Characterizations of weighted Besov and Triebel-Lizorkin spaces with variable smoothness

TL;DR

This paper investigates weighted Besov and Triebel-Lizorkin spaces with variable smoothness, establishing norm equivalences, regularity estimates for fractional evolution equations, and a generalized Sobolev embedding theorem.

Contribution

It extends the equivalence of norms to variable smoothness and weights, and provides new regularity and embedding results in this context.

Findings

Norm equivalence holds for variable smoothness and weights.

Weighted regularity estimates for time-fractional evolution equations.

Generalized Sobolev embedding theorem without weights.

Abstract

In this paper, we study different types of weighted Besov and Triebel-Lizorkin spaces with variable smoothness. The function spaces can be defined by means of the Littlewood-Paley theory in the field of Fourier analysis, while there are other norms arising in the theory of partial differential equations such as Sobolev-Slobodeckij spaces. It is known that two norms are equivalent when one considers constant regularity function spaces without weights. We show that the equivalence still holds for variable smoothness and weights, which is accomplished by making use of shifted maximal functions, Peetre's maximal functions, and the reverse H\"older inequality. Moreover, we obtain a weighted regularity estimate for time-fractional evolution equations and a generalized Sobolev embedding theorem without weights.