Two-weight norm inequalities for parabolic fractional maximal functions

TL;DR

This paper establishes two-weight norm inequalities for parabolic fractional maximal functions, providing new weak and strong-type estimates under Muckenhoupt weight conditions, advancing the understanding of parabolic harmonic analysis.

Contribution

It introduces novel two-weight inequalities and conditions for parabolic fractional maximal functions, including Sawyer-type and Muckenhoupt bump conditions, extending classical results.

Findings

Proved weak-type two-weight inequalities for parabolic fractional maximal functions.

Established that Sawyer-type conditions imply strong-type estimates.

Demonstrated strong-type estimates under Muckenhoupt bump conditions.

Abstract

We prove two-weight norm inequalities for parabolic fractional maximal functions using parabolic Muckenhoupt weights. In particular, we prove a two-weight, weak-type estimate and Fefferman-Stein type inequalities for the centered parabolic maximal function. We also prove that a parabolic Sawyer-type condition implies the strong-type estimate for the parabolic fractional maximal function. Finally, we prove the strong-type estimate for the centered parabolic maximal function assuming a stronger parabolic Muckenhoupt bump condition.