Maximal operators related to block spaces

Abstract

In this paper, we prove appropriate L p bounds for a class of maximal operators S W related to singular integrals with kernels which belong to block spaces and are supported by subvarieties. Also, we show that our condition on the kernel is optimal for the L 2 boundedness of S W . Our results improve substantially the main result obtained by L. K. Chen and H. Lin in [CL].

Weighted norm inequalities for a class of rough singular integrals

“…The study of the maximal operator M Ω has attracted the attention of many authors in recent years. For example, see [1], [2], [9], [18] and [24].…”

Rough singular integrals and maximal operators with mixed homogeneity along compound curves

“…, which was first studied by Chen and Lin [9], and subsequently by many authors (see [1]- [4], [7], [8], [20], [23], [30] et al). In particular, Al-Qassem [7] showed that M (γ ) is bounded on L p for γ ≤ p < ∞ provided that ∈ L(log + L) 1/γ S n−1 and 1 < γ ≤ 2, and bounded on L ∞ for γ = 1 (for the special case of γ = 2, also see Al-Salman's works [1], [3] in one-parameter setting, [2], [4] in multiple-parameter setting and along certain smooth surfaces formed by van der Corput type functions, which contain the class of functions F). Recently, Fan and Wu [20] generalized the results of Al-Qassem in [7] to the non-isotropic dilation setting (see [23] for the multiple-parameter case).…”

“…To prove Theorem , we need to establish the ‐boundedness of the related maximal operator defined by which is interesting itself. Historically, for and , we denote by , which was first studied by Chen and Lin , and subsequently by many authors (see –, , , , , et al.). In particular, Al‐Qassem showed that is bounded on for provided that and , and bounded on for …”

Weighted Estimates for Rough Maximal Functions with Applications to Angular Integrability

“…On the other hand, the classical maximal operator S Ω was originally introduced by Chen and Lin [21] who proved that if Ω ∈ C 1 (S −1 ), then S Ω is of type ( , ) for any > 2 /(2 −1) and the range of is best possible. Subsequently, the mapping properties of S Ω have been discussed extensively by many authors (see [22][23][24][25][26], for example). Particularly, Al-Salman [23] proved the following result.…”