Singular integrals with rough kernels on product spaces

Abstract

Suppose that Ω(x', y') ∈ L1(Sn-1 × Sm-1) is a homogeneous function of degree zero satisfying the mean zero property (1.1), and that h(s, t) is a bounded function on R × R. The singular integral operator Tf on the product space Rn × Rm(n ≥ 2, m ≥ 2) is defined by (formula) We prove that the operator Tf is bounded in Lp(Rn × Rm), p ∈ (1, ∞), provided that Ω is a function in certain block space Bq0,1(Sn-1 × Sm-1) for some q > 1. The result answers a question posed in [JL]. We also study singular integral operators along certain surfaces.

Department(s)

Mathematics

Document Type

Article

DOI

https://doi.org/10.14492/hokmj/1351001230

Keywords

Block spaces, Product spaces, Rough kernel, Singular integrals

Publication Date

1-1-1999

Journal Title

Hokkaido Mathematical Journal

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