$$ \frac{1}{c_{n}} \int_{\mathfrak{S}_{n}(\Gamma;(0,R))}\frac{\partial\Gamma_{j}(P',Q) }{\partial n_{Q}}u'(Q)\,d\sigma< \frac{\epsilon}{2}. $$

(3.10)

Since

$$\frac{1}{c_{n}} \int_{\mathfrak{S}_{n}(\Gamma;(0,R))}\frac{\partial \mathbb{G}_{\mathfrak{C}_{n}(\Gamma;(0,j))}(P,Q) }{\partial n_{Q}}u'(Q)\,d\sigma\leq H_{u}\bigl(P;\mathfrak{C}_{n}\bigl(\Gamma;(0,j)\bigr)\bigr) $$

for any $P\in\mathfrak{C}_{n}(\Gamma;(0,j))$, we have from (3.9) and (3.10) that (see [12])

$$\begin{aligned} & \mathbb{PI}_{\mathfrak{C}_{n}(\Gamma)}\bigl[u'\bigr]\bigl(P' \bigr)-H_{u}\bigl(P';\mathfrak {C}_{n}\bigl( \Gamma;(0,j)\bigr)\bigr) \\ &\quad \leq \frac{1}{c_{n}} \int_{\mathfrak{S}_{n}(\Gamma;(0,R))}\frac{\partial \Gamma_{j}(P',Q) }{\partial n_{Q}}u'(Q)\,d\sigma \\ &\qquad {} +\frac{1}{c_{n}} \int_{\mathfrak{S}_{n}(\Gamma;(R,\infty))}\mathbb {P}_{\mathfrak{C}_{n}(\Gamma)}\bigl(P',Q \bigr)u'(Q)\,d\sigma \\ &\quad < \epsilon. \end{aligned}$$

(3.11)

Here note that $H_{u}(P;\mathfrak{C}_{n}(\Gamma;(0,j)))$ is the least harmonic majorant of u on $\mathfrak{C}_{n}(\Gamma;(0,j))$ (see [13], Theorem 3.15). If h is a harmonic majorant of u on $\mathfrak{C}_{n}(\Gamma)$, then

$$H_{u}\bigl(P';\mathfrak{C}_{n}\bigl( \Gamma;(0,j)\bigr)\bigr)\leq h\bigl(P'\bigr). $$

Thus we obtain from (3.11) that

$$\mathbb{PI}_{\mathfrak{C}_{n}(\Gamma)}\bigl[u'\bigr]\bigl(P' \bigr)< h\bigl(P'\bigr)+\epsilon, $$

which gives the conclusion of Lemma 5. □

4 Proof of Theorem 1

Let $P=(r,\Theta)$ be any point of $\mathfrak{C}_{n}(\Gamma)$ and ϵ be any positive number. By the Vitali-Carathéodory theorem (see [10], p.56), we can find a lower semi-continuous function $g'(Q)$ on $\partial{\mathfrak {C}_{n}(\Gamma)}$ such that

$$ u'(Q)\leq g'(Q)\quad \mbox{on } \mathfrak{C}_{n}(\Gamma) $$

(4.1)

and

$$ \mathbb{PI}_{\mathfrak{C}_{n}(\Gamma)}\bigl[g'\bigr](P)< \mathbb{PI}_{\mathfrak {C}_{n}(\Gamma)}\bigl[u'\bigr](P)+\epsilon. $$

(4.2)

Since

$$\lim_{P\in\mathfrak{C}_{n}(\Gamma), P\rightarrow Q}u(P)\leq u'(Q)\leq g'(Q) $$

for any $Q\in\partial{\mathfrak{C}_{n}(\Gamma)}$ from (4.1), it follows from Lemma 4 that the limit $\mathscr{U}_{u}$ exists (see [11]), and if $\mathscr{U}_{u}<+\infty$, then

$$ u(P)\leq\mathbb{PI}_{\mathfrak{C}_{n}(\Gamma)}\bigl[g' \bigr](P)+M\mathscr {U}_{u}r^{\aleph^{+}}\varphi(\Theta). $$

(4.3)

Hence we have from (4.2) and (4.3) that (2.1) holds.

Next we shall assume that $h_{u}(P)$ is the least harmonic majorant of u on $\mathfrak{C}_{n}(\Gamma)$. Set $h''(P)$ is a harmonic function on $\mathfrak{C}_{n}(\Gamma)$ such that

$$ u(P)\leq h''(P)\quad \mbox{on } \mathfrak{C}_{n}(\Gamma). $$

(4.4)

Consider the harmonic function

$$h^{\ast}(P)=h_{u}(P)-h''(P)\quad \mbox{on }\mathfrak{C}_{n}(\Gamma). $$

Since

$$h^{\ast}(P)\leq h_{u}(P)\quad \mbox{on }\mathfrak{C}_{n}( \Gamma), $$

Theorem B gives that $\mathscr{U}_{{h^{\ast}}^{+}}<+\infty$. Further, from Lemma 2 we see that

$$\limsup_{P\in\mathfrak{C}_{n}(\Gamma), P\rightarrow Q}h^{\ast}(P)=\limsup_{P\in\mathfrak{C}_{n}(\Gamma), P\rightarrow Q} \bigl\{ \mathbb{PI}_{\mathfrak{C}_{n}(\Gamma)}\bigl[u'\bigr](P)-h''(P) \bigr\} \leq0 $$

for any $Q\in \partial{\mathfrak{C}_{n}(\Gamma)}$. From Theorem B and (4.4) we know

$$\mathscr{U}_{h^{\ast}}=\mathscr{U}_{h_{u}}-\mathscr{U}_{h''}= \mathscr {U}_{u}-\mathscr{U}_{h''}\leq\mathscr{U}_{u}- \mathscr{U}_{u}=0. $$

We see from Lemma 2 that $h^{\ast}(P)\leq0$ on $\mathfrak {C}_{n}(\Gamma)$, which shows that $h_{u}(P)$ is the least harmonic majorant of $u(P)$ on $\mathfrak{C}_{n}(\Gamma)$. Theorem 1 is proved.

5 Conclusions

In this article, we have obtained a new type of boundary integral behaviors of harmonic functions in a smooth cone. As an application, we also gave the least harmonic majorant of a nonnegative subharmonic function.

6 Ethics approval and consent to participate

Not applicable.

7 Consent for publication

Not applicable.

8 List of abbreviations

Not applicable.

9 Availability of data and materials

Not applicable.