Integral representation using Green function for fractional Hardy equation

TL;DR

This paper investigates the Green function for a fractional Hardy operator, establishing an integral representation for weak solutions and advancing understanding of fractional Hardy equations.

Contribution

It introduces a Green function framework for the fractional Hardy operator and demonstrates the integral representation of weak solutions, a novel approach in this context.

Findings

Derived the Green function for the fractional Hardy operator

Proved the integral representation formula for weak solutions

Enhanced theoretical understanding of fractional Hardy equations

Abstract

Our main aim is to study Green function for the fractional Hardy operator $P:=(-\Delta)^s -\frac{\theta}{|x|^{2s}}$ in $\mathbb{R}^N$, where $0<\theta<\Lambda_{N,s}$ and $\Lambda_{N,s}$ is the best constant in the fractional Hardy inequality. Using Green function, we also show that the integral representation of the weak solution holds.

Equations

$$\{\begin{array}{rl}(-\Delta {)}^{s}u-\theta \frac{u}{|x{|}^{2s}}& =\phi (x)\text{in }{\mathbb{R}}^N\\ u& \in {\dot{H}}^s({\mathbb{R}}^N),\end{array}$$

$${\left(-\Delta \right)}^{s}u(x):=c_{N,s}\text{P.V.}{\int }_{{\mathbb{R}}^N}\frac{u(x)-u(y)}{|x-y{|}^{N+2s}}\mathrm{d}y,c_{N,s}=\frac{4^s\Gamma (N/2+s)}{{\pi }^{N/2}|\Gamma (-s)|}.$$

$${\Lambda }_{N,s}:=2^{2s}\frac{{\Gamma }^2(\frac{N+2s}{4})}{{\Gamma }^2(\frac{N-2s}{4})}$$

$${\Lambda }_{N,s}{\int }_{{\mathbb{R}}^N}\frac{|u(x){|}^2}{|x{|}^{2s}}\mathrm{d}x\le {\int }_{{\mathbb{R}}^N}|\xi {|}^{2s}|\mathcal{F}(u)(\xi ){|}^2\mathrm{d}\xi .$$

$$0<\gamma \le \frac{N-2s}{2}\text{and}\theta =2^{2s}\frac{\Gamma (\frac{\gamma +2s}{2})\Gamma (\frac{N-\gamma }{2})}{\Gamma (\frac{N-\gamma -2s}{2})\Gamma (\frac{\gamma }{2})}.$$

$$0<\gamma <\frac{N-2s}{2}.$$

$$\dot{H}^{s}(\mathbb{R}^{N}):=\bigg{\{}u\in L^{2^{*}_{s}}(\mathbb{R}^{N})\,:\,\int_{\mathbb{R}^{N}}|\xi|^{2s}|\mathcal{F}(u)|^{2}\,{\rm d}\xi<\infty\bigg{\}},\quad\text{where}\;\;2^{*}_{s}=\frac{2N}{N-2s}.$$

$$\Vert u{\Vert }_{{\dot{H}}^s({\mathbb{R}}^N)}^2:={\int }_{{\mathbb{R}}^N}|\xi {|}^{2s}|\mathcal{F}(u){|}^2\mathrm{d}\xi .$$

$${\int }_{{\mathbb{R}}^N}|\xi {|}^{2s}|\mathcal{F}(u){|}^2\mathrm{d}\xi =\frac{c_{N,s}}{2}{\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}\frac{|u(x)-u(y){|}^2}{|x-y{|}^{N+2s}}\mathrm{d}x\mathrm{d}y,\forall u\in {\dot{H}}^s({\mathbb{R}}^N).$$

$$\frac{c_{N,s}}{2}{\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}\frac{(u(x)-u(y))(\psi (x)-\psi (y))}{|x-y{|}^{N+2s}}\mathrm{d}x\mathrm{d}y-\theta {\int }_{{\mathbb{R}}^N}\frac{u\psi }{|x{|}^{2s}}\mathrm{d}x={\int }_{{\mathbb{R}}^N}\phi (x)\psi (x)\mathrm{d}x.$$

$$\bigg{(}\frac{c_{N,s}}{2}\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\frac{|u(x)-u(y)|^{2}}{|x-y|^{N+2s}}\,{\rm d}x\,{\rm d}y\,-\,\theta\,\int_{\mathbb{R}^{N}}\frac{|u|^{2}}{|x|^{2s}}\,{\rm d}x\bigg{)}^{\frac{1}{2}},$$

$$H^{s}(\mathbb{R}^{N}):=\bigg{\{}u\in L^{2}(\mathbb{R}^{N})\,:\,\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\frac{|u(x)-u(y)|^{2}}{|x-y|^{N+2s}}\,{\rm d}x\,{\rm d}y<\infty\bigg{\}},$$

$$\Vert u{\Vert }_{H^s({\mathbb{R}}^N)}^2:=\Vert u{\Vert }_{L^2({\mathbb{R}}^N)}^2+{\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}\frac{|u(x)-u(y){|}^2}{|x-y{|}^{N+2s}}\mathrm{d}x\mathrm{d}y.$$

$$P:=(-\Delta {)}^{s}u-\theta \frac{u}{|x{|}^{2s}},0<\theta <{\Lambda }_{N,s}.$$

$$\tilde{S}(t)g={\int }_{{\mathbb{R}}^N}p(t,x,y)g(y)\mathrm{d}y,t>0,g\in L^2({\mathbb{R}}^N).$$

$$G:{\mathbb{R}}^N\setminus \{0\}\times {\mathbb{R}}^N\setminus \{0\}⟶[-\infty ,\infty ]$$

$$G(x,y)=G(y,x),\forall y,x\ne 0;$$

$$G(x,.)\in L_{loc}^1({\mathbb{R}}^N),\text{and}{\int }_{{\mathbb{R}}^N}\frac{|G(x,y)|}{1+|y{|}^{N+2s}}\mathrm{d}y<\infty ;$$

$$P{G}_P(x,\cdot )={\delta }_x\text{in}{\mathbb{R}}^N,$$

$$G_P(x,y):={\int }_0^{\infty }p(t,x,y)\mathrm{d}t,x,y\in {\mathbb{R}}^N\setminus \{0\},x\ne y.$$

$$x⟶{\int }_{{\mathbb{R}}^N}G(x,y)\phi (y)\mathrm{d}y,\text{a.e.}x,$$

$$\psi (x):={\int }_{{\mathbb{R}}^N}{G}_P(x,y)\phi (y)\mathrm{d}y,x\ne 0,$$

$$u(x)={\int }_{{\mathbb{R}}^N}{G}_P(x,y)\phi (y)\mathrm{d}y,x\ne 0,$$

$$P{G}_P(x,\cdot )={\delta }_x\text{in}{\mathbb{R}}^N,$$

$${\int }_{{\mathbb{R}}^N}{G}_P(x,y)\phi (y)\mathrm{d}y<\infty ,$$

$${\dot{H}}^s({\mathbb{R}}^N)\ni \psi (x)={\int }_{{\mathbb{R}}^N}{G}_P(x,y)\phi (y)\mathrm{d}y,x\ne 0.$$

$$\Phi (x,.)\in L_{loc}^1({\mathbb{R}}^N)\text{and}{\int }_{{\mathbb{R}}^N}\frac{|\Phi (x,y)|}{1+|y{|}^{N+2s}}\mathrm{d}y<\infty$$

$$P\Phi (x,\cdot )={\delta }_x\text{in}{\mathbb{R}}^N,x\ne 0,$$

$${\psi }_{\phi }:={\int }_{{\mathbb{R}}^N}\Phi (x,y)\phi (y)\mathrm{d}y\in {\dot{H}}^s({\mathbb{R}}^N)\forall 0\le \phi \in C_c^{\infty }({\mathbb{R}}^N),$$

$$\Phi_{F}(x)=\left\{\begin{array}[]{lll}a(N,s)\,|x|^{-N+2s}&\text{for}\;n\neq 2s,\\ a(N,s)\,\log|x|&\text{for}\;n=2s,\end{array}\right.$$

Full text

1. Introduction

The aim of this work is to show that the potential theoretic solution for fractional Hardy equation coincides with the weak solution and it admits an integral representation using the Green function of the fractional Hardy operator. There is a wide literature regarding problems involving the fractional Hardy potential. Avoiding to disclose the discussion we refer to the following (far from being complete) list of works and references therein [1, 2, 3, 4, 6, 7, 11, 12, 13, 14].

In this work, we consider the following fractional Hardy equation

[TABLE]

where $s\in(0,1)$ is fixed, $N>2s$, $\varphi\in C_{c}(\mathbb{R}^{N})$ and $(-\Delta)^{s}$ denotes the fractional Laplace operator which can be defined for the Schwartz class functions $\mathcal{S}(\mathbb{R}^{N})$ as follows:

[TABLE]

Here $\theta\in(0,\Lambda_{N,s})$, where

[TABLE]

denotes the sharp constant in the Hardy inequality

[TABLE]

In above inequality $\mathcal{F}(u)$ denotes the Fourier transform of $u$.

It is well-known that (see [7], [14]) if $0<\theta\leq\Lambda_{N,s}$, then there exists a unique number $\gamma$ such that

[TABLE]

Further, $\theta=\Lambda_{N,s}$ implies $\gamma=\frac{N-2s}{2}$. Therefore in this paper we consider the range

[TABLE]

This constant $\gamma$ plays an important role in the analysis of fractional Hardy equations (see [14]) and it will appear in several equations below.

Definition 1.1**.**

Let $s\in(0,1)$. We define the homogeneous fractional Sobolev space of order $s$ as

[TABLE]

In particular, $\dot{H}^{s}(\mathbb{R}^{N})$ is the completion of $C^{\infty}_{c}(\mathbb{R}^{N})$ under the norm

[TABLE]

It is also known that (see [14]) for $s\in(0,1)$, $N\geq 1$, it holds

[TABLE]

Definition 1.2**.**

We say that $u\in\dot{H}^{s}(\mathbb{R}^{N})$ is a weak solution of (1) if for every $\psi\in C_{c}^{\infty}(\mathbb{R}^{N})$ there holds

[TABLE]

Remark 1.1**.**

Since $\theta<\Lambda_{N,s}$ implies, norm in (1.3) is equivalent to

[TABLE]

it is easy to see that the weak solution of (1) is unique. **

Definition 1.3**.**

Let $s\in(0,1)$. We define the fractional Sobolev space of order $s$ as

[TABLE]

and

[TABLE]

As mentioned above we would be interested in the operator

[TABLE]

Recently, it is shown in [7] that for $0<\theta\leq\Lambda_{N,s}$, $-P$ generates a strongly continuous contraction semigroup $\{\tilde{S}(t)\}$ in $L^{2}(\mathbb{R}^{N})$. Furthermore, heat kernel of $-P$ also exists in $\mathbb{R}^{N}\setminus\{0\}$ i.e., there exists a nonnegative function $p(t,x,y)$ such that

[TABLE]

see [7] for more details.

Definition 1.4**.**

We say that a function

[TABLE]

is a Green function for the operator $P$, if

• $(i)$

[TABLE]

• $(ii)$

for $x\neq 0$ ,

[TABLE]

• $(iii)$

for any $x\neq 0$, it holds

[TABLE]

in the sense of distribution.

As usual in potential theory, we define (see also Lemma 2.1 below) the function $G_{P}(x,y)$ as follows

[TABLE]

It is also known that heat kernel is symmetric i.e., $p(t,y,z)=p(t,z,y)$ (see [7, pg 6]) and this implies $G_{P}(x,y)=G_{P}(y,x)$.

Remark 1.2**.**

It is also common in literature (see [17, Appendix D]) to define Green function using integral representation. More precisely, a symmetric $G:\mathbb{R}^{N}\setminus\{0\}\times\mathbb{R}^{N}\setminus\{0\}\longrightarrow[-\infty,\infty]$ is said to be Green function of $P$ if

• $(i)$

$G(x,\cdot)\in L^{1}_{loc}(\mathbb{R}^{N})$* a.e. $x$;*

• $(ii)$

for all $\varphi\in C_{c}(\mathbb{R}^{N})$, the function

[TABLE]

belongs to $\dot{H}^{s}(\mathbb{R}^{N})$ and it is the unique weak solution to the equation (1).

We show in Theorem 1.1 that $G_{P}$ in (1.5) does satisfy these conditions.

Below we state the main results of this paper.

Theorem 1.1**.**

Let $0<\theta<\Lambda_{N,s}$, $\varphi\in C_{c}(\mathbb{R}^{N})$ and

[TABLE]

where $G_{P}$ is as defined in (1.5). Then $\psi\in\dot{H}^{s}(\mathbb{R}^{N})$ and $P\psi=\varphi$ in $\mathbb{R}^{N}$ holds in the weak sense.

Moreover, by the previous result, we are able to show a representation formula for the weak solution to (1). We have

Corollary 1.1**.**

Let $0<\theta<\Lambda_{N,s}$ and $u$ be a weak solution of the equation (1). Then there holds

[TABLE]

where $G_{P}$ is as in Theorem 1.1.

Theorem 1.2**.**

Let $0<\theta<\Lambda_{N,s}$, $x\neq 0$ and $G_{P}$ be defined as in (1.5). Then it holds that

[TABLE]

in the sense of distribution.

Thanks to Theorem 1.2 the function $G_{P}$ defined in (1.5) is actually a Green function in the meaning of Definition 1.4. Next we show that the integral representation of solution of (1) holds for more general class of functions $\varphi$, namely we prove the following:

Theorem 1.3**.**

Let the function $\varphi$ in (1) be such that $0\leq\varphi\in L^{\frac{2N}{N+2s}}(\mathbb{R}^{N})$ and $\varphi$ be lower semi-continuous. Then there exists a unique weak solution to (1).

*Further, if *

[TABLE]

then the solution (denote it by $\psi$) will be of the form

[TABLE]

In our next theorem, we prove the uniqueness of Green function.

Theorem 1.4**.**

Let $\Phi\geq 0$ be such that for $x\neq 0$,

[TABLE]

and it holds

[TABLE]

in the sense of distribution. Further, if $\Phi(x,y)=\Phi(y,x)$, $\Phi(x,y)$ is continuous for $x,y\neq 0$, $y\neq x$ and

[TABLE]

then $\psi_{\varphi}$ satisfies (1) and $\Phi=G_{P}$ in $(\mathbb{R}^{N}\setminus\{0\})\times(\mathbb{R}^{N}\setminus\{0\})$.

It is well known that Green functions are closely related to fundamental solutions. For $\theta=0$, using Fourier transform, it can be shown (see [8, Theorem 2.3]) that

[TABLE]

where $a(N,s)$ is a suitable normalizing constant, is the fundamental solution of $(-\Delta)^{s}$ in $\mathbb{R}^{N}$. Then the Green function is given by $\Phi(x,y)=\Phi_{F}(x-y)$. Fall in [12, Lemma 4.1] (take $\alpha=\gamma-\frac{N-2s}{2}$ in [12]) proved that

[TABLE]

solves $P\tilde{\Phi}=0$ in $\mathbb{R}^{N}\setminus\{0\}$. This seems some what related to fundamental solution. The main difficulty is to guess the right candidate for the Green function. Though the heat-kernel of fractional Hardy operator is obtained recently in [7], we neither know any regularity property of the heat-kernel nor if it solves the corresponding heat equation. This is another difficulty in considering the standard approach of (local PDE) in getting the Green function. To obtain the Green function for $P$ we take the approach of semigroup theory which is also closely related to the criticality theory.

It’s well-known that the integral representation of solution using Green function of the operator is very useful in studying various properties of solutions. In the case of $\theta=0$, i.e., for the fractional Laplace operator, integral representation of solution using Green function were used in many papers, to cite few we mention [9], [10], [15], [19], [20]. In our forthcoming paper [5], we establish an asymptotic behaviour of solution for the fractional Hardy equation using the integral representation of the solution.

Notation: In this paper $C,\tilde{C},\cdots$ denote the generic constant which may vary from line to line. We denote by $B_{R}(x)$ the ball cantered at $x$ and of radius $R$. By $C_{c}(\mathbb{R}^{N})$ and $C_{c}^{\infty}(\mathbb{R}^{N})$ we denote the continuous functions with compact support and $C^{\infty}$ functions with compact support respectively. $\delta_{x}$ denotes the Dirac distribution centered at $x$.

2. Integral representation solves the equation

In this Section we aim to prove Theorem 1.1–Theorem 1.4. Towards this, first we establish two sided estimate for the Green function of fractional Hardy operator $P$. In context to the local case, recently in [18], the authors obtain sharp two sided Green function estimate for second-order elliptic operator of Fuchsian-type on $\mathbb{R}^{N}.$

2.1. Estimate for Green function

In the following, for two given non negative functions $f_{1},f_{2}$, by $f_{1}\approx f_{2}$, we mean that there exists some positive constant $C$ such that $C^{-1}{f_{1}}\leq f_{2}\leq Cf_{1}$. Now we prove the following

Lemma 2.1**.**

Let $0<\theta<\Lambda_{N,s}$ and $\gamma$ the unique solution to (1.2). Then the Green function $G_{P}$ given in (1.5) is well defined. Moreover,

[TABLE]

for $x,y\in\mathbb{R}^{N}\setminus\{0\}$, $x\not=y$.

Proof.

From [7, Theorem 1.1], it follows that heat kernel $p$ of the Hardy operator in $\mathbb{R}^{N}\setminus\{0\}$ satisfies

[TABLE]

We claim that, for $0<\theta<\Lambda_{N,s}$

[TABLE]

for all $x,y\in\mathbb{R}^{N}\setminus\{0\}$, $x\not=y$. To see this, first of all note that

[TABLE]

Therefore

[TABLE]

We deduce that

[TABLE]

Analogously, since $N-2s>2\gamma$, we obtain

[TABLE]

Finally, combining (2.3) and (2.4) with (2.2) (and (1.5)) we get the conclusion. ∎

2.2. Semigroup and quadratic forms associated with $P$

Let $\tilde{S}(t):=e^{-tP}$ be the semigroup generated by $-P$ i.e.,

[TABLE]

and

[TABLE]

Lemma 2.2**.**

Let $\varphi\in C_{c}(\mathbb{R}^{N})$ and $\tilde{S}(t)$ be defined as above. For $\alpha>0$, define $u_{\alpha}$ by

[TABLE]

Then there holds

[TABLE]

Proof.

Let $K:=\mathrm{supp}(\varphi).$ We note that $u_{\alpha}$ given in (2.6) is well defined since $\{\tilde{S}(t)\}$ is a contraction semigroup. Moreover, using (1.4) we get

[TABLE]

Fix $R>>1$ be such that $K\subset\subset B_{{R}/{2}}(0).$ We write

[TABLE]

Let $x\in\mathbb{R}^{N}\setminus B_{R}(0),$ and $y\in K,$ then $|x-y|\sim|x|.$ Therefore using the estimate of $G_{P}$ from Lemma 2.1 and Hölder inequality, we have

[TABLE]

since $N>2s$ and $\gamma\in(0,(N-2s)/2).$ For $x\in B_{R}(0)$, $x\neq 0$, using (2.7) and Lemma 2.1, we have

[TABLE]

Using Young’s inequality we see that

[TABLE]

which implies,

[TABLE]

that is,

[TABLE]

Combining we get

[TABLE]

as $\gamma\in(0,({N-2s})/{2})$ and $N>2s$. Hence (2.2) and (2.10) proves the lemma.

∎

Before going further let us introduce quadratic form associated with the Hardy operator. We first define the quadratic form $\mathcal{E}$ of $(\Delta)^{s}:=-(-\Delta)^{s},$ in the usual way as in [16] (also see [7]) :

[TABLE]

for $f\in L^{2}(\mathbb{R}^{N})$ and $S(t)$ is the semigroup generated by $(\Delta)^{s}$. Moreover, the domain of the quadratic form is given by

[TABLE]

Similarly, we define the quadratic form associated with $-P$, where $P$ is the fractional Hardy operator. Recall that $\{\tilde{S}(t)\}$ is the semigroup generated by $-P.$ We define

[TABLE]

and the domain

[TABLE]

Furthermore, we can define a bilinear form on $\mathcal{D}(\mathcal{\tilde{E}})$ as follows: for $f,g\in\mathcal{D}(\mathcal{\tilde{E}})$

[TABLE]

Also note that

[TABLE]

For more details on bilinear form see [16, Chapter 1].

This form plays a key role in our studies below. Now we recall an important lemma from [7] concerning the quadratic form of $\mathcal{E}$ and $\tilde{\mathcal{E}}.$

Lemma 2.3**.**

[7]** We have $\mathcal{D}(\mathcal{E})\subset\mathcal{D}(\tilde{\mathcal{E}})$ and

[TABLE]

Remark 2.1**.**

It is also follows from [6, Proposition 5] (also see [7, (5.2)]) that

[TABLE]

2.3. Proof of Theorem 1.1

Proof.

Recall $\{\tilde{S}(t):=e^{-tP}\}$ is the semigroup generated by $-P.$ It is known (see [7, Proposition 2.4]) that $\{\tilde{S}(t)\}$ is a $L^{2}({\mathbb{R}^{N}})$ contraction semigroup, i.e.

[TABLE]

Therefore Hille-Yosida theorem states that $(\alpha I+P)$ is invertible for every $\alpha>0$ and

[TABLE]

In particular, $R(\alpha)v\in\mathcal{D}(-P)\subset\mathcal{D}(\tilde{\mathcal{E}})$.

Define $u_{\alpha}:=R(\alpha)\varphi.$ Then by above $(P+\alpha I)R(\alpha)=\mathrm{id}$ and therefore, it holds that

[TABLE]

Thus,

[TABLE]

By Lemma 2.2, we have

[TABLE]

and $R(\alpha)\varphi\in\mathcal{D}(\tilde{\mathcal{E}})$ also implies $\mathcal{\tilde{E}}(u_{\alpha})<\infty$. Therefore, applying Remark 2.1, we obtain $u_{\alpha}\in\mathcal{D}(\mathcal{E})$. Hence $u_{\alpha}\in{H}^{s}(\mathbb{R}^{N})$ for each $\alpha>0$. Thus, by Sobolev inequality $u_{\alpha}\in\dot{H}^{s}(\mathbb{R}^{N})$ for each $\alpha>0$.

Moreover, from (2.5) it is easy to see that

[TABLE]

By (2.12), we know

[TABLE]

Combining this with Lemma 2.3 yields

[TABLE]

In particular, taking $f=u_{\alpha}$ in (2.14) and defining $K:=\mathrm{supp}(\varphi)$, it follows

[TABLE]

Since $u_{\alpha}\in\dot{H}^{s}(\mathbb{R}^{N})$, choosing $\varepsilon>0$ such that $\theta+\varepsilon<\Lambda_{N,s}$ and using Remark 1.1, we get $\{u_{\alpha}\}$ is uniformly bounded in $\dot{H}^{s}(\mathbb{R}^{N})$ and $\|\sqrt{\alpha}u_{\alpha}\|_{L^{2}(\mathbb{R}^{N})}$ is uniformly bounded. Hence there exists $u\in\dot{H}^{s}(\mathbb{R}^{N})$ and $g\in L^{2}(\mathbb{R}^{N})$ such that up to a subsequence,

[TABLE]

and

[TABLE]

as $\alpha_{k}\rightarrow 0$. This implies, for $f\in C^{\infty}_{c}(\mathbb{R}^{N})$

[TABLE]

Therefore, from (2.14), it is easy see that $u$ satisfies $Pu=\varphi$ in the weak sense. Since

[TABLE]

using dominated convergence theorem and letting $\alpha\rightarrow 0$, we obtain

[TABLE]

(Here we have used $\tilde{S}(\tau)\,|\varphi|$ as the dominant function. Using Tonelli’s theorem in the definition of $\tilde{S}(\tau)\,|\varphi|$, and arguing as in the proof of (2.9), it follows that

[TABLE]

Hence $\tilde{S}(\tau)\,|\varphi|\in L^{1}(0,\infty)$.)

We also know from the definition of $\tilde{S}(\tau)\varphi$ that

[TABLE]

where $p(.,.,.)$ is the heat kernel of $P$ as described in (2.1). Therefore, using Fubini

[TABLE]

Hence $\psi=\tilde{u}$. This implies $u_{\alpha}\to\psi$ a.e.. On the other hand, since we already proved $u_{\alpha}\to u$ a.e., it implies $u=\psi$ a.e. Hence it holds $P\psi=\phi$ in the weak sense. ∎

Proof of Corollary 1.1:

Proof.

From Theorem 1.1 we know that $\psi:=\displaystyle\int_{\mathbb{R}^{N}}G_{P}(x,y)\varphi(y)\,{\rm d}y$ is a weak solution of  (1). Moreover, from Remark 1.1, we see that weak solution of (1) is unique. Therefore we get the conclusion. ∎

Proof of Theorem 1.2

Proof.

Let $x\neq 0$. To prove the theorem we need to show that

[TABLE]

Consider $\phi_{n}\in C^{\infty}_{c}(\mathbb{R}^{N})$ such that $\phi_{n}\to\delta_{x}$ in the sense of distributions. Furthermore, we can choose that $\phi_{n}\geq 0$ and $\mathrm{supp}(\phi_{n})\subset B_{\kappa}(x)$ for all $n$, where $\kappa<\frac{|x|}{2}$. Corresponding to $\phi_{n}$, we define

[TABLE]

where $G_{P}$ is the Green function of $P$ as defined in (1.5). By Theorem 1.1, $\psi_{n}\in\dot{H}^{s}(\mathbb{R}^{N})$ and

[TABLE]

for all $f\in C^{\infty}_{c}(\mathbb{R}^{N})$. Therefore,

[TABLE]

i.e.

[TABLE]

We point out that in order to use the integration by parts formula in (2.16), we argue by density since $\psi_{n}\in\dot{H}^{s}(\mathbb{R}^{N})$ and we use the fact that $f\in C^{\infty}_{c}(\mathbb{R}^{N})$, which also implies

[TABLE]

Since the heat kernel $p(t,z,y)$ is continuous in $z$ (see [7, Lemma 4.10]), it is easy to check that $z\mapsto G_{P}(z,y)$ is continuous for $z\neq y$. Therefore, $\psi_{n}(z)\to G_{P}(z,x)$ pointwise. Clearly, RHS of (2.17)$\to f(x)$ as $n\to\infty$.

Claim: If $f\in C^{\infty}_{c}(\mathbb{R}^{N})$, then taking the limit $n\to\infty$ in (2.17) yields

[TABLE]

First we note that for $f\in C^{\infty}_{c}(\mathbb{R}^{N})$, it is easy to check

[TABLE]

We split the proof of the claim in two steps.

Step 1. We show that for each $n$

[TABLE]

Using Tonelli’s theorem, we get

[TABLE]

Now for $R>2|x|$, repeating an estimate as in the derivation of (2.9) we see that

[TABLE]

Also it is easy to see that for the choice of $R$ and $z\in\mathbb{R}^{N}\setminus B_{R}(0)$, it holds

[TABLE]

where the constant $C$ does not depend on $z$. Therefore from (2.19) and also using (2.18), we obtain

[TABLE]

This proves Step 1.

Step 2. We complete the proof of the claim in this step. Using Step 1 and Fubini’s theorem we see that

[TABLE]

We recall that, $G_{P}(z,y)=G_{P}(y,z)$. Therefore, if we can show that

[TABLE]

is continuous at $y=x\neq 0$, then we can conclude

[TABLE]

This would complete the proof of the claim.

Hence, we are left to show that $y\longmapsto\displaystyle\int_{\mathbb{R}^{N}}G_{P}(z,y)P(f)(z)\,{\rm d}z$ is continuous at $y=x\neq 0$.

To see this, first we observe that as $x\neq 0$ and $\gamma\in(0,(N-2s)/{2})$, Lemma 2.1 implies given $\varepsilon>0$ there exists $\delta,\,r_{0}\in(0,\frac{|x|}{3})$ , such that

[TABLE]

Now choose $r_{1}<\min\{r_{0},\frac{\delta}{2}\}$. Then we have

[TABLE]

Also, choose $\delta_{1}<\frac{|x|}{3}$ satisfying

[TABLE]

Now we write

[TABLE]

As we are interested in $y\to x$, let us assume $y\in B_{\kappa}(x)$. Also, by our choice,

[TABLE]

Thus using Lemma 2.1, (2.18) and (2.22) it follows that

[TABLE]

and

[TABLE]

Next, we observe from (2.18) that for $z\in B_{\frac{\delta}{2}}(x)$, $|Pf(z)|\leq C,$ where $C$ is independent of $z$. Therefore, from (2.21) and (2.20), it follows

[TABLE]

Further, by dominated convergence theorem

[TABLE]

where we use (2.18). Therefore, using (2.20), (2.23), (2.24), (2.25) and (2.26), there exists a neighborhood of $x$ such that

[TABLE]

This completes the proof of the claim as $\varepsilon$ is arbitrary. ∎

Proof of Theorem 1.3:

Proof.

Let $0\leq\varphi\in L^{\frac{2N}{N+2s}}(\mathbb{R}^{N})$ and $\varphi$ be lower semi-continuous. Therefore, there exists a sequence of bounded continuous functions $\varphi_{n}$ satisfying $0\leq\varphi_{1}\leq\varphi_{2}\leq\cdots\leq\varphi_{n}\leq\cdots\varphi$ and

[TABLE]

Furthermore, by multiplying with suitable cut-off functions we can also assume that $\varphi_{n}\in C_{c}(\mathbb{R}^{N})$.

Let

[TABLE]

Then by Theorem 1.1, $\psi_{n}\in\dot{H}^{s}(\mathbb{R}^{N})$ and satisfies

[TABLE]

in the weak sense. Since $\|\varphi_{n}\|_{L^{\frac{2N}{N+2s}}(\mathbb{R}^{N})}\leq\|\varphi\|_{L^{\frac{2N}{N+2s}}(\mathbb{R}^{N})}$, taking $\psi_{n}$ as a test function in (2.28), we obtain

[TABLE]

where $S$ is the Sobolev constant. Using Remark 1.1 on the LHS of above expressions yields $\|\psi_{n}\|_{\dot{H}^{s}(\mathbb{R}^{N})}\leq\tilde{C}\quad\forall\,n$. Therefore, there exists $\psi\in\dot{H}^{s}(\mathbb{R}^{N})$ such that $\psi_{n}\rightharpoonup\psi$ in $\dot{H}^{s}(\mathbb{R}^{N})$ and $\psi_{n}\to\psi$ a.e. From (2.28), we also have

[TABLE]

for all $f\in C^{\infty}_{c}(\mathbb{R}^{N})$. Now we take the limit $n\to\infty$ on both sides of the above expressions. For the 2nd term on the LHS we can pass the limit inside the integral sign using Vitali’s convergence theorem via Hardy inequality. Hence $\psi$ satisfies (1). By uniqueness of weak solution, (1) has unique solution as $\theta<\Lambda_{N,s}$. Further, from (2.27) we have for $x\neq 0$,

[TABLE]

Using (1.6), applying Lebesgue monotone convergence theorem the above expression yields

[TABLE]

This completes the proof.

∎

Proof of Theorem 1.4:

Proof.

Let $\Phi$ satisfy

[TABLE]

in the sense of distribution and

[TABLE]

Let $f\in C_{c}^{\infty}(\mathbb{R}^{N})$ be arbitrary and we further choose $0\leq\varphi\in C_{c}^{\infty}(\mathbb{R}^{N})$ arbitrarily, then from (2.30) we have

[TABLE]

Consequently,

[TABLE]

Claim:

[TABLE]

Assuming the claim, first we complete the proof. Using the claim, we obtain from (2.32) that

[TABLE]

Since $\psi_{\varphi}\in\dot{H}^{s}(\mathbb{R}^{N})$ and $f\in C_{c}^{\infty}(\mathbb{R}^{N})$, using an argument as in (2.15) –(2.17) (see the line just after (2.17)) we obtain

[TABLE]

Since in the above expression $f\in C_{c}^{\infty}(\mathbb{R}^{N})$ is arbitrary, we conclude $\psi_{\varphi}$ satisfies

[TABLE]

in the weak sense. Therefore, using Corollary 1.1, we have

[TABLE]

Combining (2.34) with (2.31) yields

[TABLE]

Since $\varphi\geq 0$ is an arbitrary $C_{c}^{\infty}$ function in $\mathbb{R}^{N}$ and since $G_{P}$ and $\Phi$ are continuous functions in $y$ for $y\neq x$ and $y\neq 0$ and in $L^{1}_{loc}(\mathbb{R}^{N})$ a.e. $x$ , we see from (2.35) and a density argument that

[TABLE]

which in turn, implies $G_{P}(x,\cdot)=\Phi(x,\cdot)$ in $\mathbb{R}^{N}\setminus\{x\}$. Since $x\neq 0$ is also arbitrary, $G_{P}=\Phi$ in $(\mathbb{R}^{N}\setminus\{0\})\times(\mathbb{R}^{N}\setminus\{0\})$.

Therefore, we are left to prove the claim (2.33). Since $\Phi$ and $\varphi$ are nonnegative functions, in order to justify that claim using Fubini, it’s enough to show that

[TABLE]

Using Tonelli’s theorem and (2.18), we estimate the above integration as below

[TABLE]

Since $\psi_{\varphi}\in\dot{H}^{s}(\mathbb{R}^{N})$, using Hölder inequality

[TABLE]

On the other hand, since $\frac{1}{1+|y|^{N+2s}}\in L^{\frac{2N}{N+2s}}(\mathbb{R}^{N})$ and $\psi_{\varphi}\in\dot{H}^{s}(\mathbb{R}^{N})$ implies $\psi_{\varphi}\in L^{2^{*}}_{s}(\mathbb{R}^{N})$, using Hölder inequality, we immediately get that 1st integral on the RHS of (2.3) is also finite. Hence the claim follows.

This completes the proof of the theorem.

∎

Acknowledgement: The research of A. Biswas is partially supported by an INSPIRE faculty fellowship (IFA13/MA-32) and DST-SERB grants EMR/2016/004810,

MTR/2018/000028. M. Bhakta is partially supported by the INSPIRE faculty fellowship IFA13/MA-33 and SERB MATRICS grant. D. Ganguly is partially supported by INSPIRE faculty fellowship (IFA17-MA98).