Regularity of maximal functions on Hardy–Sobolev spaces

Pérez, Carlos; Picon, Tiago; Saari, Olli; Sousa, Mateus

doi:10.1112/blms.12195

Cited by 17 publications

(11 citation statements)

References 26 publications

(49 reference statements)

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“…In the seminal paper [13], Kinnunen studied the action of the Hardy-Littlewood maximal operator on Sobolev functions, giving an elegant proof that M : W 1,p (R d ) → W 1,p (R d ) is bounded for 1 < p ≤ ∞. This work paved the way for several interesting contributions to the regularity theory of maximal operators over the past two decades, with interesting connections to potential theory and partial differential equations, see for instance [1,3,5,6,7,10,12,14,15,17,18,20,21,22,23,24,25].…”

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confidence: 83%

Gradient bounds for radial maximal functions

Carneiro,

González-Riquelme

2019

Preprint

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In this paper we study the regularity properties of certain maximal operators of convolution type at the endpoint p = 1, when acting on radial data. In particular, for the heat flow maximal operator and the Poisson maximal operator, when the initial datumis a radial function, we show that the associated maximal function u * is weakly differentiable andThis establishes the analogue of a recent result of H. Luiro for the uncentered Hardy-Littlewood maximal operator, now in a centered setting with smooth kernels. In a second part of the paper, we establish similar gradient bounds for maximal operators on the sphere S d , when acting on polar functions. Our study includes the uncentered Hardy-Littlewood maximal operator, the heat flow maximal operator and the Poisson maximal operator on S d .

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confidence: 83%

Gradient bounds for radial maximal functions

Carneiro,

González-Riquelme

2019

Preprint

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“…This work paved the way to several contributions of many researchers in this topic and its relations with other areas, see for instance [1,4,5,7,10,12,13,15,23,24,25] The most important open problem in this field is the W 1,1 -problem.…”

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confidence: 89%

Sobolev regularity of polar fractional maximal functions

González-Riquelme

2020

Nonlinear Analysis

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We study the Sobolev regularity on the sphere S d of the uncentered fractional Hardy-Littlewood maximal operator M β at the endpoint p = 1, when acting on polar data. We firstWe then prove that the mapwhen restricted to polar data. Our methods allow us to give a new proof of the continuity of the map f → |∇ M β f | from W 1,1 rad (R d ) to L q (R d ). Moreover, we prove that a conjectural local boundedness for the centered fractional Hardy-Littlewood maximal operator M β implies the continuity of the map f → |∇M β f | from W 1,1 to L q , in the context of polar functions on S d and radial functions on R d .

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“…In [21] Kinnunen and Tuominen proved the boundedness of a discrete maximal operator in the metric Hajłasz Sobolev space M 1,1 . In [27] Pérez et al proved the boundedness of certain convolution maximal operators on Hardy-Sobolev spaces Ḣ 1, p for a sharp range of exponents, including p = 1. In [29] the author proved var M d f ≤ C d var f for the dyadic maximal operator for all dimensions d.…”

Section: Introductionmentioning

confidence: 99%

Endpoint Sobolev bounds for fractional Hardy–Littlewood maximal operators

Weigt

2022

Math. Z.

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Let $$0<\alpha <d$$ 0 < α < d and $$1\le p<d/\alpha $$ 1 ≤ p < d / α . We present a proof that for all $$f\in W^{1,p}({\mathbb {R}}^d)$$ f ∈ W 1 , p ( R d ) both the centered and the uncentered Hardy–Littlewood fractional maximal operator $${\mathrm {M}}_\alpha f$$ M α f are weakly differentiable and $$ \Vert \nabla {\mathrm {M}}_\alpha f\Vert _{p^*} \le C_{d,\alpha ,p} \Vert \nabla f\Vert _p , $$ ‖ ∇ M α f ‖ p ∗ ≤ C d , α , p ‖ ∇ f ‖ p , where $$ p^* = (p^{-1}-\alpha /d)^{-1} . $$ p ∗ = ( p - 1 - α / d ) - 1 . In particular it covers the endpoint case $$p=1$$ p = 1 for $$0<\alpha <1$$ 0 < α < 1 where the bound was previously unknown. For $$p=1$$ p = 1 we can replace $$W^{1,1}({\mathbb {R}}^d)$$ W 1 , 1 ( R d ) by $$\mathrm {BV}({\mathbb {R}}^d)$$ 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 $$\alpha =0$$ α = 0 in the dyadic setting. We use that for $$\alpha >0$$ α > 0 the fractional maximal function does not use certain small balls. For $$\alpha =0$$ α = 0 the proof collapses.

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Regularity of maximal functions on Hardy–Sobolev spaces

Cited by 17 publications

References 26 publications

Gradient bounds for radial maximal functions

Gradient bounds for radial maximal functions

Sobolev regularity of polar fractional maximal functions

Endpoint Sobolev bounds for fractional Hardy–Littlewood maximal operators

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