Singular integrals with rough kernels on product spaces
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.
Block spaces, Product spaces, Rough kernel, Singular integrals
Fan, Dashan, Kanghui Guo, and Yibiao Pan. "Singular integrals with rough kernels on product spaces." Hokkaido Mathematical Journal 28, no. 3 (1999): 435-460.
Hokkaido Mathematical Journal