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Sharp two-weight inequality for fractional maximal operators

Rodrigo Bañuelos, Adam Osękowski

TL;DR

The paper addresses sharp two-weight bounds for the fractional maximal operator $\mathcal{M}^\alpha$ on tree-structured probability spaces, deriving the sharp universal bound for $\|\mathcal{M}^\alpha\|_{L^p(v)\to L^q(u)}$ under Sawyer's testing condition for $1<p\le q<\infty$ and $0\le \alpha<1$ (with sharpness established when $\alpha\ge 1/p-1/q$). The authors fuse the Bellman function method with a sharp Sobolev embedding viewpoint, constructing a four-variable Bellman function $\mathcal{B}$ tied to Bliss' inequality to control the Hardy–Littlewood–Pólya-type embedding. A sharp fractional Carleson embedding theorem is derived and used to reduce the main estimate to a Carleson condition guaranteed by the testing hypothesis, yielding the best constant $C_{p,q}$. The sharpness result is demonstrated via a carefully designed dyadic-tree example on $[0,1]$ and a localization of Bliss' inequality, establishing that the constant cannot be improved in the stated range of parameters.

Abstract

The paper is devoted to two-weight estimates for the fractional maximal operators $\mathcal{M}^α$ 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}^α\|_{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.

Sharp two-weight inequality for fractional maximal operators

TL;DR

The paper addresses sharp two-weight bounds for the fractional maximal operator on tree-structured probability spaces, deriving the sharp universal bound for under Sawyer's testing condition for and (with sharpness established when ). The authors fuse the Bellman function method with a sharp Sobolev embedding viewpoint, constructing a four-variable Bellman function tied to Bliss' inequality to control the Hardy–Littlewood–Pólya-type embedding. A sharp fractional Carleson embedding theorem is derived and used to reduce the main estimate to a Carleson condition guaranteed by the testing hypothesis, yielding the best constant . The sharpness result is demonstrated via a carefully designed dyadic-tree example on and a localization of Bliss' inequality, establishing that the constant cannot be improved in the stated range of parameters.

Abstract

The paper is devoted to two-weight estimates for the fractional maximal operators on general probability spaces equipped with a tree-like structure. For given , we study the sharp universal upper bound for the norm , where 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.
Paper Structure (5 sections, 8 theorems, 76 equations)

This paper contains 5 sections, 8 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Sobolev imbedding theorem, Bliss' inequality and a special function
  3. Proof of Theorem \ref{['mainthm']}
  4. Proof of \ref{['mainin']}
  5. Sharpness

Key Result

Theorem 1.2

Fix $1<p\leq q<\infty$, $0\leq \alpha<1$ and $L>0$. Suppose that $u$, $v$ are two weights on $(X,\mu)$ such that for any $Q\in \mathcal{T}$. Then we have the estimate If $\alpha\geq 1/p-1/q$, then the constant $C_{p,q}$ is the best possible: for any $1<p\leq q<\infty$ and any $c<C_{p,q}L$ there is a triple $(X,\mu,\mathcal{T})$ and a pair $(u,v)$ of weights satisfying the testing condition testi

Theorems & Definitions (16)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 2.1
  • Proposition 2.1
  • proof
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 3.1
  • ...and 6 more