Proof We first show ${\mathcal S}({\mathbb R}^{2n})\hookrightarrow L_w^q (L^p)({\mathbb R}^{2n})$ . To this end, for any Lebesgue measurable set $E \subset {{{\mathbb R}}^n}$ , define

$$ \begin{align*}w(E):=\int_{E}w\,dx.\end{align*} $$

By [Reference Grafakos28, Proposition 7.1.5(9)], we know that if $w\in A_q({{{\mathbb R}}^n})$ then w is doubling and satisfies

(2.1) $$ \begin{align} w(\lambda B)\le \lambda^{nq} [w]_{A_q({{{\mathbb R}}^n})} w(B) \end{align} $$

uniformly for all $\lambda \in (1,\infty )$ and all balls $B\subset {{{\mathbb R}}^n}$ , where $\lambda B$ denotes the ball with the same center as that of B but of radius $\lambda $ times of B. Then, for any $f \in {\mathcal S}({\mathbb R}^{2n})$ , we have

$$ \begin{align*}|f(x,y)|\leq \left(1+\sqrt{|x|^2+|y|^2}\right)^{-N} \rho_{N,0}(f)\end{align*} $$

for all $x,y\in {\mathbb R}^{n}$ and for some large constant N to be determined later. So,

(2.2) $$ \begin{align} \|f\|_{L_w^q (L^p)({\mathbb R}^{2n})}&\leq \rho_{N,0}(f) \left(\int_{ {{{\mathbb R}}^n} }\left(\int_{{{\mathbb R}}^n} (1+\sqrt{|x|^2+|y|^2})^{-Np}\, dx\right)^{q/p}w(y)\, dy\right)^{1/q}\\ &\leq \rho_{N,0}(f) \left(\int_{ {{{\mathbb R}}^n} }(1+|x|)^{-Np/2}\, dx\right)^{1/p} \left(\int_{ {{{\mathbb R}}^n} }(1+|y|)^{-Nq/2}w(y)\, dy\right)^{1/q}. \notag \end{align} $$

If $N>2n/p$ , then $\int _{ {{{\mathbb R}}^n} }(1+|x|)^{-Np/2}\, dx<\infty .$ Moreover, if $N>2n$ , then by (2.1), we have

(2.3) $$ \begin{align} \int_{ {{{\mathbb R}}^n} }(1+|y|)^{-Nq/2}w(y)\, dy & =\left(\int_{|y|<1}+\sum_{j=1}^{\infty}\int_{2^{j-1}\leq|y|<2^j}\right)(1+|y|)^{-Nq/2}w(y)\, dy\\ &\leq w(B({0},1))+\sum_{j=1}^{\infty}2^{-(j-1)Nq/2}w(B({0},2^j))\notag\\ &\lesssim w(B({0},1))\left(1+\sum_{j=1}^{\infty}2^{-j(N/2-n)q}\right)<\infty. \notag \end{align} $$

Altogether, we obtain $f\in L_w^q (L^p)({\mathbb R}^{2n})$ and $\|f\|_{L_w^q (L^p)({\mathbb R}^{2n})}\lesssim \rho _{N,0}(f),$ which implies

$$ \begin{align*}{\mathcal S}({\mathbb R}^{2n})\hookrightarrow L_w^q (L^p)({\mathbb R}^{2n}).\end{align*} $$

Now, we show $L_w^q (L^p)({\mathbb R}^{2n})\hookrightarrow {\mathcal S}'({\mathbb R}^{2n})$ . For any $f\in L_w^q (L^p)({\mathbb R}^{2n})$ and $\varphi \in {\mathcal S}({\mathbb R}^{2n})$ , from the Hölder inequality, we deduce

(2.4) $$ \begin{align} |\left\langle f,\, \varphi \right\rangle| &\leq\int_{ {{{\mathbb R}}^n} }\int_{ {{{\mathbb R}}^n} }|f(x,y)\varphi(x,y)|\, dx \, dy\\ &\leq \int_{ {{{\mathbb R}}^n} }\left(\int_{ {{{\mathbb R}}^n} }|f(x,y)|^{p}\, dx\right)^{1/p}w(y)^{1/q} \left(\int_{ {{{\mathbb R}}^n} }|\varphi(x,y)|^{p'}\, dx \right)^{1/p'}w(y)^{-1/q}\, dy \notag\\ &\leq \|f\|_{L_w^q (L^p)({\mathbb R}^{2n})} \left(\int_{ {{{\mathbb R}}^n} }\left(\int_{ {{{\mathbb R}}^n} }|\varphi(x,y)|^{p'}\, dx \right)^{q'/p'}w(y)^{1-q'}\, dy\right)^{1/q'}. \notag \end{align} $$

When $q\in (1,\infty )$ , by the fact that $w\in A_q({{{\mathbb R}}^n})$ if and only if $w^{1-q'}\in A_{q'}({{{\mathbb R}}^n})$ (see, for instance, [Reference Duoandikoetxea17, Proposition 7.2(2)]), and the already obtained result ${\mathcal S}({\mathbb R}^{2n})\hookrightarrow L_w^q (L^p)({\mathbb R}^{2n})$ , we find that

$$ \begin{align*} |\left\langle f,\, \varphi \right\rangle| \leq \|f\|_{L_w^q (L^p)({\mathbb R}^{2n})} \|\varphi\|_{L_{w^{1-q'}}^{q'}(L^{p'})({\mathbb R}^{2n})} \lesssim \|f\|_{L_w^q (L^p)({\mathbb R}^{2n})}\,\rho_{N,0} (\varphi) \end{align*} $$

holds when $N\in {\mathbb N}$ is sufficiently large, thereby leading to $f\in {\mathcal S}'({\mathbb R}^{2n})$ . Thus, we obtain the continuous embeddingFootnote ¹

$$ \begin{align*}L_w^q (L^p)({\mathbb R}^{2n})\hookrightarrow{\mathcal S}'({\mathbb R}^{2n})\end{align*} $$

and, hence, complete the proof of Lemma 2.1.

Proof We first show (i). Suppose that $f\in C_{c}^{\infty }({{{\mathbb R}}^n})$ . From (1.1), it follows that

$$ \begin{align*} \|\mathrm{D}_{h}^{k}f\|_{L^p({{{\mathbb R}}^n})}\le \sum_{m=0}^k \binom km \|f(\cdot+mh)\|_{L^p({{{\mathbb R}}^n})}=2^k \|f\|_{L^p({{{\mathbb R}}^n})}\lesssim 1. \end{align*} $$

Moreover, by the fact that

(2.6) $$ \begin{align} \mathrm{D}_{h}^{k}f(x) &=\int_{[0,1]^k}\sum_{j_{1}=1}^{n}\cdots \sum_{j_{k}=1}^{n} h_{j_{1}} \cdots h_{j_{k}} (\partial_{j_{1}} \cdots \partial_{j_{k}}f)(x +(t_{1}+\cdots+t_{k})h)\, dt_{1} \cdots \,dt_{k} \end{align} $$

and the Minkowski inequality, we have

$$ \begin{align*}\|\mathrm{D}_{h}^{k}f\|_{L^p({{{\mathbb R}}^n})}\lesssim|h|^k.\end{align*} $$

Therefore,

$$ \begin{align*} \|f\|_{\dot{\Lambda}_{p,\, q,\,w}^{s,\, k}({{{\mathbb R}}^n})} &\lesssim\left(\int_{{{{\mathbb R}}^n}}\frac{\min\{1, |h|^{kq}\}}{|h|^{n+sq}}w(h)\, dh\right)^{\frac{1}{q}}\\ &\approx \left(\int_0^\infty \min\{1, t^{kq}\} w(t) t^{-sq-1}\, dt\right)^{\frac{1}{q}}\\ &=\left(\int_{0}^{1}t^{(k-s)q-1}w(t)\, dt+ \int_{1}^{\infty}t^{-sq-1}w(t)\, dt\right)^{\frac{1}{q}}. \end{align*} $$

Note that $k\ge \lfloor s\rfloor +1$ , $\sigma <\lfloor s\rfloor +1-s,$ and $\delta>-s$ . On the one hand, condition (1.4) implies

(2.7) $$ \begin{align} \int_{0}^{1}t^{(k-s)q-1}w(t)\, dt &\le\int_{0}^{1}t^{(\lfloor s\rfloor +1-s)q-1}w(t)\, dt\\ &\notag=\int_{0}^{1}t^{(\lfloor s\rfloor +1-s)q}\left( \frac{w(t\cdot 1)}{w(1)}\right)w(1)\, \frac{dt}{t}\\ &\notag\lesssim\int_{0}^{1}t^{\sigma q}\, \left(\sup_{\rho \in (0, \infty)} \frac{w(t\rho)}{w(\rho)}\right)\frac{dt}{t} < \infty. \end{align} $$

On the other hand, condition (1.5) yields

(2.8) $$ \begin{align} \int_{1}^{\infty}t^{-sq-1}w(t)\, dt &\le \int_{1}^{\infty}t^{\delta q-1}w(t)\, dt\\ &\notag=\int_{1}^{\infty}t^{\delta q}\, \left( \frac{w(t\cdot 1)}{w(1)}\right)w(1)\,\frac{dt}{t}\\ &\notag\lesssim\int_{1}^{\infty}t^{\delta q}\, \left(\sup_{\rho \in (0, \infty)} \frac{w(t\rho)}{w(\rho)}\right)\frac{dt}{t}<\infty. \end{align} $$

This shows that any $f\in C_c^\infty ({{{\mathbb R}}^n})$ satisfies $\|f\|_{\dot {\Lambda }_{p,\, q,\,w}^{s,\, k}({{{\mathbb R}}^n})}<\infty $ .

Next, we show (ii). Fix $f\in {\Lambda }_{p,\, q,\,w}^{s,\, k}({{{\mathbb R}}^n})$ . Choose a function $0\le \phi \in C_c^\infty ({{{\mathbb R}}^n})$ such that $\int _{{{\mathbb R}}^{n}}\phi (x)\, dx=1$ . For any $\epsilon \in (0,\infty )$ , set

$$ \begin{align*}\phi_{\epsilon}(\cdot):=\epsilon^{-n}\phi(\epsilon^{-1}\,\cdot). \end{align*} $$

and

$$ \begin{align*}f_{\epsilon}:=\phi_{\epsilon} \ast f. \end{align*} $$

Clearly $f_{\epsilon }\in C^{\infty }({{{\mathbb R}}^n})$ . From $f \in {\Lambda }_{p,\, q,\,w}^{s,\, k}({{{\mathbb R}}^n})\subset L^p({{{\mathbb R}}^n})$ , it follows that

(2.9) $$ \begin{align} \lim_{\epsilon\to0}\|f_{\epsilon}-f\|_{L^p({{{\mathbb R}}^n})} =0. \end{align} $$

From the Young inequality and the fact $\int _{{{\mathbb R}}^n}\phi (x)\,dx=1$ , we deduce

$$ \begin{align*}\left\|{\mathrm{D}}_{h}^{k}f_{\epsilon}\right\|_{L^p({{{\mathbb R}}^n})} =\left\|\phi_{\epsilon} \ast \left( {\mathrm{D}}_{h}^{k}f\right)\right\|_{L^p({{{\mathbb R}}^n})} \leq\left\| {\mathrm{D}}_{h}^{k}f\right\|_{L^p({{{\mathbb R}}^n})}\end{align*} $$

and, hence,

$$ \begin{align*} \left\|{\mathrm{D}}_{h}^{k}(f_{\epsilon}-f)\right\|_{L^p({{{\mathbb R}}^n})} \leq \left\|{\mathrm{D}}_{h}^{k}f_{\epsilon}\right\|_{L^p({{{\mathbb R}}^n})}+ \left\|{\mathrm{D}}_{h}^{k}f\right\|_{L^p({{{\mathbb R}}^n})} \leq 2\left\|{\mathrm{D}}_{h}^{k}f\right\|_{L^p({{{\mathbb R}}^n})}. \end{align*} $$

Also, note that (2.9) and (1.1) imply

$$ \begin{align*} \lim_{\epsilon \to 0}\left\|{\mathrm{D}}_{h}^{k}(f_{\epsilon}-f)\right\|_{L^p({{{\mathbb R}}^n})} &\leq \lim_{\epsilon \to 0}\sum_{m=0}^{k}\binom{k}{m}\,\|(f_{\epsilon}-f)(\cdot+mh)\|_{L^p({{{\mathbb R}}^n})}\\ &=\sum_{m=0}^{k}\binom{k}{m}\lim_{\epsilon \to 0}\|f_{\epsilon}-f\|_{L^p({{{\mathbb R}}^n})}\\ &= 0. \end{align*} $$

Thus, by the Lebesgue dominated convergence theorem, we have

$$ \begin{align*} \lim_{\epsilon \to 0}\left(\int_{{{{\mathbb R}}^n}} \frac{\|{\mathrm{D}}_{h}^{k}(f_{\epsilon}-f) \|_{L^p({{{\mathbb R}}^n})}^{q}}{|h|^{n+sq}} w(h)\, dh \right)^{\frac{1}{q}} & =\left(\int_{{{{\mathbb R}}^n}} \frac{\lim_{\epsilon \to 0}\|{\mathrm{D}}_{h}^{k}(f_{\epsilon}-f) \|_{L^p({{{\mathbb R}}^n})}^{q}}{|h|^{n+sq}} w(h)\, dh \right)^{\frac{1}{q}}=0. \end{align*} $$

This proves

$$ \begin{align*}\lim_{\epsilon\to0}\|f_\epsilon-f\|_{\dot{\Lambda}_{p,\, q,\,w}^{s,\, k}({{{\mathbb R}}^n})} =0.\end{align*} $$

So, we have find a sequence $\{\phi _j\}_{j\in {\mathbb N}}$ in $C^\infty ({{{\mathbb R}}^n})$ such that (2.5) holds.

Now, we show that $C_c^\infty ({{{\mathbb R}}^n})$ is dense in ${\Lambda }_{p,\, q,\,w}^{s,\, k}({{{\mathbb R}}^n})\cap C^\infty ({{{\mathbb R}}^n})$ under $\|\,\cdot \,\|_{{\Lambda }_{p,\, q,\,w}^{s,\, k}({{{\mathbb R}}^n})}$ . Fix $f\in {\Lambda }_{p,\, q,\,w}^{s,\, k}({{{\mathbb R}}^n})\cap C^\infty ({{{\mathbb R}}^n})$ . Let $\eta \in C_{c}^{\infty }({{{\mathbb R}}^n})$ satisfy $\eta =1$ on $B({0},1)$ , ${\mathrm {\,supp\,}} \eta \subset B({0},2),$ and $0\leq \eta \leq 1$ . For any $N\in (0,\infty )$ , define

$$ \begin{align*}\eta_{N}(\cdot):=\eta(N^{-1}\cdot)\qquad \text{and}\qquad f_{N}:=\eta_{N}f.\end{align*} $$

It is obvious that $f_{N}\in C_{c}^{\infty }({{{\mathbb R}}^n})$ and $ f_{N}\to f$ in $L^p({{{\mathbb R}}^n})$ as $N\to \infty $ . To validate that $\{f_{N}\}_{N\in {\mathbb N}}$ converges to f with respect to the semi-norm $\|\,\cdot \,\|_{\dot {\Lambda }_{p,\, q,\,w}^{s,\, k}({{{\mathbb R}}^n})}$ , by (1.1), we write that for all $x,\ h\in {{{\mathbb R}}^n}$ ,

$$ \begin{align*} &D_h^k(f_{N}-f)(x) \\&\quad= \sum_{m=0}^k(-1)^{k-m} \binom km \left[\eta_N(x+mh)-1\right]f(x+mh)\\ &\quad=\sum_{m=0}^k(-1)^{k-m} \binom km \left[\eta_N(x+mh)-\eta_N(x)\right]f(x+mh)+\left[\eta_N(x)-1\right]D_h^kf(x), \end{align*} $$

which implies

(2.10) $$ \begin{align} |D_h^k(f_{N}-f)(x)| &\lesssim\sum_{m=0}^k\min\left\{\frac{|mh|}{N}\|\nabla \eta\|_{L^\infty({{{\mathbb R}}^n})}, \,\| \eta\|_{L^\infty({{{\mathbb R}}^n})}\right\}|f(x+mh)|+|D_h^kf(x)|\\ &\lesssim\min\{|h|,\,1\}\sum_{m=0}^k|f(x+mh)|+|D_h^kf(x)|{\nonumber} \end{align} $$

by terms of the mean value theorem. Note that (1.4) implies that (see (2.7))

$$ \begin{align*}\int_{0}^{1}t^{(1-s)q-1}w(t)\, dt<\infty.\end{align*} $$

Meanwhile, by the fact that (1.5) leads to (see (2.8))

$$ \begin{align*}\int_{1}^{\infty}t^{-sq-1}w(t)\, dt<\infty.\end{align*} $$

With these last two estimates and (2.10), we then derive

$$ \begin{align*} &\left(\int_{{{{\mathbb R}}^n}} \frac{\|\min\{|h|,\,1\}\sum_{m=0}^k|f(\cdot+mh)|+|D_h^kf| \|_{L^p({{{\mathbb R}}^n})}^{q}}{\left|h\right|{}^{n+sq}} w(h)\, dh \right)^{\frac{1}{q}}\\ &\quad\le \left(\int_{|h|\leq 1} \frac{\||h|\sum_{m=0}^k|f(\cdot+mh)| \|_{L^p({{{\mathbb R}}^n})}^{q}}{\left|h\right|{}^{n+sq}} w(h)\, dh \right)^{\frac{1}{q}}\\ &\qquad +\left(\int_{|h|>1} \frac{\|\sum_{m=0}^k|f(\cdot+mh)| \|_{L^p({{{\mathbb R}}^n})}^{q}}{|h|^{n+sq}} w(h)\, dh \right)^{\frac{1}{q}} +\|f\|_{\dot{\Lambda}_{p,\, q,\,w}^{s,\, k}({{{\mathbb R}}^n})}\\ &\quad\lesssim\|f\|_{L^p({{{\mathbb R}}^n})} \left(\int_{|h|\leq 1} |h|^{q-sq-n}w(h)\, dh\right)^{\frac{1}{q}}\\&\qquad+\|f\|_{L^p({{{\mathbb R}}^n})} \left(\int_{|h|>1} |h|^{-sq-n}w(h)\, dh\right)^{\frac{1}{q}} +\|f\|_{\dot{\Lambda}_{p,\, q,\,w}^{s,\, k}({{{\mathbb R}}^n})}\\ &\quad\approx\|f\|_{L^p({{{\mathbb R}}^n})}\left(\int_{0}^{1}t^{(1-s)q-1}w(t)\, dt\right)^{\frac{1}{q}}\\&\qquad+\|f\|_{L^p({{{\mathbb R}}^n})} \left(\int_{1}^{\infty}t^{-sq-1}w(t)\, dt\right)^{\frac{1}{q}}+\|f\|_{\dot{\Lambda}_{p,\, q,\,w}^{s,\, k}({{{\mathbb R}}^n})}\\ &\quad <\infty. \end{align*} $$

Meanwhile, we also know that for all $h\in {{{\mathbb R}}^n}$ ,

$$ \begin{align*} \lim_{N\to\infty} \|D_h^k(f_{N}-f)\|_{L^p({{{\mathbb R}}^n})} \le \sum_{m=0}^k \binom km \|[\eta_N(\cdot)-1]f(\cdot)\|_{L^p({{{\mathbb R}}^n})}=0. \end{align*} $$

From these and the Lebesgue dominated convergence theorem, we conclude that

$$ \begin{align*} \lim_{N\to\infty}\int_{{{\mathbb R}}^n} \frac{\|D_h^k(f_{N}-f) \|_{L^p({{{\mathbb R}}^n})}^{q}}{|h|^{n+sq}} w(h)\, dh=\int_{{{\mathbb R}}^n}\frac{\lim_{N\to\infty}\|D_h^k(f_{N}-f) \|_{L^p({{{\mathbb R}}^n})}^{q}}{|h|^{n+sq}} w(h)\, dh =0, \end{align*} $$

as desired. Altogether, we conclude the proof of Lemma 2.2.

Proof It is obvious that $C_c^\infty ({\mathbb R}^{2n})\subseteq L_w^q (L^p)({\mathbb R}^{2n})$ . Since w is locally integrable, we know that

$$ \begin{align*}d\mu(x,y):=w(y)\, dx\, dy\end{align*} $$

defines a $\sigma $ -finite Radon-measure on ${\mathbb R}^{2n}$ . Given any $f\in L_w^q (L^p)({\mathbb R}^{2n})$ , we have by [Reference Benedek and Panzone5, p. 313] that, for any $\epsilon>0$ , there exists a simple function

$$ \begin{align*}\varphi(x,y):=\sum_{i=1}^NC_{i}\,\mathbf{1}_{E_i}(x)\,\mathbf{1}_{F_i}(y),\end{align*} $$

where $|E_i|<\infty $ and $w(F_i)<\infty $ for all $i\in \{1,2,\dots , N\}$ , such that

(2.11) $$ \begin{align} \|f-\varphi\|_{L_w^q (L^p)({\mathbb R}^{2n})}< \epsilon. \end{align} $$

We claim that there exists a function $g\in C_c^\infty ({\mathbb R}^{2n})$ such that

(2.12) $$ \begin{align} \|\varphi-g\|_{L_w^q (L^p)({\mathbb R}^{2n})}< \epsilon. \end{align} $$

Once we have (2.12), then applying (2.11) gives

$$ \begin{align*} \|f-g\|_{L_w^q (L^p)({\mathbb R}^{2n})} \leq\|f-\varphi\|_{L_w^q (L^p)({\mathbb R}^{2n})} +\|\varphi-g\|_{L_w^q (L^p)({\mathbb R}^{2n})} <\epsilon+\epsilon =2\epsilon, \end{align*} $$

which indicates that any $f\in L_w^q (L^p)({\mathbb R}^{2n})$ can be approximated by functions in $C_c^\infty ({\mathbb R}^{2n})$ .

To show (2.12), it suffices to consider the case when $\varphi (x,y)=\mathbf {1}_E(x)\mathbf {1}_F(y)$ , where $E, F$ are measurable subsets in ${{{\mathbb R}}^n}$ with $|E|<\infty $ and $w(F)<\infty $ . To see this, by the fact that $C_c^\infty ({{{\mathbb R}}^n})$ is dense in $L^p({{{\mathbb R}}^n})$ , for any $\eta>0$ , there exists a function $\psi _1\in C_c^\infty ({{{\mathbb R}}^n})$ such that

$$ \begin{align*}\|\mathbf{1}_E-\psi_1\|_{L^p({{{\mathbb R}}^n})}<\eta\, w(F)^{-1/q}.\end{align*} $$

Since $w(x)\, dx$ is a doubling measure, by an argument similar to the unweighted case, we obtain that if

$$ \begin{align*}\begin{cases} \phi\in C_c^\infty({{{\mathbb R}}^n});\\ \int_{ {{{\mathbb R}}^n} }\phi(y) w(y)\, dy=1; \\ \phi_{t}(\cdot):=t^{-n}\phi(t^{-1}\ \cdot)\ \text{for all}\ t\in(0,\infty), \end{cases} \end{align*} $$

then any $h\in L_w^q ({{{\mathbb R}}^n})$ satisfies that

$$ \begin{align*}\int_{{{\mathbb R}}^n} \phi_{t}(\cdot-y)h(y) \mathbf{1}_{B({0},\,R)}(y)w(y)\,dy \to h\ \ \text{in}\ \ L_w^q ({{{\mathbb R}}^n})\end{align*} $$

as $R\to \infty $ and $t\to 0$ . In other words, $C_c^\infty ({{{\mathbb R}}^n})$ is dense in $L_w^q ({{{\mathbb R}}^n})$ . Consequently, for any $\eta>0$ , there exists a function $\psi _2\in C_c^\infty ({{{\mathbb R}}^n})$ , such that

$$ \begin{align*}\|\mathbf{ 1}_F-\psi_2\|_{L_w^q ({{{\mathbb R}}^n})}<\eta\|\psi_1\|_{L^p({{{\mathbb R}}^n})}^{-1}.\end{align*} $$

Let $g(x,y):=\psi _1(x)\psi _2(y)$ . Then,

$$ \begin{align*} \|\mathbf{1}_{E\times F}-g\|_{L_w^q (L^p)({\mathbb R}^{2n})} &=\left( \int_{ {{{\mathbb R}}^n} } \left( \int_{ {{{\mathbb R}}^n} } |\mathbf{1}_E(x)\mathbf{1}_F(y) -\psi_1(x)\psi_2(y)|^{p}\, dx \right)^{q/p} w(y)\, dy \right)^{1/q}\\ &\leq \left( \int_{ {{{\mathbb R}}^n} } \left( \int_{ {{{\mathbb R}}^n} } |\mathbf{1}_E(x)\mathbf{1}_F(y) -\psi_1(x)\mathbf{1}_F(y)|^{p}\, dx \right)^{q/p} w(y)\, dy \right)^{1/q}\\ &\quad + \left( \int_{ {{{\mathbb R}}^n} } \left( \int_{ {{{\mathbb R}}^n} } |\psi_1(x)\mathbf{1}_F(y) -\psi_1(x)\psi_2(y)|^{p}\, dx \right)^{q/p} w(y)\, dy \right)^{1/q}\\ &<2\eta. \end{align*} $$

This proves (2.12).

Proof Let $f \in \dot {\mathcal {L}}_{\beta }^{p,\, q,\,w}({\mathbb R}^{2n})$ . By Definition 1.4, there exists a function $\phi _0 \in L_w^q (L^p)({\mathbb R}^{2n})$ such that

$$ \begin{align*}f=(-\varDelta_{{{\mathbb R}}^{2n}})^{-\beta/2}\phi_0\ \ \mathrm{and}\ \ \|f\|_{\dot{\mathcal{L}}_{\beta}^{p,\, q,\,w}({\mathbb R}^{2n})}=\| \phi_0 \|_{L_w^q (L^p)({\mathbb R}^{2n})}.\end{align*} $$

From Lemma 2.3, for any $\epsilon>0$ , we can find a function $\phi \in C_c^\infty ({\mathbb R}^{2n})$ such that

$$ \begin{align*}\| \phi_0-\phi \|_{L_w^q (L^p)({\mathbb R}^{2n})}<\epsilon,\end{align*} $$

which implies

$$ \begin{align*} \left\|f-(-\varDelta_{{{\mathbb R}}^{2n}})^{-\beta/2}\phi\right\|_{\dot{\mathcal{L}}_{\beta}^{p,\, q,\,w}({\mathbb R}^{2n})} =\| \phi_0-\phi \|_{L_w^q (L^p)({\mathbb R}^{2n})}<\epsilon. \end{align*} $$

This ends the proof.

Proof Let $\Phi \in {\mathcal S}({\mathbb R}^{2n})$ satisfy ${\mathrm {\,supp\,}} \widehat {\Phi }\subset \{x\in {\mathbb R}^{2n}:\, c^{-1}\leq |x|\leq c\}$ for some constant $c>1$ and

$$ \begin{align*}\sum_{j\in{\mathbb Z}}\widehat{\Phi}(2^{-j}\xi)=1\quad \text{for all}\ \ \xi\neq 0.\end{align*} $$

For any $j\in {\mathbb Z}$ and $x\in {\mathbb R}^{2n}$ , let $\Phi _j(x):= 2^{2jn}\Phi (2^jx)$ . For any $\phi \in C_c^\infty ({\mathbb R}^{2n})$ and $N\in {\mathbb N}$ ,

$$ \begin{align*}\phi_N:=\sum_{|j|\leq N}\Phi_j\ast\phi\in {\mathcal S}_\infty({\mathbb R}^{2n}).\end{align*} $$

Moreover, the sequence $\{\phi _N\}_{N\in\mathbb N}$ converges to $\phi $ in ${\mathcal S}({\mathbb R}^{2n})$ (and, hence, in $L_w^q (L^p)({\mathbb R}^{2n})$ by Lemma 2.1) as $N\to \infty $ . From this and Corollary 2.4, it follows that $\{(-\varDelta _{{{\mathbb R}}^{2n}})^{-\beta /2}\phi :\, \phi \in {\mathcal S}_\infty ({\mathbb R}^{2n})\}$ is dense in $\dot {\mathcal {L}}_{\beta }^{p,\, q,\, w}({\mathbb R}^{2n})$ . Thus, to show the density of $C_c^\infty ({\mathbb R}^{2n})$ in $\dot {\mathcal {L}}_{\beta }^{p,\, q,\, w}({\mathbb R}^{2n})$ , we may as well assume that

$$ \begin{align*}f=(-\varDelta_{{{\mathbb R}}^{2n}})^{-\beta/2} g\end{align*} $$

for some $g\in {\mathcal S}_\infty ({\mathbb R}^{2n})$ , then our aim is to find a sequence $\{\varphi _k\}_{k\in {\mathbb Z}}$ in $C_c^\infty ({\mathbb R}^{2n})$ such that

(2.13) $$ \begin{align} \lim_{k\to\infty}\|\varphi_k-f\|_{\dot{\mathcal{L}}_{\beta}^{p,\, q,\, w}({\mathbb R}^{2n})}=\lim_{k\to\infty}\|(-\varDelta_{{{\mathbb R}}^{2n}})^{\beta/2}\varphi_k-g\|_{L_w^q (L^p)({\mathbb R}^{2n})}=0. \end{align} $$

Choose $\eta \in C_c^\infty ({\mathbb R}^{2n})$ satisfying $0\leq \eta \leq 1$ , $\eta =1$ if $|x|\leq 1$ and $\eta =0$ if $|x|\geq 2$ . For any $k\in {\mathbb N}$ , set

$$ \begin{align*}\varphi_k:=\eta_k(-\varDelta_{{{\mathbb R}}^{2n}})^{-\beta/2}g,\end{align*} $$

where $\eta _k(x)=\eta (x/k)$ for all $x\in {\mathbb R}^{2n}$ . Clearly, each $\varphi _k\in C_c^\infty ({\mathbb R}^{2n})$ . In order to show that such $\{\varphi _k\}_{k\in {\mathbb Z}}$ satisfies (2.13), we write $\beta =2m+s$ , where $m\in {\mathbb N}$ and $s\in [0,2)$ . Notice that

$$ \begin{align*}(-\varDelta_{{{\mathbb R}}^{2n}})^{\beta/2}\varphi_k- g=(-\varDelta_{{{\mathbb R}}^{2n}})^{s/2}(-\varDelta_{{{\mathbb R}}^{2n}})^{m} \left(\eta_k(-\varDelta_{{{\mathbb R}}^{2n}})^{-\beta/2}g\right)-g\end{align*} $$

and, if $m\neq 0$ , then

$$ \begin{align*} (-\varDelta_{{{\mathbb R}}^{2n}})^{m}\left(\eta_k(-\varDelta_{{{\mathbb R}}^{2n}})^{-\beta/2}g\right) &=\left(-\sum_{j=1}^m\partial_{x_j}^2\right)^{m}\left(\eta_k(-\varDelta_{{{\mathbb R}}^{2n}})^{-\beta/2}g\right)\\ &=(-1)^m\sum_{|\alpha|=m}\partial^{2\alpha} \left(\eta_k(-\varDelta_{{{\mathbb R}}^{2n}})^{-\beta/2}g\right)\\ &= \eta_k(-\varDelta_{{{\mathbb R}}^{2n}})^{m}\left((-\varDelta_{{{\mathbb R}}^{2n}})^{-\beta/2}g\right)\\ & \quad+\sum_{\genfrac{}{}{0pt}{}{|\alpha|=m}{0<|\gamma|\le 2m}} c_{m,\alpha,\gamma} \left(\partial^{\gamma}\eta_k\right) \partial^{2\alpha-\gamma}\left((-\varDelta_{{{\mathbb R}}^{2n}})^{-\beta/2}g\right), \end{align*} $$

where $\alpha ,\gamma $ are $2n$ -tuples of multi-indices and $c_{m,\alpha ,\gamma }$ are constants. By the Minkowski inequality, we find that the norm $\|\,\cdot \,\|_{L_w^q (L^p)({\mathbb R}^{2n})}$ in (2.13) can be controlled by the $\|\,\cdot \,\|_{L_w^q (L^p)({\mathbb R}^{2n})}$ -norm of

$$ \begin{align*}g_k:=(-\varDelta_{{{\mathbb R}}^{2n}})^{s/2}(\eta_k(-\varDelta_{{{\mathbb R}}^{2n}})^{(2m-\beta)/2}g)-g=(-\varDelta_{{{\mathbb R}}^{2n}})^{s/2}(\eta_k(-\varDelta_{{{\mathbb R}}^{2n}})^{-s/2}g)-g\end{align*} $$

and a finite linear combination of the $\|\,\cdot \,\|_{L_w^q (L^p)({\mathbb R}^{2n})}$ -norm of the following type of functions

$$ \begin{align*}h_k:=(-\varDelta_{{{\mathbb R}}^{2n}})^{s/2}((\partial^{\gamma}\eta_k)h),\end{align*} $$

where $h\in {\mathcal S}_\infty ({\mathbb R}^{2n})$ is of the form $\partial ^{2\alpha -\gamma }((-\varDelta _{{{\mathbb R}}^{2n}})^{-\beta /2}g)$ and $\gamma $ is a non-zero multi-index. Thus, to obtain (2.13), we are left to prove that the $L_w^q (L^p)({\mathbb R}^{2n})$ -norms of $g_k$ and $h_k$ tend to $0$ as $k\to \infty $ .

If $s=0$ , then $g_k=\eta _k g-g$ and $|h_k|\le k^{-|\gamma |} \|\partial ^{\gamma }\eta \|_{L^\infty ({\mathbb R}^{2n})}|h|$ , whose $L_w^q (L^p)({\mathbb R}^{2n})$ -norms obviously go to $0$ as $k\to \infty $ . Next, we consider the case when $s\in (0,2)$ .

Part 1: estimate of the $L_w^q (L^p)({\mathbb R}^{2n})$ -norm of $g_k$ . To simplify the notation, we set

$$ \begin{align*}u=(-\varDelta_{{{\mathbb R}}^{2n}})^{(2m-\beta)/2}g=(-\varDelta_{{{\mathbb R}}^{2n}})^{-s/2}g,\end{align*} $$

which belongs to ${\mathcal S}({\mathbb R}^{2n})$ by using the Fourier transform and the fact that $g\in {\mathcal S}({\mathbb R}^{2n})$ . By [Reference Silvestre40, Section 2], we have

$$ \begin{align*}(-\varDelta_{{\mathbb R}^{2n}})^{s/2} (\eta_k u)=C_{n,s}\, \text{p.v.}\int_{{\mathbb R}^{2n}}\frac{\eta_k(x) u(x)-\eta_k(y) u(y)}{|x-y|^{2n+s}}\, dy,\end{align*} $$

with $C_{n,s}$ being a positive constant depending only on n and s. Via writing

$$ \begin{align*}\eta_k(x) u(x)-\eta_k(y) u(y)&=\eta_k(x)[u(x)- u(y)] \\&\quad+[\eta_k(x)-\eta_k(y)][u(y)-u(x)]+u(x)[\eta_k(x)-\eta_k(y)], \end{align*} $$

we then have

(2.14) $$ \begin{align} g_k(x) &=\left(C_{n,s}\eta_k(x)\text{p.v.}\int_{{\mathbb R}^{2n}}\frac{u(x)- u(y)}{|x-y|^{2n+s}}\, dy-g(x)\right)\\ &\quad+C_{n,s}\text{p.v.}\int_{|x-y|<\frac{1+|x|}{2}}\frac{[\eta_k(x)-\eta_k(y)][u(y)-u(x)]}{|x-y|^{2n+s}}\, dy\notag\\ &\quad+C_{n,s}u(x)\text{p.v.}\int_{|x-y|<\frac{1+|x|}{2}}\frac{\eta_k(x)-\eta_k(y)}{|x-y|^{2n+s}}\, dy\notag\\ &\quad+C_{n,s}\text{p.v.}\int_{|x-y|\ge \frac{1+|x|}{2}}\frac{[\eta_k(x)-\eta_k(y)]u(y)}{|x-y|^{2n+s}}\, dy\notag\\ &=:G_k^1(x)+G_k^2(x)+G_k^3(x)+G_k^4(x).\notag \end{align} $$

When $k\to \infty $ , it is obvious that

(2.15) $$ \begin{align} \|G_k^1\|_{L_w^q (L^p)({\mathbb R}^{2n})}= \|\eta_k (-\varDelta_{{{{\mathbb R}}^n}})^{s/2}u -g\|_{L_w^q (L^p)({\mathbb R}^{2n})} =\|\eta_k g -g\|_{L_w^q (L^p)({\mathbb R}^{2n})}\to 0. \end{align} $$

For $G_k^2$ , note that the mean value theorem implies that

$$ \begin{align*}|\eta_k(x)-\eta_k(y)|\le k^{-1}|x-y| \|\nabla\eta\|_{L^\infty({\mathbb R}^{2n})}\end{align*} $$

and

$$ \begin{align*}|u(y)-u(x)|\le |x-y||\nabla u(x+\theta(y-x))|\end{align*} $$

hold for some constant $\theta \in (0,1)$ . If $|x-y|<\frac {1+|x|}{2}$ , then $1+|x+\theta (y-x)|\approx 1+|x|$ , which, along with the fact that $u\in {\mathcal S}_\infty $ , implies that

$$ \begin{align*}|\nabla u(x+\theta(y-x))|\lesssim (1+|x|)^{-(2n+2)}. \end{align*} $$

Thus, we obtain

$$ \begin{align*} |G_k^{2}(x)| &\le C_{n,s}\int_{|x-y|<\frac{1+|x|}{2}}\frac{|\eta_k(x)-\eta_k(y)||u(y)-u(x)|}{|x-y|^{2n+s}}\, dy\\ &\lesssim k^{-1}(1+|x|)^{-(2n+2)} \int_{|x-y|<\frac{1+|x|}{2}} |x-y|^{2-s-2n}\,dy\notag\\ & \approx k^{-1}(1+|x|)^{-(2n+s)}.\notag \end{align*} $$

According to the arguments in (2.2) and (2.3) (by taking N therein to be $2n+s$ ), we see that

(2.16) $$ \begin{align} \|(1+|\cdot|)^{-(2n+s)}\|_{L_w^q (L^p)({\mathbb R}^{2n})}<\infty \end{align} $$

and, hence,

(2.17) $$ \begin{align} \lim_{k\to\infty}\|G_k^{2}\|_{L_w^q (L^p)({\mathbb R}^{2n})}=0. \end{align} $$

Next, we consider $G_k^3$ . From the Taylor expansion formula,

$$ \begin{align*} \eta_k(x)-\eta_k(y)=\left(\frac{x-y}{k}\right)\cdot \nabla \eta\left(\frac{x}{k}\right)+O\left(\left|\frac{x-y}{k}\right|^2\right). \end{align*} $$

Observe that

$$ \begin{align*}\text{p.v.}\int_{|x-y|<\frac{1+|x|}{2}}\frac{(\frac{x-y}{k})\cdot \nabla \eta(\frac{x}{k})}{|x-y|^{2n+s}}\, dy=0,\end{align*} $$

which induces

$$ \begin{align*} |G_k^3(x)|\lesssim k^{-2}|u(x)|\int_{|x-y|<\frac{1+|x|}{2}}|x-y|^{2-s-2n}\, dy\lesssim k^{-2}(1+|x|)^{2-s}|u(x)|, \end{align*} $$

thereby leading to

(2.18) $$ \begin{align} \lim_{k\to\infty}\|G_k^{3}\|_{L_w^q (L^p)({\mathbb R}^{2n})}\lesssim \lim_{k\to\infty} k^{-2}\|(1+|\cdot|)^{2-s}u(\cdot)\|_{L_w^q (L^p)({\mathbb R}^{2n})}=0. \end{align} $$

Next, observe that the integrand of $G_k^4$ can be controlled by $\|\eta \|_{L^\infty ({\mathbb R}^{2n})}\|u\|_{L^\infty ({\mathbb R}^{2n})} |x-y|^{-(2n+s)}$ and

$$ \begin{align*}\int_{|x-y|\ge \frac{1+|x|}{2}}|x-y|^{-(2n+s)}\, dy \lesssim (1+|x|)^{-s}<\infty, \end{align*} $$

which, together with the Lebesgue dominated convergence theorem, shows that

$$ \begin{align*}\lim_{k\to\infty} G_k^{4}(x) = \int_{|x-y|\ge \frac{1+|x|}{2}}\lim_{k\to\infty}\frac{[\eta_k(x)-\eta_k(y)]u(y)}{|x-y|^{2n+s}}\, dy=0. \end{align*} $$

Moreover, note that

$$ \begin{align*} \int_{|x-y|\ge \frac{1+|x|}{2}}\frac{|u(y)|}{|x-y|^{2n+s}}\, dy &\lesssim \int_{|x-y|\ge \frac{1+|x|}{2}}\frac{|u(y)|}{(1+|x|)^{2n+s}}\, dy \lesssim (1+|x|)^{-(2n+s)}\|u\|_{L^1({\mathbb R}^{2n})}, \end{align*} $$

which, combined with (2.16) and the Lebesgue dominated convergence theorem, again yields

(2.19) $$ \begin{align} \lim_{k\to\infty}\|G_k^{4}\|_{L_w^q (L^p)({\mathbb R}^{2n})}=\left\|\lim_{k\to\infty}G_k^{4}\right\|_{L_w^q (L^p)({\mathbb R}^{2n})}=0. \end{align} $$

Substituting (2.15) and (2.17)–(2.19) into (2.14) yields that $\|g_k\|_{L_w^q (L^p)({\mathbb R}^{2n})}\to 0$ as $k\to \infty $ .

Part 2: estimate of the $L_w^q (L^p)({\mathbb R}^{2n})$ -norm of $h_k$ . Just like (2.14), we now write

$$ \begin{align*} h_k(x) &=C_{n,s}\partial^{\gamma}\eta_k(x)\text{p.v.}\int_{{\mathbb R}^{2n}}\frac{h(x)- h(y)}{|x-y|^{2n+s}}\, dy\\ &\quad+C_{n,s}\text{p.v.}\int_{|x-y|<\frac{1+|x|}{2}}\frac{[\partial^{\gamma}\eta_k(x)-\partial^{\gamma}\eta_k(y)][h(y)-h(x)]}{|x-y|^{2n+s}}\, dy\notag\\ &\quad+C_{n,s}h(x)\text{p.v.}\int_{|x-y|<\frac{1+|x|}{2}}\frac{\partial^{\gamma}\eta_k(x)-\partial^{\gamma}\eta_k(y)}{|x-y|^{2n+s}}\, dy\notag\\ &\quad+C_{n,s}\text{p.v.}\int_{|x-y|\ge \frac{1+|x|}{2}}\frac{[\partial^{\gamma}\eta_k(x)-\partial^{\gamma}\eta_k(y)]h(y)}{|x-y|^{2n+s}}\, dy\notag\\ &=:H_k^1(x)+H_k^2(x)+H_k^3(x)+H_k^4(x).\notag \end{align*} $$

For $i=2,3,4$ , the estimate of $G_k^i$ also implies that (with $\eta _k$ and u therein replaced by $\partial ^{\gamma }\eta _k$ and h, respectively)

$$ \begin{align*}\lim_{k\to\infty}\|H_k^{i}\|_{L_w^q (L^p)({\mathbb R}^{2n})}=0. \end{align*} $$

For $H_k^{1}$ , it follows from $h\in {\mathcal S}_\infty $ that $(-\varDelta _{{{\mathbb R}}^{2n}})^{s/2} h\in {\mathcal S}_\infty $ , thereby leading to

$$ \begin{align*}\|H_k^{1}\|_{L_w^q (L^p)({\mathbb R}^{2n})} &=\left\|\left(\partial^{\gamma}\eta_k\right)\left((-\varDelta_{{{\mathbb R}}^{2n}})^{s/2} h \right)\right \|_{L_w^q (L^p)({\mathbb R}^{2n})}\\ &\le k^{-|\gamma|} \|\partial^{\gamma}\eta\|_{L^\infty({\mathbb R}^{2n})}\left\|(-\varDelta_{{{\mathbb R}}^{2n}})^{s/2} h \right \|_{L_w^q (L^p)({\mathbb R}^{2n})} \to 0 \end{align*} $$

as $k\to \infty $ . This induces that $\|h_k\|_{L_w^q (L^p)({\mathbb R}^{2n})}\to 0$ when $k\to \infty $ , as desired.

Altogether, we conclude the proof of Lemma 2.5.

Proof By symmetry, it suffices to show that

$$ \begin{align*} \|f\|_{\dot{\Lambda}_{p,\, q,\, w,\, \varphi}^{s}({{{\mathbb R}}^n})}\lesssim\|f\|_{\dot{\Lambda}_{p,\, q,\, w,\, \psi}^{s}({{{\mathbb R}}^n})}. \end{align*} $$

Based on the argument in [Reference Liu, Wu, Yang and Yuan37, Proposition 2.3], there exists a function $\phi \in {\mathcal S}({{{\mathbb R}}^n})$ such that

$$ \begin{align*}{\mathrm{\,supp\,}} \hat{\phi}\subset\{x\in{{{\mathbb R}}^n}:\ c_0^{-1}\leq|x|\leq c_0\}\end{align*} $$

for some constant $c_0\in (1,\infty )$ and, moreover,

$$ \begin{align*} \int_{0}^{\infty}\overline{\hat{\phi}(t\xi)}\ \hat{\psi}(t\xi)\, \frac{dt}{t}=1\ \ {\text{for all}}\ \xi\in {{{\mathbb R}}^n} \setminus\{{0}\}. \end{align*} $$

Consequently, for any $f\in {\mathcal S}^{\prime }_\infty ({{{\mathbb R}}^n})$ , we know from [Reference Frazier, Jawerth and Weiss24, p.122, Theorem 3] that

(2.20) $$ \begin{align} f=\int_{0}^{\infty}\phi_t\ast \psi_t \ast f\, \frac{dt}{t}\quad \text{in}\,\, {\mathcal S}^{\prime}_{\infty}({{{\mathbb R}}^n}). \end{align} $$

Given any $r\in (0,\infty )$ , since $\varphi _r\in {\mathcal S}_\infty ({{{\mathbb R}}^n})$ , it follows from this last equality that

$$ \begin{align*}\varphi_r\ast f=\int_{0}^{\infty}\varphi_r \ast \phi_t\ast \psi_t \ast f\, \frac{dt}{t}\end{align*} $$

holds pointwisely. Consequently, by the Young inequality and the Hölder inequality,

$$ \begin{align*} &\|\varphi_r \ast f \|_{L^p({{{\mathbb R}}^n})}\\&\leq\int_{0}^{\infty}\|\varphi_r \ast \phi_t\ast \psi_t \ast f \|_{L^p({{{\mathbb R}}^n})}\, \frac{dt}{t}\\ &\leq\int_{0}^{\infty}\|\varphi \ast \phi_{t/r} \|_{L^{1}({{{\mathbb R}}^n})}\|\psi_t \ast f \|_{L^p({{{\mathbb R}}^n})}\, \frac{dt}{t}\\ &\leq\left(\int_{0}^{\infty}\|\varphi \ast \phi_{t/r} \|_{L^{1}({{{\mathbb R}}^n})}\, \frac{dt}{t}\right)^{1/{q'}}\left(\int_{0}^{\infty}\|\varphi \ast \phi_{t/r} \|_{L^{1}({{{\mathbb R}}^n})}\|\psi_t \ast f \|_{L^p({{{\mathbb R}}^n})}^q\, \frac{dt}{t}\right)^{1/q}. \end{align*} $$

Given any $ K,M\in {\mathbb Z}_+$ , we derive from [Reference Frazier, Jawerth and Weiss24, p.121, Lemma 2; p.122, Lemma 4] that

$$ \begin{align*}|\varphi \ast \phi_{t/r}(x)| \lesssim \begin{cases} \displaystyle \frac{t}{r}(1+|x|)^{-M},& 0<t\leq r;\\ \displaystyle \left(\frac{r}{t}\right)^{n+1+K}\left(1+\frac{r|x|}{t} \right)^{-M}\lesssim\left(\frac{r}{t}\right)^{N}\left(1+ |x|\right)^{-M},&0<r\leq t, \end{cases}\end{align*} $$

where $N=n+1+K-M$ . If we choose $M>n$ and $K>M-n-1$ , then $N>0$ and

(2.21) $$ \begin{align} \|\varphi \ast \phi_{t/r} \|_{L^{1}({{{\mathbb R}}^n})}\lesssim \min \left\{\frac{t}{r},\ \left(\frac{r}{t}\right)^N \right\}, \end{align} $$

thereby leading to

$$ \begin{align*} \int_{0}^{\infty}\|\varphi \ast \phi_{t/r} \|_{L^{1}({{{\mathbb R}}^n})}\, \frac{dt}{t}\lesssim \int_{0}^{r}\frac{t}{r}\, \frac{dt}{t}+\int_{r}^{\infty}\left(\frac{r}{t}\right)^{N}\, \frac{dt}{t}\lesssim 1. \end{align*} $$

Thus, we obtain

$$ \begin{align*}\|\varphi_r \ast f \|_{L^p({{{\mathbb R}}^n})}\lesssim \left(\int_{0}^{\infty}\|\varphi \ast \phi_{t/r} \|_{L^{1}({{{\mathbb R}}^n})}\|\psi_t \ast f \|_{L^p({{{\mathbb R}}^n})}^q\, \frac{dt}{t}\right)^{1/q},\end{align*} $$

which further implies

$$ \begin{align*} \|f\|_{\dot{\Lambda}_{p,\, q,\, w,\, \varphi}^{s}({{{\mathbb R}}^n})} &=\left(\int_{0}^{\infty}\| \varphi_r \ast f \|_{L^p({{{\mathbb R}}^n})}^{q}r^{-sq-1}w(r)\, dr\right)^{1/q}\\&\lesssim \left(\int_{0}^{\infty}r^{-sq}\left(\int_{0}^{\infty}\|\varphi \ast \phi_{t/r} \|_{L^{1}({{{\mathbb R}}^n})}\|\psi_t \ast f \|_{L^p({{{\mathbb R}}^n})}^q\, \frac{dt}{t}\right)w(r)\, \frac{dr}{r}\right)^{1/q}\\ &= \left(\int_{0}^{\infty}t^{-sq}\|\psi_t \ast f \|_{L^p({{{\mathbb R}}^n})}^qw(t)\right.\\&\quad\times\left.\left(\int_{0}^{\infty}\left(\frac{r}{t}\right)^{-sq}\|\varphi \ast \phi_{t/r} \|_{L^{1}({{{\mathbb R}}^n})}\frac{w(r)}{w(t)}\, \frac{dr}{r}\right) \frac{dt}{t}\right)^{1/q}. \end{align*} $$

We may choose K sufficiently large such that

$$ \begin{align*}N=n+1+K-M>(s+1)q.\end{align*} $$

This implies $\sigma q<N-sq$ due to $\sigma < 1$ . Further, by (2.21), (1.4), and (1.5), together with a change of variables $u=r/t$ , we deduce

$$ \begin{align*} \int_{0}^{t}\left(\frac{r}{t}\right)^{-sq}\|\varphi \ast \phi_{t/r} \|_{L^{1}({{{\mathbb R}}^n})}\frac{w(r)}{w(t)}\, \frac{dr}{r} &\lesssim\int_{0}^{t}\left(\frac{r}{t}\right)^{-sq+N}\frac{w(r)}{w(t)}\ \frac{dr}{r}\\ &\leq \int_{0}^{1} u^{-sq+N} \left(\sup_{\rho \in (0, \infty)}\frac{w(u \rho)}{w(\rho)}\right)\, \frac{du}{u}\\ &\leq\int_{0}^{1}u^{\sigma q}\, \left(\sup_{\rho \in (0, \infty)} \frac{w(u\rho)}{w(\rho)}\right)\frac{du}{u} < \infty \end{align*} $$

and

$$ \begin{align*} \int_{t}^{\infty}\left(\frac{r}{t}\right)^{-sq}\|\varphi \ast \phi_{t/r} \|_{L^{1}({{{\mathbb R}}^n})}\frac{w(r)}{w(t)}\, \frac{dr}{r} &\lesssim\int_{t}^{\infty}\left(\frac{r}{t}\right)^{-sq-1}\frac{w(r)}{w(t)}\, \frac{dr}{r}\\ &\leq \int_1^{\infty} u^{-sq-1} \left(\sup_{\rho \in (0, \infty)}\frac{w(u \rho)}{w(\rho)}\right)\, \frac{du}{u}\\ &\leq\int_1^{\infty}u^{\delta q}\, \left(\sup_{\rho \in (0, \infty)} \frac{w(u\rho)}{w(\rho)}\right)\frac{du}{u} < \infty, \end{align*} $$

where we used $N>(\sigma +s)q$ and $\delta>-s-1/q$ . Altogether, we obtain

$$ \begin{align*} \|f\|_{\dot{\Lambda}_{p,\, q,\, w,\, \varphi}^{s}({{{\mathbb R}}^n})}\lesssim \left(\int_{0}^{\infty}\| \psi_t \ast f \|_{L^p({{{\mathbb R}}^n})}^{q}t^{-sq-1}w(t)\, dt\right)^{1/q}=\|f\|_{\dot{\Lambda}_{p,\, q,\, w,\, \psi}^{s}({{{\mathbb R}}^n})}, \end{align*} $$

as desired.

Proof Observe that $Tf$ is radial, which implies

$$ \begin{align*} \|Tf\|_{L_w^q({{{\mathbb R}}^n})}=\left(\int_{ {{{\mathbb R}}^n} } |Tf(x)|^{q} w(x)\, dx\right)^{1/q} =\omega_{n-1}^{1/q}\left(\int_{0}^{\infty} |Tf(R)|^{q} w(R) R^{n-1}\, dR\right)^{1/q}. \end{align*} $$

Upon writing

$$ \begin{align*} Tf(R) &= \int_{0}^{\infty} \int_{\mathbb{S}^{n-1}} K(R, r) f(r \eta) r^{n-1}\, d\sigma(\eta) \,dr \\ &= \int_{0}^{\infty} \int_{\mathbb{S}^{n-1}} R^{-n} K(1, t)\, f(tR \eta)\, (tR)^{n-1} R \, d\sigma(\eta) \,dt \\ &= \int_{\mathbb{S}^{n-1}} \left(\int_{0}^{\infty} K(1, t)\, f(tR \eta)\, t^{n-1} \, dt\right) \,d\sigma(\eta)\\ &=: \int_{\mathbb{S}^{n-1}} T_{\eta}f(R) \,d\sigma(\eta), \end{align*} $$

we then apply the Minkowski inequality and deduce

(2.23) $$ \begin{align} \|Tf\|_{L_w^q({{{\mathbb R}}^n})} &=\omega_{n-1}^{1/q}\left(\int_0^\infty\left| \int_{\mathbb{S}^{n-1}} T_{\eta}f(R) \,d\sigma(\eta) \right|{}^qw(R)R^{n-1}\,dR\right)^{1/q}\\ &\leq\omega_{n-1}^{1/q}\int_{\mathbb{S}^{n-1}}\left(\int_{0}^{\infty} |T_{\eta}f(R)|^{q}w(R)R^{n-1}\,dR\right)^{1/q} \, d\sigma(\eta).{\nonumber} \end{align} $$

By duality, there exists a function h such that

$$ \begin{align*}\int_{0}^{\infty} |h(R)|^{q'} w(R) R^{n-1} \, dR=1\end{align*} $$

and

(2.24) $$ \begin{align} \left(\int_{0}^{\infty}|T_{\eta}f(R)|^{q}w(R)R^{n-1}\,dR\right)^{1/q} =\int_{0}^{\infty} T_{\eta}f(R) h(R)\, w(R) R^{n-1} \, dR. \end{align} $$

Notice that (2.22), the Fubini theorem and the Hölder inequality imply

$$ \begin{align*} &\int_{0}^{\infty}T_{\eta}f(R) h(R) w(R) R^{n-1} \, dR\\ &\quad\leq \int_{0}^{\infty} \left(\int_{0}^{\infty} K(1, t)\, |f(tR \eta)|\, t^{n-1} \, dt\right) |h(R)| w(R) R^{n-1} \, dR\\ &\quad= \int_{0}^{\infty} K(1, t)\,t^{n-1} \left(\int_{0}^{\infty} |f(tR \eta)\, h(R)| w(R) R^{n-1} \, dR\right)\, dt\\ &\quad\leq \int_{0}^{\infty} K(1, t)\,t^{n-1} \left(\int_{0}^{\infty} |f(tR \eta)|^{q} w(R) R^{n-1} \, dR\right)^{1/q}\, dt\\ &\quad= \int_{0}^{\infty} K(1, t)\,t^{n-1-\frac{n}{q}} \left(\int_{0}^{\infty} |f(\rho \eta)|^{q} w(t^{-1} \rho) \rho^{n-1} \, d\rho\right)^{1/q}\, dt\\ &\quad\leq \int_{0}^{\infty}K(1, t)t^{\frac{n}{q'}-1}\, \left(\sup_{\rho \in (0, \infty)} \frac{w(t^{-1}\rho)}{w(\rho)}\right)^{1/q}dt \left(\int_{0}^{\infty} |f(\rho \eta)|^{q} w(\rho) \rho^{n-1} \, d\rho\right)^{1/q}\\ &\quad=J \left(\int_{0}^{\infty} |f(\rho \eta)|^{q} w(\rho) \rho^{n-1} \, d\rho\right)^{1/q}. \end{align*} $$

This, together with (2.23), (2.24), and the Hölder inequality, further implies

$$ \begin{align*} \|Tf\|_{L_w^q({{{\mathbb R}}^n})} &\leq J \,\omega_{n-1}^{1/q} \int_{\mathbb{S}^{n-1}} \left(\int_{0}^{\infty} |f(\rho \eta)|^{q} w(\rho) \rho^{n-1} \, d\rho\right)^{1/q}\, d\sigma(\eta)\\ &\leq J \,\omega_{n-1}^{1/q} \left(\int_{\mathbb{S}^{n-1}} \int_{0}^{\infty} |f(\rho \eta)|^{q} w(\rho) \rho^{n-1} \, d\rho d\sigma(\eta)\right)^{1/q}\omega_{n-1}^{1/q'}\\ &= J \,\omega_{n-1} \|f\|_{L_w^q({{{\mathbb R}}^n})}. \end{align*} $$

Thus, we complete the proof of Lemma 2.9.

Proof We first prove (i). As in (2.20), there exists a function $\phi \in {\mathcal S}({{{\mathbb R}}^n})$ such that

$$ \begin{align*}{\mathrm{\,supp\,}} \hat{\phi}\subset \{x\in{{{\mathbb R}}^n}:\ c_0^{-1}\leq |x|\leq c_0\}\end{align*} $$

and

(2.25) $$ \begin{align}f=\int_{0}^{\infty}\phi_t\ast \varphi_t \ast f\, \frac{dt}{t}\quad \text{in}\,\, {\mathcal S}^{\prime}_{\infty}({{{\mathbb R}}^n}), \end{align} $$

where $c_0\in (1,\infty )$ is a constant. Set

$$ \begin{align*}f_0:= \int_{0}^{\infty}\phi_t\ast \varphi_t \ast f\, \frac{dt}{t}.\end{align*} $$

Clearly, $f_0$ is a Lebesgue measurable function, but may be infinite on a set of positive Lebesgue measure.

Let us calculate the semi-norm $\|f_0\|_{\dot {\Lambda }_{p,\, q,\, w}^{s}({{{\mathbb R}}^n})}$ . To this end, let $k=\lfloor s\rfloor +1$ . Then, we have

$$ \begin{align*} \mathrm{D}_{h}^{k}f_0=\int_{0}^{\infty}\mathrm{D}_{h}^{k}(\phi_t\ast \varphi_t \ast f)\, \frac{dt}{t}, \end{align*} $$

which implies

(2.26) $$ \begin{align} \|\mathrm{D}_{h}^{k}f_0 \|_{L^p({{{\mathbb R}}^n})}\leq\int_{0}^{\infty}\|\mathrm{D}_{h}^{k}(\phi_t\ast \varphi_t \ast f)\|_{L^p({{{\mathbb R}}^n})}\, \frac{dt}{t}. \end{align} $$

On the one hand, by (1.1) and the Young inequality, we obtain

(2.27) $$ \begin{align} &\|\mathrm{D}_{h}^{k}(\phi_t\ast \varphi_t \ast f) \|_{L^p({{{\mathbb R}}^n})}\\&\quad=\left\|\sum_{m=0}^{k}(-1)^{k-m}\binom{k}{m}(\phi_t\ast \varphi_t \ast f)(\cdot+mh) \right\|_{L^p({{{\mathbb R}}^n})}\notag\\ &\quad\notag\lesssim \|\phi_t\ast \varphi_t \ast f \|_{L^p({{{\mathbb R}}^n})}\notag\\&\quad\leq \|\phi_t \|_{L^{1}({{{\mathbb R}}^n})} \|\varphi_t \ast f \|_{L^p({{{\mathbb R}}^n})}\notag\\&\quad\lesssim\|\varphi_t \ast f \|_{L^p({{{\mathbb R}}^n})}.\notag \end{align} $$

On the other hand, by

$$ \begin{align*} \mathrm{D}_{h}^{k}g(x) &=\int_{[0,1]^k}\sum_{j_1=1}^{n}\cdots \sum_{j_{k}=1}^{n} h_{j_1} \cdots h_{j_{k}} (\partial_{j_1} \cdots \partial_{j_{k}}g)(x +(t_1+\cdots+t_{k})h)\, dt_1 \cdots \,dt_{k} \end{align*} $$

and the Minkowski inequality, we have

$$ \begin{align*} \|\mathrm{D}_{h}^{k}(\phi_t\ast \varphi_t \ast f) \|_{L^p({{{\mathbb R}}^n})}&\lesssim |h|^k \sum_{i_1=1}^{n}\cdots\sum_{i_k=1}^{n} \|\partial_{i_1}\cdots\partial_{i_k}(\phi_t\ast \varphi_t \ast f) \|_{L^p({{{\mathbb R}}^n})}\\ &=|h|^k \sum_{i_1=1}^{n}\cdots\sum_{i_k=1}^{n} \|(\partial_{i_1}\cdots\partial_{i_k}\phi_t)\ast (\varphi_t\ast f) \|_{L^p({{{\mathbb R}}^n})}\\ &\leq|h|^k \sum_{i_1=1}^{n}\cdots\sum_{i_k=1}^{n} \|\partial_{i_1}\cdots\partial_{i_k}\phi_t \|_{L^{1}({{{\mathbb R}}^n})}\| \varphi_t\ast f \|_{L^p({{{\mathbb R}}^n})}. \end{align*} $$

Since

$$ \begin{align*} \|\partial_{i_1}\cdots\partial_{i_k}\phi_t \|_{L^{1}({{{\mathbb R}}^n})}=t^{-k}\|(\partial_{i_1}\cdots\partial_{i_k}\phi)_{t}(x)\|_{L^1({{{\mathbb R}}^n})} \lesssim t^{-k}, \end{align*} $$

it follows that

(2.28) $$ \begin{align} \|\mathrm{D}_{h}^{k}(\phi_t\ast \varphi_t\ast f) \|_{L^p({{{\mathbb R}}^n})}&\lesssim t^{-k}|h|^k\| \varphi_t\ast f \|_{L^p({{{\mathbb R}}^n})}. \end{align} $$

Inserting (2.27) and (2.28) into (2.26) yields

(2.29) $$ \begin{align} \|{\mathrm{D}}_{h}^{k}f_0 \|_{L^p({{{\mathbb R}}^n})}&\lesssim\int_{0}^{\infty}\min\left\{1,\frac{|h|}{t}\right\}^k\|\varphi_t\ast f\|_{L^p({{{\mathbb R}}^n})}\, \frac{dt}{t}, \end{align} $$

thereby leading to

$$ \begin{align*} \|f_0\|_{\dot{\Lambda}_{p,\, q,\,w}^{s}({{{\mathbb R}}^n})} &=\left( \int_{ {{{\mathbb R}}^n} } \frac{\|{\mathrm{D}}_{h}^{k}f_0 \|_{L^p({{{\mathbb R}}^n})}^{q}}{|h|^{n+sq}} w(h)\, dh \right)^{1/q}\\ &\lesssim\left( \int_{ {{{\mathbb R}}^n} } \left(\int_{0}^{\infty}\min\left\{1,\frac{|h|}{t}\right\}^k\|\varphi_t\ast f\|_{L^p({{{\mathbb R}}^n})}\, \frac{dt}{t}\right)^{q} \frac{w(h)}{|h|^{n+sq}} \, dh \right)^{1/q}\\ &=\left(\int_{ {{{\mathbb R}}^n} }\left(\int_{0}^{\infty}\min\left\{1,\frac{|h|}{t}\right\}^k \left(\frac{t}{|h|}\right)^{s+n/q}\frac{\|\varphi_t\ast f\|_{L^p({{{\mathbb R}}^n})}}{t^{s+n/q}}\frac{\, dt}t\right)^q w(h)\, dh\right)^{1/q}. \end{align*} $$

If we set

$$ \begin{align*}F_1(y):=\frac{\|\varphi_{|y|}\ast f\|_{L^p({{{\mathbb R}}^n})}}{|y|^{s+n/q}}\end{align*} $$

and

$$ \begin{align*}K_1(|h|,|y|):=\min\left\{1,\frac{|h|}{|y|}\right\}^k \left(\frac{|y|}{|h|}\right)^{s+n/q}|y|^{-n},\end{align*} $$

then

$$ \begin{align*}\int_{0}^{\infty}\min\left\{1,\frac{|h|}{t}\right\}^k \left(\frac{t}{|h|}\right)^{s+n/q}\frac{\|\varphi_t\ast f\|_{L^p({{{\mathbb R}}^n})}}{t^{s+n/q}}\frac{\, dt}t=\frac{1}{\omega_{n-1}}\int_{ {{{\mathbb R}}^n} }K_1(|h|,|y|)F_1(y)\, dy\end{align*} $$

and, hence,

(2.30) $$ \begin{align}\|f_0\|_{\dot{\Lambda}_{p,\, q,\,w}^{s}({{{\mathbb R}}^n})}\lesssim \left(\int_{ {{{\mathbb R}}^n} }\left(\frac{1}{\omega_{n-1}}\int_{ {{{\mathbb R}}^n} }K_1(|h|,|y|)F_1(y)\, dy\right)^q w(h)\, dh\right)^{1/q}. \end{align} $$

Clearly, $K_1$ is homogeneous of degree $-n$ . Moreover, by (1.4) and (1.5), we have

$$ \begin{align*} &\int_{0}^{\infty}K_1(1, t)\,t^{\frac n{q'}-1} \left(\sup_{\rho \in (0, \infty)} \frac{w(t^{-1}\rho)}{w(\rho)}\right)^{\frac 1 q }dt\\ &\quad=\int_{0}^{1}t^{s} \left(\sup_{\rho \in (0, \infty)} \frac{w(t^{-1}\rho)}{w(\rho)}\right)^{\frac 1 q }\,\frac{dt}{t}+\int_1^{\infty}t^{s-k} \left(\sup_{\rho \in (0, \infty)} \frac{w(t^{-1}\rho)}{w(\rho)}\right)^{\frac 1 q }\,\frac{dt}{t}\\ &\quad=\int_1^{\infty}t^{-s} \left(\sup_{\rho \in (0, \infty)} \frac{w(t\rho)}{w(\rho)}\right)^{\frac 1 q }\,\frac{dt}{t}+\int_{0}^{1}t^{k-s} \left(\sup_{\rho \in (0, \infty)} \frac{w(t\rho)}{w(\rho)}\right)^{\frac 1 q }\,\frac{dt}{t}\\ &\quad\leq\left(\int_1^{\infty}t^{\delta q} \left(\sup_{\rho \in (0, \infty)} \frac{w(t\rho)}{w(\rho)}\right)\,\frac{dt}{t}\right)^{\frac 1 q }\left(\int_1^{\infty}t^{-(s+\delta) q'}\,\frac{dt}{t}\right)^{\frac{1}{q'}}\\ &\quad\quad+\left(\int_{0}^{1}t^{\sigma q} \left(\sup_{\rho \in (0, \infty)} \frac{w(t\rho)}{w(\rho)}\right)\,\frac{dt}{t}\right)^{\frac 1 q }\left(\int_0^1t^{(k-s-\sigma) q'}\,\frac{dt}{t}\right)^{\frac{1}{q'}}\\ &\quad<\infty, \end{align*} $$

which, together with Lemma 2.9, implies

(2.31)

We still need to show that $f_0\in \dot {\Lambda }_{p,\, q,\, w}^{s}({{{\mathbb R}}^n})$ . To achieve this, we only need to validate that $f_0$ can be approximated by $C_c^\infty ({{{\mathbb R}}^n})$ -functions under $\|\,\cdot \,\|_{\dot {\Lambda }_{p,\, q,\, w}^{s}({{{\mathbb R}}^n})}$ . Indeed, for any $\epsilon \in (0,1)$ , define

$$ \begin{align*}f_{\epsilon}:= \int_{\epsilon}^{1/\epsilon}\phi_t\ast \varphi_t \ast f\, \frac{dt}{t}.\end{align*} $$

After a revisit of the above proof of $\|f_0\|_{\dot {\Lambda }_{p,\, q,\,w}^{s}({{{\mathbb R}}^n})}<\infty $ , we derive from (2.26) and (2.29)–(2.31) that

$$ \begin{align*}\|f_0\|_{\dot{\Lambda}_{p,\, q,\,w}^{s}({{{\mathbb R}}^n})}^q\le \int_{ {{{\mathbb R}}^n} } \frac{\big(\int_{0}^{\infty}\|\mathrm{D}_{h}^{k}(\phi_t\ast \varphi_t \ast f)\|_{L^p({{{\mathbb R}}^n})}\, \frac{dt}{t}\big)^{q}}{|h|^{n+sq}} w(h)\, dh<\infty, \end{align*} $$

which directly gives

(2.32) $$ \begin{align}\|f_\epsilon\|_{\dot{\Lambda}_{p,\, q,\, w}^{s}({{{\mathbb R}}^n})}^q \le \int_{ {{{\mathbb R}}^n} } \frac{\big(\int_{\epsilon}^{1/\epsilon}\|\mathrm{D}_{h}^{k}(\phi_t\ast \varphi_t \ast f)\|_{L^p({{{\mathbb R}}^n})}\, \frac{dt}{t}\big)^{q}}{|h|^{n+sq}} w(h)\, dh <\infty. \end{align} $$

Meanwhile, observing that

$$ \begin{align*}\|\mathrm{D}^k_h(f_0-f_\epsilon)\|_{L^p({{{\mathbb R}}^n})} \le \left(\int_{0}^{\epsilon} +\int_{1/\epsilon}^{\infty} \right)\|\mathrm{D}^k_h(\phi_t\ast \varphi_t \ast f)\|_{L^p({{{\mathbb R}}^n})}\, \frac{dt}{t},\end{align*} $$

and applying the Lebesgue dominated convergence theorem to

$$ \begin{align*}\int_{ {{{\mathbb R}}^n} } \frac{\big(\int_{0}^{\epsilon} \|\mathrm{D}_{h}^{k}(\phi_t\ast \varphi_t \ast f)\|_{L^p({{{\mathbb R}}^n})}\, \frac{dt}{t} +\int_{1/\epsilon}^{\infty} \|\mathrm{D}^k_h(\phi_t\ast \varphi_t \ast f)\|_{L^p({{{\mathbb R}}^n})}\, \frac{dt}{t}\big)^{q}}{|h|^{n+sq}} w(h)\, dh, \end{align*} $$

we arrive at

(2.33) $$ \begin{align} \lim_{\epsilon\to0}\|f_0-f_\epsilon\|_{\dot{\Lambda}_{p,\, q,\, w}^{s}({{{\mathbb R}}^n})} =0. \end{align} $$

Moreover, by the Young inequality and the Hölder inequality, we have

$$ \begin{align*} \|f_\epsilon\|_{L^p({{{\mathbb R}}^n})}&\leq \int_{\epsilon}^{1/\epsilon}\|\phi_t\ast \varphi_t \ast f\|_{L^p({{{\mathbb R}}^n})}\, \frac{dt}{t}\\ &\leq \int_{\epsilon}^{1/\epsilon}\|\phi_t\|_{L^{1}({{{\mathbb R}}^n})}\|\varphi_t \ast f\|_{L^p({{{\mathbb R}}^n})}\, \frac{dt}{t}\\ &\leq \left(\int_{\epsilon}^{1/\epsilon}\| \varphi_t \ast f \|_{L^p({{{\mathbb R}}^n})}^{q}t^{-sq}w(t)\, \frac{dt}t\right)^{1/q}\left(\int_{\epsilon}^{1/\epsilon}\|\phi_t\|_{L^{1}({{{\mathbb R}}^n})}^{q'}t^{sq'}w(t)^{1-q'}\, \frac{dt}t\right)^{1/q'}\\ &\leq \|f\|_{\dot{\Lambda}_{p,\, q,\, w,\, \varphi}^{s}({{{\mathbb R}}^n})}\|\phi\|_{L^1({{{\mathbb R}}^n})}\left(\int_{\epsilon}^{1/\epsilon}t^{sq'}w(t)^{1-q'}\, \frac{dt}t\right)^{1/q'}. \end{align*} $$

Note that $w\in A_q({{{\mathbb R}}^n})$ means $w^{1-q'}\in A_{q'}({{{\mathbb R}}^n})$ . This, together with the radial property of w, implies

$$ \begin{align*} \int_{\epsilon}^{1/\epsilon}t^{sq'-1}w(t)^{1-q'}\, dt\approx \int_{\epsilon\leq |x|\leq \frac{1}{\epsilon}}|x|^{sq'-n}w(x)^{1-q'}\, dx<\infty \end{align*} $$

and, hence,

$$ \begin{align*}f_\epsilon\in L^p({{{\mathbb R}}^n}).\end{align*} $$

By this and (2.32), we see that $f_\epsilon \in {\Lambda }_{p,\, q,\, w}^{s}({{{\mathbb R}}^n})$ . Further, it follows from Lemma 2.2 that there exists a sequence $\{f_{\epsilon ,j}\}_j\subset C_c^\infty ({{{\mathbb R}}^n})$ such that

(2.34) $$ \begin{align} \lim_{j\to\infty}\|f_{\epsilon,j}-f_\epsilon\|_{\dot{\Lambda}_{p,\, q,\, w}^{s}({{{\mathbb R}}^n})}\le\lim_{j\to\infty}\|f_{\epsilon,j}-f_\epsilon\|_{\Lambda_{p,\, q,\, w}^{s}({{{\mathbb R}}^n})}=0. \end{align} $$

Combining (2.33) and (2.34), we deduce that $f_0$ can be approximated by $C_c^\infty ({{{\mathbb R}}^n})$ -functions $\{f_{\epsilon ,j}\}_{\epsilon ,j}$ under the semi-norm $\|\,\cdot \,\|_{\dot {\Lambda }_{p,\, q,\, w}^{s}({{{\mathbb R}}^n})}$ . This finishes the proof of (i).

Next, we show (ii). To this end, we choose a radial function $\psi \in {\mathcal S}_\infty ({{{\mathbb R}}^n})$ satisfying ${ \mathrm {\,supp\,}} \hat {\psi }\subset \{x\in {{{\mathbb R}}^n}:\ c_0^{-1}\leq |x|\leq c_0\}$ for some constant $c_0\in (1,\infty )$ . For any $z\in {{{\mathbb R}}^n}$ , define

$$ \begin{align*}\Psi(z)=\sum_{m=1}^{k}(-1)^{k-m}\binom{k}{m}m^{-n}\psi\left(\frac{z}{m}\right).\end{align*} $$

Clearly, $\Psi \in {\mathcal S}_\infty ({{{\mathbb R}}^n})$ and ${ \mathrm {\,supp\,}} \hat {\Psi }\subset \{x\in {{{\mathbb R}}^n}:\ (kc_0)^{-1}\leq |x|\leq c_0\}.$ Due to Lemma 2.8, it suffices to show that for any $f\in \dot {\Lambda }_{p,\, q,\, w}^{s}({{{\mathbb R}}^n})$ there is

$$ \begin{align*}\|f\|_{\dot{\Lambda}_{p,\, q,\, w,\,\Psi}^{s}({{{\mathbb R}}^n})}\lesssim\|f\|_{\dot{\Lambda}_{p,\, q,\, w}^{s}({{{\mathbb R}}^n})}.\end{align*} $$

Indeed, by the cancellation condition of $\psi $ , we write

$$ \begin{align*} \Psi_t\ast f(x)&=\int_{ {{{\mathbb R}}^n} }f(x-z)\Psi_t(z)\ dz\\ &=\int_{ {{{\mathbb R}}^n} }f(x-z)\sum_{m=1}^{k}(-1)^{k-m}\binom{k}{m}m^{-n} \psi_t\left(\frac{z}{m}\right)\ dz\\ &=\int_{ {{{\mathbb R}}^n} }\sum_{m=1}^{k}(-1)^{k-m}\binom{k}{m}f(x+my)\psi_t(y)\ dy\\ &=\int_{ {{{\mathbb R}}^n} }\sum_{m=0}^{k}(-1)^{k-m}\binom{k}{m}f(x+my)\psi_t(y)\ dy\\ &=\int_{ {{{\mathbb R}}^n} }\mathrm{D}_{y}^{k}f(x)\psi_t(y)\ dy, \end{align*} $$

which, together with the Minkowski inequality, further implies

$$ \begin{align*} \|\Psi_t\ast f \|_{L^p({{{\mathbb R}}^n})}^q&=\left(\int_{ {{{\mathbb R}}^n} }\left|\int_{ {{{\mathbb R}}^n} }\mathrm{D}_{y}^{k}f(x)\psi_t(y)\ dy\right|{}^p\ dx\right)^{q/p}\\ &\leq\left(\int_{ {{{\mathbb R}}^n} }\left(\int_{ {{{\mathbb R}}^n} }|\mathrm{D}_{y}^{k}f(x)|^p|\psi_t(y)|^p\ dx\right)^{1/p}\ dy\right)^{q}\\ &=\left(\int_{ {{{\mathbb R}}^n} }|\psi_t(y)|\, \|\mathrm{D}_{y}^{k}f \|_{L^p({{{\mathbb R}}^n})}\ dy\right)^{q}. \end{align*} $$

Thus,

$$ \begin{align*} \|f\|_{\dot{\Lambda}_{p,\, q,\, w,\, \Psi}^{s}({{{\mathbb R}}^n})} &=\left(\int_{0}^{\infty}\| \Psi_t \ast f \|_{L^p({{{\mathbb R}}^n})}^{q}t^{-sq-1}w(t)\, dt\right)^{1/q}\\ &\leq\left(\int_{0}^{\infty}\left(\int_{ {{{\mathbb R}}^n} }|\psi_t(y)|\, \|\mathrm{D}_{y}^{k}f \|_{L^p({{{\mathbb R}}^n})}\ dy\right)^{q}t^{-sq-1}w(t)\, dt\right)^{1/q}\\ &=\left(\frac{1}{\omega_{n-1}}\int_{ {{{\mathbb R}}^n} }\left(\int_{ {{{\mathbb R}}^n} }|\psi_{|x|}(y)|\, \|\mathrm{D}_{y}^{k}f \|_{L^p({{{\mathbb R}}^n})}\ dy\right)^{q}|x|^{-sq-n}w(x)\, dx\right)^{1/q}\\ &\approx\left(\int_{ {{{\mathbb R}}^n} }\left(\int_{ {{{\mathbb R}}^n} }|\psi_{|x|}(y)|\left(\frac{|y|}{|x|}\right)^{s+\frac{n}{q}}\,\frac{\|\mathrm{D}_{y}^{k}f \|_{L^p({{{\mathbb R}}^n})}}{|y|^{s+\frac{n}{q}}} \ dy\right)^{q}w(x)\, dx\right)^{1/q}. \end{align*} $$

Since $\psi \in {\mathcal S}_\infty ({{{\mathbb R}}^n})$ , it follows that for all $x,y\in {{{\mathbb R}}^n}$ ,

$$ \begin{align*}\left|\psi_{|x|}(y)\right|\left(\frac{|y|}{|x|}\right)^{s+n/q}\lesssim |x|^{-n}\left(\frac{|y|}{|x|}\right)^{s+n/q}\left(1+\frac{|y|}{|x|}\right)^{-N},\end{align*} $$

where $N>s+n+\sigma $ . For any $x,\,y\in {{{\mathbb R}}^n}$ , upon setting

$$ \begin{align*}F_2(y):=\frac{\|\mathrm{D}_{y}^{k}f \|_{L^p({{{\mathbb R}}^n})}}{|y|^{s+n/q}}\end{align*} $$

and

$$ \begin{align*}K_2(|x|,|y|):=|x|^{-n}\left(\frac{|y|}{|x|}\right)^{s+n/q}\left(1+\frac{|y|}{|x|}\right)^{-N},\end{align*} $$

we then arrive at the estimate

$$ \begin{align*}\|f\|_{\dot{\Lambda}_{p,\, q,\,w}^{s}({{{\mathbb R}}^n})}\lesssim \left(\int_{ {{{\mathbb R}}^n} }\left(\int_{ {{{\mathbb R}}^n} }K_2(|x|,|y|)F_2(y)\, dy\right)^q w(x)\, dx\right)^{1/q}.\end{align*} $$

Clearly, $K_2$ is homogeneous of degree $-n$ . Moreover, from (1.4) and (1.5), it follows that

$$ \begin{align*} &\int_{0}^{\infty}K_2(1, t)\,t^{\frac n{q'}-1} \left(\sup_{\rho \in (0, \infty)} \frac{w(t^{-1}\rho)}{w(\rho)}\right)^{\frac 1 q }dt\\ &\quad\leq\int_{0}^{1} t^{s+n}\left(\sup_{\rho \in (0, \infty)}\frac{w(t^{-1}\rho)}{w(\rho)}\right)^{\frac 1 q }\frac{dt}{t} +\int_1^{\infty}t^{s+n-N}\left(\sup_{\rho \in (0, \infty)}\frac{w(t^{-1}\rho)}{w(\rho)}\right)^{\frac 1 q }\frac{dt}{t}\\ &\quad=\int_1^{\infty} t^{-s-n}\left(\sup_{\rho \in (0, \infty)}\frac{w(t\rho)}{w(\rho)}\right)^{\frac 1 q }\frac{dt}{t}+\int_{0}^{1} t^{N-s-n}\left(\sup_{\rho \in (0, \infty)}\frac{w(t\rho)}{w(\rho)}\right)^{\frac 1 q }\frac{dt}{t}\\ &\quad\leq\left(\int_1^{\infty} t^{\delta q}\left(\sup_{\rho \in (0, \infty)}\frac{w(t\rho)}{w(\rho)}\right)\frac{dt}{t}\right)^{\frac 1 q }\left(\int_1^{\infty}t^{-(\delta+s+n)q'}\frac{dt}{t}\right)^{\frac{1}{q'}}\\ &\quad\quad+\left(\int_{0}^{1} t^{\sigma q}\left(\sup_{\rho \in (0, \infty)}\frac{w(t\rho)}{w(\rho)}\right)\frac{dt}{t}\right)^{\frac 1 q }\left(\int_{0}^{1}t^{(N-s-n-\sigma)q'}\frac{dt}{t}\right)^{\frac{1}{q'}}\\ &\quad<\infty, \end{align*} $$

where in the last step we used $N>s+n+\sigma $ and $\delta>-s$ . We then have by Lemma 2.9 that

$$ \begin{align*} \|f\|_{\dot{\Lambda}_{p,\, q,\, w,\, \Psi}^{s}({{{\mathbb R}}^n})} \lesssim\left( \int_{ {{{\mathbb R}}^n} } \frac{\|{\mathrm{D}}_{y}^{k}f \|_{L^p({{{\mathbb R}}^n})}^{q}}{|y|^{n+sq}} w(y)\, dy \right)^{\frac 1 q }=\|f\|_{\dot{\Lambda}_{p,\, q,\, w}^{s}({{{\mathbb R}}^n})}. \end{align*} $$

This finishes the proof of (ii) and, hence, Theorem 2.10.

Proof Let $e_1:=(1,0,\dots ,0)$ be the unit vector in ${{{\mathbb R}}^n}$ . After a rotation, we then have

$$ \begin{align*} \int_{ {{{\mathbb R}}^n} } G_{\alpha}(|h| \xi) \left(e^{2 \pi i h \cdot \xi}-1\right)^{k} \,\frac{dh}{|h|^n} =\int_{ {{{\mathbb R}}^n} } G_{\alpha}(|h| e_1) \left(e^{2 \pi i h\cdot e_1}-1\right)^{k} \,\frac{dh}{|h|^n}, \end{align*} $$

which is independent of $\xi $ if the integral converges. For all $h\in {{{\mathbb R}}^n}$ , observe that

$$ \begin{align*}\left|\left(e^{2 \pi i h\cdot e_1}-1\right)^{k}\right|\lesssim \min\{1, |h|^k\}.\end{align*} $$

Note that $\hat {G}_{\alpha }$ is radial, so does $G_{\alpha }$ . Due to this reason, we may write $G_\alpha (x)$ as $G_\alpha (r)$ whenever $|x|=r$ . Thus,

$$ \begin{align*} \left|\int_{ {{{\mathbb R}}^n} } G_{\alpha}(|h| e_1) \left(e^{2 \pi i h\cdot e_1}-1\right)^{k} \,\frac{dh}{|h|^n}\right|&\lesssim \int_{ {{{\mathbb R}}^n} } |G_{\alpha}(|h| e_1)|\min\{1, |h|^k\}\,\frac{dh}{|h|^n}\\ &\lesssim \int_{0}^{\infty}G_{\alpha}(r)\min\{1, r^k\}\, \frac{dr}{r}. \end{align*} $$

According to the two formulae (1.2.11) and (1.2.12) in [Reference Grafakos29, Proposition 1.2.5], we know that when $|\xi |<1$ ,

$$ \begin{align*} G_{\alpha}(\xi)\approx \begin{cases} \displaystyle |\xi|^{\alpha-n}+1+O(|\xi|^{\alpha-n+2})\approx|\xi|^{\alpha-n}&\quad\mathrm{if}\ \ \alpha\in(0,n);\\ \displaystyle \log{|\xi|^{-1}}+1+O(|\xi|^2)\lesssim\log{|\xi|^{-1}}&\quad\mathrm{if}\ \ \alpha=n;\\ \displaystyle 1+O(|\xi|^{\alpha-n})\lesssim1&\quad\mathrm{if}\ \ \alpha\in(n,\infty), \end{cases} \end{align*} $$

and when $|\xi |\ge 1$ ,

$$ \begin{align*}G_{\alpha}(\xi)\lesssim e^{-\pi|\xi|}.\end{align*} $$

From these, we deduce that

$$ \begin{align*} \int_{0}^{1}G_{\alpha}(r)\min\{1, r^k\}\, \frac{dr}{r}\lesssim \begin{cases} \displaystyle \int_{0}^{1} r^{\alpha-n}\, r^{k-1}\, dr \lesssim 1&\quad\mathrm{if}\ \ \alpha\in(0,n);\\ \displaystyle \int_{0}^{1} \log \frac{1}{r}\, r^{k-1}\, dr \lesssim 1&\quad\mathrm{if}\ \ \alpha=n;\\ \displaystyle \int_{0}^{1} r^{k-1}\, dr\lesssim1&\quad\mathrm{if}\ \ \alpha\in(n,\infty), \end{cases} \end{align*} $$

and

$$ \begin{align*} \int_1^{\infty}G_{\alpha}(r)\min\{1, r^k\}\, \frac{dr}{r} \lesssim \int_1^{\infty} e^{-\pi r}\, \frac{dr}{r} \lesssim 1. \end{align*} $$

Combining the last two formulae derives that $\int _{ {{{\mathbb R}}^n} } G_{\alpha }(|h| \xi ) (e^{2 \pi i h \cdot \xi }-1)^{k} \,\frac {dh}{|h|^n}$ converges to a constant which independent of $\xi $ . This constant will be denoted by $c_{\alpha }$ .

Proof of Theorem 1.6(i) under $\beta \in (0,2n)$

Since $\dot {\Lambda }_{p,\, q,\,w}^{s}({{{\mathbb R}}^n})$ is defined as the completion of $C_c^\infty ({{{\mathbb R}}^n})$ under the semi-norm $\|\,\cdot \,\|_{\dot {\Lambda }_{p,\, q,\,w}^{s}({{{\mathbb R}}^n})}$ , we may as well assume that $g\in C_c^\infty ({{{\mathbb R}}^n})\cap \dot {\Lambda }_{p,\, q,\,w}^{s}({{{\mathbb R}}^n})$ . Let $k:=\lfloor s\rfloor +1$ . For any $x,y\in {{{\mathbb R}}^n}$ , define

(3.1) $$ \begin{align} f(x, y):=|y|^{-\beta} \mathrm{D}_{y}^{k} g(x) \end{align} $$

and

(3.2) $$ \begin{align} F(x,y)&:= \int_{ {{{\mathbb R}}^n} }\int_{ {{{\mathbb R}}^n} } \left(|x-z|^{2}+|y-h|^{2}\right)^{\frac{\beta-2n}{2}} |h|^{-\beta} \mathrm{D}_{h}^{k} g(z)\, dz \, dh. \end{align} $$

Before going further, we are about to show that for all $\varphi \in {\mathcal S}({\mathbb R}^{2n})$ ,

(3.3) $$ \begin{align} \int_{ {{{\mathbb R}}^n} }\int_{ {{{\mathbb R}}^n} }|F(x,y)||\varphi(x,y)|\, dx\, dy<\infty. \end{align} $$

Because $\varphi \in {\mathcal S}({\mathbb R}^{2n})$ , the integral in (3.3) is bounded by a constant multiple of

$$ \begin{align*}{\mathcal J}:=\int_{ {{{\mathbb R}}^n} }\int_{{{{\mathbb R}}^n} }\int_{ {{{\mathbb R}}^n} }\int_{ {{{\mathbb R}}^n} } \frac{\left(|z|+|y-h|\right)^{\beta-2n} |h|^{-\beta} |\mathrm{D}_{h}^{k} g(x-z)|}{(1+|x|)^N(1+|y|)^N}\, dz \, dh\, dx\, dy, \end{align*} $$

where $N\in {\mathbb N}$ can be any large integer. To obtain (3.3), it suffices to prove that ${\mathcal J}<\infty $ holds for sufficiently large N.

Since $g\in C_c^\infty ({{{\mathbb R}}^n})$ , we take a large number $R\in (1,\infty )$ such that ${ \mathrm {\,supp\,}} g\subset B({0},R)$ . Via splitting the integral domain of ${\mathcal J}$ into annulus, we find that

$$ \begin{align*} {\mathcal J} &\le\iint_{|x|+|y|<R}\int_{ {{{\mathbb R}}^n} }\int_{ {{{\mathbb R}}^n} } \frac{\left(|z|+|y-h|\right)^{\beta-2n} |h|^{-\beta} |\mathrm{D}_{h}^{k} g(x-z)|}{(1+|x|)^N(1+|y|)^N}\, dz \, dh\, dx\, dy\\ &\quad +\sum_{j=1}^\infty\iint_{2^{j-1}R\leq|x|+|y|<2^jR}\int_{ {{{\mathbb R}}^n} }\int_{ {{{\mathbb R}}^n} } \frac{\left(|z|+|y-h|\right)^{\beta-2n} |h|^{-\beta} |\mathrm{D}_{h}^{k} g(x-z)|}{(1+|x|)^N(1+|y|)^N}\, dz \, dh\, dx\, dy\\ &=:\mathrm{I}_0+\sum_{j=1}^\infty \mathrm{I}_j. \end{align*} $$

If $2^{j-1}R\leq |x|+|y|$ for some $j\ge 1$ , then either $|x|\ge 2^{j-2}R$ or $|y|\ge 2^{j-2}R$ , whatever the cases we always have

$$ \begin{align*} \mathrm{I}_j &\lesssim(2^{j}R)^{-\frac N2}\iint_{|x|+|y|<2^jR}\int_{{{{\mathbb R}}^n}}\int_{ {{{\mathbb R}}^n} } \frac{\left(|z|+|y-h|\right)^{\beta-2n} |h|^{-\beta} |\mathrm{D}_{h}^{k} g(x-z)|}{(1+|x|)^{N/2}(1+|y|)^{N/2}}\, dz \, dh\, dx\, dy. \end{align*} $$

Fix $j\in \{0,1,2,\dots \}$ . We now consider the cases $|h|\ge 2^{j+1}R$ and $|h|< 2^{j+1}R$ , respectively.

• • If $|x|+|y|<2^jR$ and $|h|\ge 2^{j+1}R$ , then $|y-h|\approx |h|$ , which, together with (1.1) and $g\in C_c^\infty ({\mathbb R}^{n})$ , implies that

  (3.4) $$ \begin{align} \mathrm{Z}_1&:=\int_{|h| \ge 2^{j+1}R} \int_{{{{\mathbb R}}^n}} \left(|z|+|y-h|\right)^{\beta-2n} |h|^{-\beta} \left|\mathrm{D}_{h}^{k} g(x-z)\right|\, dz \, dh\\ &\lesssim \sum_{m=0}^{k} \binom{k}{m} \int_{|h|\geq 2^{j+1}R}\int_{{{{\mathbb R}}^n}} |h|^{-2n} |g(x-z+mh)|\, dz \, dh \notag\\ &\lesssim \|g\|_{L^1({{{\mathbb R}}^n})} \int_{|h|\geq 2^{j+1}R}|h|^{-2n}\, dh \notag\\ &\lesssim 1. \notag \end{align} $$

• • If $|x|+|y|<2^jR$ and $|h|<2^{j+1}R$ , then by (1.1), we have ${ \mathrm {\,supp\,}} (\mathrm {D}_{h}^{k}g)\subseteq B({0}, k2^{j+2}R)$ , which implies that $\mathrm {D}_{h}^{k}g(x-z)\neq 0$ only if $|x-z|<k2^{j+2}R$ , thereby leading to $|z|<k2^{j+3}R.$ Moreover, by (2.6) and $g\in C_c^{\infty }({\mathbb R}^{n})$ , we see that

  $$ \begin{align*} \left|\mathrm{D}_{h}^{k}g(x)\right| &\le\sum_{j_1=1}^{n}\cdots\sum_{j_{k}=1}^{n} |h|^k\int_{[0,1]^k} \left|(\partial_{j_1} \cdots \partial_{j_{k}}g)(x +(t_1+\cdots+t_{k})h)\right|\, dt_1 \cdots \,dt_{k}\\ &\lesssim|h|^k{\nonumber} \end{align*} $$
  holds uniformly in $x\in {{{\mathbb R}}^n}$ . Thus, we obtain
  $$ \begin{align*} \mathrm{Z}_2&:=\int_{|h| < 2^{j+1}R} \int_{{{{\mathbb R}}^n}} \left(|z|+|y-h|\right)^{\beta-2n} |h|^{-\beta} \left|\mathrm{D}_{h}^{k} g(x-z)\right|\, dz \, dh\\ &\lesssim\int_{|z| <k2^{j+3}R} \int_{\genfrac{}{}{0pt}{}{|h| < 2^{j+1}R}{|y-h|\ge |h|/2}} (|z|+|h|)^{\beta-2n}|h|^{k-\beta} \, dh\,dz\\&\quad+ \int_{|z| <k2^{j+3}R}\int_{\genfrac{}{}{0pt}{}{|h| < 2^{j+1}R}{|y-h|< |h|/2}} (|z|+|y-h|)^{\beta-2n}|h|^{k-\beta} \, dh \, dz. \end{align*} $$
  Denote these last two double-integrals by $\mathrm {Z}_{2,1}$ and $\mathrm {Z}_{2,2}$ , respectively. Clearly,
  $$ \begin{align*} \mathrm{Z}_{2,1}\lesssim \int_0^{2^{j+1}R} \left(\int_0^{k2^{j+3}R} (s+t)^{\beta-2n} s^{n-1} t^{k+n-\beta-1}\,ds\right)\,dt. \end{align*} $$
  Considering the integrals over the domain $s\le t$ and $s>t$ , we then get
  $$ \begin{align*} \mathrm{Z}_{2,1} &\lesssim \int_0^{2^{j+1}R} \left(\int_0^t s^{n-1} t^{k-n-1}\,ds\right)\,dt + \int_0^{2^{j+1}R} \left(\int_t^{k2^{j+3}R} s^{\beta-n-1} t^{k+n-\beta-1}\,ds\right)\,dt\\ &\lesssim \int_0^{2^{j+1}R} t^{k-1}\,dt+\begin{cases} \int_0^{2^{j+1}R} t^{k-1}\,dt &\text{as}\ 0<\beta<n\\ \int_0^{2^{j+1}R} \left(\log\frac{k{2^{j+3}R}}{t}\right)t^{k-1}\,dt &\text{as}\ \beta=n\\ [k2^{j+3}R]^{\beta-n}\int_0^{2^{j+1}R} t^{k+n-\beta-1} \,dt &\text{as}\ n<\beta<2n \end{cases} \\ &\lesssim(2^j R)^k, \end{align*} $$
  where for the case $n<\beta <2n$ the integral converges because of the fact
  $$ \begin{align*}k+n-\beta=\lfloor s\rfloor+1+n-\left(s+\frac nq\right)>0.\end{align*} $$
  Concerning $\mathrm {Z}_{2,2}$ , observe that if $|y-h|< |h|/2$ then $|y|/2<|h|<2|y|$ , which induces
  $$ \begin{align*} \mathrm{Z}_{2,2}& \lesssim |y|^{k-\beta} \int_{|z| <k2^{j+3}R} \int_{|y-h|< |y|} (|z|+|y-h|)^{\beta-2n} \, dh \, dz\\ &\approx |y|^{k-\beta} \int_{|z| <k2^{j+3}R} \int_{|w|< |y|} (|z|+|w|)^{\beta-2n} \, dw \, dz\\ &\approx |y|^{k-\beta} \int_0^{2^{j+1}R}\left(\int_0^{k2^{j+3}R} (s+\tau )^{\beta-2n} s^{n-1} \tau^{n-1}\,ds \right)\,d\tau. \end{align*} $$
  In a way similar to the estimate of $\mathrm {Z}_{2,2}$ , it holds that
  $$ \begin{align*} \mathrm{Z}_{2,2}& \lesssim |y|^{k-\beta} (2^j R)^\beta. \end{align*} $$
  A combination of the estimates of $\mathrm {Z}_{2,1}$ and $\mathrm {Z}_{2,2}$ gives
  (3.5) $$ \begin{align} \mathrm{Z}_{2}\lesssim (2^j R)^k+|y|^{k-\beta} (2^j R)^\beta. \end{align} $$

Next, upon taking $N>2(n+k+\beta )$ , we utilize (3.4) and (3.5) to derive

$$ \begin{align*} \mathrm{I}_j&\lesssim (2^{j-2}R)^{-N/2} \int_{|x|<2^j R}\int_{|y|<2^j R} \frac{(2^j R)^k+|y|^{k-\beta} (2^j R)^\beta}{(1+|x|)^{N/2}(1+|y|)^{N/2}}\,dy\,dx\\&\lesssim (2^j R)^{k-N/2}+(2^j R)^{\beta-N/2}. \end{align*} $$

Note that this last estimate is still true when $j=0$ . Summing in j, we arrive at the conclusion

$$ \begin{align*}{\mathcal J} \le \sum_{j=0}^\infty \mathrm{I}_j \lesssim \sum_{j=0}^\infty \left((2^jR)^{k-N/2}+(2^j R)^{\beta-N/2}\right)\lesssim 1.\end{align*} $$

Thus, we obtain the desired estimate in (3.3).

Next, we continue with the proof of Theorem 1.6(i). Observe that the proof of (3.3) (especially the estimate of $\mathrm {I}_0$ ) also implies that the restricted function $F(\cdot , 0)$ is locally integrable on ${{{\mathbb R}}^n}$ . Given any $\psi \in C_c^\infty ({{{\mathbb R}}^n})$ , it follows from the Fubini theorem that

$$ \begin{align*} \left\langle F(\cdot, 0),\, \psi \right\rangle &=\int_{{{{\mathbb R}}^n}}\left(\int_{ {{{\mathbb R}}^n} }\int_{ {{{\mathbb R}}^n} } \left(|x-z|^{2}+|h|^{2}\right)^{\frac{\beta-2n}{2}} |h|^{-\beta} \mathrm{D}_{h}^{k} g(z)\, dz \, dh\right)\psi(x)\, dx\\ &=\int_{{{{\mathbb R}}^n}}\left(\int_{ {{{\mathbb R}}^n} }\int_{ {{{\mathbb R}}^n} } \left[\left(\frac{|x-z|}{|h|}\right)^{2}+1\right]^{\frac{\beta-2n}{2}}|h|^{-n} \mathrm{D}_{h}^{k} g(z)\,\psi(x)\, dz \, dx\right)\,\frac{dh}{|h|^n}. \end{align*} $$

Let $G_{2n-\beta }$ be the kernel of the Bessel potential on ${{{\mathbb R}}^n}$ as in Lemma 3.1. For any $x\in {{{\mathbb R}}^n}$ , set

$$ \begin{align*}\phi(x)=\hat{G}_{2n-\beta}(x)=(1+|x|^2)^{-\frac{2n-\beta}{2}}.\end{align*} $$

Via using the dilation $\phi _t(x)=t^{-n}\phi (t^{-1}x)$ for all $t\in (0,\infty )$ and $x\in {{{\mathbb R}}^n}$ , we then have

$$ \begin{align*}\int_{ {{{\mathbb R}}^n} } \left[\left(\frac{|x-z|}{|h|}\right)^{2}+1\right]^{\frac{\beta-2n}{2}}|h|^{-n} \mathrm{D}_{h}^{k} g(z)\, dz =\left(\phi_{|h|}*\mathrm{D}_{h}^{k}g\right)(x) \end{align*} $$

and, hence,

$$ \begin{align*} &\int_{ {{{\mathbb R}}^n} }\left(\int_{ {{{\mathbb R}}^n} } \left[\left(\frac{|x-z|}{|h|}\right)^{2}+1\right]^{\frac{\beta-2n}{2}}|h|^{-n} \mathrm{D}_{h}^{k} g(z)\, dz\right)\psi(x) \, dx\\ &\quad= \left\langle \phi_{|h|}*\mathrm{D}_{h}^{k}g,\, \psi \right\rangle\\ &\quad= \left\langle \left(\phi_{|h|}*\mathrm{D}_{h}^{k}g\right)^\wedge,\, \psi^\vee \right\rangle\\ &\quad= \left\langle \widehat{\phi_{|h|}}\widehat{\mathrm{D}_{h}^{k}g},\, \psi^\vee \right\rangle\\ &\quad= \left\langle \widehat{\phi}(|h|\cdot)\left(e^{2 \pi i h \cdot (\cdot)}-1\right)^{k} \,\hat{g}(\cdot),\, \psi^\vee(\cdot) \right\rangle\\ &\quad=\int_{{{{\mathbb R}}^n}} G_{2n-\beta}(|h|\xi)\,\left(e^{2 \pi i h \cdot \xi}-1\right)^{k} \,\hat{g}(\xi)\, \psi^{\vee}(\xi)\,d\xi, \end{align*} $$

where the last step is due to $\hat {\phi }(\xi )=G_{2n-\beta }(-\xi )$ so that $\hat {\phi }(|h|\xi )=G_{2n-\beta }(-|h|\xi )=G_{2n-\beta }(|h|\xi )$ . Further, an application of Lemma 3.1 yields

$$ \begin{align*} \left\langle F(\cdot, 0),\, \psi \right\rangle &=\int_{{{{\mathbb R}}^n}}\left(\int_{{{{\mathbb R}}^n}} G_{2n-\beta}(|h|\xi)\,\left(e^{2 \pi i h \cdot \xi}-1\right)^{k} \,\hat{g}(\xi)\, \psi^{\vee}(\xi)\,d\xi\right)\,\frac{dh}{|h|^n}\\ &=\int_{{{{\mathbb R}}^n}}\left(\int_{{{{\mathbb R}}^n}} G_{2n-\beta}(|h|\xi)\,\left(e^{2 \pi i h \cdot \xi}-1\right)^{k} \,\hat{g}(\xi)\, \psi^{\vee}(\xi)\,\frac{dh}{|h|^n}\right)\,d\xi\\ &=c_{2n-\beta}\int_{{{{\mathbb R}}^n}} \hat{g}(\xi)\, \psi^{\vee}(\xi)\,d\xi\\ &=c_{2n-\beta} \left\langle g,\, \psi \right\rangle, \end{align*} $$

where $c_{2n-\beta }$ is the constant determined in Lemma 3.1. Thus,

(3.6)

Define the extension operator $\mathcal E$ via

(3.7) $$ \begin{align} \mathcal{E}(g):=\left(c_{2n-\beta}\right)^{-1}\left(\frac{\Gamma(\frac{2n-\beta}2)}{2^{\beta}\pi^{n} \Gamma(\frac \beta 2)}\right)^{-1} I_{\beta}^{(2n)}\ast f, \end{align} $$

which is an element in ${\mathcal S}_\infty '({\mathbb R}^{2n})$ by using (3.3) and (1.3) in Remark 1.5. Applying (3.1),(3.2) and (3.6),(3.7) yields that for a. e. $x\in {{{\mathbb R}}^n}$ ,

(3.8) $$ \begin{align} \mathcal{E}(g)(x,{0})=\left(c_{2n-\beta}\right)^{-1}\left(\frac{\Gamma(\frac{2n-\beta}2)}{2^{\beta}\pi^{n} \Gamma(\frac \beta 2)}\right)^{-1} (I_{\beta}^{(2n)}\ast f)(x,0)=\left(c_{2n-\beta}\right)^{-1} F(x,0)=g(x). \end{align} $$

Since $g \in C_c^\infty ({{{\mathbb R}}^n})$ , it follows that f in (3.1) belongs to $ L_w^q (L^p)({\mathbb R}^{2n})$ , which implies both

$$ \begin{align*}\mathcal{E}g\in \dot{\mathcal{L}}_{\beta}^{p,\, q,\,w}({\mathbb R}^{2n})\end{align*} $$

and

$$ \begin{align*} c_{2n-\beta} \left(\frac{\Gamma(\frac{2n-\beta}2)}{2^{\beta}\pi^{n} \Gamma(\frac \beta 2)}\right)\|\mathcal{E}g\|_{\dot{\mathcal{L}}_{\beta}^{p,\, q,\,w}({\mathbb R}^{2n})} =\|f\|_{L_w^q (L^p)({\mathbb R}^{2n})} =\|g\|_{\dot{\Lambda}_{p,\, q,\,w}^{s}({{{\mathbb R}}^n})}. \end{align*} $$

Summarizing all, we conclude that Theorem 1.6(i) holds under $\beta \in (0,2n)$ .

Proof Assume without loss of generality that the norm $\|\phi (\cdot , \xi )\|_{L^p( {{{\mathbb R}}^n} )}$ is finite; otherwise items (i)–(iii) hold trivially.

Given a locally integrable function f on ${\mathbb R}^{2n}$ , we denote by $\widetilde {\mathrm {D}_{h}^{k}}f$ the k-order difference for the first n variables of f, that is, for all $x,y,h\in {{{\mathbb R}}^n}$ ,

(3.9) $$ \begin{align} \widetilde{\mathrm{D}_{h}^{k}} f(x,y):=\sum_{m=0}^k (-1)^{k-m} \binom km f(x+mh,y). \end{align} $$

With this notation, we apply the Minkowski inequality to write

(3.10) $$ \begin{align} \left\|\mathrm{D}_{h}^{k} g_{\xi}\right\|_{L^p( {{{\mathbb R}}^n} )} &= \left(\int_{ {{{\mathbb R}}^n} } \left| \int_{ {{{\mathbb R}}^n} } \phi(z, \xi)\, \widetilde{\mathrm{D}_{h}^{k} }(I_{\beta}^{(2n)})(x-z,\xi) \, dz\right|{}^{p}\, dx\right)^{1/p}\\\notag &= \left(\int_{ {{{\mathbb R}}^n} } \left| \int_{ {{{\mathbb R}}^n} } \phi(x-z, \xi)\, \widetilde{\mathrm{D}_{h}^{k} }(I_{\beta}^{(2n)})(z,\xi) \, dz\right|{}^{p}\, dx\right)^{1/p}\\\notag &\le\int_{{{{\mathbb R}}^n}} \left(\int_{{{{\mathbb R}}^n}}|\phi(x-z,\xi)|^p\,dx\right)^{1/p}\left|\widetilde{\mathrm{D}_h^k}(I_\beta^{(2n)})(z,\xi)\right|\,dz\\\notag &=\|\phi(\cdot, \xi)\|_{L^p( {{{\mathbb R}}^n} )}\int_{{{{\mathbb R}}^n}}\left|\widetilde{\mathrm{D}_h^k}(I_\beta^{(2n)})(z,\xi)\right|\, dz. \end{align} $$

In order to obtain the desired estimates in (i)–(iii), we only need to deal with

$$ \begin{align*}\mathrm{G}:=\int_{{{{\mathbb R}}^n}}\left|\widetilde{\mathrm{D}_h^k}(I_\beta^{(2n)})(z,\xi)\right|\, dz.\end{align*} $$

Before further continuation, observe that for all $z,\xi ,h\in {{{\mathbb R}}^n}$ the identities (3.9) and (2.6) imply not only

(3.11) $$ \begin{align} \left|\widetilde{\mathrm{D}_{h}^{k}}(I_{\beta}^{(2n)})(z,\xi)\right| &= \left|\sum_{m=0}^{k} (-1)^{k-m}\, \binom{k}{m}\, I_{\beta}^{(2n)}(z+mh,\xi)\right|\lesssim \sum_{m=0}^{k} (|z+mh|+|\xi|)^{\beta-2n}, \end{align} $$

but also

(3.12) $$ \begin{align} &\left|\widetilde{\mathrm{D}_{h}^{k}}(I_{\beta}^{(2n)})(z,\xi)\right|\\ &\quad\le |h|^k \sum_{j_1=1}^{n}\cdots\sum_{j_{k}=1}^{n} \int_{[0,1]^k} \left|\left(\partial_{x_{j_1}} \cdots \partial_{x_{j_{k}}}I_{\beta}^{(2n)}\right)(z +(t_1+\cdots+t_{k})h,\xi)\right|\, dt_1 \cdots \,dt_{k}\notag\\ &\quad\lesssim|h|^k\int_{[0,1]^k} (|z+(t_1+\cdots+t_{k})h|+|\xi|)^{\beta-2n-k}\, dt_1 \cdots \,dt_{k}.\notag \end{align} $$

Next, upon utilizing (3.11) and (3.12), we estimate $\mathrm {G}$ .

• • If $|\xi |>k|h|$ , then the integrand in (3.12) satisfies

  $$ \begin{align*} (|z+(t_1+\cdots+t_{k})h|+|\xi|)^{\beta-2n-k} &\approx \begin{cases} |\xi|^{\beta-2n-k} &\ \text{as} \ |z|<2|\xi|;\\ |z|^{\beta-2n-k} &\ \text{as} \ |z|\ge 2|\xi|, \end{cases} \end{align*} $$
  and, hence,
  $$ \begin{align*} \mathrm{G}&\lesssim |h|^{k}\left(\int_{|z|<2|\xi|}|\xi|^{\beta-2n-k} \,dz+ \int_{|z|\ge 2|\xi|} |z|^{\beta-2n-k}\,dz\right)\\ & \approx |h|^{k}|\xi|^{\beta-n-k}\\ &\approx \begin{cases} |\xi|^{\beta-n}\min\left\{1,\,\frac{|h|}{|\xi|}\right\}^{k}&\ \text{as} \ \beta\in(0,n);\\ \min\left\{\log\left(e+\frac{|h|}{|\xi|}\right),\,\left(\frac{|h|}{|\xi|}\right)^k\right\}&\ \text{as} \ \beta=n;\\ |h|^{\beta-n}\min\left\{1,\,\frac{|h|}{|\xi|}\right\}^{n+k-\beta}&\ \text{as} \ \beta\in(n,2n). \end{cases} \end{align*} $$
  From this and (3.10), it follows that (i)–(iii) hold when $|\xi |>k|h|$ .

• • If $\beta \in (0,n)$ and $|\xi |\le k|h|$ , then applying (3.11) yields

  $$ \begin{align*} \mathrm{G} &\lesssim \sum_{m=0}^{k} \int_{ {{{\mathbb R}}^n} } (|z+mh|+|\xi|)^{\beta-2n}\, dz{\nonumber}\\ &\lesssim \sum_{m=0}^{k} \left(\int_{|z+mh| \leq |\xi|} |\xi|^{\beta-2n}\, dz +\int_{|z+mh|> |\xi|}|z+mh|^{\beta-2n}\, dz\right){\nonumber}\\ &\approx |\xi|^{\beta-n}\\ &\approx |\xi|^{\beta-n}\min\left\{1,\,\frac{|h|}{|\xi|}\right\}^{k}. \end{align*} $$
  This, together with (3.10), induces (i) under the situation $|\xi |\le k|h|$ .

• • Now, let $\beta \in [n,2n)$ and $|\xi |\le k|h|$ . On the one hand, if we assume further that $|z|\ge 2k|h|$ , then for $t_1,\dots , t_k\in [0,1]$ , we have

  $$ \begin{align*} |z+(t_1+\cdots+t_{k})h|+|\xi| \approx |z|+|\xi| \approx |z|, \end{align*} $$
  which, along with (3.12) and the fact $\beta -n-k<0$ , implies
  (3.13) $$ \begin{align} \int_{|z|\ge 2k|h|} \left|\widetilde{\mathrm{D}_h^k}(I_\beta^{(2n)})(z,\xi)\right|\, dz &\lesssim |h|^k \int_{|z|\ge 2k|h|}|z|^{\beta-2n-k} \, dz\\ & \lesssim \begin{cases} |h|^{\beta-n} &\ \text{as}\ \ \beta\in(n,2n);\\ 1 &\ \text{as}\ \ \beta=n. \end{cases}\notag \end{align} $$
  If $|z|<2k|h|$ , then for all $m\in \{0,1,\dots , k\}$ we have $|z+mh|<3k|h|$ , so that applying (3.11) and a change of variables $u=(z+mh)/|\xi |$ derives
  (3.14) $$ \begin{align} \int_{|z|< 2k|h|} \left|\widetilde{\mathrm{D}_h^k}(I_\beta^{(2n)})(z,\xi)\right|\, dz &\lesssim \sum_{m=0}^{k} \int_{|z|<2k|h|} \left(|z+mh|+|\xi|\right)^{\beta-2n}\, dz\\ &\lesssim |\xi|^{\beta-n} \int_{|u|< \frac{3k|h|}{|\xi|} } (|u|+1)^{\beta-2n}\, du\notag\\ &\lesssim \begin{cases} |\xi|^{\beta-n} \left(\frac{|h|}{|\xi|}\right)^{\beta-n} &\ \text{as}\ \ \beta\in(n,2n);\\ \log\left(\frac{3k|h|}{|\xi|}\right) &\ \text{as}\ \ \beta=n. \end{cases}\notag \end{align} $$
  From (3.13) and (3.14), we deduce that (ii) and (iii) hold when $|\xi |\le k|h|$ .

Altogether, we conclude the proof of Proposition 3.2.

Proof of Theorem 1.6(ii) under $\beta \in (0,2n)$

Let $f \in \dot {\mathcal {L}}_{\beta }^{p,\, q,\,w}({\mathbb R}^{2n})$ , where $p\in [1,\infty )$ and $q\in (1,\infty )$ . Since $\beta \in (0,2n)$ , we consider first the case $f=(-\varDelta )_{{{\mathbb R}}^{2n}}^{-\beta /2}\phi $ for some $\phi \in L_w^q (L^p)({\mathbb R}^{2n})$ such that

$$ \begin{align*}I_{\beta}^{(2n)} \ast \phi\in L_{\text{\,loc}}^1({\mathbb R}^{2n})\cap {\mathcal S}_\infty'({\mathbb R}^{2n})\end{align*} $$

and

$$ \begin{align*}(-\varDelta)_{{{\mathbb R}}^{2n}}^{-\beta/2}\phi=I_{\beta}^{(2n)} \ast \phi\quad \text{in}\ {\mathcal S}_\infty'({\mathbb R}^{2n}). \end{align*} $$

By modifying f at a null set, we may assume that f is pointwisely defined on ${\mathbb R}^{2n}$ . In this case, we can write

$$ \begin{align*}f(x,{0})= I_{\beta}^{(2n)} \ast \phi(x,0) = \int_{ {{{\mathbb R}}^n} } \int_{ {{{\mathbb R}}^n} } \phi(z, \xi) I_{\beta}^{(2n)}(x-z,\xi)\, dz\,d\xi\quad \text{for all}\ x\in{{{\mathbb R}}^n}.\end{align*} $$

Assume for the moment that we have proved

(3.15) $$ \begin{align} \|f(\cdot, {0})\|_{\dot{\Lambda}_{p,\, q,\,w}^{s}({{{\mathbb R}}^n})} \lesssim \|f\|_{\dot{\mathcal{L}}_{\beta}^{p,\, q,\,w}({\mathbb R}^{2n})}. \end{align} $$

After we have obtained (3.15), then we immediately have $f(\cdot , {0})\in \dot {\Lambda }_{p,\, q,\,w}^{s}({{{\mathbb R}}^n})$ . Indeed, by Lemma 2.5, there exists a sequence $\{\varphi _j\}_{j\in {\mathbb N}}$ in $ C_c^\infty ({\mathbb R}^{2n})$ such that $\varphi _j\to f$ in $\dot {\mathcal {L}}_{\beta }^{p,\, q,\,w}({\mathbb R}^{2n})$ , which, together with (3.15), implies that $\varphi _j(\cdot ,0)\to f(\cdot ,0)$ in $\dot {\Lambda }_{p,\, q,\,w}^{s}({{{\mathbb R}}^n})$ . Then, upon defining the restriction operator $\mathcal {R}$ via

(3.16) $$ \begin{align} \mathcal{R}f(\cdot) := f(\cdot,{0})= I_{\beta}^{(2n)} \ast \phi(\cdot,0), \end{align} $$

we find that

(3.17) $$ \begin{align} \|\mathcal{R}f\|_{\dot{\Lambda}_{p,\, q,\,w}^{s}({{{\mathbb R}}^n})} = \|f(\cdot, {0})\|_{\dot{\Lambda}_{p,\, q,\,w}^{s}({{{\mathbb R}}^n})} \lesssim \|f\|_{\dot{\mathcal{L}}_{\beta}^{p,\, q,\,w}({\mathbb R}^{2n})}. \end{align} $$

For a general $f\in \dot {\mathcal {L}}_{\beta }^{p,\, q,\,w}({\mathbb R}^{2n})$ , by Corollary 2.4, there exists a sequence $\{\phi _j\}_{j\in {\mathbb N}}\subset C_c^\infty ({\mathbb R}^{2n})$ such that $\{I_{\beta }^{(2n)} \ast \phi _j\}_{j\in {\mathbb N}}$ converges to f in $\dot {\mathcal {L}}_{\beta }^{p,\, q,\,w}({\mathbb R}^{2n})$ . For any $j\in {\mathbb N}$ and $x\in {{{\mathbb R}}^n}$ , let

$$ \begin{align*}g_j(x):=(I_{\beta}^{(2n)} \ast \phi_j)(x,0).\end{align*} $$

For any $k,j\in {\mathbb N}$ , by (3.16) and (3.17), we have

$$ \begin{align*} \|g_j-g_k\|_{\dot{\Lambda}_{p,\, q,\,w}^{s}({{{\mathbb R}}^n})} &=\|(I_{\beta}^{(2n)} \ast (\phi_j-\phi_k))(\cdot,0)\|_{\dot{\Lambda}_{p,\, q,\,w}^{s}({{{\mathbb R}}^n})} \\&\lesssim \|I_{\beta}^{(2n)} \ast (\phi_j-\phi_k)\|_{\dot{\mathcal{L}}_{\beta}^{p,\, q,\,w}({\mathbb R}^{2n})}, \end{align*} $$

which tends to $0$ as $k,j\to \infty $ since $\{I_{\beta }^{(2n)} \ast \phi _j\}_{j\in {\mathbb N}}$ converges to f in $\dot {\mathcal {L}}_{\beta }^{p,\, q,\,w}({\mathbb R}^{2n})$ . Thus, $\{g_j\}_{j\in {\mathbb N}}$ is a Cauchy sequence in $\dot {\Lambda }_{p,\, q,\,w}^{s}({{{\mathbb R}}^n})$ . So, it is reasonable to define

(3.18) $$ \begin{align} \mathcal{R}f:=\lim_{j\to \infty}g_j\quad \text{in}\quad \dot{\Lambda}_{p,\, q,\,w}^{s}({{{\mathbb R}}^n}). \end{align} $$

We need to explain that $\mathcal {R}f$ in (3.18) is a uniquely defined element in $\dot {\Lambda }_{p,\, q,\,w}^{s}({{{\mathbb R}}^n})$ . Indeed, if $\{\psi _j\}_{j\in {\mathbb N}}\subset C_c^\infty ({\mathbb R}^{2n})$ is another sequence such that $\{I_{\beta }^{(2n)} \ast \psi _j\}_{j\in {\mathbb N}}$ converges to f in $\dot {\mathcal {L}}_{\beta }^{p,\, q,\,w}({\mathbb R}^{2n})$ , then we use (3.17) to deduce

$$ \begin{align*} &\lim_{j\to \infty}\|(I_{\beta}^{(2n)} \ast \phi_j)(\cdot,0)-(I_{\beta}^{(2n)} \ast \psi_j)(\cdot,0)\|_{\dot{\Lambda}_{p,\, q,\,w}^{s}({{{\mathbb R}}^n})}\\ &\quad\lesssim \lim_{j\to \infty}\|I_{\beta}^{(2n)} \ast \phi_j-I_{\beta}^{(2n)} \ast \psi_j\|_{\dot{\mathcal{L}}_{\beta}^{p,\, q,\,w}({\mathbb R}^{2n})}\\ &\quad\leq \lim_{j\to \infty}\left(\|I_{\beta}^{(2n)} \ast \phi_j-f\|_{\dot{\mathcal{L}}_{\beta}^{p,\, q,\,w}({\mathbb R}^{2n})}+\|f-I_{\beta}^{(2n)} \ast \psi_j\|_{\dot{\mathcal{L}}_{\beta}^{p,\, q,\,w}({\mathbb R}^{2n})}\right)\\ &\quad =0. \end{align*} $$

Moreover, for $\mathcal {R}f$ in (3.18), we have

$$ \begin{align*} \|\mathcal{R}f\|_{\dot{\Lambda}_{p,\, q,\,w}^{s}({{{\mathbb R}}^n})} = \lim_{j\to \infty}\|g_j\|_{\dot{\Lambda}_{p,\, q,\,w}^{s}({{{\mathbb R}}^n})} &= \lim_{j\to \infty}\|(I_{\beta}^{(2n)} \ast \phi_j)(\cdot, 0)\|_{\dot{\Lambda}_{p,\, q,\,w}^{s}({{{\mathbb R}}^n})}\\ &\lesssim \lim_{j\to \infty}\|I_{\beta}^{(2n)} \ast \phi_j\|_{\dot{\mathcal{L}}_{\beta}^{p,\, q,\,w}({\mathbb R}^{2n})} = \|f\|_{\dot{\mathcal{L}}_{\beta}^{p,\, q,\,w}({\mathbb R}^{2n})}. \end{align*} $$

Further, by (3.2) and (3.8), we see that $ \mathcal {R}\mathcal {E}=id $ on $\dot {\Lambda }_{p,\, q,\,w}^{s}({{{\mathbb R}}^n})$ .

To finish the proof of Theorem 1.6(ii) under $\beta \in (0,2n)$ , we still need to validate (3.15). To simplify the notation, for any $\xi ,x\in {{{\mathbb R}}^n}$ , set $g(x)=f(x,0)$ and

$$ \begin{align*}g_{\xi}(x):=\int_{ {{{\mathbb R}}^n} } \phi(z, \xi) I_{\beta}^{(2n)}(x-z,\xi)\, dz.\end{align*} $$

Let $k:=\lfloor s\rfloor +1$ . Then, for any $h\in {{{\mathbb R}}^n}$ , by the expression of g and the Minkowski inequality, we have

$$ \begin{align*} \left\|\mathrm{D}_{h}^{k}g\right\|_{L^p( {{{\mathbb R}}^n} )} =\left(\int_{ {{{\mathbb R}}^n} } \left|\int_{ {{{\mathbb R}}^n} } \mathrm{D}_{h}^{k}g_{\xi}(x)\, d\xi\right|{}^{p}\, dx \right)^{1/p} \leq\int_{ {{{\mathbb R}}^n} } \left\|\mathrm{D}_{h}^{k}g_{\xi}\right\|_{L^p( {{{\mathbb R}}^n} )}\, d\xi. \end{align*} $$

Consequently,

(3.19) $$ \begin{align} \|g\|_{\dot{\Lambda}_{p,\, q,\,w}^{s}({{{\mathbb R}}^n})} &= \left(\int_{ {{{\mathbb R}}^n} } \left\| {\mathrm{D}}_{h}^{k}g \right\|_{L^p( {{{\mathbb R}}^n} )}^{q} |h|^{-n-sq} w(h)\, dh\right)^{1/q}\\ &\leq\left(\int_{ {{{\mathbb R}}^n} }\left(\int_{ {{{\mathbb R}}^n} } \left\|\mathrm{D}_{h}^{k}g_{\xi}\right\|_{L^p( {{{\mathbb R}}^n} )}\, d\xi\right)^{q} |h|^{-n-sq} w(h)\, dh\right)^{1/q}\notag\\ &=\left(\int_{ {{{\mathbb R}}^n} }\left(\int_{ {{{\mathbb R}}^n} } |h|^{-\beta} \left\|\mathrm{D}_{h}^{k}g_{\xi}\right\|_{L^p( {{{\mathbb R}}^n} )}\, d\xi\right)^{q} w(h)\, dh\right)^{1/q},\notag \end{align} $$

where in the last step we used the fact $\beta =s+n/q$ . Observe that the integrand $|h|^{-\beta }\|\mathrm {D}_{h}^{k}g_{\xi }\|_{L^p( {{{\mathbb R}}^n} )}$ can be estimated by utilizing Proposition 3.2. Indeed, upon setting

$$ \begin{align*}\begin{cases} K_1(|h|, |\xi|): =|h|^{-n}\left(\frac{|h|}{|\xi|}\right)^{n-\beta}\min\left\{1,\frac{|h|}{|\xi|}\right\}^{k}\quad &\text{as}\ \beta\in(0,n);\\ K_2(|h|, |\xi|): =|h|^{-n}\min\left\{\log\left(e+\frac{|h|}{|\xi|}\right),\,\left(\frac{|h|}{|\xi|}\right)^k\right\}\quad &\text{as}\ \beta=n;\\ K_3(|h|, |\xi|): =|h|^{-n}\min\left\{1,\frac{|h|}{|\xi|}\right\}^{n+k-\beta}\quad &\text{as}\ \beta\in(n, 2n), \end{cases} \end{align*} $$

we apply Proposition 3.2 and find that the last line of (3.19) can be controlled by a constant multiple of

$$ \begin{align*} \left(\int_{ {{{\mathbb R}}^n} }\left(\int_{ {{{\mathbb R}}^n} }K_i(|h|,|\xi|)\left\|\phi(\cdot, \xi)\right\|_{L^p( {{{\mathbb R}}^n} )}\, d\xi\right)^q w(h)\, dh\right)^{1/q}\quad\text{for}\ i\in\{1,2,3\}. \end{align*} $$

Thus, the proof of (3.15) falls into validating that for $i\in \{1,2,3\}$ ,

(3.20) $$ \begin{align} &\left(\int_{ {{{\mathbb R}}^n} }\left(\int_{ {{{\mathbb R}}^n} }K_i(|h|,|\xi|)\left\|\phi(\cdot, \xi)\right\|_{L^p( {{{\mathbb R}}^n} )}\, d\xi\right)^q w(h)\, dh\right)^{1/q}\\ &\qquad \lesssim \left(\int_{ {{{\mathbb R}}^n} } \|\phi(\cdot, \xi)\|_{L^p( {{{\mathbb R}}^n} )}^{q} w(\xi)\, d\xi\right)^{1/q}=\|\phi\|_{L_w^q (L^p)({\mathbb R}^{2n})}. \notag \end{align} $$

Below, we prove (3.20) for $i=1,2,3$ , respectively.

• • Proof of (3.20) for $i=1$ . Since $K_1$ is homogeneous of degree $-n$ , it follows from Lemma 2.9 that (3.20) holds provided that $K_1$ satisfies (2.22).

  Let us show that $K_1$ satisfies (2.22). Note that $\beta =s+\frac nq$ . So, after a change of variables $r=t^{-1}$ , we find that

  $$ \begin{align*} &\int_{0}^{\infty}K_1(1, t)t^{\frac n{q'}-1}\, \left(\sup_{\rho \in (0, \infty)} \frac{w(t^{-1}\rho)}{w(\rho)}\right)^{\frac 1 q }\,dt\\ &\quad = \int_{0}^{\infty}t^{s} \min\left\{1,t^{-1}\right\}^{k}\, \left(\sup_{\rho \in (0, \infty)} \frac{w(t^{-1}\rho)}{w(\rho)}\right)^{\frac 1 q }\,\frac{dt} t\\ &\quad=\int_{0}^{\infty}r^{-s} \min\left\{1,r\right\}^{k}\, \left(\sup_{\rho \in (0, \infty)} \frac{w(r\rho)}{w(\rho)}\right)^{\frac 1 q }\,\frac{dr} r\\ &\quad =\int_0^1 r^{k-s} \left(\sup_{\rho \in (0, \infty)} \frac{w(r\rho)}{w(\rho)}\right)^{\frac 1 q }\,\frac{dr} r + \int_1^\infty r^{-s} \left(\sup_{\rho \in (0, \infty)} \frac{w(r\rho)}{w(\rho)}\right)^{\frac 1 q }\,\frac{dr} r. \end{align*} $$
  By (1.4) and (1.5), together with $\sigma <\lfloor s\rfloor +1-s=k-s$ and $\delta>-s$ , we obtain
  (3.21) $$ \begin{align} &\int_0^1 r^{k-s} \left(\sup_{\rho \in (0, \infty)} \frac{w(r\rho)}{w(\rho)}\right)^{\frac 1 q }\,\frac{dr} r\\ &\quad \le \left(\int_{0}^{1}r^{\sigma q}\, \left(\sup_{\rho \in (0, \infty)} \frac{w(r\rho)}{w(\rho)}\right)\frac{dr}{r}\right)^{\frac 1 q} \left(\int_{0}^{1}r^{(k-s-\sigma)q'}\, \frac{dr}{r}\right)^{\frac 1 {q'}} <\infty\notag \end{align} $$
  and
  (3.22) $$ \begin{align} &\int_1^\infty r^{-s} \left(\sup_{\rho \in (0, \infty)} \frac{w(r\rho)}{w(\rho)}\right)^{\frac 1 q }\,\frac{dr} r\\ &\quad \le \left(\int_1^{\infty}r^{\delta q}\, \left(\sup_{\rho \in (0, \infty)} \frac{w(r\rho)}{w(\rho)}\right)\frac{dr}{r}\right)^{\frac 1q}\left(\int_1^{\infty}r^{-(s+\delta) q'}\, \frac{dr}{r}\right)^{\frac 1 {q'}}<\infty.\notag \end{align} $$
  This proves that $K_1$ satisfies (2.22) and, hence, (3.20) holds.

• • Proof of (3.20) for $i=2$ . The proof is similar to the case $i=1$ . Indeed, observing that $K_2$ is homogeneous of degree $-n$ , we only need to validate that $K_2$ satisfies (2.22), so that Lemma 2.9 can be applied to show that (3.20) holds. To this end, using $\beta =n$ and $\frac n {q'}=\beta -\frac nq=s$ , we then write

  $$ \begin{align*} &\int_{0}^{\infty}K_2(1, t)t^{\frac n{q'}-1}\, \left(\sup_{\rho \in (0, \infty)} \frac{w(t^{-1}\rho)}{w(\rho)}\right)^{\frac 1 q }\,dt\\ &\quad = \int_{0}^{\infty}t^{s} \min\left\{\log(e+t^{-1}),\,t^{-k}\right\}\, \left(\sup_{\rho \in (0, \infty)} \frac{w(t^{-1}\rho)}{w(\rho)}\right)^{\frac 1 q }\,\frac{dt} t\\ &\quad=\int_{0}^{\infty}r^{-s} \min\left\{\log(e+r),\,r^{k}\right\}\, \left(\sup_{\rho \in (0, \infty)} \frac{w(r\rho)}{w(\rho)}\right)^{\frac 1 q }\,\frac{dr} r\\ &\quad \approx \int_0^1 r^{k-s} \left(\sup_{\rho \in (0, \infty)} \frac{w(r\rho)}{w(\rho)}\right)^{\frac 1 q }\,\frac{dr} r + \int_1^\infty r^{-s} \log(e+r) \left(\sup_{\rho \in (0, \infty)} \frac{w(r\rho)}{w(\rho)}\right)^{\frac 1 q }\,\frac{dr} r. \end{align*} $$
  Clearly, (3.21) remains valid. Instead of (3.22), we now also have
  $$ \begin{align*} &\int_1^\infty r^{-s} \log(e+r)\left(\sup_{\rho \in (0, \infty)} \frac{w(r\rho)}{w(\rho)}\right)^{\frac 1 q }\,\frac{dr} r\\ &\quad \le \left(\int_1^{\infty}r^{\delta q}\, \left(\sup_{\rho \in (0, \infty)} \frac{w(r\rho)}{w(\rho)}\right)\frac{dr}{r}\right)^{\frac 1q}\left(\int_1^{\infty}r^{-(s+\delta) q'} \log(e+r)\, \frac{dr}{r}\right)^{\frac 1 {q'}}<\infty,\notag \end{align*} $$
  by terms of (1.5) and the fact $\delta>-s$ . Thus, $K_2$ satisfies (2.22). We obtain (3.20).

• • Proof of (3.20) for $i=3$ . In this case, write the left-hand side of (3.20) as

  $$ \begin{align*} & \left(\int_{ {{{\mathbb R}}^n} }\left(\frac1{|h|^n}\int_{|\xi|\leq|h|} \|\phi(\cdot, \xi)\|_{L^p( {{{\mathbb R}}^n} )}\, d\xi\right)^q w(h)\, dh\right)^{1/q}\\ &\quad +\left(\int_{ {{{\mathbb R}}^n} }\left(\int_{|\xi|>|h|} |h|^{-n}\left(\frac{|h|}{|\xi|}\right)^{n+k-\beta} \|\phi(\cdot, \xi)\|_{L^p( {{{\mathbb R}}^n} )}\, d\xi\right)^q w(h)\, dh\right)^{1/q} =:\mathrm{I}+\mathrm{II}. \end{align*} $$

  Let $\Phi (\xi ):=\|\phi (\cdot , \xi )\|_{L^p( {{{\mathbb R}}^n} )}$ and ${\mathcal M}$ be the classical Hardy–Littlewood maximal function on ${{{\mathbb R}}^n}$ , that is,

  $$ \begin{align*}\mathcal{M}\Phi(z):=\sup_{B\ni z}\frac1{|B|}\int_B|\Phi(x)|\,dx\quad\text{for all}\ z\in{{{\mathbb R}}^n}.\end{align*} $$

  Then, using $q\in (1,\infty )$ , $w \in A_{q}( {{{\mathbb R}}^n} )$ and the boundedness of ${\mathcal M}$ on $L^q_w({{{\mathbb R}}^n})$ , we derive

  $$ \begin{align*} \mathrm{I} &\lesssim \left( \int_{ {{{\mathbb R}}^n} } [{\mathcal M} \Phi(h)]^{q} w(h)\, dh\right)^{1/q} \lesssim \left(\int_{ {{{\mathbb R}}^n} } [\Phi(h)]^{q} w(h)\, dh\right)^{1/q} =\|\phi\|_{L_w^q (L^p)({\mathbb R}^{2n})}. \end{align*} $$

  For $\mathrm {II}$ , upon setting

  $$ \begin{align*}K_3^\prime(|h|, |\xi|): =|h|^{-n}\min\left\{1,\frac{|h|}{|\xi|}\right\}^{n+k-\beta}{\mathbf 1}_{|\xi|>|h|}\end{align*} $$
  and using $\beta =s+\frac n q$ , we find that
  $$ \begin{align*} &\int_{1}^{\infty}K_3^\prime(1, t)t^{\frac n{q'}-1}\, \left(\sup_{\rho \in (0, \infty)} \frac{w(t^{-1}\rho)}{w(\rho)}\right)^{\frac 1 q }\,dt\\ &\quad = \int_1^{\infty} t^{n-\beta+s} t^{-(n+k-\beta)}\, \left(\sup_{\rho \in (0, \infty)} \frac{w(t^{-1}\rho)}{w(\rho)}\right)^{\frac 1 q }\,\frac{dt} t\\ &\quad=\int_{0}^1r^{k-s} \, \left(\sup_{\rho \in (0, \infty)} \frac{w(r\rho)}{w(\rho)}\right)^{\frac 1 q }\,\frac{dr} r, \end{align*} $$
  which is finite by terms of (3.21). From this, the fact that $K_3'$ is homogeneous of degree $-n$ and Lemma 2.9, it follows that
  $$ \begin{align*}\mathrm{II}\lesssim \left(\int_{ {{{\mathbb R}}^n} } \|\phi(\cdot, \xi)\|_{L^p( {{{\mathbb R}}^n} )}^{q} w(\xi)\, d\xi\right)^{1/q}=\|\phi\|_{L_w^q (L^p)({\mathbb R}^{2n})}.\end{align*} $$

  Combining the estimates of $\mathrm {I}$ and $\mathrm {II}$ shows that (3.20) holds for $i=3$ .

Summarizing all, we conclude the proof of Theorem 1.6(ii) under $\beta \in (0,2n)$ .

Proof For any $f \in \dot {\mathcal {L}}_{\beta }^{p,\, q,\,w}({\mathbb R}^{2n})$ , by Definition 1.4, there exists a function $\phi \in L_w^q (L^p)({\mathbb R}^{2n})$ such that $f=(-\varDelta _{{{\mathbb R}}^{2n}})^{-\beta /2}\phi $ . From this and

$$ \begin{align*}(-\varDelta_{{{\mathbb R}}^{2n}})^{\gamma/2} f=(-\varDelta_{{{\mathbb R}}^{2n}})^{\gamma/2}\,(-\varDelta_{{{\mathbb R}}^{2n}})^{-\beta/2}\,\phi=(-\varDelta_{{{\mathbb R}}^{2n}})^{-(\beta-\gamma)/2}\phi,\end{align*} $$

we derive

$$ \begin{align*}(-\varDelta_{{{\mathbb R}}^{2n}})^{\gamma/2} f \in \dot{\mathcal{L}}_{\beta-\gamma}^{p,\, q,\,w}({\mathbb R}^{2n})\end{align*} $$

and

$$ \begin{align*} \| (-\varDelta_{{{\mathbb R}}^{2n}})^{\gamma/2} f \|_{\dot{\mathcal{L}}_{\beta-\gamma}^{p,\, q,\,w}({\mathbb R}^{2n})}=\| \phi \|_{L_w^q (L^p)({\mathbb R}^{2n})}=\|f\|_{\dot{\mathcal{L}}_{\beta}^{p,\, q,\,w}({\mathbb R}^{2n})}, \end{align*} $$

as desired.

Proof Associated with the function $\varphi $ , define a function $\psi $ via

$$ \begin{align*}\hat{\psi}(\xi):=|2\pi\xi|^{\gamma}\hat{\varphi}(\xi)\quad \text{for all}\ \xi\in{{{\mathbb R}}^n}.\end{align*} $$

Clearly, $\psi \in {\mathcal S}_\infty ({{{\mathbb R}}^n})$ and ${ \mathrm {\,supp\,}} \hat {\psi }\subset \{x\in {{{\mathbb R}}^n}:\ c^{-1}\leq |x|\leq c\}$ . Note that the Fourier transform gives

$$ \begin{align*} \left(\varphi_t \ast \left((-\varDelta_{{{{\mathbb R}}^n}})^{\gamma/2} g\right)\right)^{\wedge}(\xi)=\widehat{\varphi_t}(\xi)\,(2\pi |\xi|)^{\gamma}\,\hat{g}(\xi) =t^{-\gamma}\,\widehat{\psi_t}(\xi)\,\hat{g}(\xi)=t^{-\gamma} (\psi_t\ast g)^{\wedge}(\xi), \end{align*} $$

which implies

$$ \begin{align*}\varphi_t \ast \left((-\varDelta_{{{{\mathbb R}}^n}})^{\gamma/2} g\right)=t^{-\gamma} (\psi_t\ast g).\end{align*} $$

By this and Lemma 2.8, we obtain

$$ \begin{align*} \| (-\varDelta_{{{{\mathbb R}}^n}})^{\gamma/2} g \|_{\dot{\Lambda}_{p,\, q,\,w, \, \varphi}^{s-\gamma}({{{\mathbb R}}^n})} &= \left(\int_{0}^{\infty}\left\| \varphi_t \ast \left((-\varDelta_{{{{\mathbb R}}^n}})^{\gamma/2} g\right) \right\|_{L^p({{{\mathbb R}}^n})}^{q}t^{-(s-\gamma)q-1}w(t)\, dt\right)^{1/q}\\ &= \left(\int_{0}^{\infty}\|\psi_t\ast g\|_{L^p({{{\mathbb R}}^n})}^{q}t^{-sq-1}w(t)\, dt\right)^{1/q}\\ &= \| g \|_{\dot{\Lambda}_{p,\, q,\,w,\,\psi}^{s}({{{\mathbb R}}^n})}\approx\| g \|_{\dot{\Lambda}_{p,\, q,\,w,\,\varphi}^{s}({{{\mathbb R}}^n})}. \end{align*} $$

This ends the proof.

Proof of Theorem 1.6 under $\beta \in [2n,\infty )$

If $\beta \geq 2n$ , then there exists a unique $k\in {\mathbb N}$ such that $2n+k-1\leq \beta <2n+k$ , which implies $\beta -k\in [2n-1,2n)$ . Moreover, the facts $\beta -k\geq 2n-1$ and $\beta =s+n/q$ imply that $s>k$ .

First, we show (i). Suppose that $g\in \dot {\Lambda }_{p,\, q,\,w}^{s}({{{\mathbb R}}^n})$ . By Definiton 1.1, we may as well assume that $g\in C_c^\infty ({{{\mathbb R}}^n})$ . From Lemma 3.4 and Theorem 2.10(ii), it follows that

(3.23) $$ \begin{align} \| (-\varDelta_{{{{\mathbb R}}^n}})^{k/2} g \|_{\dot{\Lambda}_{p,\, q,\,w,\,\varphi}^{s-k}({{{\mathbb R}}^n})} \approx \|g\|_{\dot{\Lambda}_{p,\, q,\,w,\,\varphi}^{s}({{{\mathbb R}}^n})}\lesssim \|g\|_{\dot{\Lambda}_{p,\, q,\,w}^{s}({{{\mathbb R}}^n})}, \end{align} $$

where $\varphi $ is as in Lemma 3.4.

If k is even, say $k=2m$ for some $m\in {\mathbb N}$ , then, we have,

$$ \begin{align*}(-\varDelta_{{{{\mathbb R}}^n}})^{k/2} g =(-\varDelta_{{{{\mathbb R}}^n}})^{m} g\in C_c^\infty({{{\mathbb R}}^n}),\end{align*} $$

so that Theorem 2.10(i) and Remark 2.11 imply

(3.24) $$ \begin{align} \| (-\varDelta_{{{{\mathbb R}}^n}})^{k/2} g \|_{\dot{\Lambda}_{p,\, q,\,w}^{s-k}({{{\mathbb R}}^n})}\lesssim \| (-\varDelta_{{{{\mathbb R}}^n}})^{k/2} g \|_{\dot{\Lambda}_{p,\, q,\,w,\,\varphi}^{s-k}({{{\mathbb R}}^n})} \end{align} $$

and, hence, $(-\varDelta _{{{{\mathbb R}}^n}})^{k/2} g \in {\dot {\Lambda }_{p,\, q,\,w}^{s-k}({{{\mathbb R}}^n})}.$

If k is odd, then we write $k=2m+1$ and, hence,

$$ \begin{align*}(-\varDelta_{{{{\mathbb R}}^n}})^{k/2} g = (-\varDelta_{{{{\mathbb R}}^n}})^{1/2}\left( (-\varDelta_{{{{\mathbb R}}^n}})^{m} g\right)\in (-\varDelta_{{{{\mathbb R}}^n}})^{1/2}\left( C_c^\infty({{{\mathbb R}}^n})\right).\end{align*} $$

For any $f\in {\mathcal S}({{{\mathbb R}}^n})$ , it is known that (see [Reference Silvestre40, Section 2]) the function $(-\varDelta _{{{{\mathbb R}}^n}})^{1/2} f$ satisfies

$$ \begin{align*}|(-\varDelta_{{{{\mathbb R}}^n}})^{1/2}f(x)|\lesssim (1+|x|)^{-(n+1)} \end{align*} $$

uniformly in $x\in {{{\mathbb R}}^n}$ . This in turn gives that $(-\varDelta _{{{{\mathbb R}}^n}})^{k/2} g \in L^p({{{\mathbb R}}^n})$ , which, along with Remark 2.11 and Theorem 2.10(i), again implies

(3.25) $$ \begin{align} \| (-\varDelta_{{{{\mathbb R}}^n}})^{k/2} g \|_{\dot{\Lambda}_{p,\, q,\,w}^{s-k}({{{\mathbb R}}^n})}\lesssim \| (-\varDelta_{{{{\mathbb R}}^n}})^{k/2} g \|_{\dot{\Lambda}_{p,\, q,\,w,\,\varphi}^{s-k}({{{\mathbb R}}^n})}. \end{align} $$

From this and Lemma 2.2, we deduce that $(-\varDelta _{{{{\mathbb R}}^n}})^{k/2} g $ can be approximated by $C_c^\infty ({{{\mathbb R}}^n})$ -functions under the semi-norm $\|\,\cdot \,\|_{\dot {\Lambda }_{p,\, q,\,w}^{s-k}({{{\mathbb R}}^n})}$ , which induces $(-\varDelta _{{{{\mathbb R}}^n}})^{k/2} g \in {\dot {\Lambda }_{p,\, q,\,w}^{s-k}({{{\mathbb R}}^n})}.$

Moreover, since $\beta -k\in (0,2n)$ , it follows from the already proved argument of Theorem 1.6 in Section 3.1 that

$$ \begin{align*}\mathcal{E}\left((-\varDelta_{{{{\mathbb R}}^n}})^{k/2} g\right)\in\dot{\mathcal{L}}_{\beta-k}^{p,\, q,\,w}({\mathbb R}^{2n}),\end{align*} $$

where $\mathcal {E}$ is the extension operator defined in Section 3.1. Thus, it makes sense to define the extension operator $\widetilde {\mathcal {E}}$ by

$$ \begin{align*} \widetilde{\mathcal{E}}g:=(-\varDelta_{{{\mathbb R}}^{2n}})^{-k/2} \mathcal{E}\left((-\varDelta_{{{{\mathbb R}}^n}})^{k/2} g\right). \end{align*} $$

By this, together with Lemma 3.3 and (3.23)–(3.25), we derive $\widetilde {\mathcal {E}}g\in \dot {\mathcal {L}}_{\beta }^{p,\, q,\,w}({\mathbb R}^{2n})$ with

$$ \begin{align*} \|\widetilde{\mathcal{E}}g\|_{\dot{\mathcal{L}}_{\beta}^{p,\, q,\,w}({\mathbb R}^{2n})} &=\left\|(-\varDelta_{{{\mathbb R}}^{2n}})^{-k/2}\mathcal{E}\left((-\varDelta_{{{{\mathbb R}}^n}})^{k/2} g\right)\right\|_{\dot{\mathcal{L}}_{\beta}^{p,\, q,\,w}({\mathbb R}^{2n})}\\ &=\left\|\mathcal{E}\left((-\varDelta_{{{{\mathbb R}}^n}})^{k/2} g\right)\right\|_{\dot{\mathcal{L}}_{\beta-k}^{p,\, q,\,w}({\mathbb R}^{2n})}\\&= \left\| (-\varDelta_{{{{\mathbb R}}^n}})^{k/2} g \right\|_{\dot{\Lambda}_{p,\, q,\,w}^{s-k}({{{\mathbb R}}^n})}\lesssim \| g \|_{\dot{\Lambda}_{p,\, q,\,w}^{s}({{{\mathbb R}}^n})}. \end{align*} $$

This proves that (i) holds for $\beta \in [2n,\infty )$ .

Now, we show (ii). Let $f \in \dot {\mathcal {L}}_{\beta }^{p,\, q,\,w}({\mathbb R}^{2n})$ . Then, $f=(-\varDelta _{{{\mathbb R}}^{2n}})^{-\beta /2}\phi $ for some $\phi \in L_w^q (L^p)({\mathbb R}^{2n})$ . Moreover, it follows from Lemma 3.3 that

$$ \begin{align*}(-\varDelta_{{{\mathbb R}}^{2n}})^{-(\beta-k)/2} \phi=(-\varDelta_{{{\mathbb R}}^{2n}})^{k/2} f \in \dot{\mathcal{L}}_{\beta-k}^{p,\, q,\,w}({\mathbb R}^{2n}).\end{align*} $$

Since $\beta -k\in (0,2n)$ , by the already proved argument of Theorem 1.6(ii) in Section 3.2, we obtain

$$ \begin{align*}\mathcal{R}\left((-\varDelta_{{{\mathbb R}}^{2n}})^{k/2} f\right) =\mathcal{R}\left((-\varDelta_{{{\mathbb R}}^{2n}})^{-(\beta-k)/2} \phi\right)\in\dot{\Lambda}_{p,\, q,\,w}^{s-k}({{{\mathbb R}}^n})\end{align*} $$

and

(3.26) $$ \begin{align} \left\|\mathcal{R}\left((-\varDelta_{{{\mathbb R}}^{2n}})^{k/2} f\right)\right\|_{\dot{\Lambda}_{p,\, q,\,w}^{s-k}({{{\mathbb R}}^n})} &=\left\|\mathcal{R}\left((-\varDelta_{{{\mathbb R}}^{2n}})^{-(\beta-k)/2} \phi\right)\right\|_{\dot{\Lambda}_{p,\, q,\,w}^{s-k}({{{\mathbb R}}^n})}\\ &\lesssim \left\|(-\varDelta_{{{\mathbb R}}^{2n}})^{-(\beta-k)/2} \phi\right\|_{\dot{\mathcal{L}}_{\beta-k}^{p,\, q,\,w}({\mathbb R}^{2n})}\notag\\ & = \left\|(-\varDelta_{{{\mathbb R}}^{2n}})^{k/2} f\right\|_{\dot{\mathcal{L}}_{\beta-k}^{p,\, q,\,w}({\mathbb R}^{2n})} \approx \| f\|_{\dot{\mathcal{L}}_{\beta}^{p,\, q,\,w}({\mathbb R}^{2n})}. \notag \end{align} $$

Further, from Lemma 3.4 and Theorem 2.10(ii), we derive

(3.27) $$ \begin{align} \left\|(-\varDelta_{{{{\mathbb R}}^n}})^{-k/2}\mathcal{R}\left((-\varDelta_{{{\mathbb R}}^{2n}})^{k/2} f\right)\right\|_{\dot{\Lambda}_{p,\, q,\,w,\, \varphi}^{s}({{{\mathbb R}}^n})} &\lesssim\left\|\mathcal{R}\left((-\varDelta_{{{\mathbb R}}^{2n}})^{k/2} f\right)\right\|_{\dot{\Lambda}_{p,\, q,\,w,\, \varphi}^{s-k}({{{\mathbb R}}^n})}\\ &\lesssim \left\|\mathcal{R}\left((-\varDelta_{{{\mathbb R}}^{2n}})^{k/2} f\right)\right\|_{\dot{\Lambda}_{p,\, q,\,w}^{s-k}({{{\mathbb R}}^n})}{\nonumber} \end{align} $$

and, hence,

$$ \begin{align*}(-\varDelta_{{{{\mathbb R}}^n}})^{-k/2}\mathcal{R}\left((-\varDelta_{{{\mathbb R}}^{2n}})^{k/2} f\right)\in \dot{\Lambda}_{p,\, q,\,w,\, \varphi}^{s}({{{\mathbb R}}^n}).\end{align*} $$

Next, by Theorem 2.10(i), there exists a Lebesgue measurable function $g_0\in \dot {\Lambda }_{p,\, q,\, w}^{s}({{{\mathbb R}}^n})$ such that

$$ \begin{align*}g_0=(-\varDelta_{{{{\mathbb R}}^n}})^{-k/2}\mathcal{R}\left((-\varDelta_{{{\mathbb R}}^{2n}})^{k/2} f\right)\ \ \text{in}\ \, {\mathcal S}^{\prime}_\infty({{{\mathbb R}}^n})\end{align*} $$

and

(3.28) $$ \begin{align} \|g_0\|_{\dot{\Lambda}_{p,\, q,\,w}^{s}({{{\mathbb R}}^n})} \lesssim \left\|(-\varDelta_{{{{\mathbb R}}^n}})^{-k/2}\mathcal{R}\left((-\varDelta_{{{\mathbb R}}^{2n}})^{k/2} f\right)\right\|_{\dot{\Lambda}_{p,\, q,\,w,\, \varphi}^{s}({{{\mathbb R}}^n})}. \end{align} $$

Thus, when $\beta \in [2n+k-1,2n+k)$ , it makes sense to define the restriction operator $\widetilde {\mathcal {R}}$ on $\dot {\mathcal {L}}_{\beta }^{p,\, q,\,w}({\mathbb R}^{2n})$ by

$$ \begin{align*} \widetilde{\mathcal{R}}f:=g_0. \end{align*} $$

By (3.26)-(3.27)-(3.28), we know that $\widetilde {\mathcal {R}}f\in \dot {\Lambda }_{p,\, q,\,w}^{s}({{{\mathbb R}}^n})$ and

$$ \begin{align*} \|\widetilde{\mathcal{R}}f\|_{\dot{\Lambda}_{p,\, q,\,w}^{s}({{{\mathbb R}}^n})}\lesssim \| f \|_{\dot{\mathcal{L}}_{\beta}^{p,\, q,\,w}({\mathbb R}^{2n})}. \end{align*} $$

Furthermore, by the fact that $\mathcal {R}\mathcal {E}=id$ under $\beta \in (0,2n)$ , we deduce that, for any $g\in \dot {\Lambda }_{p,\, q,\,w}^{s}({{{\mathbb R}}^n})$ with $2n+k-1\leq \beta <2n+k$ ,

$$ \begin{align*} \widetilde{\mathcal{R}}\widetilde{\mathcal{E}}g= \widetilde{\mathcal{R}} \left( (-\varDelta_{{{\mathbb R}}^{2n}})^{-k/2} \mathcal{E}\left((-\varDelta_{{{{\mathbb R}}^n}})^{k/2} g\right)\right) = (-\varDelta_{{{{\mathbb R}}^n}})^{-k/2}\,\mathcal{R}\,\mathcal{E}\,(-\varDelta_{{{{\mathbb R}}^n}})^{k/2} g=g, \end{align*} $$

where all the equalities hold in ${\mathcal S}^{\prime }_\infty ({{{\mathbb R}}^n})$ .

Altogether, we conclude the proof of Theorem 1.6 under $\beta \in [2n,\infty )$ .