Poincaré inequalities for the maximal function

Saari, Olli

doi:10.2422/2036-2145.201705_001

Cited by 15 publications

(13 citation statements)

References 34 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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].…”

mentioning

confidence: 82%

Gradient bounds for radial maximal functions

Carneiro,

González-Riquelme

2019

Preprint

View full text Add to dashboard Cite

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 .

show abstract

mentioning

confidence: 82%

Gradient bounds for radial maximal functions

Carneiro,

González-Riquelme

2019

Preprint

View full text Add to dashboard Cite

show abstract

“…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

View full text Add to dashboard Cite

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 .

show abstract

“…In fact, he showed the stronger result with M defined by averages of f as opposed to |f |. Further work in this direction can be found in [5,8,15,24,27,28,32].…”

Section: Introductionmentioning

confidence: 99%

Geometric maximal operators and BMO on product bases

Dafni,

Gibara,

Yue

2020

Preprint

View full text Add to dashboard Cite

We consider the problem of the boundedness of maximal operators on BMO on shapes in R n . We prove that for bases of shapes with an engulfing property, the corresponding maximal function is bounded from BMO to BLO, generalising a known result of Bennett for the basis of cubes. When the basis of shapes does not possess an engulfing property but exhibits a product structure with respect to lower-dimensional shapes coming from bases that do possess an engulfing property, we show that the corresponding maximal function is bounded from BMO to a space we define and call rectangular BLO. G.D. was partially supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada, the Centre de recherches mathématiques (CRM), and the Fonds de recherche du Québec -Nature et technologies (FRQNT). R.G. was partially supported by the Centre de recherches mathématiques (CRM), the Institut des sciences mathématiques (ISM), and the Fonds de recherche du Québec -Nature et technologies (FRQNT). H.Y. was partially supported by GCSU Faculty Development Funds.

show abstract

Poincaré inequalities for the maximal function

Cited by 15 publications

References 34 publications

Gradient bounds for radial maximal functions

Gradient bounds for radial maximal functions

Sobolev regularity of polar fractional maximal functions

Geometric maximal operators and BMO on product bases

Contact Info

Product

Resources

About