Endpoint Sobolev bounds for fractional Hardy–Littlewood maximal operators
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A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
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2022-07
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Mcode
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Language
en
Pages
21
2317-2337
2317-2337
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MATHEMATISCHE ZEITSCHRIFT, Volume 301, issue 3
Abstract
Let 0 < α< d and 1 ≤ p< d/ α. We present a proof that for all f∈ W1,p(Rd) both the centered and the uncentered Hardy–Littlewood fractional maximal operator M αf are weakly differentiable and ‖∇Mαf‖p∗≤Cd,α,p‖∇f‖p, where p∗=(p-1-α/d)-1. In particular it covers the endpoint case p= 1 for 0 < α< 1 where the bound was previously unknown. For p= 1 we can replace W1 , 1(Rd) by BV (Rd). 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 α= 0 in the dyadic setting. We use that for α> 0 the fractional maximal function does not use certain small balls. For α= 0 the proof collapses.Description
Funding Information: I would like to thank my supervisor, Juha Kinnunen, for all of his support. I would like to thank Olli Saari for introducing me to this problem. I am also thankful for the discussions with Juha Kinnunen, Panu Lahti and Olli Saari who made me aware of a version of the coarea formula [, Theorem 3.11], which was used in the first draft of the proof, and for discussions with David Beltran, Cristian González-Riquelme and Jose Madrid, in particular about the centered fractional maximal operator. The author has been supported by the Vilho, Yrjö and Kalle Väisälä Foundation of the Finnish Academy of Science and Letters. Publisher Copyright: © 2022, The Author(s).
Keywords
Dyadic cubes, Fractional maximal function, Variation
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Citation
Weigt , J 2022 , ' Endpoint Sobolev bounds for fractional Hardy–Littlewood maximal operators ' , MATHEMATISCHE ZEITSCHRIFT , vol. 301 , no. 3 , pp. 2317-2337 . https://doi.org/10.1007/s00209-022-02969-x