Carleson operators on doubling metric measure spaces

TL;DR

This paper develops a general framework for Carleson operators on doubling metric measure spaces, extending classical results beyond Euclidean spaces and providing new $L^p$ bounds with computer-verified proofs.

Contribution

It introduces an axiomatic approach to modulation functions on doubling metric measure spaces and proves $L^p$ bounds for the associated Carleson operators, generalizing existing results.

Findings

Established $L^p$ bounds for Carleson operators in general metric spaces.

Provided a computer-verified proof using Lean and mathlib.

Extended classical Carleson operator results to a broader setting.

Abstract

Doubling metric measure spaces provide a natural framework for singular integral operators. In contrast, the study of maximally modulated singular integral operators, the so-called Carleson operators, has largely been limited to Euclidean space with modulation functions such as polynomials defined by algebraic means. We present a general axiomatic approach to modulation functions on doubling metric measure spaces and prove $L^p$ bounds for the corresponding Carleson operators in Theorem 1.1 and Theorem 1.2. This generalizes classical and modern results on Carleson operators. In addition to the proofs presented here, our main results have been computer verified using the language Lean and the library mathlib, as documented in the sibling communication arXiv:2405.06423.