2-microlocal spaces associated with Besov type and Triebel–Lizorkin type spaces

Abstract

In this paper we introduce and investigate new 2-microlocal spaces associated with Besov type and Triebel–Lizorkin type spaces. We establish characterizations of these function spaces via the $\varphi$–transform, the atomic and molecular decomposition and the wavelet decomposition. As applications we consider boundedness of the Calder$\acute{\mathrm{o}}$n–Zygmund operator and the pseudo–differential operator on the function spaces.

Appendix

We will prove the Lemma used in the proof of Theorem 1. For the notations see the proof of Theorem 1.

Lemma A

We have, for a dyadic cube P with $l(P)=2^{-j}$,

• (i)
  $$\begin{aligned} \sup (f)^*(P) \approx \inf _{\gamma }(f)^*(P) \end{aligned}$$

  if $\gamma$ is sufficently large,

• (ii)
  $$\begin{aligned} \inf _{\gamma }(f)(P)\chi _{P}\le 2^{\gamma L}\sum \limits _{R\subset P, l(R) =2^{-(\gamma +j)}}t_{\gamma }^*(R)\chi _{R} \end{aligned}$$
• (iii)
  $$\begin{aligned} c({\dot{e}}^{s'}_{pq})(P) \approx c^*({\dot{e}}^{s'}_{pq})(P), \ \ c(\tilde{{\dot{e}}}^{s'}_{pq})^{\sigma }_{x_{0}}(P) \approx c^*(\tilde{{\dot{e}}}^{s'}_{pq})^{\sigma }_{x_{0}}(P). \end{aligned}$$

Proof

(i) is just [6], Lemma A.4].

(ii) Let $R_{0}$ and R in P be cubes with $l(R_{0})=l(R)=2^{-(\gamma +j)}$. It is sufficient to show

$$\begin{aligned} t_{\gamma }(R_{0}) \le C2^{\gamma L}t_{\gamma }^*(R). \end{aligned}$$

Since

$$\begin{aligned} 1 \le C2^{\gamma L}(1+ 2^{\gamma +j}|x_{R}-x_{R_{0}}|)^{-L}, \end{aligned}$$

we have

$$\begin{aligned} t_{\gamma }(R_{0})\le & {} Ct_{\gamma }(R_{0})2^{\gamma L} (1+2^{\gamma +j}|x_{R}-x_{R_{0}}|)^{-L} \\\le & {} C2^{\gamma L}\sum \limits _{l(R')=2^{-(\gamma +j)}}t_{\gamma }(R') (1+2^{\gamma +j}|x_{R}-x_{R'}|)^{-L}=C2^{\gamma L}t_{\gamma }^*(R) \end{aligned}$$

(iii) It is sufficient to prove

$$\begin{aligned} c^*({\dot{e}}^{s'}_{pq})(P) \le C c({\dot{e}}^{s'}_{pq})(P) \end{aligned}$$

since $|c(P)| \le c^*(P)$.

Using the maximal operator $M_{t}\ (0< t \le 1)$ as in the proof of Lemma 1 and the Fefferman-Stein vector valued inequality, we have

$$\begin{aligned} c^*({\dot{f}}^{s'}_{pq})(P)= & {} \left| \left| \left\{ \sum \limits _{i \ge j} \left( 2^{is'}\sum \limits _{l(R)=2^{-i}}|c^*(R)|\chi _{R}\right) ^{q} \right\} ^{1/q}\right| \right| _{L^p(P)}\\\le & {} C\left| \left| \left\{ \sum \limits _{i \ge j} (2^{is'}\right. \right. \right. \\&\left. \left. \left. \times \sum \limits _{l(R)=2^{-i}}\sum \limits _{l(R')=2^{-i}}|c(R')| (1+2^{i}|x_{R}-x_{R'}|)^{-L}\chi _{R})^{q} \right\} ^{1/q}\right| \right| _{L^p(P)}\\\le & {} C\left| \left| \left\{ \sum \limits _{i \ge j} \left( 2^{is'}\sum \limits _{l(R)=2^{-i}}M_{t}\left( \sum \limits _{l(R')=2^{-i}}|c(R')| \chi _{R'}\right) \chi _{R}\right) ^{q}\right\} ^{1/q}\right| \right| _{L^p(P)} \\\le & {} C\left| \left| \left\{ \sum \limits _{i \ge j} \left( \sum \limits _{l(R')=2^{-i}}2^{is'}|c(R')|\chi _{R'}\right) ^{q} \right\} ^{1/q}\right| \right| _{L^p(P)} =Cc({\dot{f}}^{s'}_{pq})(P) \end{aligned}$$

if $0< t < \min (p,q)$, $L > n/t$ and $0< p< \infty , 0< q \le \infty$. For the B-type case, we obtain the same result by the same argument as the above. Moreover, for $p=\infty$ case, we have the same result. We also have same result for $c(\tilde{{\dot{e}}}^{s'}_{pq})^{\sigma }_{x_{0}}(P)$ by same way as the above.