Endpoint Sobolev bounds for fractional Hardy–Littlewood maximal operators

Julian Weigt

doi:10.1007/s00209-022-02969-x

Endpoint Sobolev bounds for fractional Hardy–Littlewood maximal operators

Julian Weigt^*

^*Corresponding author for this work

Department of Mathematics and Systems Analysis

Research output: Contribution to journal › Article › Scientific › peer-review

10 Citations (Scopus)

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Abstract

Let 0 < α< d and 1 ≤ p< d/ α. We present a proof that for all f∈ W¹^,^p(R^d) 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 W^{1 , 1}(R^d) by BV (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 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.

Original language	English
Pages (from-to)	2317-2337
Number of pages	21
Journal	Mathematische Zeitschrift
Volume	301
Issue number	3
Early online date	18 Feb 2022
DOIs	https://doi.org/10.1007/s00209-022-02969-x
Publication status	Published - Jul 2022
MoE publication type	A1 Journal article-refereed

Funding

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.

Keywords

Dyadic cubes
Fractional maximal function
Variation

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10.1007/s00209-022-02969-xLicence: CC BY

Endpoint Sobolev bounds for fractional Hardy–Littlewood maximal operatorsFinal published version, 315 KBLicence: CC BY

Cite this

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title = "Endpoint Sobolev bounds for fractional Hardy–Littlewood maximal operators",

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.",

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author = "Julian Weigt",

note = "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 Gonz{\'a}lez-Riquelme and Jose Madrid, in particular about the centered fractional maximal operator. The author has been supported by the Vilho, Yrj{\"o} and Kalle V{\"a}is{\"a}l{\"a} Foundation of the Finnish Academy of Science and Letters. Publisher Copyright: {\textcopyright} 2022, The Author(s).",

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N1 - 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).

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N2 - 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.

AB - 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.

KW - Dyadic cubes

KW - Fractional maximal function

KW - Variation

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DO - 10.1007/s00209-022-02969-x

M3 - Article

AN - SCOPUS:85124734711

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EP - 2337

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

IS - 3

ER -

Endpoint Sobolev bounds for fractional Hardy–Littlewood maximal operators

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