This paper is concerned with rectangular summation of multiple Fourier series in matrix weighted

Let

In this paper we are interested in convergence properties of multiple trigonometric series in

defines the rectangular partial sum operator for the trigonometric system. We define the action of

It is well known (see, e.g., [

At a first glance, the study of vector-valued operators like

A natural question to pose is whether

is an isometric isomorphism between

It was proved by the present author [

The structure of this paper is as follows. In Section

In this section we introduce a matrix Muckenhoupt

The scalar

More recently, Hunt-Muckenhoupt-Wheeden type results for matrix weights have been considered. The matrix

Since our goal is to study operators of product type related to rectangular trigonometric partial sums, the condition given by (

Let

for any

For

The similarity of conditions (

Let

We refer the reader to Roudenko [

The following Lemma reveals why one can expect

Let

For any rectangles

Suppose

For (a), we notice that whenever

Now we turn to the proof of (b). It suffices to consider

where the constant is independent of

By repeating the argument from the

We can now prove the following result that explains how to get from a bounded convolution operator on

Suppose that

are uniformly bounded on

induce a uniformly bounded family of operators on

Suppose that

According to Lemma

We define

Notice that

An integration yields

Similarly,

It follows that the family

We now turn to a converse type result to Proposition

We need the following Lemma which gives an estimate of the norm of integral operators on

Suppose

The proof of Lemma

We can now give a proof of Proposition

Let

induce a uniformly bounded family

We have to estimate

By assumption, the kernel

Let a rectangle

Notice that for

For notational convenience we put

The plan of attack is to use the simple fact that

We wish to estimate the operator norm of

Notice that estimate (

According to Lemma

so

with constant

This section contains applications of the results of Section

We begin by studying the univariate Dirichlet kernel. The Hilbert transform

We lift

Treil and Volberg completely characterized when the Hilbert transform

Let

We recall that the univariate Dirichlet kernel

and for

We have the following lemma which follows easily from Theorem

Let

We let

and the result follows.

For

We notice that

Let

It is easy to verify that vectors of trigonometric polynomials are dense in

Corollary

The rectangular Cesàro summation is given by

with the product Féjer kernel given by

Notice that

The corresponding product kernel is

We now conclude by stating the main result, which summarizes the results obtained in the present paper. The theorem shows that uniform boundedness of the rectangular operators

Let

the operators

the operators

the operators

We first notice that each of the univariate kernels