Search results for "maximal function"

Maximal function estimates and self-improvement results for Poincaré inequalities

2018

Our main result is an estimate for a sharp maximal function, which implies a Keith–Zhong type self-improvement property of Poincaré inequalities related to differentiable structures on metric measure spaces. As an application, we give structure independent representation for Sobolev norms and universality results for Sobolev spaces. peerReviewed

Weighted norm inequalities in a bounded domain by the sparse domination method

2019

AbstractWe prove a local two-weight Poincaré inequality for cubes using the sparse domination method that has been influential in harmonic analysis. The proof involves a localized version of the Fefferman–Stein inequality for the sharp maximal function. By establishing a local-to-global result in a bounded domain satisfying a Boman chain condition, we show a two-weight p-Poincaré inequality in such domains. As an application we show that certain nonnegative supersolutions of the p-Laplace equation and distance weights are p-admissible in a bounded domain, in the sense that they support versions of the p-Poincaré inequality.

Rectifiability and singular integrals

1995

Smoothing properties of the discrete fractional maximal operator on Besov and Triebel-Lizorkin spaces

2013

Motivated by the results of Korry, and Kinnunen and Saksman, we study the behaviour of the discrete fractional maximal operator on fractional Hajlasz spaces, Hajlasz-Besov, and Hajlasz-Triebel-Lizorkin spaces on metric measure spaces. We show that the discrete fractional maximal operator maps these spaces to the spaces of the same type with higher smoothness. Our results extend and unify aforementioned results. We present our results in a general setting, but they are new already in the Euclidean case.

On the regularity of the Hardy-Littlewood maximal operator on subdomains of ℝn

2010

AbstractWe establish the continuity of the Hardy-Littlewood maximal operator on W1,p(Ω), where Ω ⊂ ℝn is an arbitrary subdomain and 1 < p < ∞. Moreover, boundedness and continuity of the same operator is proved on the Triebel-Lizorkin spaces Fps,q (Ω) for 1 < p,q < ∞ and 0 < s < 1.

The variation of the maximal function of a radial function

2017

We study the problem concerning the variation of the Hardy-Littlewood maximal function in higher dimensions. As the main result, we prove that the variation of the non-centered Hardy-Littlewood maximal function of a radial function is comparable to the variation of the function itself.

Self-improvement of pointwise Hardy inequality

2019

We prove the self-improvement of a pointwise p p -Hardy inequality. The proof relies on maximal function techniques and a characterization of the inequality by curves.

Maximal potentials, maximal singular integrals, and the spherical maximal function

2014

We introduce a notion of maximal potentials and we prove that they form bounded operators from L to the homogeneous Sobolev space Ẇ 1,p for all n/(n − 1) < p < n. We apply this result to the problem of boundedness of the spherical maximal operator in Sobolev spaces.

Weighted Hardy Spaces of Quasiconformal Mappings

2019

We establish a weighted version of the $H^p$-theory of quasiconformal mappings.

Functions of One Variable

2019

A classical result of Fatou gives that every bounded holomorphic function on the disc has radial limits for almost every point in the torus, and the limit function belongs to the Hardy space H_\infty of the torus. This property is no longer true when we consider vector-valued functions. The Banach spaces X for which this property is satisfied are said to have the analytic Radon-Nikodym property (ARNP). Some important equivalent reformulations of ARNP are studied in this chapter. Among others, X has ARNP if and only if each X-valued H_p- function f on the disc has radial limits almost everywhere on the torus (and not only H_\infty-functions). Even more, in this case each such f has non-tange…