Weighted Norm Inequalities for the Local Sharp Maximal Function

Abstract

Several weighted rearrangement inequalities for uncentered and centered local sharp functions are proved. These results are applied to obtain new weighted weak-type and strong-type estimates for singular integrals. A self-improving property of sharp function inequalities is established.

Commutators of singular integrals on generalized $L^p$ spaces with variable exponent

“…This inequality was proved in [15,Theorem 3]. It follows easily by putting T f in place of ϕ in Theorem 2.6 and by using Proposition 2.3 along with Theorem 2.4.…”

“…Our proof of Part (a) is motivated by an analog of the FeffermanStein theorem on the sharp maximal function for L p(·) (R n ) proved recently by Diening and Růžička [9,Theorem 3.6]. To prove Part (a), we combine a little bit more elaborate version of the latter result, based on the so-called local sharp maximal function and on a duality inequality due to the second author [15,Theorem 1], with a sharp function inequality for commutators due to Strömberg (see [11]) and Pérez [22,Lemma 3.1].…”

Sharp Maximal Inequalities and Its Application to Some Bilinear Estimates

“…Moreover, this maximal function was used for the extension problem of functions belonging to Triebel-Lizorkin spaces [9] and the modified sharp function was used to characterize Besov spaces and Triebel-Lizorkin spaces with smoothness index s > 0 [34,37]. See also [3,20,25,36], for several variants of the sharp maximal functions.…”

“…For example, by the duality argument [20,35], by using the non-increasing rearrangement [1] and by a good λ-inequality [15,28]. Similarly, the weighted inequality is shown by these techniques, see, for example [11].…”

Weighted norm inequalities for maximal singular integrals with nondoubling measures

“…0,s and M , 0,s in the setting of Euclidean spaces were first introduced by John [5] and then rediscovered by Strömberg [15] and Lerner [7,8]. It is easy to verify that for any cube Q x and ε > 0,…”