Singular integrals with mixed homogeneity

Singular Integrals with Mixed Homogeneity in Product Spaces

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 alo...

Boundedness of Marcinkiewicz integrals with mixed homogeneity along compound surfaces

*Correspondence: [email protected] 1School of Mathematical Sciences, Xiamen University, Xiamen, 361005, China Full list of author information is available at the end of the article Abstract In this note we establish the Lp boundedness of Marcinkiewicz integrals with mixed homogeneity along compound surfaces, which improve and extend some previous results. The main ingredient is to presen...

Singular Integrals

In this paper I will a t tempt to describe the subject as it has developed in the last fifteen years, outlining the methods by which its problems have been approached and discussing its connections with other branches of Analysis. I t will perhaps be best to start by considering certain classical situations which lead naturally to singular integrals and which contain the seeds of some of the me...

Multi-parameter singular integrals

L Bounds for Singular Integrals and Maximal Singular Integrals with Rough Kernels

Convolution type Calderón-Zygmund singular integral operators with rough kernels p.v. Ω(x)/|x| are studied. A condition on Ω implying that the corresponding singular integrals and maximal singular integrals map L → L for 1 < p < ∞ is obtained. This condition is shown to be different from the condition Ω ∈ H1(Sn−1).