Singular Integrals on Product Spaces with Mixed Norms

( School of Mathematical Sciences, Nankai University, Tianjin 300071, China)

Abstract: In this paper, we prove the boundedness of the singular integral operators on product spaces with mixed norms and obtain the endpoint weak-type estimates.

Keywords: Singular integral operators; Mixed norms; Product spaces

§1. Introduction

As is well known, Calder´on and Zygmund in [7] and [6] or [8, Corollary5.8] gave the boundedness of certain convolution operators on Rnwhich generalized Hilbert transform on R1.

Suppose that the operatorTgiven by exists, thenTcan be extended to an operator which is bounded onLp(Rn), 1＜p＜∞, and is weak(1,1).

In this paper, we consider the two parameter family of dilation (x,y)→(δ1x,δ2y),x∈Rn,y ∈Rm,δi ＞0, instead of the usual one-parameter dilations. For example, the double Hilbert transformDon R2,

By iteration argument, it is easy to obtain the boundedness properties ofDonLp(Rn×Rm), for 1＜p＜∞. But, if we consider operatorTf=f*kwherekis defined on Rn×Rmand satisfies all analogous estimates to those satisfied by 1/xybut cannot be written in the formk1(x)k2(x),the arguments which deal withDfail.

In 1982, Fefferman [9] gave a singular integral on product space which kernel satisfied”cancellation”and”size”properties if the kernelk(x,y)cannot be written in the formk1(x)k2(y).

Supposek(x,y) is a function on Rn×Rm, locally integrable away from the cross{x=0}∪{y=0}. Define

with the usual modifications made when pi=∞for some i∈{1,...,n}.

Benedek and Panzone in [5] first studied the Lebesgue space with mixed norm and proved that such spaces have similar properties as ordinary Lebesgue spaces, Related works refer to [4,10,13,17].

Recently, many works have been done for lebesgue spaces with mixed norms. For example,Fernandez [10] studied the vector-valued singular integral operators with product kernels and obtained the boundedness ofDonLp(Lq). Stefanov and Torres [18] proved the boundedness of Calder´on-Zygmund operators on mixed norm spaces. Ho [12] studied the strong maximal operator on mixed norm spaces. Kurtz [14] proved some classical operators (the strong maximal function, the double Hilbert transform, and the singular integral operators) on mixed norm spaces with product weights are bounded.

In this paper, we consider the singular integral operators on product spaces with mixed norms. The purpose of us is to prove the boundedness of the operatorTwith kernelk(x,y)(cannot be written in the formk1(x)k2(y)) onLp(Lq)(Rn×Rm) for all 1＜p,q ＜∞and obtain the endpoint weak-type estimates. Specifically, we prove the following.

Suppose the kernel ˆkN∈is given by

§2. Preliminaries

In this section, we collect some preliminary results which are used in the proof.

A vector-valued extension of the theory is well known, we will need the following version of the original result of Benedek, Calder´on and Panzone. The proof of this theory refers to [11] .

§3. Proof of Theorem 1.1

In this section, we give a proof for the main result.

whereΩ1andΩ2with zero average on Sn-1and Sm-1, s′=s/|s|and t′=t/|t|. If we let f(x,y)=f(x)f(y)and f(x,y)∈L1(Rn)(Lp(Rm)), then Tf(x,y)=T1f(x)T2f(y). The fourier transform of T1f(x)cannot be continuous at zero, so T1f(x)/∈L1(Rn), that is Tf(x,y)/∈L1(Rn)(Lp(Rm)).Remark 3.2.[9, section 6] Gave us some examples which satisfies the assumption of Theorem 1.1.