Some remarks on extrapolation with "flat" weights

TL;DR

This paper establishes an extrapolation theorem for operators, showing that weak boundedness on flat weights near a certain p-value implies boundedness on a neighborhood of p, leading to strong type estimates.

Contribution

It introduces a new extrapolation result for operators under weak assumptions, focusing on flat weights and extending boundedness properties.

Findings

Weak boundedness on flat weights implies boundedness on a neighborhood of p.

Strong type estimates follow from weak boundedness under the given conditions.

Results apply to general operators with minimal assumptions.

Abstract

We prove an extrapolation result for general operators under some weak assumptions on the boundedness of the operator. In particular, we show that if the operator is weakly bounded on some L^{p_{0}}(w), for all "flat" weights, w in A_{p_{0}}, 1 < p_{0} <\infty, then for p in some small neighborhood around p_{0}, and all "flat" A_{p} weights, w, the operator is weakly bounded on L^{p}(w), and as a result we get strong type estimates for the operator.