Weighted inequalities for singular integral operators on the half-line

TL;DR

This paper establishes weighted inequalities for singular integral operators on the half-line, extending classical results to broader weight classes and applying these to extrapolate regularity results for various Cauchy problems.

Contribution

It introduces weighted estimates for singular integrals with relaxed kernel conditions, unifying and extending recent results in the field.

Findings

Weighted estimates hold for a broad class of weights including Muckenhoupt and Sawyer's conditions.

Applications to extrapolation of maximal $L^p$ regularity in various Banach spaces.

Extension and unification of recent results by Auscher, Axelsson, Chill, and Fiorenza.

Abstract

We prove weighted estimates for singular integral operators which operate on function spaces on a half-line. The class of admissible weights includes Muckenhoupt weights and weights satisfying Sawyer's one-sided conditions. The kernels of the operators satisfy relaxed Dini conditions. We apply the weighted estimates to extrapolation of maximal $L^p$ regularity of first order, second order and fractional order Cauchy problems into weighted rearrangement invariant Banach function spaces. In particular, we provide extensions, as well as a unification of recent results due to Auscher and Axelsson, and Chill and Fiorenza.