Endpoint Sobolev Bounds for Fractional Hardy-Littlewood Maximal Operators

Let $0<\alpha <d$ and $1\le p<d\mathrm{/}\alpha$. 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 $\mathrm{B}\mathrm{V}({\mathbb{R}}^d)$. 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.