Endpoint Sobolev Bounds for Fractional Hardy-Littlewood Maximal Operators

TL;DR

This paper proves endpoint Sobolev bounds for fractional Hardy-Littlewood maximal operators, establishing weak differentiability and gradient bounds for functions in Sobolev and BV spaces, including previously unknown endpoint cases.

Contribution

It provides the first proof of Sobolev bounds for fractional maximal operators at the endpoint p=1, extending known results to this critical case.

Findings

Weak differentiability of fractional maximal operators for all relevant p and alpha

Gradient bounds in Sobolev spaces for fractional maximal operators

Extension of results to the endpoint p=1 in BV spaces

Abstract

Let $0<\alpha<d$ and $1\leq p<d/\alpha$. We present a proof that for all $f\in W^{1,p}(\mathbb{R}^d)$ both the centered and the uncentered Hardy-Littlewood fractional maximal operator $\mathcal M_\alpha f$ are weakly differentiable and $ \|\nabla\mathcal M_\alpha f\|_{p^*} \leq C_{d,\alpha,p} \|\nabla f\|_p , $ where $ p^* = (p^{-1}-\alpha/d)^{-1} . $ In particular it covers the endpoint case $p=1$ for $0<\alpha<1$ where the bound was previously unknown. For $p=1$ we can replace $W^{1,1}(\mathbb{R}^d)$ by $\mathrm{BV}(\mathbb{R}^d)$. The ingredients used are a pointwise estimate for the gradient of the fractional maximal function, the layer cake formula, a Vitali type argument, a reduction from balls to dyadic cubes, the coarea formula, a relative isoperimetric inequality and an earlier established result for $\alpha=0$ in the dyadic setting. We use that for $\alpha>0$ the fractional maximal function does not use certain small balls. For $\alpha=0$ the proof collapses.