Marcinkiewicz integrals on product spaces

Abstract

Abstract.We prove the L p boundedness of the Marcinkiewicz integral operators 1. Introduction. Marcinkiewicz integrals have been studied by many authors, dating back to the investigations of such operators by Zygmund on the circle and by Stein on R n .We shall be primarily concerned with Marcinkiewicz integrals on the product space R n × R m , since the more general setting of R n 1 × · · · × R n k can be handled similarly (see Section 4).For n, m ≥ 2, x ∈ R n \{0}, y ∈ R m \{0}, we let x = x/|x| and y = y/|y|. Let Ω ∈ L 1 (S n−1 × S m−1 ) be a function satisfying the following cancellation conditions:

Parabolic Marcinkiewicz integrals on product spaces and extrapolation

“…This result was improved (for p D 2) in [13] in which the author established that M is bounded on L 2 .R n R m / for all 2 L.log L/.S n 1 S m 1 /. Recently, Al-Qaseem et al found in [14] that the boundedness of M is obtained under the condition 2 L.log L/.S n 1 S m 1 / for 1 < p < 1. Furthermore, they proved that the exponent 1 is the best possible.…”

“…We prove Theorem 1.1 by applying the same approaches as in [5,14], which have their roots in [20]. Let us assume that h 2 .R C R C / for some > 1.…”

Rough Marcinkiewicz integrals on product spaces

“…M. Stein proved that if Ω ∈ Lip α (S n−1 ), (0 < α ≤ 1), then μ Ω is bounded on L p for all 1 < p ≤ 2 ( [20]). Subsequently, the study of the L p boundedness of μ Ω under various conditions on the function Ω has been studied by many authors ( [2], [3], [4], [5], [7], [8], [9], [17], among others). A particular result that is closely related to our work is the boundedness result of μ Ω obtained by Chen-Fan-Pan in ( [9]).…”

Boundedness of Marcinkiewicz Integrals on Product Spaces and Extrapolation

“…For example, the author of [14] proved that M Ω,c is bounded for all 1 < p < ∞ provided that Ω ∈ L(log L) 2 (S n−1 × S m−1 ). For more information about the importance and the recent advances on the study of such operators, the readers are refereed (for instance to [1], [3], [13], [15], [16], [29], [30], as well as [31], and the references therein).…”