Pseudo-multipliers and Smooth Molecules on Hermite Besov and Hermite Triebel–Lizorkin Spaces

Abstract

We obtain new molecular decompositions and molecular synthesis estimates for Hermite Besov and Hermite Triebel–Lizorkin spaces and use such tools to prove boundedness properties of Hermite pseudo-multipliers on those spaces. The notion of molecule we develop leads to boundedness of pseudo-multipliers associated to symbols of Hörmander-type adapted to the Hermite setting on spaces for which the smoothness allowed includes non-positive values; in particular, we obtain continuity results for such operators on Lebesgue and Hermite local Hardy spaces. As a byproduct of our results on boundedness properties of pseudo-multipliers, we show that Hermite Besov spaces and Hermite Triebel–Lizorkin spaces are closed under non-linearities.

Appendices

Appendix A. Estimates for $\varphi _j(\sqrt{{\mathcal {L}}})$

In this appendix, we state and prove Lemma A.1, which is used in the proofs of Lemmas 3.3 and 3.4, and Theorem 5.12.

Lemma A.1

Let $\{\varphi _j\}_{j\in {\mathbb {N}}_0}$ be an admissible system. If $\eta \ge 1$, $\varepsilon \ge 4,$ $\gamma \in {\mathbb {N}}_0^n$ and $K\ge 0,$ it holds that

$$\begin{aligned}&|\partial _y^\gamma \varphi _j(\sqrt{{\mathcal {L}}}) (x,y)| +|\partial _x^\gamma \varphi _j(\sqrt{{\mathcal {L}}}) (x,y)| \nonumber \\&\quad \lesssim \frac{2^{j(n+|\gamma |)}}{(1+2^j|x-y|)^\eta } \mathrm {e}_{\varepsilon 4^j}(x) e_{\varepsilon 4^j}(y)\quad \forall x,y\in {\mathbb {R}}^n, j\in {\mathbb {N}}_0, \end{aligned}$$

(5.11)

and, for $|\gamma |\le K,$

$$\begin{aligned} \Big |\int _{{\mathbb {R}}^n}(x-y)^\gamma \varphi _j(\sqrt{{\mathcal {L}}})(x,y)\,dy\Big | \lesssim 2^{-j|\gamma |}\Big (\frac{1+|x|}{2^j}\Big )^{K-|\gamma |} \mathrm {e}_{\varepsilon 4^j}(x)\quad \forall x\in {\mathbb {R}}^n,j\in {\mathbb {N}}_0. \end{aligned}$$

(5.12)

Remark A.2

Note that by the symmetry of the kernels $\varphi _j(\sqrt{{\mathcal {L}}})(x,y)$, (5.12) also holds with dx in place of dy on the left hand side, and y in place of x on the right hand side.

Remark A.3

Lemma A.1 holds true, with the same proof, for a family $\{\varphi _j\}_{j\in {\mathbb {N}}_0}$ where $\varphi _j(\lambda )=\varphi (2^{-j}\lambda )$ and $\varphi$ is a smooth function supported in [0, c] for some $c>0$ that satisfies $\varphi ^{(k)}(0)=0$ for all $k\in {\mathbb {N}}.$

Proof of Lemma A.1

Regarding (5.11), recall that

$$\begin{aligned} \varphi _j(\sqrt{{\mathcal {L}}})(x,y)=\sum _{k\in {\mathbb {N}}_0}\varphi _j(\sqrt{\lambda _k}){\mathbb {P}}_k(x,y). \end{aligned}$$

Then (5.11) can be proved employing the same ideas in the proof of Theorem 4.3 through the use of Lemmas B.1 and B.2 presented in Appendix B.

We turn to the proof of (5.12). Assume $|\gamma |\le K$ and fix $x\in {\mathbb {R}}^n$ and $j\in {\mathbb {N}}_0.$ Set $B=B(x,\varrho (x))$ where the function $\varrho (\cdot )$ is defined in (4.2). Let $\chi$ be a function in $C^\infty ({\mathbb {R}}^n)$ supported in 2B that satisfies $\chi =1$ on B,  $0\le \chi \le 1$ and

$$\begin{aligned} \Vert \chi ^{(\nu )}\Vert _\infty \le \frac{C}{\varrho (x)^{|\nu |}}\qquad \forall \nu \in {\mathbb {N}}^n_0. \end{aligned}$$

We split the integral into two terms:

$$\begin{aligned} \int _{{\mathbb {R}}^n}(x-y)^\gamma \varphi _j(\sqrt{{\mathcal {L}}})(x,y)\,dy&= \int _{{\mathbb {R}}^n}(1-\chi (y))(x-y)^\gamma \varphi _j(\sqrt{{\mathcal {L}}})(x,y)\,dy \\&\qquad +\int _{{\mathbb {R}}^n} \chi (y)(x-y)^\gamma \varphi _j(\sqrt{{\mathcal {L}}})(x,y)\,dy \\&=:I + II. \end{aligned}$$

To estimate I we use the bounds from (5.11) with $\eta >n+K$ and recall that $|\gamma |\le K$ to obtain

$$\begin{aligned} |I|&\lesssim \mathrm {e}_{\varepsilon 4^j(x)} \int _{B^c} \frac{(2^j|x-y|)^{|\gamma |-K}2^{j(n-|\gamma |)}}{(1+2^j|x-y|)^{\eta -K}}\,dy \\&\le \Big (\frac{1+|x|}{2^j}\Big )^{K-|\gamma |}2^{-j|\gamma |}\mathrm {e}_{\varepsilon 4^j(x)} \int _{{\mathbb {R}}^n}\frac{2^{jn}}{(1+2^j|x-y|)^{\eta -K}}\,dy \\&\lesssim \Big (\frac{1+|x|}{2^j}\Big )^{K-|\gamma |}2^{-j|\gamma |}\mathrm {e}_{\varepsilon 4^j(x)}. \end{aligned}$$

For the second term we have, by employing the Cauchy-Schwarz inequality,

$$\begin{aligned} |II|&=\Big | \sum _{k\in {\mathbb {N}}_0}\varphi _j(\sqrt{\lambda _k}) \sum _{|\xi |=k}h_\xi (x)\int _{{\mathbb {R}}^n}\chi (y) (y-x)^\gamma h_\xi (y)\,dy\Big |\\&\le \Vert \varphi \Vert _\infty \Big (\sum _{k\in I_j} \sum _{|\xi |=k} h_\xi (x)^2\Big )^{1/2}\Big (\sum _{k\in I_j}\sum _{|\xi |=k} \Big | \int _{{\mathbb {R}}^n} (y-x)^\gamma \chi (y) h_\xi (y)\,dy\Big |^2\Big )^{1/2}, \end{aligned}$$

where we recall that $k\in I_j$ means $\frac{1}{2}4^{j-2}-\lfloor n/2 \rfloor \le k\le \frac{1}{2}4^j -\lceil n/2\rceil$.

To estimate the second factor we note that for any $N \in {\mathbb {N}}_0,$ it holds that

$$\begin{aligned} \Big | \int _{{\mathbb {R}}^n} (y-x)^\gamma \chi (y) h_\xi (y)\,dy\Big |&=\lambda ^{-N}_{|\xi |} \Big |\int _{{\mathbb {R}}^n} {\mathcal {L}}_y^N\big [ (y-x)^\gamma \chi (y)\big ] h_\xi (y)\,dy\Big | \\&\le \lambda ^{-N}_{|\xi |} \Big \Vert {\mathcal {L}}^N \big [(\cdot -x)^\gamma \chi (\cdot )\big ] \Big \Vert _{L^2(2B)} \Vert h_\xi \Vert _{L^2(2B)} \\&\sim (1+|\xi |)^{-N} \Big \Vert {\mathcal {L}}^N \big [(\cdot -x)^\gamma \chi (\cdot )\big ] \Big \Vert _{L^2(2B)} \Vert h_\xi \Vert _{L^2(2B)}. \end{aligned}$$

Repeated application of the Leibniz’ rule gives, with the sum running over indices such that $|a|+|b|\le 2N,$ $\beta +\nu =b$ and $|\beta | \le |\gamma |,$

$$\begin{aligned} {\mathcal {L}}^N \big [(\cdot -x)^\gamma \chi (\cdot )\big ](y) = \sum _{a,b,\beta ,\nu ,\gamma } C_{a,b,\beta ,\nu } \,y^a (y-x)^{\gamma -\beta }\chi ^{(\nu )}(y) \end{aligned}$$

so that

$$\begin{aligned} \Big \Vert {\mathcal {L}}^N \big [(\cdot -x)^\gamma \chi (\cdot )\big ] \Big \Vert _{L^2(2B)}&\sim \sum _{a,b,\beta ,\nu ,\gamma } \Big (\int _{2B} \big | |y|^{|a|} |y-x|^{|\gamma |-|\beta |}|\chi ^{(\nu )}(y)| \big |^2\,dy\Big )^{1/2} \\&\lesssim \sum _{a,b,\beta ,\nu ,\gamma } \varrho (x)^{|\gamma |-|\beta |-|\nu | +n/2} \sup _{y\in 2B}|y|^{|a|} \\&\lesssim \sum _{\begin{array}{c} |a|+|b|\le 2N \end{array}} (1+|x|)^{|a|+|b|-|\gamma | -n/2} \\&\lesssim (1+|x|)^{2N-|\gamma |-n/2}. \end{aligned}$$

Inserting this into the estimate for II leads to

$$\begin{aligned} |II|&\lesssim \Big (\sum _{k\in I_j} \sum _{|\xi |=k} h_\xi (x)^2\Big )^{1/2}\Big (\sum _{k\in I_j}\sum _{|\xi |=k} \bigg |\frac{(1+|x|)^{2N-|\gamma |-n/2}}{(1+|\xi |)^N}\bigg |^2 \Vert h_\xi \Vert _{L^2(2B)}^2\Big )^{1/2} \\&\lesssim \Big (\frac{1+|x|}{2^j}\Big )^{2N-|\gamma |-n/2}2^{-j(|\gamma |+n/2)}\Big (\sum _{k\in I_j} \sum _{|\xi |=k} h_\xi (x)^2\Big )^{1/2}\Big (\sum _{k\in I_j}\sum _{|\xi |=k} \Vert h_\xi \Vert _{L^2(2B)}^2\Big )^{1/2}\\&\lesssim \Big (\frac{1+|x|}{2^j}\Big )^{2N-|\gamma |-n/2}2^{-j(|\gamma |+n/2)} \Big ({\mathbb {Q}}_{4^j}(x,x)\Big )^{1/2} \Big (\int _{2B} {\mathbb {Q}}_{4^j}(y,y) \,dy\Big )^{1/2}. \end{aligned}$$

where in the last line we used that $\sum _{k\le 4^j}\sum _{|\xi |=k} h_\xi (y)^2= {\mathbb {Q}}_{4^j}(y,y).$ We next apply the bounds (2.3) to get

$$\begin{aligned} |II|&\lesssim \Big (\frac{1+|x|}{2^j}\Big )^{2N-|\gamma |-n/2}2^{-j(|\gamma |+n/2)} \big (2^{jn}\mathrm {e}_{\varepsilon 4^j}(x)^2\big )^{1/2} 2^{jn/2}|2B|^{1/2} \\&\sim \Big (\frac{1+|x|}{2^j}\Big )^{2N-|\gamma |-n}2^{-j|\gamma |}\mathrm {e}_{\varepsilon 4^j}(x)\\&\le \Big (\frac{1+|x|}{2^j}\Big )^{K-|\gamma |}2^{-j|\gamma |}\mathrm {e}_{\varepsilon 4^j}(x) \end{aligned}$$

by choosing N appropriately depending on whether $\frac{1+|x|}{2^j}$ is larger or smaller than 1. $\square$

Appendix B. Useful Identities and Estimates

In this appendix, we present identities and estimates used in the proof of Theorem 4.3.

Lemma B.1

• (a) Suppose that

  $$\begin{aligned} {\mathbb {F}}(x,y) = \sum _{k\in {\mathbb {N}}_0} f(x,y,k) \,{\mathbb {P}}_k(x,y). \end{aligned}$$

  If $N\in {\mathbb {Z}}_+,$ it holds that

  $$\begin{aligned} 2^N(x_i- y_i)^N {\mathbb {F}}(x,y) = \sum _{\frac{N}{2}\le \ell \le N}c_{\ell , N}\sum _{k\in {\mathbb {N}}_0} \triangle _k^\ell f(x,y,k)\big (A^{(y)}_i-A^{(x)}_i\big )^{2\ell -N}{\mathbb {P}}_k(x,y), \end{aligned}$$

  (5.13)

  where $c_{\ell ,N} = (-4)^{N-\ell }(2N-2\ell -1)!!\left( {\begin{array}{c}N\\ 2\ell -N\end{array}}\right)$.

• (b) If $N, M\in {\mathbb {Z}}_+,$ it holds that

  $$\begin{aligned} x_i^M \big (A^{(x)}_i-A^{(y)}_i\big )^N&= \sum _{k=0}^M\genfrac(){0.0pt}1{M}{k} \tfrac{N!}{(N-k)!}\big (A^{(x)}_i-A^{(y)}_i\big )^{N-k}x_i^{M-k} \end{aligned}$$

  (5.14)

  and

  $$\begin{aligned} (x_i-y_i)^N\big (A^{(x)}_i\big )^M&=\sum _{k=0}^M\genfrac(){0.0pt}1{M}{k} \tfrac{N!}{(N-k)!}\big (A^{(x)}_i\big )^{M-k}(x_i-y_i)^{N-k}, \end{aligned}$$

  (5.15)

  where $\frac{N!}{(N-k)!}$ is defined to be 0 whenever $N<k$.

• (c) If $\beta \in {\mathbb {N}}_0^n$ and $k\in {\mathbb {N}}_0$, it holds that

  $$\begin{aligned} x^\beta \,{\mathbb {P}}_k(x,y)= \sum _{\omega \le \beta } \sum _{|\xi |=k}b_{\omega ,\beta }(\xi ) h_{\xi +\beta -2\omega }(x) h_\xi (y),\end{aligned}$$

  (5.16)

  where $b_{\omega ,\beta }(\xi )=\prod _{i=1}^n b_{\omega _i,\beta _i}(\xi _i)$ with $b_{\omega _i,\beta _i}(\xi _i)=0$ if $\xi _i+\beta _i-2\omega _i<0$ and $b_{\omega _i,\beta _i}(\xi _i)\sim \xi _i^{\beta _i/2}$ otherwise.

• (d) If $\xi ,\alpha \in {\mathbb {N}}_0^n,$ $m\in {\mathbb {N}}_0$ and $i\in \{1,\dots , n\},$ it holds that

  $$\begin{aligned} \big |\big (A^{(x)}_i\big )^m h_\xi (x)\big |&\le \big [2(\xi _i+m)+2\big ]^{\frac{m}{2}} | h_{\xi +m e_i}(x)| \end{aligned}$$

  (5.17)

  and

  $$\begin{aligned} \big |\big (A^{(x)}\big )^\alpha h_\xi (x)\big |&\le \big [2(|\xi |+|\alpha |)+2\big ]^{\frac{|\alpha |}{2}} | h_{\xi +\alpha }(x)|. \end{aligned}$$

  (5.18)

Proof of Lemma B.1

The identity in part (a) can be found in [30, Lemma 8] and [34, p.72] with k as a function of $k$ only. However, it can be checked that the proof also works when k depends on both x and y. Part (b) is from [30, Lemma 9]. Part (c) is [30, equation (6.14)] with $\mu =0$. In part (d), estimate (5.17) follows from [30, equation (6.5)]:

$$\begin{aligned} \big (A^{(x)}_i\big )^m h_\xi (x) = \prod _{r=0}^{m-1}\sqrt{2(\xi _i+r)+2}\,\, h_{\xi +me_i}(x). \end{aligned}$$

The inequality (5.18) follows from applying (5.17) repeatedly. $\square$

Lemma B.2

• (a) If $\ell \in {\mathbb {N}}_0,$ it holds that

  $$\begin{aligned} \triangle _k^\ell \big (f(k)\,g(k)\big ) = \sum _{r=0}^\ell \left( {\begin{array}{c}\ell \\ r\end{array}}\right) \triangle _k^r f(k)\,\triangle _k^{\ell -r}g(k+r). \end{aligned}$$

  (5.19)

• (b) If $\alpha \in {\mathbb {N}}_0^n$, it holds that

  $$\begin{aligned} A^\alpha (fg) =\sum _{\nu \le \alpha } \left( {\begin{array}{c}\alpha \\ \nu \end{array}}\right) (-1)^\nu \partial ^\nu f \, A^{\alpha -\nu } g. \end{aligned}$$

  (5.20)

Proof of Lemma B.2

Part (a) is well known. For part (b), first note that the following representation for Hermite derivatives holds:

$$\begin{aligned} A_i^m = (-1)^me^{x_i^2/2} \partial _i^m e^{-x_i^2/2}\qquad \forall m\in {\mathbb {N}}_0. \end{aligned}$$

(5.21)

This identity can be obtained by direct calculation for $m=1$ and by induction for all m. We next show that (5.21) gives

$$\begin{aligned} A_i^{\alpha _i} (fg) = \sum _{\nu _i=0}^{\alpha _i} \left( {\begin{array}{c}\alpha _i\\ \nu _i\end{array}}\right) (-1)^{\nu _i} \partial _i^{\nu _i} f \, A_i^{\alpha _i-\nu _i} g. \end{aligned}$$

(5.22)

Indeed, by (5.21) and the Leibniz rule for differentiation we obtain

$$\begin{aligned} A_i^{\alpha _i} (fg)&= (-1)^{\alpha _i}e^{x_i^2/2} \partial _i^{\alpha _i} \big ( e^{-x_i^2/2}fg\big ) \\&= (-1)^{\alpha _i}e^{x_i^2/2} \sum _{\nu _i=0}^{\alpha _i} \left( {\begin{array}{c}\alpha _i\\ \nu _i\end{array}}\right) \partial _i^{\nu _i} f \, \cdot \partial _i^{\alpha _i-\nu _i}\big (e^{-x_i^2/2} g\big )\\&=\sum _{\nu _i=0}^{\alpha _i} \left( {\begin{array}{c}\alpha _i\\ \nu _i\end{array}}\right) (-1)^{\nu _i}\partial _i^{\nu _i} f \, \cdot (-1)^{\alpha _i-\nu _i}e^{x_i^2/2}\partial _i^{\alpha _i-\nu _i}\big (e^{-x_i^2/2} g\big ). \end{aligned}$$

Equality (5.22) follows by applying (5.21) again. The identity (5.20) then follows by applying (5.22) to each component $1\le i\le n$. $\square$

Appendix C. Remarks About Hermite Besov and Triebel–Lizorkin Spaces

In this appendix, we present some embeddings of Hermite Besov and Hermite Triebel–Lizorkin spaces. The embeddings stated in Corollary C.2, a consequence of Theorem C.1, are used in the proof of Corollary 5.13. In addition, we comment on the Fatou property of Hermite Besov and Hermite Triebel–Lizorkin spaces, which is also used in the proof of Corollary 5.13.

Theorem C.1

• (a) If $\alpha \in {\mathbb {R}}, \varepsilon >0,$ $0<q\le \infty ,$ $0<q_1\le \infty ,$ and $0<p\le \infty$ for Besov spaces or $0<p<\infty$ for Triebel–Lizorkin spaces, it holds that

  $$\begin{aligned} A^{p,q}_{\alpha +\varepsilon }({\mathcal {L}})\hookrightarrow A^{p,q_1}_{\alpha }({\mathcal {L}}). \end{aligned}$$

• (b) If $0<q\le \infty ,$ $0<p<\infty$ and $\alpha \in {\mathbb {R}},$ it holds that

  $$\begin{aligned} B^{p,\min (p,q)}_\alpha ({\mathcal {L}})\hookrightarrow F^{p,q}_\alpha ({\mathcal {L}})\hookrightarrow B^{p,\max (p,q)}_\alpha ({\mathcal {L}}). \end{aligned}$$

• (c) If $0<q\le \infty ,$ $0<p<p_1<\infty$ and $\alpha ,\alpha _1\in {\mathbb {R}}$ are such that $\alpha _1<\alpha ,$ it holds that

  $$\begin{aligned} A^{p,q}_{\alpha }({\mathcal {L}})\hookrightarrow A^{p_1,q}_{\alpha _1}({\mathcal {L}})\quad \text {if}\quad \alpha -\frac{n}{p}=\alpha _1-\frac{n}{p_1}. \end{aligned}$$

The proofs of the embeddings stated in Theorem C.1 are the same as those in the Euclidean setting; see [36, p.47, Proposition 2] for (a) and (b) and [30, Propositions 6 and 7] for (c).

Corollary C.2

Let $0<q\le \infty .$ If $1<p<\infty$ and $\varepsilon >0,$ then $A^{p,q}_\varepsilon ({\mathcal {L}})\hookrightarrow L^p({\mathbb {R}}^n);$ if $0<p\le 1$ and $\varepsilon >n(\frac{1}{p}-1),$ there exists $p_1> 1$ such that $A^{p,q}_\varepsilon ({\mathcal {L}})\hookrightarrow L^{p_1}({\mathbb {R}}^n).$

Proof

Let $0<q\le \infty .$

Case $1<p<\infty$ and $\varepsilon >0:$ Taking $\alpha =0$ in part (a) of Theorem C.1 we obtain that $F^{p,q}_{\varepsilon }({\mathcal {L}})\hookrightarrow F^{p,2}_{0}({\mathcal {L}})=L^p({\mathbb {R}}^n).$ Using parts (a) and (b) of Theorem C.1 we have $B^{p,q}_{\alpha +\varepsilon }({\mathcal {L}})\hookrightarrow B^{p,\min (p,2)}_{\alpha }({\mathcal {L}})\hookrightarrow F^{p,2}_\alpha ({\mathcal {L}});$ then $\alpha =0$ implies $B^{p,q}_{\varepsilon }({\mathcal {L}})\hookrightarrow F^{p,2}_0({\mathcal {L}})=L^p({\mathbb {R}}^n).$

Case $0<p\le 1$ and $\varepsilon >n(\frac{1}{p}-1):$ Let $p_1> 1$ be such that $p_1>p$ and $\varepsilon >n(\frac{1}{p}-\frac{1}{p_1});$ such $p_1$ exists since $\varepsilon >n(\frac{1}{p}-1)\ge 0.$ Setting $\alpha _1=\varepsilon -n(\frac{1}{p}-\frac{1}{p_1}),$ part (c) of Theorem C.1 and the previous case imply that

$$\begin{aligned} A^{p,q}_{\varepsilon }({\mathcal {L}})\hookrightarrow A^{p_1,q}_{\alpha _1}({\mathcal {L}})\hookrightarrow L^{p_1}({\mathbb {R}}^n). \end{aligned}$$

$\square$

Next, we comment about the Fatou property for Hermite Besov and Hermite Triebel–Lizorkin spaces.

Let ${\mathcal {A}}$ be a quasi-Banach space such that ${\mathscr {S}}({\mathbb {R}}^n)\hookrightarrow {\mathcal {A}}\hookrightarrow {\mathscr {S}}'({\mathbb {R}}^n).$ The space ${\mathcal {A}}$ is said to have the Fatou property if for every sequence $\{f_j\}_{j\in {\mathbb {N}}}\subset {\mathcal {A}}$ that converges in ${\mathscr {S}}'({\mathbb {R}}^n),$ as $j\rightarrow \infty ,$ and that satisfies $\liminf _{j\rightarrow \infty } \Vert f_j\Vert _{{\mathcal {A}}}<\infty ,$ it follows that $\lim _{j\rightarrow \infty }f_j\in {\mathcal {A}}$ and $\Vert \lim _{j\rightarrow \infty }f_j\Vert _{{\mathcal {A}}}\lesssim \liminf _{j\rightarrow \infty } \Vert f_j\Vert _{{\mathcal {A}}},$ where the implicit constant is independent of $\{f_j\}_{j\in {\mathbb {N}}}.$

It can be shown, using standard proofs (see for instance [37, p.48, Proposition 2.8]), that $A^{p,q}_{\alpha }({\mathcal {L}})$ posses the Fatou property for any $\alpha \in {\mathbb {R}},$ $0<q\le \infty ,$ $0<p\le \infty$ for Besov spaces and $0<p<\infty$ for Triebel–Lizorkin spaces. This is due to the following facts: (1) if $f,g\in L^p({\mathbb {R}}^n)$ and $|f|\le |g|$ pointwise a.e., then $\Vert f\Vert _{L^p}\le \Vert g\Vert _{L^p};$ (2) if $\{f_j\}_{j\in {\mathbb {N}}}\subset L^p({\mathbb {R}}^n)$ and $f_j\ge 0$ poinwise a.e., then $\Vert \liminf _{j\rightarrow \infty } f_j\Vert _{L^p}\le \liminf _{j\rightarrow \infty }\Vert f_j\Vert _{L^p};$ (3) if $f_j\rightarrow f$ in ${\mathscr {S}}'({\mathbb {R}}^n)$ then, for any $k\in {\mathbb {N}}_0$ and any admissible pair $(\varphi _0,\varphi _j),$ $\varphi _k(\sqrt{{\mathcal {L}}})f_j\rightarrow \varphi _k(\sqrt{{\mathcal {L}}})f$ pointwise as $j\rightarrow \infty .$

Appendix D. Operator Norm

The following result about the operator norm of pseudo-multipliers is used in the proof of Corollary 5.13.

Lemma D.1

Let $m\in {\mathbb {R}},$ $\rho \ge 0,$ $\delta \ge 0,$ ${\mathcal {N}},{\mathcal {K}}\in {\mathbb {N}}_0,$ $\alpha , {\tilde{\alpha }}\in {\mathbb {R}},$ $0<p< \infty ,$ $0<q< \infty ,$ and $0<{\tilde{p}}\le \infty$ for Besov spaces or $0<{\tilde{p}}<\infty$ for Triebel–Lizorkin spaces. If $T_\sigma$ is bounded from $A^{p,q}_\alpha ({\mathcal {L}})$ to $A^{{\tilde{p}},{\tilde{q}}}_{{\tilde{\alpha }}}({\mathcal {L}})$ for all $\sigma \in S^{m,{\mathcal {K}},{\mathcal {N}}}_{\rho ,\delta },$ it holds that

$$\begin{aligned} \Vert T_\sigma \Vert _{A^{p,q}_\alpha \rightarrow A^{{\tilde{p}},{\tilde{q}}}_{{\tilde{\alpha }}}}\lesssim \mathop {\max _{0\le |\nu |\le {\mathcal {N}}}}_{0\le \kappa \le {\mathcal {K}}} \mathop {\sup _{x\in {\mathbb {R}}^n}}_{k\in {\mathbb {N}}_0}| \partial ^\nu _x \triangle ^\kappa _k\sigma (x,k)| (1+\sqrt{k})^{-m+2\rho \kappa -\delta |\nu |} \quad \forall \sigma \in S^{m,{\mathcal {K}},{\mathcal {N}}}_{\rho ,\delta }. \end{aligned}$$

Proof

We follow ideas from the proof of [2, Lemma 2.6]. Set

$$\begin{aligned} \Vert \sigma \Vert _{S^{m,{\mathcal {K}},{\mathcal {N}}}_{\rho ,\delta }}= \mathop {\max _{0\le |\nu |\le {\mathcal {N}}}}_{0\le \kappa \le {\mathcal {K}}} \mathop {\sup _{x\in {\mathbb {R}}^n}}_{k\in {\mathbb {N}}_0}| \partial ^\nu _x \triangle ^\kappa _k\sigma (x,k)| (1+\sqrt{k})^{-m+2\rho \kappa -\delta |\nu |} \quad \forall \sigma \in S^{m,{\mathcal {K}},{\mathcal {N}}}_{\rho ,\delta }; \end{aligned}$$

then $S^{m,{\mathcal {K}},{\mathcal {N}}}_{\rho ,\delta }$ is a Banach space with the norm $\Vert \cdot \Vert _{S^{m,{\mathcal {K}},{\mathcal {N}}}_{\rho ,\delta }},$ Define the linear operator

$$\begin{aligned} {\mathcal {U}}: S^{m,{\mathcal {K}},{\mathcal {N}}}_{\rho ,\delta }\rightarrow L(A^{p,q}_\alpha ({\mathcal {L}}), A^{{\tilde{p}},{\tilde{q}}}_{{\tilde{\alpha }}}({\mathcal {L}})), \quad {\mathcal {U}}(\sigma )=T_\sigma , \end{aligned}$$

where $L(A^{p,q}_\alpha ({\mathcal {L}}), A^{{\tilde{p}},{\tilde{q}}}_{{\tilde{\alpha }}}({\mathcal {L}}))$ is the quasi-Banach space of all linear bounded operators from $A^{p,q}_\alpha ({\mathcal {L}})$ to $A^{{\tilde{p}},{\tilde{q}}}_{{\tilde{\alpha }}}({\mathcal {L}})$ with the usual operator norm. We will show that the graph of ${\mathcal {U}}$ is closed; as a consequence of the Closed Graph Theorem, it follows that ${\mathcal {U}}$ is continuous and therefore the desired result follows.

Let $\{(\sigma _j,T_{\sigma _j})\}_{j\in {\mathbb {N}}}$ be a sequence in the graph of ${\mathcal {U}}$ that converges to $(\sigma , T)\in S^{m,{\mathcal {K}},{\mathcal {N}}}_{\rho ,\delta }\times L(A^{p,q}_\alpha ({\mathcal {L}}), A^{{\tilde{p}},{\tilde{q}}}_{{\tilde{\alpha }}}({\mathcal {L}}))$ in the product topology. We will show that $T(f)=T_\sigma (f)$ for all $f\in {\mathscr {S}}({\mathbb {R}}^n);$ assuming the latter, since ${\mathscr {S}}({\mathbb {R}}^n)$ is dense in $A^{p,q}_\alpha ({\mathcal {L}})$ and $T_\sigma , T\in L(A^{p,q}_\alpha ({\mathcal {L}}), A^{{\tilde{p}},{\tilde{q}}}_{{\tilde{\alpha }}}({\mathcal {L}})),$ it follows that $T_\sigma =T.$ As a consequence, the graph of ${\mathcal {U}}$ is closed.

Given $f\in {\mathscr {S}}({\mathbb {R}}^n)$ and N sufficiently large, using the definition of $\Vert \cdot \Vert _{S^{m,{\mathcal {K}},{\mathcal {N}}}_{\rho ,\delta }},$ [30, Lemma 3] and that $h_\xi$ are bounded uniformly in $\xi$ by [34, Lemma 1.5.2, p.27], we obtain

$$\begin{aligned} |T_\sigma (f)(x)-T_{\sigma _j}(f)(x)|&=\left| \sum _{k\in {\mathbb {N}}_0} (\sigma (x,\lambda _k)-\sigma _j(x,\lambda _k)){\mathbb {P}}_k(f)(x)\right| \\&\le \Vert \sigma -\sigma _j\Vert _{S^{m,{\mathcal {K}},{\mathcal {N}}}_{\rho ,\delta }} \sum _{k\in {\mathbb {N}}_0} (1+\sqrt{\lambda _k})^m \sum _{|\xi |=k} |\langle f,h_\xi \rangle | |h_\xi (x)|\\&\lesssim \Vert \sigma -\sigma _j\Vert _{S^{m,{\mathcal {K}},{\mathcal {N}}}_{\rho ,\delta }} \sum _{\xi \in {\mathbb {N}}_0^n} (1+\sqrt{|\xi |})^m \frac{1}{(1+|\xi |)^N}\\&\lesssim \Vert \sigma -\sigma _j\Vert _{S^{m,{\mathcal {K}},{\mathcal {N}}}_{\rho ,\delta }}, \end{aligned}$$

which implies that $T_{\sigma _j}(f)$ converges to $T_{\sigma }(f)$ uniformly in ${\mathbb {R}}^n.$ On the other hand, we have

$$\begin{aligned} \Vert T_{\sigma _j}(f)-T(f)\Vert _{A^{{\tilde{p}},{\tilde{q}}}_\alpha }\lesssim \Vert T_{\sigma _j}-T\Vert _{A^{p,q}_\alpha \rightarrow A^{{\tilde{p}},{\tilde{q}}}_{{\tilde{\alpha }}}} \Vert f\Vert _{A^{p,q}_\alpha }\rightarrow 0. \end{aligned}$$

The above implies that $T_{\sigma _j}(f)$ converges to $T_{\sigma }(f)$ in ${\mathscr {S}}'({\mathbb {R}}^n)$ and $T_{\sigma _j}(f)$ converges to T(f) in ${\mathscr {S}}'({\mathbb {R}}^n)$ for all $f\in {\mathscr {S}}({\mathbb {R}}^n)$ (for the latter see [30, Proposition 4, p. 385 and Section 5, p. 392], which state that $A^{{\tilde{p}},{\tilde{q}}}_\alpha ({\mathcal {L}})\hookrightarrow {\mathscr {S}}'({\mathbb {R}}^n)$). Therefore $T_{\sigma }(f)=T(f)$ for all $f\in {\mathscr {S}}({\mathbb {R}}^n),$ as desired. $\square$