Sharp weighted bounds without testing or extrapolation

TL;DR

This paper presents a concise proof of sharp weighted bounds for sparse operators and Calderón-Zygmund operators, avoiding traditional techniques like two weight inequalities and extrapolation, with applicability to fractional integrals.

Contribution

It introduces a novel, simplified proof method for weighted bounds that bypasses common complex techniques, extending to fractional integral operators.

Findings

Sharp weighted bounds for sparse operators established

Bounds hold for all p in (1, ∞)

Applicable to fractional integral operators

Abstract

We give a short proof of the sharp weighted bound for sparse operators that holds for all $p$, $1<p<\infty$. By recent developments this implies the bounds hold for any Calder\'on-Zygmund operator. The novelty of our approach is that we avoid two techniques that are present in other proofs: two weight inequalities and extrapolation. Our techniques are applicable to fractional integral operators as well.