Sharp two-weight inequality for fractional maximal operators

TL;DR

This paper establishes sharp two-weight inequalities for fractional maximal operators on probability spaces with a tree structure, using Bellman functions to connect to Sobolev embedding theorems.

Contribution

It provides the first sharp universal bounds for fractional maximal operators in a general probabilistic setting with arbitrary weights.

Findings

Derived the sharp universal upper bound for the operator norm.

Connected the problem to the classical Sobolev embedding theorem.

Applied the Bellman function method to a new setting.

Abstract

The paper is devoted to two-weight estimates for the fractional maximal operators $\mathcal{M}^\alpha$ on general probability spaces equipped with a tree-like structure. For given $1<p\leq q<\infty$, we study the sharp universal upper bound for the norm $ \|\mathcal{M}^\alpha\|_{L^p(v)\to L^q(u)}$, where $(u,v)$ is an arbitrary pair of weights satisfying the Sawyer testing condition. The proof is based on the abstract Bellman function method, which reveals an unexpected connection of the above problem with the sharp version of the classical Sobolev imbedding theorem.