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- W4287642283 abstract "Let $0<alpha<d$ and $1leq p<d/alpha$. We present a proof that for all $fin W^{1,p}(mathbb{R}^d)$ both the centered and the uncentered Hardy-Littlewood fractional maximal operator $mathcal M_alpha f$ are weakly differentiable and $ |nablamathcal 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." @default.
- W4287642283 created "2022-07-25" @default.
- W4287642283 creator A5031814290 @default.
- W4287642283 date "2020-10-12" @default.
- W4287642283 modified "2023-09-26" @default.
- W4287642283 title "Endpoint Sobolev Bounds for Fractional Hardy-Littlewood Maximal Operators" @default.
- W4287642283 doi "https://doi.org/10.48550/arxiv.2010.05561" @default.
- W4287642283 hasPublicationYear "2020" @default.
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