Variable Triebel–Lizorkin-Type Spaces

Abstract

In this paper, we study Triebel–Lizorkin-type spaces with variable smoothness and integrability. We show that our space is well-defined, i.e., independent of the choice of basis functions and we obtain their atomic characterization. Moreover, the Sobolev embeddings for these function spaces are obtained.

Appendix

Here we present more technical proofs of the Lemmas.

Proof of Lemma 3

By the scaling argument, we see that it suffices to consider when $\left\| \left( f_{v}\right) _{v}\right\| _{L_{p(\cdot )}^{\tau (\cdot )}(\ell ^{q(\cdot )})}\le 1$ and show that for any dyadic cube P

$$\begin{aligned} \left\| \left( \sum _{v=v_{P}^{+}}^{\infty }\left| \frac{\eta _{v,m}*\left| f_{v}\right| }{|P|^{\tau (\cdot )}}\right| ^{q(\cdot )} \right) ^{1/q(\cdot )}\chi _{P}\right\| _{p(\cdot )}\lesssim 1. \end{aligned}$$

(22)

We will study two cases in particular.

Case 1.$|P|>1$. Let $Q_{v}\subset P$ be a cube, with $l(Q_{v})=2^{-v}$ and $x\in Q_{v}\subset P$. We have

$$\begin{aligned}&\eta _{v,m}*\left| f_{v}\right| (x) \nonumber \\&\quad =2^{vn}\int _{{\mathbb {R}}^{n}}\frac{\left| f_{v}(z)\right| }{\left( 1+2^{v}\left| x-z\right| \right) ^{m}}\hbox {d}z \nonumber \\&\quad =\int _{3Q_{v}}\cdot \cdot \cdot \hbox {d}z+\sum _{k=(k_{1},\ldots ,k_{n})\in {\mathbb {Z}} ^{n},\max _{i=1,\ldots ,n}|k_{i}|\ge 2}\int _{Q_{v}+kl(Q_{v})}\cdot \cdot \cdot \hbox {d}z \nonumber \\&\quad =J_{v}^{1}(f_{v}\chi _{3Q_{v}})(x)+\sum _{k\in {\mathbb {Z}}^{n},|k|\ge 2}J_{v,k}^{2}(f_{v}\chi _{Q_{v}+kl(Q_{v})})(x). \end{aligned}$$

(23)

Let $z\in Q_{v}+kl(Q_{v})$ with $k\in {\mathbb {Z}}^{n}$ and $|k|>4\sqrt{n}$. Then $\left| x-z\right| \ge \left| k\right| 2^{-v-1}$ and

$$\begin{aligned} J_{v,k}^{2}(f_{v}\chi _{Q_{v}+kl(Q_{v})})(x)\lesssim \left| k\right| ^{-m}M_{Q_{v}+kl(Q_{v})}\left( f_{v}\right) . \end{aligned}$$

Let $d>0$ be such that $\tau ^{+}<d<\tau ^{-}\min \left( p^{-},q^{-}\right)$ . Therefore, the left-hand side of (22) is bounded by

$$\begin{aligned}&\sum _{k\in {\mathbb {Z}}^{n},|k|\le 4\sqrt{n}}\left\| \left( \sum _{v=0}^{\infty }\left| \frac{g_{v,k}}{|P|^{\tau (\cdot )}}\right| ^{q(\cdot )}\right) ^{1/q(\cdot )}\chi _{P}\right\| _{p(\cdot )} \nonumber \\&\quad +\sum _{k\in {\mathbb {Z}}^{n},|k|>4\sqrt{n}}\left| k\right| ^{w\frac{ \tau ^{+}}{d}-m+n(1+\frac{1}{t^{-}})\tau ^{+}}(\log \left| k\right| )^{\frac{\tau ^{+}}{d}}\left\| \left( \sum _{v=0}^{\infty }\left| \frac{ b_{k}g_{v,k}}{|P|^{\tau (\cdot )}}\right| ^{q(\cdot )}\right) ^{1/q(\cdot )}\chi _{P}\right\| _{p(\cdot )},\nonumber \\ \end{aligned}$$

(24)

where

$$\begin{aligned} g_{v,k}=\left\{ \begin{array}{lll} M_{3Q_{v}}\left( f_{v}\right) , &{} \text {if} &{} k=0 \\ M_{Q_{v}+kl(Q_{v})}\left( f_{v}\right) , &{} \text {if} &{} 0<|k|\le 4\sqrt{n}\\ M_{Q_{v}+kl(Q_{v})}\left( \left| k\right| ^{-n\left( 1+1/t\left( \cdot \right) \right) \tau \left( \cdot \right) }f_{v}\right) , &{} \text {if} &{} |k|>4, \end{array} \right. \end{aligned}$$

with $b_{k}=(|k|^{w}\log \left| k\right| )^{-\tau (\cdot )/d}$ and $\frac{1}{d}=\frac{1}{p(\cdot )\tau (\cdot )}+\frac{1}{t(\cdot )}$. By similarity we only estimate the second quasi-norm in (24) . This term is bounded if and only if

$$\begin{aligned} \left\| \left( \sum _{v=0}^{\infty }\left| \frac{(b_{k}g_{v,k})^{d/\tau (\cdot )}}{ |P|^{d}}\right| ^{q(\cdot )\tau (\cdot )/d}\right) ^{d/\tau (\cdot )q(\cdot )}\chi _{P}\right\| _{p(\cdot )\tau (\cdot )/d}\lesssim 1. \end{aligned}$$

(25)

By Hölder’s inequality,

$$\begin{aligned} \left| Q(x,\left| k\right| 2^{1-v})\right| M_{Q(x,\left| k\right| 2^{1-v})}\left( \left| k\right| ^{-n\left( 1+1/t\left( \cdot \right) \right) d}\left| f_{v}\right| ^{d/\tau \left( \cdot \right) }\right) \end{aligned}$$

can be estimated from above by

$$\begin{aligned} \Big \Vert \frac{\left| f_{v}\right| ^{1/\tau (\cdot )}}{|Q(x,\left| k\right| 2^{1-v})|}\chi _{Q(x,\left| k\right| 2^{1-v})}\Big \Vert _{p(\cdot )\tau (\cdot )}^{d}\Big \Vert |Q(x,\left| k\right| 2^{1-v})|\left| k\right| ^{-n\left( 1+1/t(\cdot )\right) }\chi _{Q(x,\left| k\right| 2^{1-v})}\Big \Vert _{t(\cdot )}^{d}. \end{aligned}$$

The second quasi-norm is bounded. The first term is bounded if and only if

$$\begin{aligned} \Big \Vert \frac{f_{v}}{|Q(x,\left| k\right| 2^{1-v})|^{\tau (\cdot )}} \chi _{Q(x,\left| k\right| 2^{1-v})}\Big \Vert _{p(\cdot )}\lesssim 1, \end{aligned}$$

which follows since $\left\| \left( f_{v}\right) _{v}\right\| _{L_{p(\cdot )}^{\tau (\cdot )}(\ell ^{q(\cdot )})}\le 1$. Therefore we can apply Theorem 1,

$$\begin{aligned}&\Big (M_{Q(x,\left| k\right| 2^{1-v})}\left( \left| k\right| ^{-n\left( 1+1/t\left( \cdot \right) \right) \tau \left( \cdot \right) }f_{v}\right) \Big )^{d/\tau \left( x\right) } \\&\quad \le M_{Q(x,\left| k\right| 2^{1-v})}\Big (\left| \left| k\right| ^{-n\left( 1+1/t\left( \cdot \right) \right) \tau \left( \cdot \right) }f_{v}\right| ^{d/\tau \left( \cdot \right) }\Big )+\min (1,\left| k\right| ^{ns}2^{n\left( 1-v\right) s})\omega (x) \end{aligned}$$

for any $s>0$ large enough, where

$$\begin{aligned} \omega (x)= & {} (e+\left| x\right| )^{-s}+M_{Q(x,\left| k\right| 2^{1-v})}((e+\left| \cdot \right| )^{-s})\left( x\right) \\\le & {} (e+\left| x\right| )^{-s}+{\mathcal {M}}\left( (e+\left| x\right| )^{-s}\right) (x) \\= & {} h(x). \end{aligned}$$

Observing that $Q_{v}+kl(Q_{v})\subset Q(x,\left| k\right| 2^{1-v})$ . Therefore $\left( g_{v,k}\right) ^{d/\tau (\cdot )}$ can be estimated by

$$\begin{aligned} c\text { }M_{Q(x,\left| k\right| 2^{1-v})}\left( \left| k\right| ^{-nd}\left| f_{v}\right| ^{d/\tau \left( \cdot \right) }\right) +\sigma _{v,k}\text { }h, \end{aligned}$$

where

$$\begin{aligned} \sigma _{v,k}=\left\{ \begin{array}{lll} 1 &{} \text {if} &{} 2^{v}\le 2\left| k\right| \\ \left| k\right| ^{ns}2^{-vns} &{} \text {if} &{} 2^{v}>2\left| k\right| . \end{array} \right. \end{aligned}$$

Therefore, the quantity $\Big (\cdot \cdot \cdot \Big )^{d/\tau (\cdot )q(\cdot )}$ of the term (25) is bounded by

$$\begin{aligned}&|k|^{-w}\left( \sum _{v=0}^{\infty }\left| M_{Q(\cdot ,\left| k\right| 2^{1-v})}\left( \frac{\left| f_{v}\right| ^{d/\tau \left( \cdot \right) }}{\left| k\right| ^{nd}|P|^{d}}\chi _{Q(\cdot ,\left| k\right| 2^{-v+1})}\right) \right| ^{q(\cdot )\tau (\cdot )/d}\right) ^{d/\tau (\cdot )q(\cdot )} \nonumber \\&\quad +\frac{1}{\log \left| k\right| }\left( \sum _{v=0}^{\infty }\left( \sigma _{v,k}\left| h\right| \right) ^{q(\cdot )\tau (\cdot )/d}\right) ^{d/\tau (\cdot )q(\cdot )}. \end{aligned}$$

(26)

The first term is bounded by

$$\begin{aligned} \left( \sum _{v=0}^{\infty }\left| \eta _{v,w}*\left( \frac{\left| f_{v}\right| ^{d/\tau \left( \cdot \right) }}{\left| k\right| ^{nd}|P|^{d}}\chi _{Q(\cdot ,\left| k\right| 2^{1-v})}\right) \right| ^{q(\cdot )\tau (\cdot )/d}\right) ^{d/\tau (\cdot )q(\cdot )}. \end{aligned}$$

where $w>n$. Applying Theorem 2, the $L^{p(\cdot )\tau (\cdot )/d}$-norm of this expression is bounded by

$$\begin{aligned} \left\| \left( \sum _{v=0}^{\infty }\left| \frac{\left| k\right| ^{-n\tau (\cdot )}f_{v}}{|P|^{\tau (\cdot )}}\right| ^{q(\cdot )}\chi _{Q(\cdot ,\left| k\right| 2^{1-v})}\right) ^{d/\tau (\cdot )q(\cdot )}\right\| _{p(\cdot )\tau (\cdot )/d}. \end{aligned}$$

Observing that $Q(\cdot ,\left| k\right| 2^{-v+1})\subset Q(c_{P},\left| k\right| 2^{-v_{P}+1})$ and the measure of the last cube is greater than 1, the last norm is bounded by 1. The second sum in (26) can be estimated by

$$\begin{aligned} c\left| h\right| \left( \sum _{v\ge 0,2^{v}\le 2\left| k\right| }\frac{1}{\log \left| k\right| }+\sum _{v\ge 0,2^{v}>2\left| k\right| }\Big (\frac{2^{v}}{\left| k\right| } \Big )^{-ns}\right) \lesssim \left| h\right| . \end{aligned}$$

This expression, with power $d/\tau (\cdot )q(\cdot )$, in the $L^{p(\cdot )\tau (\cdot )/d}$-norm is bounded by 1. Therefore, the second sum in (24) is bounded by taking m large enough such that $m>2n\tau ^{+}+n+w$, with $w>n$.

Case 2.$|P|\le 1$. As before,

$$\begin{aligned} \eta _{v,m}*\left| f_{v}\right| (x)\lesssim J_{v}^{1}(f_{v}\chi _{3P})(x)+\sum _{k\in {\mathbb {Z}}^{n},|k|\ge 2}J_{v,k}^{2}(f_{v}\chi _{P+kl(P)})(x). \end{aligned}$$

We see that

$$\begin{aligned} J_{v}^{1}(f_{v}\chi _{3P})(x)\lesssim \eta _{v,m}*\left( \left| f_{v}\right| \chi _{3P}\right) (x). \end{aligned}$$

Since $\tau$ is log-Hölder continuous,

$$\begin{aligned} \left| P\right| ^{-\tau (x)}\le c\left| P\right| ^{-\tau (y)}(1+2^{v_{P}}\left| x-y\right| )^{c_{\log }\left( \tau \right) }\le c\left| P\right| ^{-\tau (y)}(1+2^{v}\left| x-y\right| )^{c_{\log }\left( \tau \right) } \end{aligned}$$

for any $x\in P$ and any $y\in 3P$. Hence

$$\begin{aligned} \left| P\right| ^{-\tau (x)}J_{v}^{1}(f_{v}\chi _{3P})(x)\lesssim \eta _{v,m-c_{\log }\left( \tau \right) }*\left( \left| P\right| ^{-\tau (\cdot )}\left| f_{v}\right| \chi _{3P}\right) (x). \end{aligned}$$

Also, we have

$$\begin{aligned} \left| P\right| ^{-\tau (x)}J_{v,k}^{2}(f_{v}\chi _{P+kl(P)})(x)\lesssim \eta _{v,m-c_{\log }\left( \tau \right) }*\left( \left| P\right| ^{-\tau (\cdot )}\left| f_{v}\right| \chi _{P+kl(P)}\right) (x) \end{aligned}$$

for $x\in P$ and $z\in P+kl(P)$ with $k\in {\mathbb {Z}}^{n}$ and $|k|>4\sqrt{n}$. Our estimate follows by Theorem 2. The proof is complete. $\square$

Proof of Lemma 15

Obviously, $\left\| \lambda \right\| _{{\mathfrak {f}}_{p\left( \cdot \right) ,q\left( \cdot \right) }^{\alpha \left( \cdot \right) ,\tau (\cdot )}}\le \left\| \lambda _{r,d}^{*}\right\| _{{\mathfrak {f}}_{p\left( \cdot \right) ,q\left( \cdot \right) }^{\alpha \left( \cdot \right) ,\tau (\cdot )}}$. Let us prove the converse inequality. By the scaling argument, it suffices to consider the case $\left\| \lambda \right\| _{{\mathfrak {f}}_{p\left( \cdot \right) ,q\left( \cdot \right) }^{\alpha \left( \cdot \right) ,\tau (\cdot )}}\le 1$ and show that the modular of a the sequence on the left-hand side is bounded. It suffices to prove that

$$\begin{aligned} \left\| \left( \sum _{v=v_{P}^{+}}^{\infty }\left| \frac{\sum _{m\in {\mathbb {Z}} ^{n}}2^{v(\alpha (\cdot ) +n/2)}\lambda _{v,m,r,d}^{*}\chi _{v,m}}{|P|^{\tau (\cdot ) }}\right| ^{q(\cdot )}\right) ^{1/q(\cdot )}\chi _{P}\right\| _{p(\cdot )}\lesssim 1 \end{aligned}$$

(27)

for any dyadic cube $P\in {\mathcal {Q}}$. For each $k\in {\mathbb {N}}_{0}$ we define $\varOmega _{k}:=\{h\in {\mathbb {Z}}^{n}:2^{k-1}<2^{v}\left| 2^{-v}h-2^{-v}m\right| \le 2^{k}\}$ and $\varOmega _{0}:=\{h\in {\mathbb {Z}} ^{n}:2^{v}\left| 2^{-v}h-2^{-v}m\right| \le 1\}$. Then for any $x\in Q_{v,m}\cap P$, $\sum _{h\in {\mathbb {Z}}^{n}}\frac{2^{vr\alpha \left( x\right) }|\lambda _{v,h}|^{r}}{(1+2^{v}|2^{-v}h-2^{-v}m|)^{d}}$ can be rewritten as

$$\begin{aligned}&\sum _{k=0}^{\infty }\sum _{h\in \varOmega _{k}}\frac{2^{vr\alpha \left( x\right) }\left| \lambda _{v,h}\right| ^{r}}{\left( 1+2^{v}\left| 2^{-v}h-2^{-v}m\right| \right) ^{d}} \nonumber \\&\quad \lesssim \sum _{k=0}^{\infty }2^{-dk}\sum _{h\in \varOmega _{k}}2^{vr\alpha \left( x\right) }\left| \lambda _{v,h}\right| ^{r} \nonumber \\&\quad =\sum _{k=0}^{\infty }2^{(n-d)k+(v-k)n+vr\alpha \left( x\right) }\int \limits _{\cup _{z\in \varOmega _{k}}Q_{v,z}}\sum _{h\in \varOmega _{k}}\left| \lambda _{v,h}\right| ^{r}\chi _{v,h}(y)\hbox {d}y. \end{aligned}$$

(28)

Let $x\in Q_{v,m}\cap P$ and $y\in \cup _{z\in \varOmega _{k}}Q_{v,z}$, then $y\in Q_{v,z}$ for some $z\in \varOmega _{k}$ and $2^{k-1}<2^{v}\left| 2^{-v}z-2^{-v}m\right| \le 2^{k}$. From this it follows that y is located in some cube $Q(x,2^{k-v+3})$. In addition, from the fact that

$$\begin{aligned} \left| y_{i}-(c_{P})_{i}\right| \le \left| y_{i}-x_{i}\right| +\left| x_{i}-(c_{P})_{i}\right| \le 2^{k-v+2}+2^{-v_{P}-1}<2^{k-v_{P}+3},\text { }i=1,\ldots ,n, \end{aligned}$$

we have y is located in some cube $Q(c_{P},2^{k-v_{P}+4})$. Since $\alpha$ is log-Hölder continuous we can prove that

$$\begin{aligned} 2^{v(\alpha \left( x\right) -\alpha \left( y\right) )}\lesssim \left\{ \begin{array}{ccc} 2^{c_{\log }(\alpha )k} &{} \text {if} &{} k<\max (0,v-h_{n}) \\ 2^{(\alpha ^{+}-\alpha ^{-})k} &{} \text {if} &{} k\ge \max (0,v-h_{n}), \end{array} \right. \end{aligned}$$

with $h_{n}\in {\mathbb {N}}_{0}$ and $c>0$ not depending on v and k. Therefore, (28) does not exceed

$$\begin{aligned}&c\sum _{k=0}^{\infty }2^{(n-d+a)k+(v-k)n}\int \limits _{Q(x,2^{k-v+3})}2^{v\alpha \left( y\right) r}\sum _{h\in \varOmega _{k}}\left| \lambda _{v,h}\right| ^{r}\chi _{v,h}(y)\chi _{{Q(c_{P},2^{k-v_{P}+4})}}(y)\hbox {d}y \\&\quad \lesssim \sum _{k=0}^{\infty }2^{(n-d+a)k}M_{{Q(x,2^{k-v+3})}}\left( \sum _{h\in \varOmega _{k}}2^{v\alpha \left( \cdot \right) r}\left| \lambda _{v,h}\right| ^{r}\chi _{v,h}\chi _{{Q(c_{P},2^{k-v_{P}+4})}}\right) , \end{aligned}$$

where $a=\max \left( c_{\log }(\alpha ),\alpha ^{+}-\alpha ^{-}\right)$. To prove (27) we can distinguish two cases:

Case 1.$|P|>1$. The left-hand side of (27) is bounded by 1 if

$$\begin{aligned} c\sum _{k=0}^{\infty }2^{\varrho k}\left\| \left( \sum _{v=0}^{\infty }\left( \frac{ M_{{Q(\cdot ,2^{k-v+3})}}(2^{-kn\left( r+1/t\left( \cdot \right) \right) \tau (\cdot ) }g_{v,k,v_{P}})}{|P|^{r\tau (\cdot ) }} \right) ^{q\left( \cdot \right) /r}\right) ^{r/q (\cdot ) }\chi _{P} \right\| _{p(\cdot )/r}\lesssim 1, \nonumber \\ \end{aligned}$$

(29)

with

$$\begin{aligned} g_{v,k,v_{P}}=\sum _{h\in \varOmega _{k}}2^{v(\alpha \left( \cdot \right) +n/2)r}\left| \lambda _{v,h}\right| ^{r}\chi _{v,h}\chi _{{ Q(c_{P},2^{k-v_{P}+4})}} \end{aligned}$$

and

$$\begin{aligned} \varrho =n-d+a+nr\tau ^{+}+n\frac{\tau ^{+}}{t^{-}}. \end{aligned}$$

Let us prove that

$$\begin{aligned} \left\| \left( \sum _{v=0}^{\infty }\left( \frac{\omega _{k}M_{{Q(\cdot ,2^{k-v+3})} }(2^{-kn\left( r+1/t\left( \cdot \right) \right) \tau \left( \cdot \right) }g_{v,k,v_{P}})}{|P|^{r\tau (\cdot ) }}\right) ^{q (\cdot ) /r}\right) ^{r/q (\cdot ) }\chi _{P}\right\| _{p(\cdot )/r}\lesssim 1 \end{aligned}$$

for any $k\in {\mathbb {N}}_{0}$ and any $P\in {\mathcal {Q}}$, where

$$\begin{aligned} \omega _{k}=\frac{1}{2^{(w-n-\frac{n\sigma }{t^{+}})k}+2^{kns}},\quad w,s>0. \end{aligned}$$

This is equivalent to

$$\begin{aligned} \left\| \left( \sum _{v=0}^{\infty }\left( \frac{\left( \omega _{k}M_{{Q(\cdot ,2^{k-v+3})}}(2^{-kn\left( r+1/t\left( \cdot \right) \right) \tau (\cdot ) }g_{v,k,v_{P}})\right) ^{\frac{\sigma }{\tau \left( \cdot \right) }}}{|P|^{r\sigma }}\right) ^{\frac{q\left( \cdot \right) \tau \left( \cdot \right) }{r\sigma }}\right) ^{\frac{\sigma r}{q\left( \cdot \right) \tau \left( \cdot \right) }}\right\| _{\frac{p(\cdot )\tau \left( \cdot \right) }{ r\sigma }}\lesssim 1, \nonumber \\ \end{aligned}$$

(30)

with $\sigma >0$ such that $\tau ^{+}<\sigma <\frac{\tau ^{-}\min \left( p^{-},q^{-}\right) }{r}$. By Hölder’s inequality,

$$\begin{aligned}&|{Q(x,2^{k-v+3})}|M_{{Q(x,2^{k-v+3})}}\left( 2^{-kn\left( r+1/t\left( \cdot \right) \right) \sigma }\left| g_{v,k,v_{P}}\right| ^{\sigma /\tau \left( \cdot \right) }\right) \\&\quad \lesssim \Big \Vert \frac{(g_{v,k,v_{P}})^{1/\tau (\cdot )}}{|{Q(x,2^{k-v+3})} |^{r}}\chi _{{Q(x,2^{k-v+3})}}\Big \Vert _{p(\cdot )\tau (\cdot )/r}^{\sigma } \\&\qquad \times \Big \Vert |{Q(x,2^{k-v+3})}|^{r}2^{-kn\left( r+1/t\left( \cdot \right) \right) }\chi _{{Q(x,2^{k-v+3})}}\Big \Vert _{t(\cdot )}^{\sigma }, \end{aligned}$$

where $\frac{1}{\sigma }=\frac{r}{p(\cdot )\tau (\cdot )}+\frac{1}{t(\cdot )}$. The second quasi-norm is bounded. The first quasi-norm is bounded if and only if

$$\begin{aligned} \left\| \frac{\left( g_{v,k,v_{P}}\right) ^{1/r}}{|{Q(x,2^{k-v+3})} |^{\tau (\cdot )}}\chi _{{Q(x,2^{k-v+3})}}\right\| _{p(\cdot )}\lesssim 1, \end{aligned}$$

which follows since $\left\| \lambda \right\| _{{\mathfrak {f}}_{p\left( \cdot \right) ,q\left( \cdot \right) }^{\alpha \left( \cdot \right) ,\tau (\cdot )}}\le 1$. Hence we can apply Theorem 1 to estimate

$$\begin{aligned} \left( M_{{Q(x,2^{k-v+3})}}(2^{-kn\left( r+1/t\left( \cdot \right) \right) \tau \left( \cdot \right) }g_{v,k,v_{P}})\right) ^{\sigma /\tau \left( x\right) } \end{aligned}$$

by

$$\begin{aligned} c\text { }M_{{Q(x,2^{k-v+3})}}\left( \left( 2^{-kn\left( r+1/t\left( \cdot \right) \right) \tau \left( \cdot \right) }g_{v,k,v_{P}}\right) ^{\sigma /\tau \left( \cdot \right) }\right) +\min \left( {2^{n(k-v)s}},1\right) h\left( x\right) , \end{aligned}$$

where $s>0$ is large enough and h is the same function as in the proof of Lemma 3. Hence the term in (30), with $\sum _{v=0}^{k+3}$ in place of $\sum _{v=0}^{\infty }$ is bounded by

$$\begin{aligned} c\sum _{v=0}^{k+3}\frac{1}{2^{(w-n-\frac{n\sigma }{t^{+}})k}+2^{kns}}\Big \Vert \frac{M_{{Q(\cdot ,2^{k-v+3})}}\left( g_{v,k,v_{P}}\right) ^{\frac{\sigma }{ \tau \left( \cdot \right) }}}{|{Q(\cdot ,2^{k-v_{P}+3})}|^{r\sigma }}\Big \Vert _{ \frac{p(\cdot )\tau \left( \cdot \right) }{r\sigma }}+\frac{c\left( k+4\right) ^{s}}{2^{(w-n-\frac{n\sigma }{t^{+}})k}+2^{kns}}. \end{aligned}$$

Since ${\mathcal {M}}$ is bounded in $L^{p(\cdot )\tau \left( \cdot \right) /r\sigma }$ the last norm is bounded by

$$\begin{aligned} \left\| {\mathcal {M}}\left( \frac{\left( g_{v,k,v_{P}}\right) ^{\frac{\sigma }{ \tau \left( \cdot \right) }}}{|{Q(\cdot ,2^{k-v_{P}+3})}|^{r\sigma }}\right) \right\| _{\frac{p(\cdot )\tau \left( \cdot \right) }{r\sigma }}\lesssim \Big \Vert \frac{\left( g_{v,k,v_{P}}\right) ^{\frac{\sigma }{\tau \left( \cdot \right) }}}{|{Q(\cdot ,2^{k-v_{P}+3})}|^{r\sigma }}\Big \Vert _{\frac{p(\cdot )\tau \left( \cdot \right) }{r\sigma }}. \end{aligned}$$

This term is bounded by constant if and only if

$$\begin{aligned} \left\| \frac{\left( g_{v,k,v_{P}}\right) ^{1/r}}{|{Q(c_{P},2^{k-v_{P}+3})} |^{\tau \left( \cdot \right) }}\right\| _{p(\cdot )}\lesssim 1, \end{aligned}$$

which follows since $|{Q(\cdot ,2^{k-v_{P}+3})}|\ge 1$. Now, with $w>n$, the term in (30), with $\sum _{v=k+4}^{\infty }$ in place of $\sum _{v=0}^{\infty }$ is bounded by

$$\begin{aligned}&\frac{c\text { }2^{(w-n-\frac{n\sigma }{t^{+}})k}}{2^{(w-n-\frac{n\sigma }{ t^{+}})k}+2^{kns}}\Big \Vert \Big (\sum _{v=k+4}^{\infty }\Big (\frac{\eta _{v,w}*\left( g_{v,k,v_{P}}\right) ^{\frac{\sigma }{\tau \left( \cdot \right) }}}{|{Q(\cdot ,2^{k-v_{P}+3})}|^{r\sigma }}\Big )^{\frac{q\left( \cdot \right) \tau \left( \cdot \right) }{r\sigma }}\Big )^{\frac{\sigma r}{ q\left( \cdot \right) \tau \left( \cdot \right) }}\Big \Vert _{\frac{p(\cdot )\tau \left( \cdot \right) }{r\sigma }} \\&\qquad +\frac{c\text { }2^{kns}}{2^{(w-n-\frac{n\sigma }{t^{+}})k}+2^{kns}} \\&\quad \lesssim 1, \end{aligned}$$

by Theorem 2. Therefore our estimate (29) follows by taking $0<s<\frac{w-n}{n}$ and the fact that

$$\begin{aligned} d>n\left( r+\frac{1}{t^{-}}\right) \tau ^{+}+a+w. \end{aligned}$$

Case 2.$|P|<1$. We have

$$\begin{aligned} \left| P\right| ^{-\tau (x)}\le c\left| P\right| ^{-\tau (y)}(1+2^{v_{P}}\left| x-y\right| )^{c_{\log }\left( \tau \right) }\le c\left| P\right| ^{-\tau (y)}(1+2^{v}\left| x-y\right| )^{c_{\log }\left( \tau \right) } \end{aligned}$$

for any $x,y\in {\mathbb {R}}^{n}$. Hence,

$$\begin{aligned} \frac{\eta _{v,w}*g_{v,k,v_{P}}}{|P|^{\tau (\cdot )r}}\lesssim \eta _{v,w-c_{\log }\left( \tau \right) }*\big (\frac{g_{v,k,v_{P}}}{|P|^{\tau (\cdot )r}}\big ),\quad k\in {\mathbb {N}}_{0}. \end{aligned}$$

Our estimate follows by Theorem 2. The proof is complete. $\square$