Rough Marcinkiewicz integrals on product spaces

Abstract

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

Marcinkiewicz integral operators along twisted surfaces

“…Here, we say that the surface determined by Γ is of standard type if Γ(x, y) = (Γ 1 (x), Γ 2 (y)) for some suitable mappings Γ 1 : R n → R n and Γ 2 : R m → R m . If Γ is of standard type, then the L p mapping properties of the corresponding operator and its related maximal operator are relatively well understood [1,6,7,5,24]. When Γ(u, v) = (u, v) and h = 1, we denote M Γ,Ω,h by M Ω .…”

“…The condition Ω ∈ L(log + L)(S n−1 ×S m−1 ) is known to be optimal in the sense that the operator M Ω may fail to be bounded on L p if Ω is assumed to be in L(log + L) 1− (S n−1 ×S m−1 ) for some > 0 [1]. For further results concerning the classical operator M Ω , we advise readers to consult [2,6,13,17], among others. However, for general surfaces that are not of standard type, the L p boundedness of the operator M Γ P,Q ,Ω,h is still unknown.…”