On approximate differentiability of the maximal function

Hajłasz, Piotr; Malý, Jan

doi:10.1090/s0002-9939-09-09971-7

Cited by 54 publications

(31 citation statements)

References 20 publications

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“…We begin with showing that harmonic functions are Lipschitz in the large scale, that is assuming that the distance between points is not too small. For the similar result in the setting of maximal operators see Lemma 8 in Haj lasz-Malý [13]. In the previous section, see Theorem 4.2, we show, among other results, the Lipschitz regularity for harmonic functions imposing the 1-annular decay condition on the measure.…”

Section: The Lipschitz Regularity and Uniform Measures Weak Upper Grsupporting

confidence: 74%

Harmonic functions on metric measure spaces

2018

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We introduce and study strongly and weakly harmonic functions on metric measure spaces defined via the mean value property holding for all and, respectively, for some radii of balls at every point of the underlying domain. Among properties of such functions we investigate various types of Harnack estimates on balls and compact sets, weak and strong maximum principles, comparison principles, the Hölder and the Lipshitz estimates and some differentiability properties. The latter one is based on the notion of a weak upper gradient. The Dirichlet problem for functions satisfying the mean value property is studied via the dynamical programming method related to stochastic games. We employ the Perron method to construct a harmonic function with continuous boundary data. Finally, we discuss and prove the Liouville type theorems.Our results are obtained for various types of measures: continuous with respect to a metric, doubling, uniform, measures satisfying the annular decay condition. Relations between such measures are presented as well. The presentation is illustrated by examples.

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Section: The Lipschitz Regularity and Uniform Measures Weak Upper Grsupporting

confidence: 74%

Harmonic functions on metric measure spaces

2018

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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.…”

mentioning

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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“…The value of the approximate derivative of M β f is a simple computation which can be obtained arguing as in [11] or [20], and has its roots in the work of Luiro [17]. The stronger statement in (ii) regarding the a.e.…”

Section: 2mentioning

confidence: 99%

Endpoint Sobolev Continuity of the Fractional Maximal Function in Higher Dimensions

Beltran

Madrid

2020

International Mathematics Research Notices

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We establish continuity mapping properties of the non-centered fractional maximal operator M β in the endpoint input space W 1,1 (R d ) for d ≥ 2 in the cases for which its boundedness is known. More precisely, we provef is radial and for 1 ≤ β < d for general f . The results for 1 ≤ β < d extend to the centered counterpart M c β . Moreover, if d = 1, we show that the conjectured boundedness of that map for M c β implies its continuity. Af 1,q ≤ C f 1,p . At the endpoint p = 1, one cannot expect boundedness of M β from W 1,1 to W 1, d d−β to hold, as M β fails to be bounded at the level of Lebesgue spaces. However, one Date: June 4, 2019. 2010 Mathematics Subject Classification. 42B25, 46E35.

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On approximate differentiability of the maximal function

Cited by 54 publications

References 20 publications

Harmonic functions on metric measure spaces

Harmonic functions on metric measure spaces

Sobolev regularity of polar fractional maximal functions

Endpoint Sobolev Continuity of the Fractional Maximal Function in Higher Dimensions

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