On weighted estimates for Stein's maximal function (original) (raw)

L 2 -ESTIMATES FOR SOME MAXIMAL FUNCTIONS

SUNTO. Si presentono alcuni risultati sugli operatori massimali associati alle misure superficiali in R n . E. M. Stein [9] ha iniziato lo studio di quest'argomento, dimostrando una disuguaglianza a priori per la funzione massimale sferica, mediante l'utilizzo delle funzioni "g". In [3], M. Cowling e G. Mauceri hanno generalizzato il risultato di Stein. Qui si dà una dimostrazione diversa e forse più semplice della stima a priori, utilizzando la transformata di Mellin, come suggerito in un altro lavoro di Cowling e Mauceri [2].

A Sharp Estimate for the Hardy-Littlewood Maximal Function

Arxiv preprint math/ …, 1997

arXiv:math/9704211v2 [math.FA] 3 Dec 1999 A SHARP ESTIMATE FOR THE HARDY-LITTLEWOOD MAXIMAL FUNCTION Loukas Grafakos∗, Stephen Montgomery-Smith∗∗, and Olexei Motrunich∗∗∗ University of Missouri, Columbia and Princeton University Abstract. ...

On an inequality of Sagher and Zhou concerning Stein’s lemma

Collectanea mathematica, 2009

We provide two alternative proofs of the following formulation of Stein's lemma obtained by Sagher and Zhou [6]: there exists a constant A > 0 such that for any measurable set E ⊂ [0, 1], |E| = 0, there is an integer N that depends only on E such that for any square-summable real-valued sequence {c k } ∞ k=0 we have:

On Rough maximal inequalities: an extension of Fefferman-Stein results

Matematychni Studii

We prove some vector-valued inequalities for a rough maximal operator on Lebesgue spaces. These results are an extension of Fefferman-Stein (1971) and Sawano (2006) since the rough maximal operator is a generalization of the Hardy-Littlewood maximal operator and also a fractional maximal operator, respectively. We also establish some vector-valued inequalities for a rough maximal operator on Morrey spaces. 1. Introduction. Let f be a measurable function on R n. Let also Ω be a homogeneous function of degree zero on R n. For α ∈ [0, n), the rough maximal operator M Ω,α maps the function f to the rough maximal function M Ω,α f which is given by M Ω,α f (x) := sup r>0 r α−n ∫ B(x,r) |Ω(x − y)||f (y)|dy, x ∈ R n where B(x, r) denotes the open ball centered in x with radius r > 0. For α ∈ (0, n), the operator M 1,α is known as a fractional maximal operator. The operator M 1,0 is well known as the Hardy-Littlewood maximal operator. We write f to denote a sequence of measurable functions {f j } ∞ j=1 on R n. We also write M Ω,α f to denote the rough maximal function sequence {M Ω,α f j } ∞ j=1. We use a b to indicate that there exists c > 0 such that a ≤ cb. We express Fefferman-Stein maximal inequalities using this notations as follows.

Best constants for uncentered maximal functions

arXiv (Cornell University), 1994

We precisely evaluate the operator norm of the uncentered Hardy-Littlewood maximal function on L p (R 1). Consequently, we compute the operator norm of the "strong" maximal function on L p (R n), and we observe that the operator norm of the uncentered Hardy-Littlewood maximal function over balls on L p (R n) grows exponentially as n → ∞. For a locally integrable function f on R n , let (M n f)(x) = sup B∋x * Research partially supported by the NSF.

Weighted estimates for fractional maximal functions related to spherical means

Bulletin of The Australian Mathematical Society, 2002

We prove weighted LP-L q estimates for the maximal operators M a , given by M a f = sup \t a nt * / | , where fit denotes the normalised surface measure on the sphere of oo centre 0 and radius t in R d . The techniques used involve interpolation and the Mellin transform. To do this, we also prove weighted IP-IP estimates for the operators of convolution with the kernels |-|~a~"'.

New Proofs of Weighted Inequalities for the One-Sided Hardy-Littlewood Maximal Functions

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.