A characterisation of the Besov-Lipschitz and Triebel-Lizorkin spaces using Poisson like kernels

TL;DR

This paper provides a comprehensive characterization of Besov-Lipschitz and Triebel-Lizorkin spaces using non-smooth kernels, maximal function estimates, and fractional derivatives of the Poisson kernel.

Contribution

It introduces a new characterization method for these function spaces employing minimal conditions on kernels and advanced harmonic analysis tools.

Findings

Characterization of spaces using non-smooth kernels

Application of fractional derivatives of the Poisson kernel

Refinement of Calderon reproducing formula

Abstract

We give a complete characterisation of the spaces $\dot{B}^{\alpha}_{p,q}$ and $\dot{F}^{\alpha}_{p,q}$ by using a non-smooth kernel satisfying near minimal conditions. The tools used include a Stromberg-Torchinsky type estimate for certain maximal functions and the concept of a distribution of finite growth, inspired by Stein. Moreover, our exposition also makes essential use of a number of refinements of the well-known Calderon reproducing formula. The results are then applied to obtain the characterisation of these spaces via a fractional derivative of the Poisson kernel.