Non-smooth atomic decomposition of variable 2-microlocal Besov-type and Triebel-Lizorkin-type spaces

TL;DR

This paper develops non-smooth atomic decompositions and characterizations via maximal functions for variable 2-microlocal Besov-type and Triebel-Lizorkin-type spaces, extending classical and Morrey spaces.

Contribution

It introduces non-smooth atomic decompositions and maximal function characterizations for these advanced function spaces with variable exponents.

Findings

Atomic decompositions established for the spaces.

Maximal function characterizations provided.

Pointwise multiplier results derived.

Abstract

In this paper we provide non-smooth atomic decompositions of 2-microlocal Besov-type and Triebel-Lizorkin-type spaces with variable exponents $B^{\mathrm{\boldsymbol{\omega}}, \phi}_{p(\cdot),q(\cdot)}(\mathbb{R}^n)$ and $F^{\mathrm{\boldsymbol{\omega}}, \phi}_{p(\cdot),q(\cdot)}(\mathbb{R}^n)$. Of big importance in general, and an essential tool here, are the characterizations of the spaces via maximal functions and local means, that we also present. These spaces were recently introduced by Wu at al. and cover not only variable 2-microlocal Besov and Triebel-Lizorkin spaces $B^{\mathrm{\boldsymbol{\omega}}}_{p(\cdot),q(\cdot)}(\mathbb{R}^n)$ and $F^{\mathrm{\boldsymbol{\omega}}}_{p(\cdot),q(\cdot)}(\mathbb{R}^n)$, but also the more classical smoothness Morrey spaces $B^{s, \tau}_{p,q}(\mathbb{R}^n)$ and $F^{s,\tau}_{p,q}(\mathbb{R}^n)$. Afterwards, we state a pointwise multipliers assertion for this scale.