Endpoint Sobolev Bounds for Fractional Hardy-Littlewood Maximal Operators

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.