On 2-microlocal spaces with all exponents variable

TL;DR

This paper investigates the properties of variable exponent 2-microlocal Besov and Triebel--Lizorkin spaces, focusing on lifting, embeddings, and Fourier multipliers, while refining recent statements in the field.

Contribution

It provides new insights and improvements on the properties of variable exponent 2-microlocal function spaces, including clarifications of recent results.

Findings

Established key properties like lifting and embeddings for these spaces

Analyzed Fourier multiplier behavior in variable exponent contexts

Corrected and refined previous statements in the literature

Abstract

In this paper we study various key properties for $2$-microlocal Besov and Triebel--Lizorkin spaces with all exponents variable, including the lifting property, embeddings and Fourier multipliers. We also clarify and improve some statements recently published.