Gradient of solution of the Poisson equation in the unit ball and related operators

TL;DR

This paper calculates the bounds of an integral operator related to the gradient of solutions to the Poisson equation in a unit ball, providing insights into its behavior in different function spaces.

Contribution

It determines the exact $L^1 o L^1$ and $L^{f ext{infty}} o L^{ extbf ext{infty}}$ norms of the operator associated with the Poisson equation's gradient in the unit ball.

Findings

Calculated $L^1 o L^1$ norm of the operator.

Calculated $L^{ ext{infty}} o L^{ ext{infty}}$ norm of the operator.

Provided explicit bounds for the operator in these norms.

Abstract

In this paper we determine the $L^1\to L^1$ and $L^{\infty}\to L^\infty$ norms of an integral operator $\mathcal{N}$ related to the gradient of the solution of Poisson equation in the unit ball with vanishing boundary data in sense of distributions.

Equations

$$\Vert T\Vert =\sup \{\Vert Tx\Vert :\Vert x\Vert =1\}.$$

$$G(x,y)=c_n\left(\frac{1}{|x-y{|}^{n-2}}-\frac{1}{[x,y{]}^{n-2}}\right),$$

$$c_n=\frac{1}{(n-2){\omega }_{n-1}},$$

$$[x,y]:=|x|y|-y/|y||=|y|x|-x/|x||.$$

$$P(x,\eta )=\frac{1-|x{|}^2}{|x-\eta {|}^n},|x|<1,\eta \in S^{n-1}.$$

$$\left\{\begin{array}[]{rr}\triangle u(x)&=g,x\in\Omega\\ &u|_{\partial\Omega}=f\end{array}\right.$$

$$u(x)=P[f](x)-\mathcal{G}[g](x):={\int }_{S^{n-1}}P(x,\eta )f(\eta )d\sigma (\eta )-{\int }_{B^n}G(x,y)g(y)dy,$$

$$\left\{\begin{array}[]{rr}\triangle u(x)&=g,x\in\Omega\\ &u|_{\partial\Omega}=0\end{array}\right.$$

$$u(x)=-\mathcal{G}[g](x)=-{\int }_{\mathbf{B}}G(x,y)g(y)dy,|x|<1.$$

$${\nabla }_{x}G(x,y)=c_n(2-n)\left(\frac{x-y}{|x-y{|}^n}-\frac{|y{|}^{2}x-y}{[x,y{]}^n}\right),$$

$$\mathcal{D}[g](x):=\nabla u(x)=\frac{1}{{\omega }_{n-1}}{\int }_{\mathbf{B}}\left(\frac{x-y}{|x-y{|}^n}-\frac{|y{|}^{2}x-y}{[x,y{]}^n}\right)g(y)dy.$$

$$\mathcal{N}[f](x)=\frac{1}{{\omega }_{n-1}}{\int }_{\mathbf{B}}\left|\frac{x-y}{|x-y{|}^n}-\frac{|y{|}^{2}x-y}{[x,y{]}^n}\right|f(y)dy.$$

$$\sum _{n=0}^{\infty }\frac{(a{)}_n(b{)}_n}{n!(c{)}_n}{t}^n,\text{mbox}for|t|<1,$$

$${\int }_0^{\pi }\frac{{\sin }^{\mu -1}t}{(1+r^2-2r\cos t{)}^{\nu }}dt=\mathrm{B}\left(\frac{\mu }{2},\frac{1}{2}\right)F\left(\nu ,\nu +\frac{1-\mu }{2};\frac{1+\mu }{2},r^2\right)$$

$$k(p-m-k)⩾0(⩽0)$$

$$\Gamma (p)\Gamma (m)⩾(⩽)\Gamma (p-k)\Gamma (m+k).$$

$$T_{x}y=\frac{(1-|x{|}^2)(y-x)-|y-x{|}^{2}x}{[x,y{]}^2},$$

$$|T_{x}y|=\left|\frac{x-y}{[x,y]}\right|$$

$$dy={\left(\frac{1-|x{|}^2}{[z,-x{]}^2}\right)}^{n}dz.$$

$$\Vert \mathcal{N}{\Vert }_{L^{\infty }\to L^{\infty }}=\frac{2n{\pi }^{n/2}}{(n+1)\Gamma (n/2)}.$$

$$\Vert \mathcal{N}{\Vert }_{L^{\infty }\to L^{\infty }}=\underset{x\in \mathbf{B}}{\sup }{\int }_{\mathbf{B}}\left|\frac{x-y}{|x-y{|}^n}-\frac{|y{|}^{2}x-y}{[x,y{]}^n}\right|dy.$$

$$\underset{x\in \mathbf{B}}{\sup }K(x),$$

$$K(x)={\int }_{\mathbf{B}}\left|\frac{x-y}{|x-y{|}^n}-\frac{|y{|}^{2}x-y}{[x,y{]}^n}\right|dy.$$

$$T_{-x}(z)=\frac{(1-|x{|}^2)(y+x)+x|z+x{|}^2}{[z,-x{]}^2}.$$

$$x-T_{-x}(z)=\frac{(1-|x{|}^2)(-x|z{|}^2-z)}{[z,-x{]}^2},\left|x-T_{-x}(z)\right|=|z|\frac{1-|x{|}^2}{[z,-x]}.$$

$$\begin{array}{rl}& \left|\frac{x-y}{|x-y{|}^n}-\frac{|y{|}^{2}x-y}{[x,y{]}^n}\right|\\ & =\frac{1}{|x-y{|}^n}\left|(x-y)-(|y{|}^{2}x-y){\left|\frac{x-y}{[x,y]}\right|}^n\right|\\ & =\frac{1}{|x-y{|}^n}\left|(x-y)-(|y{|}^{2}x-y)|z{|}^n\right|\\ & =\frac{1}{|x-T_{-x}z{|}^n}\left|(x-T_{-x}z)-(|T_{-x}z{|}^{2}x-T_{-x}z)|z{|}^n\right|\\ & =\frac{[z,-x{]}^n}{(1-|x{|}^2{)}^n|z{|}^n}\left|\frac{(1-|x{|}^2)(-x|z{|}^2-z)}{[z,-x{]}^2}+\frac{(1-|x{|}^2)(z+x)}{[z,-x{]}^2}|z{|}^n\right|\\ & =\frac{[z,-x{]}^n}{(1-|x{|}^2{)}^n|z{|}^n}\frac{(1-|x{|}^2)}{[z,-x{]}^2}|z|\left|(-x|z|-z/|z|)+|z{|}^{n-1}(z+x)\right|\\ & =\frac{[z,-x{]}^{(n-2)}}{(1-|x{|}^2{)}^{(n-1)}|z{|}^{n-1}}\left||z{|}^{n-1}(z+x)-(x|z|+z/|z|)\right|.\end{array}$$

$$\begin{array}{rl}I=& \underset{x\in B^n}{\sup }K(x)\\ & =\underset{x\in B^n}{\sup }(1-|x{|}^2){\int }_0^{1}dr{\int }_{\mathbf{S}}\frac{\left|rx(r^{n-2}-1)+\xi (r^n-1)\right|}{|rx+\xi {|}^{n+2}}d\xi .\end{array}$$

$$\left|rx(r^{n-2}-1)+\xi (r^n-1)\right|⩽(1-r^{n-2})|rx+\xi |+r^{n-2}-r^n,$$

$$\frac{\left|rx(r^{n-2}-1)+\xi (r^n-1)\right|}{|rx+\xi {|}^{n+2}}⩽\frac{(1-r^{n-2})|rx+\xi |}{|rx+\xi {|}^{n+2}}+\frac{r^{n-2}-r^n}{|rx+\xi {|}^{n+2}}.$$

$$I⩽\underset{x\in B^n}{\max }(1-|x{|}^2){\int }_0^{1}dr{\int }_{\mathbf{S}}\frac{\left((1-r^{n-2})|rx+\xi |+r^{n-2}-r^n\right)}{|rx+\xi {|}^{n+2}}d\xi .$$

Full text

1. Introduction and Notation

We denote by $\mathbf{B}={B}^{n}$ and $\mathbf{S}={S}^{n-1}$ the unit ball and the unit sphere in $\mathbf{R}^{n}$ respectively. We will assume that $n>2$ (the case $n=2$ has been already treated in [10, 11]). By the vector norm $|\cdot|$ we consider the standard Euclidean distance $|x|=(\sum_{i=1}^{n}x_{i}^{2})^{\frac{1}{2}}.$

The norm of an operator $T:X\rightarrow Y$ defined on the normed space $X$ with image in the normed space $Y$ is defined as

[TABLE]

Let $G$ be the Green function, i.e., the function

[TABLE]

where

[TABLE]

where $\omega_{n-1}$ is the Hausdorff measure of $S^{n-1}$ and

[TABLE]

The Poisson kernel $P$ is defined

[TABLE]

We are going to consider the Poisson equation

[TABLE]

where $f:S^{n-1}\rightarrow\mathbf{R}$ is bounded integrable function on the unit sphere $S^{n-1},$ and $g:B^{n}\rightarrow\mathbf{R}$ is a continuous function.

The solution of the equation in the sense of distributions is given by

[TABLE]

$|x|<1.$ Here $d\sigma$ is the normalized Lebesgue $n-1$ dimensional measure of the unit sphere $\mathbf{S}=S^{n-1}$.

Our main focus of observation is related to the special case of Poisson equation with Dirichlet boundary condition

[TABLE]

where $g\in L^{\infty}(B^{n}).$ The weak solution is then given by

[TABLE]

The problem of estimating the norm of the operator $\mathcal{G}$ in case of various $L^{p}-$ spaces was established by both authors in [12].

Since

[TABLE]

this naturally induces the differential operator related to Poisson equation

[TABLE]

Related to the problem of estimating the norm of the operator $\mathcal{D}$, we are going to observe the operator $\mathcal{N}:L^{\infty}(B^{n})\rightarrow L^{\infty}(B^{n})$ defined by

[TABLE]

The main goal of our paper is related to estimating various norms of the integral operator and $\mathcal{N}$. Then we use those results to obtain some norm estimates of the operator $\mathcal{D}.$ The compressive study of this problem for $n=2$ has been done by the first author in [10, 11] and by Dostanić in [5, 6]. For related results we refer to the papers [3, 4].

1.1. Gauss hypergeometric function

Through the paper we will often use the properties of the hypergeometric functions. First of all, the hypergeometric function $F(a,b,c,t)={{}_{2}F_{1}}(a,b;c;t)$ is defined by the series expansion

[TABLE]

and by the continuation elsewhere. Here $(a)_{n}$ denotes shifted factorial, i.e. $(a)_{n}=a(a+1)...(a+n-1)$ and $a$ is any real number.

The following identity will be used in proving the main results of this paper:

[TABLE]

(see, e.g., Prudnikov, Brychkov and Marichev [14, 2.5.16(43)]), where $\mathrm{B}$ is the beta function.

By using the Chebychev’s inequality one can easily obtain the following inequality for Gamma function (see [8]).

Proposition 1.1**.**

Let $m$, $p$ and $k$ be real numbers with $m,p>0$ and $p>k>-m$: If

[TABLE]

then we have

[TABLE]

1.2. Möbius transformations of the unit ball

The set of isometries of the hyperbolic unit ball $B^{n}$ is a Kleinian subgroup of all Möbius transformations of the extended space $\overline{\mathbf{R}}^{n}$ onto itself denoted by $\mathbf{Conf}(\mathbf{B})=\mathbf{Isom}(\mathbf{B})$. We refer to the Ahlfors’ book [2] for detailed survey to this class of important mappings.

[TABLE]

and

[TABLE]

[TABLE]

2. The $L^{\infty}$ norm of the operator $\mathcal{N}$

In this section we are going to find the norm of the operator $\mathcal{N},$ defined in (1.6), and by using this we estimate the norm of operator $\mathcal{D}$.

Theorem 2.1**.**

Let ${\mathcal{N}}:L^{\infty}(\mathbf{B})\rightarrow L^{\infty}(\mathbf{B})$ be the operator defined in (1.6). Then

[TABLE]

Proof.

At the beginning, let us note that

[TABLE]

So, we need to find

[TABLE]

where

[TABLE]

Now we are going to use the change of variables $y=T_{-x}z(T_{x}y=z),$ where $T_{-x}:\mathbf{B}\rightarrow\mathbf{B}$ is the Möbius transform defined by

[TABLE]

Now we use the following relations $|T_{x}(y)|=\left|\frac{x-y}{[x,y]}\right|$ and

[TABLE]

We have that

[TABLE]

According to the identity (LABEL:ide), we have

[TABLE]

Further we have the following simple inequality

[TABLE]

and thus

[TABLE]

So

[TABLE]

Then we have

[TABLE]

By (1.7) we obtain

[TABLE]

In view of

[TABLE]

we then infer

[TABLE]

Hence

[TABLE]

with

[TABLE]

Here

[TABLE]

where $A_{0}(r)=1-r^{n}$, $r=|x|$, and for $m\geqslant 1$

[TABLE]

Thus

[TABLE]

where

[TABLE]

and

[TABLE]

Then $a_{m}<0$ if and only if

[TABLE]

Then by (1.9) we have

[TABLE]

and so

[TABLE]

Since the last expression increases in $m$ because

[TABLE]

we have

[TABLE]

So

[TABLE]

what we needed to prove.

∎

Corollary 2.2**.**

Let ${\mathcal{D}}$ be the mapping defined in (1.5) and $v=\nabla u={\mathcal{D}}g,$ $g\in L^{\infty}(B^{n}).$ Then

[TABLE]

Proof.

At the beginning, let us note that

[TABLE]

For $x\in\mathbf{B}$ we have

[TABLE]

So, we obtain the upper estimate for the gradiente of $u,$ i.e.

[TABLE]

∎

3. The $L^{1}$ norm of the operator $\mathcal{N}$

In the sequel let us state a well-known result related to the Riesz potential.

Let $\Omega$ be a domain of $R^{n},$ and let $|\Omega|$ be its volume. For $\mu\in(0,1]$ define the operator $V_{\mu}$ on the space $L^{1}(\Omega)$ by the Riesz potential

[TABLE]

The operator $V_{\mu}$ is defined for any $f\in L^{1}(\Omega),$ and $V_{\mu}$ is bounded on $L^{1}(\Omega),$ or more generally we have the next lemma.

Lemma 3.1**.**

([9], pp. 156-159]). Let $V_{\mu}$ be defined on the $L^{p}(\Omega)$ with $p>0.$ Then $V_{\mu}$ is continuous as a mapping $V_{\mu}:L^{p}(\Omega)\rightarrow L^{q}(\Omega),$ where $1\leq q\leq\infty,$ and

[TABLE]

Moreover, for any $f\in L^{p}(\Omega)$

[TABLE]

Theorem 3.2**.**

The norm of the operator $\mathcal{N}:L^{1}\to L^{1}$ is $\frac{1}{n-2}$.

Corollary 3.3**.**

Let $g\in L^{1}(\mathbf{B})$ and $v=\nabla u=\mathcal{D}[g]$. Then

[TABLE]

In order to prove Theorem 3.2 we need the following lemma

Lemma 3.4**.**

Let $H(x,y)=\frac{y-x}{|y-x|^{n}}-\frac{|x|^{2}y-x}{[y,x]^{n}}$, and let

[TABLE]

Then

[TABLE]

Proof of Lemma 3.4.

We need to find

[TABLE]

We will show that its supremum is achieved for $x=0$. We first have

[TABLE]

Further we have

[TABLE]

We use the change of variables $z=T_{x}y,\enspace\mbox{i.e.}\enspace T_{-x}z=y,$ where $T_{x}y$ is the Möbius transform

[TABLE]

We obtain

[TABLE]

Assume without loss of generality that $x=|x|e_{1}$. Further for $\xi=(\xi_{1},...,\xi_{n}),$

[TABLE]

where

[TABLE]

with

[TABLE]

In the first appearance of hypergemetric function we used (1.7).

On the other hand similarly we prove that

[TABLE]

where

[TABLE]

Hence

[TABLE]

This implies (3.1). ∎

Proof of Theorem 3.3 and Corollary 3.2.

Since

[TABLE]

where $\mathcal{N}^{\ast}$ is appropriate adjoint operator and

[TABLE]

we have

[TABLE]

So Theorem 3.3 follows from Lemma 3.4.

On the other hand, Corollary 3.2 follows from the following inequality

[TABLE]

∎

At this point let us point out the fact that

[TABLE]

where $L^{p}(\mathbf{B},\mathbf{R}^{n})$ is the appropriate Lebesgue space of vector-functions. By $\|\mathcal{D}\|_{p}$ we denote the norm of the operator $\mathcal{D}.$

By using the Ries-Thorin interpolation theorem we obtain the next estimates of the norm for the operators $\mathcal{N}$ and $\mathcal{D}$

Corollary 3.5**.**

Let us denote by $\|\mathcal{N}\|_{i}:=\|{\mathcal{N}}\|_{L^{i}\rightarrow L^{i}}$, $i\in\{1,\infty\}$. Then

[TABLE]

Conjecture 3.6**.**

We know that ${\mathcal{D}},\mathcal{N}$ maps $L^{p}(\mathbf{B})$ into $L^{\infty}(\mathbf{B})$ for $p>n$. We have that

[TABLE]

and

[TABLE]

where

[TABLE]

and

[TABLE]

Then we conjecture that

[TABLE]

and

[TABLE]