Stability of Localized Integral Operators on Weighted $L^p$ spaces

TL;DR

This paper investigates the stability of localized integral operators with mild singularities and regularity on weighted L^p spaces, showing stability extends across different p and weight classes.

Contribution

It proves that stability of such operators on one weighted L^p space implies stability on all weighted L^{p'} spaces with corresponding Muckenhoupt weights.

Findings

Stability extends from one weighted L^p space to all others.

Operators with mild singularity and Holder regularity exhibit this stability property.

Results apply to operators like the Bessel potential operator.

Abstract

In this paper, we consider localized integral operators whose kernels have mild singularity near the diagonal and certain Holder regularity and decay off the diagonal. Our model example is the Bessel potential operator ${\mathcal J}_\gamma, \gamma>0$. We show that if such a localized integral operator has stability on a weighted function space $L^p_w$ for some $p\in [1, \infty)$ and Muckenhoupt $A_p$-weight $w$, then it has stability on weighted function spaces $L^{p'}_{w'}$ for all $1\le p'<\infty$ and Muckenhoupt $A_{p'}$-weights $w'$.