Singular integrals with mixed homogeneity in product spaces

Abstract

Let Ω ∈ L(logL+)2(Sn−1 × Sm−1) (n, m 2) satisfy some cancellation conditions. We prove the Lp boundedness (1 < p < ∞ ) of the singular integral T f (x1,x2) = p. v. ∫ ∫ Rn×Rm Ω(y1,y ′ 2)h(ρ1(y1),ρ2(y2)) ρα 1 (y1)ρ β 2 (y2) f (x1 − y1,x2 − y2)dy1 dy2, where ρ1 , ρ2 are some metrics which are homogeneous with respect to certain non-isotropic dilations. We also study the above singular integral along some surfaces. Mathematics subject classification (2010): 42B20, 42B25.