Rough Marcinkiewicz integral operators

Abstract

We study the Marcinkiewicz integral operator M𝒫f(x)=(∫−∞∞|∫|y|≤2tf(x−𝒫(y))(Ω(y)/|y|n−1)dy|2dt/22t)1/2, where 𝒫 is a polynomial mapping from ℝn into ℝd and Ω is a homogeneous function of degree zero on ℝn with mean value zero over the unit sphere Sn−1. We prove an Lp boundedness result of M𝒫 for rough Ω

Parabolic Marcinkiewicz integrals on product spaces and extrapolation

“…Al-Qassem in [15], found that M ;h is bounded on R n R m (1 < p < 1) provided that h is a bounded radial function and is a function in certain block space B .0;0/ q .S n 1 S m 1 / for q > 1. He also established the optimality of the condition in the sense that the space B .0;0/ q .S n 1 S m 1 / cannot be replaced by B .0;"/ q .S n 1 S m 1 / for any 1 < " < 0.…”

Rough Marcinkiewicz integrals on product spaces

“…By an elementary procedure, choose two collections of C ∞ functions {ω (1) k } k∈Z and {ω (2) k } k∈Z on (0, ∞) with the following properties:…”

“…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).…”