Endpoint Sobolev Bounds for Fractional Hardy-Littlewood Maximal Operators

Abstract

Let $0<\alpha <d$ and $1\le p<d/\alpha$. 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 $\Vert \nabla {\mathcal{M}}_{\alpha }f{\Vert }_{p^{\ast }}\le C_{d,\alpha ,p}\Vert \nabla f{\Vert }_p,$ where $p^{\ast }=(p^{-1}-\alpha /d{)}^{-1}.$ 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.

arXiv commentExplicitely write the proof for the centered and the uncentered operator. Changed some formulations