Sharp weighted estimates for multi-frequency Calder\'on-Zygmund operators

TL;DR

This paper establishes sharp weighted bounds for multi-frequency Calderón-Zygmund operators using sparse domination, revealing how bounds depend on the number of frequencies and weight characteristics.

Contribution

It introduces new weighted estimates for multi-frequency Calderón-Zygmund operators with bounds depending explicitly on frequency count and weight class.

Findings

Bound T in weighted spaces depends on N^{|1/r - 1/2|}

Bounds involve the _{p/r} characteristic of the weight

Results apply to operators with Dini-continuous modulus of continuity

Abstract

In this paper we study weighted estimates for the multi-frequency $\omega-$Calder\'{o}n-Zygmund operators $T$ associated with the frequency set $\Theta=\{\xi_1,\xi_2,\dots,\xi_N\}$ and modulus of continuity $\omega$ satisfying the usual Dini condition. We use the modern method of domination by sparse operators and obtain bounds $\|T\|_{L^p(w)\rightarrow L^p(w)}\lesssim N^{|\frac{1}{r}-\frac{1}{2}|}[w]_{\mathbb{A}_{p/r}}^{max(1,\frac{1}{p-r})},~1\leq r<p<\infty,$ for the exponents of $N$ and $\mathbb{A}_{p/r}$ characteristic $[w]_{\mathbb{A}_{p/r}}$.