In this paper, we prove the boundedness of multilinear Littlewood-Paley square operators and their commutators on weighted Morrey spaces, then give weak-type $$L\log L$$ estimates for g-functions Marcinkiewicz integrals spaces in form corollaries.

We deal with some applications of Marcinkiewicz integrals to problems related to harmonic measure on the one hand, and to removability problems for Sobolev spaces and quasiconformal mappings on the other hand. Techniques of this type have been used by Carleson, Jones, Makarov, Smirnov... to address these problems, and as a general program to understand harmonic functions on complicated domains ...

Let L = −∆ + V (x) be a Schrödinger operator, where ∆ is the Laplacian on R, while nonnegative potential V (x) belonging to the reverse Hölder class. We establish the boundedness of the commutators of Marcinkiewicz integrals with rough kernel associated with schrödinger operator on vanishing generalized Morrey spaces.

In this paper, we establish an Lp boundedness result of a class of Marcinkiewicz integral operators on product domains with rough kernels.

We establish a weighted Lp boundedness of a parametric Marcinkiewicz integral operator ρ Ω,h if Ω is allowed to be in the block space B (0,−1/2) q (Sn−1) for some q > 1 and h satisfies a mild integrability condition. We apply this conclusion to obtain the weighted Lp boundedness for a class of the parametric Marcinkiewicz integral operators ∗,ρ Ω,h,λ and ρ Ω,h,S related to the Littlewood-Paley ...

Stein proved that ifΩ∈ Lipα(Sn−1), (0<α≤ 1), then μΩ is bounded on Lp for all 1<p ≤ 2 [18]. Since then, the study of the Lp boundedness of μΩ under various conditions on the function Ω has attracted the attention of many authors ([1, 4, 5, 7, 10, 13], among others). In particular, Chen et al. in [8] studied the Lp boundedness of μΩ under the following condition on the function Ω which was intro...