Extremals for fractional order Hardy–Sobolev–Maz’ya inequality

Abstract

In this article, we derive the existence of positive solution of a semi-linear, non-local elliptic PDE, involving a singular perturbation of the fractional laplacian, coming from the fractional Hardy–Sobolev–Maz’ya inequality, derived in this paper. We also derive symmetry properties and a precise asymptotic behaviour of solutions.

Appendix

1.1 A density property

Main aim of this section is to derive Lemma 4.1. The arguments are modifications of those in [12], where the Muckenhoupt $A_1$ properties of the weights have been used crucially. We will sketch the proof by pointing out main steps. First, let us define

$$\begin{aligned} \mathcal {W} := \left\{ u\in L^{2^*}\left( {\mathbb {R}}^{N};\frac{1}{|x'|^{{\tilde{\alpha }}2^*}}\right) : \int _{{\mathbb {R}}^{N}}\int _{{\mathbb {R}}^{N}}\frac{\left| u(x)-u(y) \right| ^2 dxdy}{\left| x-y \right| ^{N+2s}|x'|^{{\tilde{\alpha }}}|y'|^{{\tilde{\alpha }}}}< \infty \right\} \!, \end{aligned}$$

endowed with the following norm

$$\begin{aligned} \left| \left| u \right| \right| _{\mathcal {W}} := \left[ \left[ u\right] \right] _{s,{\tilde{\alpha }},{\mathbb {R}}^{N}} + \left| \left| u \right| \right| _{2^*,{\tilde{\alpha }},{\mathbb {R}}^{N}} . \end{aligned}$$

Here,

$$\begin{aligned} \left| \left| u \right| \right| _{2^*,{\tilde{\alpha }},{\mathbb {R}}^{N}} : = \left( \int _{{\mathbb {R}}^{N}} \frac{\left| u(x) \right| ^{2^*}}{|x'|^{{\tilde{\alpha }}2^*}}dx\right) ^{\frac{1}{2^*}}, \end{aligned}$$

and the semi-norm $\left[ \left[ u\right] \right] _{s,{\tilde{\alpha }},{\mathbb {R}}^{N}}$ is same, as defined in Sect. 4. We also, define the following:

$$\begin{aligned} \text {when }&N' = 2N, \ w(z,z) = (z,z), \ \Theta (X) =|x'|^{{\tilde{\alpha }}}|y'|^{{\tilde{\alpha }}}, \ X=(x,y); \ x,y,z\in {\mathbb {R}}^{N}, \nonumber \\ \text {and when }&N' = N, \ w(z)= z,\ \Theta (X) = |x'|^{{\tilde{\alpha }}2^*},\ X=x ;\ x,y\in {\mathbb {R}}^{N}. \end{aligned}$$

(8.1)

Next, we will prove the following lemma.

Lemma 8.1

Let $u\in C_c^\infty ({\mathbb {R}}^{N})$. Also, we consider $\eta _\epsilon$, defined by the following

$$\begin{aligned} \eta _\epsilon (x')={\left\{ \begin{array}{ll} 0 , \text { if } |x'|<\epsilon ^2 \\ \frac{\ln \left( \frac{|x'|}{\epsilon ^2}\right) }{|\ln \epsilon |} , \text { if } \epsilon ^2\le |x'|\le \epsilon \\ 1 , \text { if } |x'|>\epsilon . \end{array}\right. } \end{aligned}$$

Then, for any $0<{\tilde{\alpha }}\le (k-2s)/2$, the following are true

• (i) $\int _{{\mathbb {R}}^{N}}\int _{{\mathbb {R}}^{N}} \frac{\left| u(x)-u(y) \right| ^2}{\left| x-y \right| ^{N+2s}|x'|^{{\tilde{\alpha }}}|y'|^{{\tilde{\alpha }}}} dxdy<\infty ,$

• (ii) $\displaystyle \lim _{\epsilon \rightarrow 0} \int _{{\mathbb {R}}^{N}}\int _{{\mathbb {R}}^{N}} \frac{u^2(x)\left| \eta _\epsilon (x')-\eta _\epsilon (y') \right| ^2}{\left| x-y \right| ^{N+2s}|x'|^{{\tilde{\alpha }}}|y'|^{{\tilde{\alpha }}}}dxdy =0$.

In particular, $\eta _\epsilon u\in C_c^{0,1}({\mathbb {R}}^{N}_k)$ converges to u under the semi norm $\left[ [.]\right] _{s,{\tilde{\alpha }},{\mathbb {R}}^{N}}$, i.e $u\in \mathcal {{\dot{H}}}^{s,{\tilde{\alpha }}}({\mathbb {R}}^{N})$.

Proof

We will only prove (ii). One can easily check that (i) holds in fact for $u\in C_c^{0,1}({\mathbb {R}}^{N})$. Notice that

$$\begin{aligned}&\int _{{\mathbb {R}}^{N}}\int _{{\mathbb {R}}^{N}} \frac{u^2(x)\left| \eta _\epsilon (x')-\eta _\epsilon (y') \right| ^2}{\left| x-y \right| ^{N+2s}|x'|^{{\tilde{\alpha }}}|y'|^{{\tilde{\alpha }}}}dxdy= \int _{{\mathbb {R}}^k}\int _{{\mathbb {R}}^k} \frac{\left| \eta _\epsilon (x')-\eta _\epsilon (y') \right| ^2}{|x'|^{{\tilde{\alpha }}}|y'|^{{\tilde{\alpha }}}}dx'dy' \nonumber \\&\int _{{\mathbb {R}}^{N-k}} u^2(x',x'') \int _{{\mathbb {R}}^{N-k}} \frac{dy''}{\left( |x'-y'|^2+|x''-y''|^2\right) ^{\frac{N+2s}{2}}} dx'' \nonumber \\&\quad \le C \int _{{\mathbb {R}}^k}\int _{{\mathbb {R}}^k} \frac{\left| \eta _\epsilon (x')-\eta _\epsilon (y') \right| ^2}{\left| x'-y' \right| ^{k+2s}|x'|^{{\tilde{\alpha }}}|y'|^{{\tilde{\alpha }}}}dx'dy', \end{aligned}$$

(8.2)

where (and for the rest of the proof) $C>0$ is constant depending on $N,k,s, {\tilde{\alpha }}$, $\left| \left| u \right| \right| _{L^\infty ({\mathbb {R}}^{N})}$ and ${\text {supp}}u$. We define

$$\begin{aligned} I_\epsilon (x',y')&:= \frac{\left| \eta _\epsilon (x')-\eta _\epsilon (y') \right| ^2}{\left| x'-y' \right| ^{k+2s}|x'|^{{\tilde{\alpha }}}|y'|^{{\tilde{\alpha }}}}dx'dy' \text { and } H_\epsilon := \int _{{\mathbb {R}}^{N}}\int _{{\mathbb {R}}^{N}} I_\epsilon (x',y') dx'dy'. \end{aligned}$$

Then, in view of the (8.2), it is enough to show $H_\epsilon = o(1)$ as $\epsilon \rightarrow 0.$ We define

$$\begin{aligned} H_{\epsilon ,1}&:= \int _{|x'|<\epsilon ^2}\int _{\epsilon ^2<|y'|<\epsilon }I_\epsilon , \ H_{\epsilon ,2} : = \int _{|x'|>\epsilon }\int _{\epsilon ^2<|y'|<\epsilon } I_\epsilon , \\ H_{\epsilon ,3}&:= \int _{|x'|<\epsilon ^2} \int _{|y'|>\epsilon }, \ H_{\epsilon ,4} : = \int _{\epsilon ^2<|x'|<\epsilon }\int _{\epsilon ^2<|y'|<\epsilon } I_\epsilon . \end{aligned}$$

Then using the symmetry of $I_\epsilon$ we have

$$\begin{aligned} H_\epsilon : = 2H_{\epsilon ,1}+ 2H_{\epsilon ,2}+2H_{\epsilon ,3} +H_{\epsilon ,4}. \end{aligned}$$

We will show $H_{\epsilon ,m} = o(1)$ as $\epsilon \rightarrow 0$ for $m=1,2,3,4$.

Step 1 In this step will estimate $H_{\epsilon ,1}.$ For this we define

$$\begin{aligned} F_{x'}:&=\{\epsilon ^2<|y'|<\epsilon \} \cap \{y': |y'-x'|\ge \epsilon ^2/2\} , \text { and } \\ F'_{x'} :&= \{\epsilon ^2<|y'|<\epsilon \} \cap \{y': |y'-x'|< \epsilon ^2/2\}. \text { Then }\\ H_{\epsilon ,1}&= \int _{|x'|<\epsilon ^2} \int _{F_{x'}}I_\epsilon + \int _{|x'|<\epsilon ^2}\int _{F'_{x'}} I_\epsilon . \end{aligned}$$

We first consider

$$\begin{aligned} \int _{|x'|<\epsilon ^2}\int _{F_{x'}'}I_\epsilon&\le \frac{C}{|\ln \epsilon |^2} \int _{|x'|<\epsilon ^2} \frac{1}{|x'|^{2{\tilde{\alpha }}}} \int _{F'_{x'}} \frac{\left| \ln |y'|-\ln |x'| \right| ^2}{|x'-y'|^{k+2s}} dy' dx' \nonumber \\&\le \int _0^1 \frac{C}{|\ln \epsilon |^2} \int _{|x'|<\epsilon ^2} \frac{1}{|x'|^{2{\tilde{\alpha }}}} \int _{F'_{x'}} \frac{dy'}{|x'-y'|^{k+2s-2}\left| y'+r(x'-y') \right| ^2} dx' dr \nonumber \\&\le \frac{C}{|\ln \epsilon |^2} \int _{|x'|<\epsilon ^2} \frac{1}{|x'|^{2{\tilde{\alpha }}}} \int _{F'_{x'}} \frac{dy'}{|x'-y'|^{k+2s-2}|y'|^2} dx' \nonumber \\&\le \frac{C}{\epsilon ^4|\ln \epsilon |^2} \int _{|x'|<\epsilon ^2} \frac{1}{|x'|^{2{\tilde{\alpha }}}} \int _{\{|x'-y'|<\frac{\epsilon ^2}{2}\}} \frac{dy'}{|x'-y'|^{k+2s-2}} dx' \nonumber \\&\le \frac{C}{\epsilon ^4|\ln \epsilon |^2} \epsilon ^{4s} \epsilon ^{4-4s} = o(1), \text { as } \epsilon \rightarrow 0 , \end{aligned}$$

(8.3)

where in the last inequality we have used the fact that, for small $\epsilon >0$, $\epsilon ^{k-2{\tilde{\alpha }}} \le \epsilon ^{4s}$, for any $0<2{\tilde{\alpha }}\le k-2s$. Next, we consider

$$\begin{aligned} \int _{|x'|<\epsilon ^2} \int _{F_{x'}} I_\epsilon&\le \frac{C}{|\ln \epsilon |^2} \int _{|x'|\le \epsilon ^2}\frac{1}{|x'|^{{\tilde{\alpha }}}} \int _{F_{x'}} \frac{\ln ^2(\frac{|y'|}{\epsilon ^2})}{\left| x'-y' \right| ^{k+2s}|y'|^{{\tilde{\alpha }}}}dy' dx' \nonumber \\&\le \frac{C\epsilon ^{k-2s-2{\tilde{\alpha }}}}{|\ln \epsilon |^2} \int _{|x'|\le 1}\frac{1}{|x'|^{{\tilde{\alpha }}}} \int _{\begin{array}{c} \{1<|y'|<\frac{1}{\epsilon }\} \\ \cap \{|x'-y'|>\frac{1}{2}\} \end{array}} \frac{\ln ^2|y'|}{\left| x'-y' \right| ^{k+2s}|y'|^{{\tilde{\alpha }}}}dy' dx' \nonumber \\&\le o(1)+ \frac{C}{|\ln \epsilon |^2} \int _{|x'|<1} \int _{\begin{array}{c} \{2<|y'|<\frac{1}{\epsilon }\} \\ \cap \{|x'-y'|>\frac{1}{2}\} \end{array}}\frac{\ln ^2|y'|}{\left| x'-y' \right| ^{k+2s}|y'|^{{\tilde{\alpha }}}}dy' dx'\nonumber \\&\le o(1) +\frac{C}{|\ln \epsilon |^2}\int _{\{2<|y'|<\epsilon ^{-1}\}} \frac{\ln ^2|y'|dy'}{|y'|^{k+2s+\frac{k-2s}{2}}} \nonumber \\&\le o(1) +\frac{C}{|\ln \epsilon |^2}\int _{\{2<|y'|<\epsilon ^{-1}\}} \frac{\ln ^2|y'|dy'}{|y'|^{k+2s}} = o(1), \text { as } \epsilon \rightarrow 0. \end{aligned}$$

(8.4)

Hence, combining (8.3) and (8.4) we have $H_{\epsilon ,1} = o(1),$ as $\epsilon \rightarrow 0$.

Step 2 In this step, we will show that $H_{\epsilon ,m} =o(1)$, as $\epsilon \rightarrow 0$ for $m=2,3$. In fact, we will show this, only for the case $m=2$. The assertion, for the case, $m=3$, will follow similarly and much more easily.

By a change of variable we get

$$\begin{aligned} H_{\epsilon ,2}&\le \frac{1}{|\ln \epsilon |^2} \int _{\epsilon<|y'|<1}\frac{1}{|y'|^{2{\tilde{\alpha }}}}\int _{|x'|>1} \frac{\left| \ln \frac{|y'|}{\epsilon }-\ln \frac{1}{\epsilon } \right| ^2}{|x'-y'|^{k+2s}} dy'dx' \nonumber \\&\le H'_{\epsilon ,2}+H''_{\epsilon ,2}, \end{aligned}$$

(8.5)

where

$$\begin{aligned} H'_{\epsilon ,2}&:= \frac{C}{|\ln \epsilon |^2} \int _0^1\int _{\epsilon<|y'|<1} \frac{1}{|y'|^{2{\tilde{\alpha }}}} \int _{\{|x'|>1\}\cap \{|x'-y'|\le \frac{1}{2}\}} \frac{dx'}{|x'-y'|^{k+2s-2}|x'+r(x'-y')|^2}dy'dr \\&\le \frac{C}{|\ln \epsilon |^2} \int _{\epsilon<|y'|<1} \frac{1}{|y'|^{2{\tilde{\alpha }}}}\int _{\{|x'-y'|\le \frac{1}{2}\}} \frac{dx'}{|x'-y'|^{k+2s-2}}dy' = o(1), \text { as } \epsilon \rightarrow 0, \end{aligned}$$

and

$$\begin{aligned} H''_{\epsilon ,2}&:= \frac{C}{|\ln \epsilon |^2} \int _{\epsilon<|y'|<1}\frac{1}{|y'|^{2{\tilde{\alpha }}}}\int _{\{|x'|>1\}\cap \{|x'-y'|\ge \frac{1}{2}\}} \frac{\left| \ln \frac{|y'|}{\epsilon }-\ln \frac{1}{\epsilon } \right| ^2}{|x'-y'|^{k+2s}} dy'dx' \\&\le \frac{C}{|\ln \epsilon |^2} \int _{\epsilon<|y'|<1} \frac{\ln ^2|y'|}{|y'|^{2{\tilde{\alpha }}}}dy' = \frac{C}{|\ln \epsilon |^2}\int _\epsilon ^1 \frac{\ln ^2r}{r^{1-\alpha '}} dr \\&= o(1), \text { as } \epsilon \rightarrow 0, \end{aligned}$$

where $\alpha ' = k-2{\tilde{\alpha }}\ge 2s.$ Hence, from (8.5) we have $H_{\epsilon ,2}= o(1),$ as $\epsilon \rightarrow 0$.

Step 3 In this step, we will show that $H_{\epsilon ,4} =o(1)$, as $\epsilon \rightarrow 0$. Similarly, considering different regions, we see that, it is enough to show the following:

$$\begin{aligned} H_{\epsilon ,4,1}: = \frac{1}{|\ln \epsilon |^2} \int \int \limits _{F}\frac{\left| \ln |x'|-\ln |y'| \right| ^2}{\left| x'-y' \right| ^{N+2s}|x'|^{{\tilde{\alpha }}}|y'|^{{\tilde{\alpha }}}}dx'dy' = o(1), \text { as } \epsilon \rightarrow 0, \end{aligned}$$

(8.6)

where F is defined as follows:

$$\begin{aligned} F: = \{(x',y'): \epsilon ^2< |x'|\le |y'|<\epsilon \text { and } |y'|<2|x'|\}. \end{aligned}$$

Clearly, $F\subset \{(x',y'): \epsilon ^2< |x'|\le |y'|<\epsilon \text { and } |x'- y'|<3|x'|\}$. So, using, $\ln r \le r-1$, for $r\ge 1$, we estimate

$$\begin{aligned} H_{\epsilon ,4,1}&\le \frac{1}{|\ln \epsilon |^2} \int _{\epsilon ^2<|x'|<\epsilon }\frac{1}{|x'|^{k-2s+2}} \int _{|x'-y'|<3|x'|} \frac{dy'}{|x'-y'|^{k+2s-2}}dx' \\&= \frac{C}{|\ln \epsilon |^2}\int _{\epsilon ^2<|x'|<\epsilon } \frac{dx'}{|x'|^k} = o(1), \text { as } \epsilon \rightarrow 0. \end{aligned}$$

Combining Step 1, Step 2 and Step 3 we conclude the lemma. $\square$

In light of the Lemma 8.1, it is enough to prove that $C_c^\infty ({\mathbb {R}}^{N})$ is dense in $\mathcal {W}$ to conclude Lemma 4.1. The following Lemma shows that, we can approximate $u\in \mathcal {W}$ by a sequence of compactly supported functions lying in $\mathcal {W}$.

Lemma 8.2

Let $u\in \mathcal {W}$, $0<{\tilde{\alpha }}\le \frac{k-2s}{2}$ and $\eta \in C_c^\infty \left( B_2^N(0);[0,1]\right)$ such that $\eta = 1$ in $B_1^N(0)$ and $\eta _j(x) = \eta (x/j)$. Then

$$\begin{aligned} \displaystyle {\lim _{j\rightarrow \infty }} \left[ \left| \left| u-\eta _ju \right| \right| _{2^*,{\tilde{\alpha }},{\mathbb {R}}^{N}} + \left[ \left[ u-\eta _ju\right] \right] _{s,{\tilde{\alpha }},{\mathbb {R}}^{N}}\right] =0. \end{aligned}$$

Proof

We define

$$\begin{aligned} I_j : = \int _{{\mathbb {R}}^{N}}\int _{{\mathbb {R}}^{N}\setminus B_j^N(0)} \frac{|u(y)|^2\left| \eta _j(x)-\eta _j(y) \right| ^2}{|x-y|^{N+2s}|x'|^{{\tilde{\alpha }}}|y'|^{{\tilde{\alpha }}}} dy dx. \end{aligned}$$

Since, $\eta = 1$ on $B_1^N(0),$ so, to prove the Lemma, it is enough to prove, $\displaystyle {\lim _{j\rightarrow \infty }}I_j = 0$. We define the following sets

$$\begin{aligned} D_{j,0}&: = \{(x,y) \in {\mathbb {R}}^{N}\times ({\mathbb {R}}^{N}\setminus B_j^N(0)) : |x|\le |y|/2\},\\ D_{j,1}&: = \{(x,y) \in {\mathbb {R}}^{N}\times ({\mathbb {R}}^{N}\setminus B_j^N(0)) : |x|\ge |y|/2 \text { and } |x-y|\ge j\}, \\ D_{j,2}&: = \{(x,y) \in {\mathbb {R}}^{N}\times ({\mathbb {R}}^{N}\setminus B_j^N(0)) : |x|\ge |y|/2 \text { and } |x-y|\le j\}. \end{aligned}$$

For $m=0,1,2$, we write

$$\begin{aligned} I_{j,m} : = \int \int _{D_{j,m}} \frac{|u(y)|^2\left| \eta _j(x)-\eta _j(y) \right| ^2}{|x-y|^{N+2s}|x'|^{{\tilde{\alpha }}}|y'|^{{\tilde{\alpha }}}} dy dx. \end{aligned}$$

Then

$$\begin{aligned} I_j = I_{j,0}+I_{j,1}+I_{j,2}. \end{aligned}$$

(8.7)

Now, we break

$$\begin{aligned} \frac{|u(y)|^2\left| \eta _j(x)-\eta _j(y) \right| ^2}{|x-y|^{N+2s}|x'|^{{\tilde{\alpha }}}|y'|^{{\tilde{\alpha }}}} = \frac{|\eta _j(x)-\eta _j(y)|^2}{|x-y|^{2s+2\sigma _m}} \frac{|u(y)|^2}{|x-y|^{N-2\sigma _m|x'|^{{\tilde{\alpha }}}|y'|^{{\tilde{\alpha }}}}}, \end{aligned}$$

where $\sigma _0 = s,$ $\sigma _1 \in (0,s)$ and $s<\sigma _2<1$ such that $\frac{N(N-2\sigma _2)}{N-2s}>\max \{N-k,k\}.$ We denote, $\sigma '_m := \frac{N-2\sigma _2}{N-2s}$. Then using Hölder inequality

$$\begin{aligned} I_{j,m} \le \left( \int \int _{D_{j,m}} \frac{|\eta _j(x)-\eta _j(y)|^{\frac{N}{s}}}{\left| x-y \right| ^{N+\sigma _m\frac{N}{s}}}dxdy\right) ^{\frac{2s}{N}} \left( \int \int _{D_{j,m}} \frac{|u(y)|^{2^*} }{|x-y|^{N\sigma '_m }|x'|^{{\tilde{\alpha }}} |y'|^{{\tilde{\alpha }}}}dx dy \right) ^{\frac{N-2s}{N} }. \end{aligned}$$

(8.8)

Clearly,

$$\begin{aligned} \int \int _{D_{j,m}} \frac{|\eta _j(x)-\eta _j(y)|^{\frac{N}{s}}}{\left| x-y \right| ^{N+\sigma _m\frac{N}{s}}}dxdy \le j^{(s-\sigma _m)\frac{N}{s}} \int _{{\mathbb {R}}^{N}}\int _{{\mathbb {R}}^{N}} \frac{|\eta _j(x)-\eta _j(y)|^{\frac{N}{s}}}{\left| x-y \right| ^{N+\sigma _m\frac{N}{s}}}dxdy \le C j^{\frac{(s-\sigma _m)N}{s}}. \end{aligned}$$

(8.9)

Now, we consider

$$\begin{aligned} \int \int _{D_{j,0}} \frac{|u(y)|^{2^*} }{|x-y|^{N\sigma '_0 }|x'|^{{\tilde{\alpha }}} |y'|^{{\tilde{\alpha }}}}dx dy&\le \int _{|y|>j} \frac{|u(y)|^{2^*}}{|y'|^\frac{2^*{{\tilde{\alpha }}}}{2}} \int _{|x|<\frac{|y|}{2} } \frac{dx}{|x-y|^N|y'|^\frac{2^*{{\tilde{\alpha }}}}{2}} dy\nonumber \\&\le C\int _{|y|>j} \frac{|u(y)|^{2^*}}{|y'|^{\frac{2^*{{\tilde{\alpha }}}}{2}}|y|^{\frac{{{\tilde{\alpha }}}2^*}{2}}}dy \le C \int _{|y|>j} \frac{|u(y)|^{2^*}}{|y'|^{\alpha 2^*}}dy. \end{aligned}$$

(8.10)

Since, $N\sigma '_1>N>N-k$, we estimate

$$\begin{aligned} \int \int _{D_{j,1}} \frac{|u(y)|^{2^*} }{|x-y|^{N\sigma '_1 }|x'|^{{\tilde{\alpha }}} |y'|^{{\tilde{\alpha }}}}dx dy&\le \int _{|y|>j} \int _{\begin{array}{c} \{|x-y|>j\} \\ \cap \{|x'|\le \frac{|y'|}{2} \} \end{array}} \frac{|u(y)|^{2^*} }{|x-y|^{N\sigma '_1}|x'|^{{\tilde{\alpha }}} |y'|^{{\tilde{\alpha }}}}dx dy\nonumber \\&\quad + \int _{|y|>j} \int _{\begin{array}{c} \{|x-y|>j\} \\ \cap \{|x'|\ge \frac{|y'|}{2}\} \end{array}}\frac{|u(y)|^{2^*} }{|x-y|^{N\sigma '_1 }|x'|^{{\tilde{\alpha }}} |y'|^{{\tilde{\alpha }}}}dx dy \nonumber \\&\le C \int _{|y|>j} \frac{|u(y)|^{2^*}}{|y'|^{\frac{{{\tilde{\alpha }}}2^*}{2}}} \int _{|x'|<\frac{|y'|}{2}} \frac{dx'}{|x'-y'|^{N\sigma '_1-N+k} |x'|^{\frac{{{\tilde{\alpha }}}2^*}{2}}} dy' \nonumber \\&\quad + C\int _{|y|>j}\frac{|u(y)|^{2^*}}{|y'|^{2^*{{\tilde{\alpha }}}}} \int _{|x-y|>j} \frac{dx}{|x-y|^{N\sigma '_1}} dy \nonumber \\&\le C\int _{|y|>j} \frac{|u(y)|^{2^*}}{|y'|^{{{\tilde{\alpha }}}2^*}} \frac{dy}{|y'|^{N\sigma '_1-N}} \nonumber \\&\quad + C\frac{1}{j^{N\sigma '_1-1}}\int _{|y|>j} \frac{|u(y)|^{2^*}}{|y'|^{2^*{{\tilde{\alpha }}}}}dy \nonumber \\&\le \frac{C}{j^{\frac{2N(s-\sigma _1)}{N-2s}}} \int _{|y|>j} \frac{|u(y)|^{2^*}}{|y'|^{2^*{{\tilde{\alpha }}}}}dy. \end{aligned}$$

(8.11)

Similarly, using $N>N{\bar{\sigma }}_2> \max \{ N-k,k\}$, we can derive

$$\begin{aligned} \int \int _{D_{j,2}} \frac{|u(y)|^{2^*} }{|x-y|^{N\sigma '_1 }|x'|^{{\tilde{\alpha }}} |y'|^{{\tilde{\alpha }}}}dx dy \le \frac{C}{j^{\frac{2N(s-\sigma _2)}{N-2s}}} \int _{|y|>j} \frac{|u(y)|^{2^*}}{|y'|^{2^*{{\tilde{\alpha }}}}}dy. \end{aligned}$$

(8.12)

Hence, plugging (8.9), (8.10), (8.11) and (8.12) into (8.8) and then using (8.7) we get

$$\begin{aligned} I_j \le C \left| \left| u \right| \right| ^2_{L^{2^*}\left( {\mathbb {R}}^{N}\setminus B_j^N(0); \frac{1}{|x'|^{2{{\tilde{\alpha }}}}}\right) } \rightarrow 0 , \text { as } j \rightarrow \infty . \end{aligned}$$

This proves the lemma. $\square$

The next proposition is a reminiscence of the fact, that $\Theta$ is in $A_1$. Although, in this case, the proof is a direct consequence of Proposition 4.1 and 4.2 of [12].

Proposition 8.3

There exists a constant $C>0$ such that for every $X\in {\mathbb {R}}^{N}_k\times {\mathbb {R}}^{N}_k$, when $N'=2N$ and $X\in {\mathbb {R}}^{N}_k$, when $N'=N$, the following inequality is true

$$\begin{aligned} \displaystyle {\sup _{r>0}}\frac{1}{r^N}\int _{B^N_r(0)}\frac{dz}{\Theta \left( X+w(z)\right) } \le \frac{C}{\Theta (X)}. \end{aligned}$$

Using Proposition 8.3 and the fact, that the measure $\frac{dX}{\Theta (X)}$, is finite on compact sets of ${\mathbb {R}}^{N'}$, we can derive the following lemma which is related to the boundedness of the maximal operator.

Lemma 8.4

Let $q>1$ and $V:{\mathbb {R}}^{N'}\rightarrow {\mathbb {R}}$ be a measurable function. Then, for any $r>0$,

$$\begin{aligned} \int _{{\mathbb {R}}^{N'}} \left( \frac{1}{r^n}\int _{B_r^{N'}(0)} \left| V(X-w(z)) \right| dz\right) ^q\frac{dX}{\Theta (X)} \le C \int _{{\mathbb {R}}^{N'}} \frac{|V(X)|^q}{\Theta (X)}, \end{aligned}$$

for some constant $C>0$.

Next, for $V:{\mathbb {R}}^{N'}:\rightarrow {\mathbb {R}}$ measurable, we define the following operator

$$\begin{aligned} V\star \eta _0 (X) := \int _{{\mathbb {R}}^{N}} V(X-w(z))\eta _0(z)dz, \end{aligned}$$

where $\eta _0$ is a radially symmetric mollifier in ${\mathbb {R}}^{N}$, with $\eta _0\ge 0$ and ${\text {supp}}\eta _0 \subset B_1^N(0)$. Notice that, when $N'= N$, $V\star \eta _0$ coincides with the usual convolution operator $V*\eta _0$. As a consequence of Lemma 8.4, we could control appropriate weighted $L^p$ norm of $V\star \eta _0$. More precisely, we could derive the following proposition.

Proposition 8.5

There exists a constant $C>0$, such that for any measurable function $V: {\mathbb {R}}^{N'}\rightarrow {\mathbb {R}}$ we have

$$\begin{aligned} \int _{{\mathbb {R}}^{N'}} \left| V\star \eta _0 \right| ^p \frac{dX}{\Theta (X)} \le C\int _{{\mathbb {R}}^{N'}}|V(X)|^p\frac{dX}{\Theta (X)}, \end{aligned}$$

where $p=2$, when $N'=2N$ and $p=2^*$, when $N'=N$.

1.2 Proof of Lemma 4.1

Proof

We define

$$\begin{aligned} L^p({\mathbb {R}}^{N'};\Theta ) := \left\{ V:{\mathbb {R}}^{N'}\rightarrow {\mathbb {R}}\text { measurable }: \int _{{\mathbb {R}}^{N'}} |V(X)|^p \frac{dX}{\Theta (X)} <\infty \right\} , \end{aligned}$$

where p is defined in the Proposition 8.5. Then, since $\frac{dX}{\Theta (X)}$ is finite on compact sets of ${\mathbb {R}}^{N'}$, so using Lusin’ s theorem and Proposition 8.5, we can prove that $C_c^\infty ({\mathbb {R}}^{N'})$ is dense in $L^p({\mathbb {R}}^{N};\Theta )$. As a consequence of this density and Proposition 8.2 and the fact, that for any $u\in \mathcal {W}$ and $\eta \in C_c^\infty ({\mathbb {R}}^{N})$, $V^u*\eta = V^{u*\eta }$, we can prove that $C_c^\infty ({\mathbb {R}}^{N})$ is dense in $\mathcal {W}$, where $V^u(x,y): = \frac{u(x)-u(y)}{\left| x-y \right| ^{\frac{N}{2}+s}}$, for $x,y \in {\mathbb {R}}^{N}$. This proves that, $\mathcal {{\dot{H}}}^{s,{\tilde{\alpha }}}({\mathbb {R}}^{N}) = \mathcal {W},$ which is exactly what we wanted to prove in Lemma 4.1. $\square$