On weighted estimates for Stein's maximal function

AMERICAN MATHEMATICAL …

arXiv (Cornell University), 2023

Journal of Functional Analysis, 2007

For a family of weight functions invariant under a finite reflection group, the boundedness of a maximal function on the unit sphere is established and used to prove a multiplier theorem for the orthogonal expansions with respect to the weight function on the unit sphere. Similar results are also established for the weighted space on the unit ball and on the standard simplex.

Proceedings of the American Mathematical Society, 2014

We introduce a notion of maximal potentials and we prove that they form bounded operators from L p to the homogeneous Sobolev spaceẆ 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.

1999

The purpose of this paper is to prove the L p (R n ; dd) boundedness, for p &gt; 1, of the non-centered Hardy-Littlewood maximal operator associated with the Gaussian measure dd = e ?jxj 2 dx. Let dd = e ?jxj 2 dx be a Gaussian measure in Euclidean space R n. We consider the non-centered maximal function deened by Mf(x) = sup x2B 1 (B) Z B jfj dd; where the supremum is taken over all balls B in R n containing x. P. Sjj ogren 2] proved that M is not of weak type (1,1) with respect to dd. A more general result was obtained by A. Vargas 3], who characterized those radial and strictly positive measures for which the corresponding maximal operator is of weak type (1,1). However, these papers leave open the question of the L p (dd) boundedness of M for p &gt; 1 and n &gt; 1: The main result in this paper is Theorem 1 M is a bounded operator on L p (dd) for p &gt; 1, that is, there exists a constant C = C(n; p) such that for f 2 L p (dd); kMfk L p (dd) Ckfk L p (dd) : We denote S n?1

Proceedings of the American Mathematical Society, 1993

In this note we give a simple proof of the characterization of the weights for which the one-sided Hardy-Littlewood maximal functions apply I-PiV) into weak-LP{U) and a direct proof of the characterization of the weights for which the one-sided Hardy-Littlewood maximal functions apply LP(W) into LP(W). Recently [8, 5], the good weights for these operators have been characterized. In particular the following results were proved. Theorem 1 [8, 5]. Let U and V be nonnegative measurable functions. The following are equivalent. (a) There exists a constant C > 0 such that for all X > 0 and every fi £ U(V), I U<^f\fi\pV. J{x : M+f{x)>X} W 7k (b) (U, V) satisfies A+, i.e., there exists a nonnegative real number A such that rs(riLv){iC¥-"^")"'A Vr>1-M~U(x)<AV(x) a.e.ifp=l.

Mathematische Annalen, 2006

We discuss the manner in which one might expect directional maximal functions to control the Fourier extension operator via L 2 weighted inequalities. We prove a general inequality of this type for the extension operator restricted to circles in the plane.

Analysis Mathematica, 2018

We study weighted boundedness of Hardy-Littlewood-type maximal function involving Orlicz functions. We also obtain some sufficient conditions for the weighted boundedness of the Hardy-Littlewood maximal function of the upper-half plane. 1 |Q I |ω,α Q I |f (z)| p ω(z)dV α (z)

Journal of Geometric Analysis, 2012

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 of R are Lip α (M f) ≤ (1 + α) −1 Lip α (f), α ∈ (0, 1]. On R, the best bound for Lipschitz functions is Lip(M f) ≤ (√ 2 − 1) Lip(f). In higher dimensions, we determine the asymptotic behavior, as d → ∞, of the norm of the maximal operator associated to cross-polytopes, euclidean balls and cubes, that is, ℓ 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.

Iranian Journal of Science and Technology, Transactions A: Science, 2021

In this note besides two abstract versions of the Vitali Covering Lemma an abstract Hardy-Littlewood Maximal Inequality, generalizing weak type (1, 1) maximal function inequality, associated to any outer measure and a family of subsets on a set is introduced. The inequality is (effectively) satisfied if and only if a special numerical constant called Hardy-Littelwood maximal constant is finite. Two general sufficient conditions for the finiteness of this constant are given and upper bounds for this constant associated to the family of (centered) balls in homogeneous spaces, family of dyadic cubes in Euclidean spaces, family of admissible trapezoids in homogeneous trees, and family of Calderón-Zygmund sets in (ax + b)-group, are considered. Also a very simple application to find some nontrivial estimates about mass density in Classical Mechanics is given. MSC 2020. 42B25, 43A99.