Gradient bounds for radial maximal functions

Carneiro, Emanuel; González-Riquelme, Cristian

doi:10.5186/aasfm.2021.4631

Cited by 8 publications

(11 citation statements)

References 28 publications

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“…Example 5.2. Let d = 1 and g = 1 [0,3) + 1 [5,8) and h = 1 [2,3) + 1 [5,6) . Then for f = g, h, g + h we have var Mf = Mf However since maximal operators are pointwise subadditive, one might hope to find a modification of var M that is subadditive.…”

Section: Further Approachesmentioning

confidence: 99%

“…That means from (8) one could conclude var Mf ≤ C d var f . But even for the dyadic maximal operator (8) is still an open question. Note that in Example 5.2 we have var h = 4 so it is not a counterexample against (8).…”

Section: Further Approachesmentioning

confidence: 99%

“…But even for the dyadic maximal operator (8) is still an open question. Note that in Example 5.2 we have var h = 4 so it is not a counterexample against (8). Maybe ( 8) is still close enough to var Mh ≤ C d var h for characteristic functions to make use of [27].…”

Section: Further Approachesmentioning

confidence: 99%

“…We focus on the endpoint p = 1. For example in [12] Carneiro and Svaiter and in [8] and Carneiro and González-Riquelme consider convolution maximal operators associated to certain partial differential equations. They prove ∇Mf L 1 (R d ) ≤ C d ∇f L 1 (R d ) for d = 1, and for d > 1 if f is radial.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Variation of the dyadic maximal function

Weigt¹

2020

Preprint

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Section: Further Approachesmentioning

confidence: 99%

Section: Further Approachesmentioning

confidence: 99%

Section: Further Approachesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Variation of the dyadic maximal function

Weigt¹

2020

Preprint

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show abstract

“…There are also a couple of promising new results by J. Weigt, solving the total variation version of this question for characteristic functions of sets of finite perimeter [30], and its analogue for the dyadic maximal operator [31]. Related works in this topic include [7,8,11,13,16,19,25,26,27].…”

mentioning

confidence: 99%

Sunrise strategy for the continuity of maximal operators

Carneiro¹,

González-Riquelme²,

Madrid³

2020

Preprint

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In this paper we address the W 1,1 -continuity of several maximal operators at the gradient level. A key idea in our global strategy is the decomposition of a maximal operator, with the absence of strict local maxima in the disconnecting set, into "lateral" maximal operators with good monotonicity and convergence properties. This construction is inspired in the classical sunrise lemma in harmonic analysis. A model case for our sunrise strategy considers the uncentered Hardy-Littlewood maximal operator M acting on W 1,1 rad (R d ), the subspace of W 1,1 (R d ) consisting of radial functions. In dimension d ≥ 2 it was recently established by H. Luiro that the map f → ∇ M f), and we show that such map is also continuous. Further applications of the sunrise strategy in connection with the W 1,1 -continuity problem include nontangential maximal operators on R d acting on radial functions when d ≥ 2 and general functions when d = 1, and the uncentered Hardy-Littlewood maximal operator on the sphere S d acting on polar functions when d ≥ 2 and general functions when d = 1.

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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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Gradient bounds for radial maximal functions

Cited by 8 publications

References 28 publications

Variation of the dyadic maximal function

Variation of the dyadic maximal function

Sunrise strategy for the continuity of maximal operators

Endpoint Sobolev bounds for fractional Hardy–Littlewood maximal operators

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