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.
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The author would like to thank Prof. Yoshihiro Sawano for his encouragement and many helpful remarks. The author thanks the referee for his/her valuable comments and his/her constructive suggestions.
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Appendix
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
Since
we have
(iii) It is sufficient to prove
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
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.
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Saka, K. 2-microlocal spaces associated with Besov type and Triebel–Lizorkin type spaces. Rev Mat Complut 35, 923–962 (2022). https://doi.org/10.1007/s13163-021-00412-z
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DOI: https://doi.org/10.1007/s13163-021-00412-z
Keywords
- Wavelet
- Besov space
- Triebel–Lizorkin space
- Calder\(\acute{\mathrm{o}}\)n–Zygmund operator
- Pseudo–differential operator
- \(\varphi \) transform
- atomic and molecular decompostion
- 2-microlocal space