Regularity of the Hardy–Littlewood maximal operator on block decreasing functions

Aldaz, J. M.; Lázaro, J. Pérez

doi:10.4064/sm194-3-3

Cited by 11 publications

(13 citation statements)

References 14 publications

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“…In [AlPe], part of the project outlined in the previous paragraph is carried out for functions of bounded variation and d = 1 (the situation when d > 1 is still not well understood, cf. [AlPe2] and [AlPe3]). Unlike the Hölder case, here a qualitative gain in regularity does occur (cf.…”

Section: Introductionmentioning

confidence: 98%

Optimal Bounds on the Modulus of Continuity of the Uncentered Hardy–Littlewood Maximal Function

2010

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We obtain sharp bounds for the modulus of continuity of the uncentered maximal function in terms of the modulus of continuity of the given function, via integral formulas. Some of the results deduced from these formulas are the following: The best constants for Lipschitz and Hölder functions on proper subintervals ofIn higher dimensions, we determine the asymptotic behavior, as d → ∞, of the norm of the maximal operator associated with cross-polytopes, Euclidean balls, and cubes, that is, p balls for p = 1, 2, ∞. We do this for arbitrary moduli of continuity. In the specific case of Lipschitz and Hölder functions, the operator norm of the maximal operator is uniformly bounded by 2 −α/q , where q is the conjugate exponent of p = 1, 2, and as d → ∞ the norms approach this bound. When p = ∞, best constants are the same as when p = 1.

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Section: Introductionmentioning

confidence: 98%

Optimal Bounds on the Modulus of Continuity of the Uncentered Hardy–Littlewood Maximal Function

2010

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“…The latter proof turned out to be much more complicated. In [APL09] Aldaz and Pérez Lázaro have proven the gradient bound for the uncentered maximal operator for block decreasing functions in W 1,1 (R d ) and any dimension d. In [Lui18] Luiro has done the same for radial functions. More endpoint results are available for related maximal operators, for example convolution maximal operators [CS13,CGR19], fractional maximal operators [KS03, CM17, CM17, BM19, BRS19, Wei21, HKKT15], and discrete maximal operators [CH12], as well as maximal operators on different spaces, such as in the metric setting [KT07] and on Hardy-Sobolev spaces [PPSS18].…”

Section: Introductionmentioning

confidence: 99%

The Variation of the Uncentered Maximal Operator with respect to Cubes

Weigt

2021

Preprint

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“…For the uncentered Hardy-Littlewood maximal function Hajłasz's and Onninen's question already also has a positive answer for all dimensions d in several special cases. For radial functions Luiro proved it in [24], for block decreasing functions Aldaz and Pérez Lázaro proved it in [2] and for characteristic functions the author proved it in [30]. As a first step towards weak differentiability, Hajłasz and Malý proved in [15] that for f ∈ L 1 (R d ) the centered Hardy-Littlewood maximal function is approximately differentiable.…”

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 the Hardy–Littlewood maximal operator on block decreasing functions

Cited by 11 publications

References 14 publications

Optimal Bounds on the Modulus of Continuity of the Uncentered Hardy–Littlewood Maximal Function

Optimal Bounds on the Modulus of Continuity of the Uncentered Hardy–Littlewood Maximal Function

The Variation of the Uncentered Maximal Operator with respect to Cubes

Endpoint Sobolev bounds for fractional Hardy–Littlewood maximal operators

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