Summation of Multiple Fourier Series in Matrix Weighted 𝐿_𝑝-Spaces

Abstract

This paper is concerned with rectangular summation of multiple Fourier series in matrix weighted 𝐿_𝑝-spaces. We introduce a product Muckenhoupt 𝐴_𝑝 condition for matrix weights 𝑊 and prove that rectangular Fourier partial sums converge in the corresponding matrix weighted space 𝐿_𝑝(𝕋^𝑑;𝑊), 1<𝑝<∞, if and only if the weight satisfies the product Muckenhoupt 𝐴_𝑝 condition. The same result is shown to hold true for other summation methods such as Cesàro and summation with the Jackson kernel.

1. Introduction

Let ℳ be the family of nonnegative-definite 𝑚×𝑚 complex-valued matrices. A (periodic) matrix weight is by definition an integrable map 𝑊∶𝕋^𝑑→ℳ. For a measurable vector-valued function 𝐟=(𝑓₁,…,𝑓_𝑚)^𝑇∶𝕋^𝑑→ℂ^𝑚, let 𝑓_{𝐿𝑝(𝕋𝑑;𝑊)}∶=_𝕋𝑑𝑊^1/𝑝(𝑡)𝐟(𝑡)^𝑝𝑑𝑡^1/𝑝,1≤𝑝<∞, (1) where |⋅| denotes the usual norm on ℂ^𝑚. We let 𝐿_𝑝(𝕋^𝑑;𝑊) denote the family {𝐟∶𝕋^𝑑→ℂ^𝑚∶‖𝐟‖_{𝐿𝑝(𝕋}^𝑑_;𝑊)<∞}, and 𝐿_𝑝(𝕋^𝑑;𝑊) becomes a Banach space when we factorize over 𝒩={𝐟∶‖𝐟‖_{𝐿𝑝(𝕋}^𝑑_;𝑊)=0}.

In this paper we are interested in convergence properties of multiple trigonometric series in 𝐿_𝑝(𝕋^𝑑;𝑊) and how specific convergence properties of trigonometric series can be related to properties of the weight 𝑊. To be more specific, let 𝐷_𝑛(𝑡)=∑_{|𝑘|≤𝑛}𝑒^{−2𝜋𝑖𝑘𝑡} denote the univariate Dirichlet kernel, and for 𝐍∈ℕ^𝑑 we define the rectangular kernel 𝐷_𝐍(𝑡)∶=∏^𝑑_𝑗=1𝐷_𝑁𝑗(𝑡_𝑗). Then 𝑆_𝐍𝑓∶=𝑓∗𝐷_𝐍∶=_𝕋𝑑𝑓(𝑡)𝐷_𝐍(⋅−𝑡)𝑑𝑡,𝑓∈𝐿₁𝕋^𝑑, (2)

defines the rectangular partial sum operator for the trigonometric system. We define the action of 𝑆_𝐍 on vector-valued functions 𝐟 by letting it act separately on each coordinate function; that is, 𝑆_𝐍(𝐟)_𝑗=𝑆_𝐍𝑓_𝑗,𝑗=1,2,…𝑚. (3)

It is well known (see, e.g., [1, Theorem 3.5.7]) that ‖𝑓−𝑆_𝐍(𝑓)‖_{𝐿𝑝(𝕋}^𝑑₎→0, as min_𝑗𝑁_𝑗→+∞, for 1<𝑝<∞. An immediate corollary is that we have convergence of the partial sums 𝑆_𝐍(𝐟), in 𝐿_𝑝(𝕋^𝑑;Id), for 1<𝑝<∞ in the vector-valued case. However, it is not obvious what can be said about convergence of 𝑆_𝐍(𝐟) in 𝐿_𝑝(𝕋^𝑑;𝑊) for a general matrix weight 𝑊. The main result of the preset paper completely characterizes the special class of weights that allow convergence; 𝑆_𝐍(𝐟) converges if and only if the weight 𝑊 satisfies a certain matrix Muckenhoupt 𝐴_𝑝 product condition. Moreover, the characterization relies solely on certain localization properties of the Dirichlet kernels shared by many other summation kernels. So, in addition, we prove that the rectangular Cesàro means and approximation using the Jackson kernels converge in 𝐿_𝑝(𝕋^𝑑;𝑊) if and only if the weight 𝑊 satisfies the mentioned matrix Muckenhoupt 𝐴_𝑝 product condition.

At a first glance, the study of vector-valued operators like 𝑆_𝐍 on 𝐿_𝑝(𝕋^𝑑;𝑊) may seem artificial, but let us mention one important application to finitely generated shift invariant spaces where such mappings appear naturally. For a finite set of functions {𝑓_𝑘}^𝑚_𝑘=1 in 𝐿₂(ℝ^𝑑), the associated shift invariant space 𝑆 is given by 𝑆∶=Span𝑓_𝑘(⋅−𝑙)∶𝑙∈ℤ^𝑑,𝑘=1,2,…,𝑚⊂𝐿₂ℝ^𝑑. (4)

A natural question to pose is whether 𝐵∶=Span{𝑓_𝑘(⋅−𝑙)∶𝑙∈ℤ^𝑑,𝑘=1,2,…,𝑚} forms some sort of “stable” generating system for 𝑆. Here stable can mean a Schauder basis, or even some weaker notion such as a block Schauder basis. Consider the vector valued system 𝒯∶={𝐞_𝑗𝑒^{2𝜋𝑖𝑘⋅𝑡}∶𝑘∈ℤ^𝑑,𝑗=1,…,𝑚}, where {𝐞_𝑗}^𝑚_𝑗=1 is the standard basis for ℂ^𝑚, and let 𝐺 the 𝑚×𝑚 Gram matrix be given by 𝐺_𝑖,𝑗∶=_{𝑘∈ℤ𝑑}𝑓_𝑖(⋅−𝑘)𝑓_𝑗(⋅−𝑘), (5) where the Fourier transform is given by 𝑓(𝜉)∶=∫_ℝ𝑑𝑓(𝑡)𝑒^{−2𝜋𝑖𝑡⋅𝜉}𝑑𝑡. One can show (see [2]) that the map 𝑈∶𝐿₂(𝕋^𝑑;𝐺)→𝑆, given by 𝑈(𝜏)∶=^𝑁_ℓ=1𝜏_ℓ(𝜉)𝜓_ℓ^∨, (6)

is an isometric isomorphism between 𝐿₂(𝕋^𝑑;𝐺) and 𝑆, satisfying 𝑈(𝐞_𝑗𝑒^{−2𝜋𝑖𝑘⋅𝑡})=𝑓_𝑗(⋅−𝑘). Hence, the metric properties of 𝐵 in 𝑆 are equivalent to the properties of 𝒯 in 𝐿₂(𝕋^𝑑;𝐺). For example, 𝑈 will map {𝑆_𝐍} to a corresponding rectangular partial sum operators for 𝐵 on 𝑆. The same correspondence holds true for the other summation methods related to 𝒯 such as rectangular Cesàro means and approximation using the Jackson kernels converge in 𝐿_𝑝(𝕋^𝑑;𝑊) that will be discussed below.

It was proved by the present author [2] that the rectangular partial sums {𝑈𝑆_𝐍} converge in 𝑆 precisely when {𝑆_𝐍} are uniformly bounded on 𝐿₂(𝕋;𝐺) which happens exactly when 𝐺 is a Muckenhoupt 𝐴₂ product matrix weight.

The structure of this paper is as follows. In Section 2 we introduce a product 𝐴_𝑝 condition for matrix weights. Then necessary and sufficient conditions for a convolution operator of product type (such as 𝑆_𝐍) to be bounded on 𝐿_𝑝(𝕋^𝑑;𝑊) are given. Section 3 contains applications of the results in Section 2 to convolution operators induced by rectangular Dirichlet, Fejér, and Jackson kernels.

2. The Muckenhoupt Condition and Operators on 𝐿_𝑝(𝕋^𝑑;𝑊)

In this section we introduce a matrix Muckenhoupt 𝐴_𝑝 product condition suitable for dealing with convolution operators of product type such as the partial sum operator 𝑆_𝐍 defined by (3). A sufficient condition for convolution operators of product type bounded on 𝐿_𝑝(𝕋^𝑑;𝑊) is given in Proposition 5, while a converse type result is considered in Proposition 7. We prove that convolution operators with “nicely localized” kernels can only be uniformly bounded on 𝐿_𝑝(𝕋^𝑑;𝑊) when 𝑊 satisfies the product 𝐴_𝑝 condition.

The scalar 𝐴_𝑝 condition was introduced by Muckenhoupt [3], and it was proved by Hunt et al. in their seminal paper [4] that the 𝐴_𝑝 condition on a weight 𝑤 is necessary and sufficient for the Hilbert transform to be bounded on the weighted space 𝐿_𝑝(𝕋;𝑤).

More recently, Hunt-Muckenhoupt-Wheeden type results for matrix weights have been considered. The matrix 𝐴_𝑝 condition was introduced by Nazarov et al. [5–7] and they showed that it is the right condition for “standard” singular integral operators to be bounded on 𝐿_𝑝(𝕋^𝑑;𝑊). The 𝐴_𝑝 condition (1<𝑝<∞) for weights 𝑊∶ℝ^𝑑→ℳ was originally stated in terms of dual matrices and average, but it was shown by Roudenko [8] to be equivalent to sup_𝐵𝐵𝐵𝑊^1/𝑝(𝑥)𝑊^−1/𝑝𝑦^𝑝′𝑑𝑦|𝐵|^𝑝/𝑝′𝑑𝑥|𝐵|^𝑝′/𝑝, (7) where the sup is taken over all open balls in ℝ^𝑑 and 𝑝^′=𝑝/(𝑝−1) is the conjugate exponent.

Since our goal is to study operators of product type related to rectangular trigonometric partial sums, the condition given by (7) is not the appropriate one. The periodic weights satisfying (7) are well behaved when it comes to the study of square or spherical partial sum operators for trigonometric series. Let us therefore introduce a new and slightly modified Muckenhoupt condition. Inspired by (7), we let ℛ(𝑑) denote the family of all rectangles in ℝ^𝑑 of the form 𝑅=𝐼₁×𝐼₂×⋯×𝐼_𝑑, with 𝐼_𝑗 being a bounded open interval in ℝ. Then we consider the following more restrictive subclass of matrix weights.

Definition 1. Let 𝑊∶𝕋^𝑑→ℳ be a periodic matrix weight. For 1<𝑝<∞, let 𝑝^′=𝑝/(𝑝−1) denote the conjugate exponent to 𝑝. We say that 𝑊 belongs to the matrix Muckenhoupt (product) class 𝑃𝐴_𝑝 provided there exists a uniform constant 𝑐_𝑊 such that 𝐴_𝑊𝑅,𝑝∶=_𝑅𝑅𝑊^1/𝑝(𝑥)𝑊^−1/𝑝𝑦^𝑝′𝑑𝑦|𝑅|^𝑝/𝑝′𝑑𝑥|𝑅|≤𝑐_𝑊, (8)
for any 𝑅∈ℛ(𝑑).

Remark 2. For 𝑑=1, Definition 1 reduces to the standard matrix 𝐴_𝑝 condition on 𝕋, which we denote by 𝐴_𝑝(𝕋). In the scalar case (i.e., 𝑚=1), Definition 1 reduces to the known product 𝐴_𝑝 condition for scalar weights, which has a long history; see [9] and references therein.
The similarity of conditions (8) and (7) implies that many results for matrix 𝐴_𝑝 weights have straightforward analogs in the product case; the proofs can be “translated verbatim.” Let us state the following lemma which will be needed below.

Lemma 3. Let 𝑊∶𝕋^𝑑→ℳ be a matrix weight. Then the following statements are equivalent for 1<𝑝<∞: (i)𝑊∈𝑃𝐴_𝑝; (ii)𝑊^{−𝑝′/𝑝}∈𝑃𝐴_𝑝′; (iii)∫_𝑅(∫_𝑅‖𝑊^1/𝑝(𝑥)𝑊^−1/𝑝(𝑦)‖^𝑝(𝑑𝑥/|𝑅|))^𝑝′/𝑝(𝑑𝑦/|𝑅|)≤𝑐_𝑊, for all𝑅∈ℛ(𝑑).

We refer the reader to Roudenko [8] for the proof of Lemma 3 in the nonproduct case.

The following Lemma reveals why one can expect 𝐴𝑃_𝑝 to be useful for product operators. A weight in 𝐴𝑃_𝑝 is uniformly in 𝐴_𝑝(𝕋) in each of its 𝑑 variables.

Lemma 4. Let 𝑊∶𝕋^𝑑→ℳ be a matrix weight, and let 1<𝑝<∞. Then the following holds.(a)For any rectangles 𝑅⊆𝑅⊂ℛ(𝑑), 𝐴_𝑊≤𝑅𝑅,𝑝|𝑅|^𝑝𝐴_𝑊𝑅,𝑝. (9)(b)Suppose 𝑊∈𝑃𝐴_𝑝; then the univariate weight 𝜉_𝑗→𝑊(𝜉), obtained by fixing the variables 𝜉_𝑘, 𝑘≠𝑗, is uniformly in 𝔸_𝑝(𝕋) for a.e. (𝜉₁,…,𝜉_𝑗−1,𝜉_𝑗+1,…,𝜉_𝑑)∈𝕋^𝑑−1.

Proof. For (a), we notice that whenever 𝑅⊆𝑅⊂ℛ(𝑑), 𝐴_𝑊=𝑅,𝑝_𝑅𝑅𝑊^1/𝑝(𝑥)𝑊^−1/𝑝(𝑡)^𝑝′𝑑𝑡|𝑅|^𝑝/𝑝′𝑑𝑥|𝑅|≤𝑅|𝑅|^𝑝𝑅𝑅𝑊^1/𝑝(𝑥)𝑊^−1/𝑝(𝑡)^𝑝′𝑑𝑡𝑅^𝑝/𝑝′𝑑𝑥𝑅=𝑅|𝑅|^𝑝𝐴_𝑊𝑅,𝑝. (10)
Now we turn to the proof of (b). It suffices to consider 𝑊(𝑡)∶=𝑊(𝑡,𝜉₂,…,𝜉_𝑑) for (𝜉₂,…,𝜉_𝑑)∈𝕋^𝑑−1 fixed. Given an interval 𝐼⊂ℝ, we form 𝑅_𝜀=𝐼_𝜀(𝜉₂)×⋯×𝐼_𝜀(𝜉_𝑑), where 𝐼_𝜀(𝜉_𝑗) is an interval of length 2𝜀 centered at 𝜉_𝑗. First suppose 𝑝≤𝑝^′. Since 𝑊∈𝑃𝐴_𝑝, there exists a constant 𝑐_𝑊 independent of 𝐼×𝑅_𝜀 such that 1𝑅_𝜀^2𝑅_𝜀^𝑅_𝜀^𝐼𝐼𝑊^1/𝑝(𝑡,𝐮)𝑊^−1/𝑝×(𝑤,𝐯)^𝑝′𝑑𝑤|𝐼|^𝑝/𝑝′𝑑𝑡|𝐼|≤𝑑𝐮𝑑𝐯_{𝑅𝜀𝐼𝑅𝜀𝐼}𝑊^1/𝑝(𝑡,𝐮)𝑊^−1/𝑝(𝑤,𝐯)^𝑝′×𝑑𝑤𝑑𝐯|𝐼|⋅𝑅_𝜀^𝑝/𝑝′𝑑𝑡𝑑𝐮|𝐼|⋅𝑅_𝜀=𝐴_𝑊𝑅,𝑝≤𝑐_𝑊, (11)where we have used the continuous embedding 𝐿₁(𝑅_𝜀;𝑑𝐯/|𝑅_𝜀|)↪𝐿_𝑝/𝑝′(𝑅_𝜀;𝑑𝐯/|𝑅_𝜀|). Hence, by Lebesgue's differentiation theorem, for almost every (𝜉₂,…,𝜉_𝑑)∈𝕋^𝑑−1, 𝑐_𝑊≥lim_𝜀→0+𝐴_𝑊𝐼×𝑅_𝜀=,𝑝_𝐼𝐼𝑊^1/𝑝(𝑡)𝑊^−1/𝑝(𝑤)^𝑝′𝑑𝑤|𝐼|^𝑝/𝑝′𝑑𝑡|𝐼|=𝐴_𝑊𝐼,𝑝, (12)
where the constant is independent of 𝐼 and (𝜉₂,…,𝜉_𝑑). It follows that 𝑊 is uniformly in 𝔸_𝑝(𝕋) for a.e. (𝜉₂,…,𝜉_𝑑)∈𝕋^𝑑−1. In the case 𝑝^′<𝑝, we use Lemma 3 to conclude that 𝑊^{−𝑝′/𝑝}∈𝑃𝐴_𝑝′, which implies the following estimate:_{𝑅𝜀𝐼𝑅𝜀𝐼}𝑊^1/𝑝(𝑡,𝐮)𝑊^−1/𝑝(𝑤,𝐯)^𝑝×𝑑𝑡𝑑𝐮|𝐼|⋅𝑅_𝜀^𝑝′/𝑝𝑑𝑤𝑑𝐯|𝐼|⋅𝑅_𝜀≤𝑐_𝑊. (13)
By repeating the argument from the 𝑝≤𝑝^′ case, we conclude that 𝑊^{−𝑝′/𝑝} is uniformly in 𝐴_𝑝′(𝕋) which again by Lemma 3 implies that 𝑊 is uniformly in 𝐴_𝑝(𝕋).

We can now prove the following result that explains how to get from a bounded convolution operator on 𝐿_𝑝(𝕋;𝑊) to a bounded convolution operator on 𝐿_𝑝(𝕋^𝑑;𝑊), for 𝑊∈𝐴𝑃_𝑝, simply by forming the natural product kernel.

Proposition 5. Suppose that {𝐾_𝑁}_𝑁≥0 is a sequence of convolution kernels defined on 𝕋 for which the corresponding operators 𝑇_𝑁𝑓∶=_𝕋𝑓(𝑡)𝐾_𝑁(⋅−𝑡)𝑑𝑡 (14)
are uniformly bounded on 𝐿_𝑝(𝕋;𝑊) whenever 𝑊∈𝐴_𝑝(𝕋). Then the associated product convolution kernels 𝐾_𝐍(𝜉)=^𝑑_𝑗=1𝐾_𝑁𝑗𝜉_𝑗,𝐍=𝑁₁,…,𝑁_𝑑∈ℕ^𝑑₀,𝜉∈𝕋^𝑑, (15)
induce a uniformly bounded family of operators on 𝐿_𝑝(𝕋^𝑑;𝑊) for 𝑊∈𝑃𝐴_𝑝.

Proof. Suppose that 𝑊∈𝑃𝐴_𝑝. In the case 𝑑=1, there is nothing to prove. We focus on the case 𝑑=2; the reader can easily verify that the argument below generalizes to any 𝑑≥3.
According to Lemma 4(b), 𝑊_𝜉1∶=𝑊(𝜉₁,⋅) and 𝑊_𝜉2∶=𝑊(⋅,𝜉₂) satisfy uniform Muckenhoupt 𝐴_𝑝-conditions a.e. on 𝕋. Pick any 𝑓∈𝐿_𝑝(𝕋²,𝑊). By Fubini's theorem, 𝑓_𝜉1∶=𝑓(𝜉₁,⋅)∈𝐿_𝑝(𝕋,𝑊_𝜉1) and 𝑓_𝜉2∶=𝑓(⋅,𝜉₂)∈𝐿_𝑝(𝕋,𝑊_𝜉2) for a.e. [𝜉₁] and [𝜉₂], respectively.
We define 𝑇¹_𝑁𝑓∶=𝐾_𝑁∗𝑓_𝜉2∶=_𝕋𝑓_𝜉2(𝑡)𝐾_𝑁()𝑇⋅−𝑡𝑑𝑡,_2𝑀𝑓∶=𝐾_𝑀∗𝑓_𝜉1∶=_𝕋𝑓_𝜉1(𝑡)𝐾_𝑀(⋅−𝑡)𝑑𝑡. (16)
Notice that 𝑇_𝑁,𝑀𝑓=𝑇¹_𝑁𝑇²_𝑀𝑓. By assumption, _𝕋𝑊_𝜉^1/𝑝₁𝜉₂𝑇²_𝑀𝑓_𝜉1𝜉₂^𝑝𝑑𝜉₂≤𝐶_𝕋𝑊_𝜉^1/𝑝₁𝜉₂𝑓_𝜉1𝜉₂^𝑝𝑑𝜉₂,a.e.𝜉₁. (17)
An integration yields _𝕋𝕋𝑊^1/𝑝𝜉₁,𝜉₂𝑇²_𝑀𝑓𝜉₁,𝜉₂^𝑝𝑑𝜉₂𝑑𝜉₁≤𝐶_𝕋𝕋𝑊^1/𝑝𝜉₁,𝜉₂𝑓𝜉₁,𝜉₂^𝑝𝑑𝜉₂𝑑𝜉₁. (18)
Similarly, 𝑇_𝑁,𝑀𝑓^𝑝_{𝐿2𝑝(𝕋,𝑊)}=_𝕋𝕋𝑊^1/𝑝𝜉₁,𝜉₂𝑇¹_𝑁𝑇²_𝑀𝑓𝜉₁,𝜉₂^𝑝𝑑𝜉₁𝑑𝜉₂≤𝐶_𝕋𝕋𝑊^1/𝑝𝜉₁,𝜉₂𝑇²_𝑀𝑓𝜉₁,𝜉₂^𝑝𝑑𝜉₁𝑑𝜉₂≤𝐶²_𝕋𝕋𝑊^1/𝑝𝜉₁,𝜉₂𝑓^𝑝𝑑𝜉₁𝑑𝜉₂. (19)
It follows that the family {𝑇_𝐍}_𝐍∈ℕ²₀ is uniformly bounded on 𝐿_𝑝(𝕋²;𝑊).

We now turn to a converse type result to Proposition 5. Proposition 7 will show that well-localized trigonometric convolution kernels of product type can only be uniformly bounded on 𝐿_𝑝(𝕋^𝑑;𝑊) when 𝑊∈𝐴𝑃_𝑝.

We need the following Lemma which gives an estimate of the norm of integral operators on 𝐿₂(𝕋^𝑑;𝑊) with nice compactly supported kernels.

Lemma 6. Suppose 𝑆𝑓(𝜉)=∫_𝕋𝑑𝑆(𝜉,𝜂)𝑓(𝜂)𝑑𝜂 is an integral operator with a scalar kernel 𝑆(𝜉,𝜂) that satisfies |𝑆(𝜉,𝜂)|≤𝛼|𝑅|⁻¹𝜒_𝑅×𝑅 for some bounded rectangle 𝑅⊂ℝ^𝑑. For 1<𝑝<∞, there exists a constant 𝐶_𝑑 independent of the particular choice of 𝑆 such that the norm of 𝑆 on 𝐿_𝑝(𝕋^𝑑;𝑊) is at most 𝐶_𝑑⋅𝛼⋅𝐴_𝑊(𝑅,𝑝), with 𝐴_𝑊(𝑅,𝑝) given by (8). Moreover, the kernel 𝛼|𝑅|⁻¹𝜒_𝑅×𝜒_𝑅 induces an operator with norm at least 𝐶_𝑑⁻¹⋅𝛼⋅𝐴_𝑊(𝑅,𝑝) on 𝐿_𝑝(𝕋^𝑑;𝑊).

The proof of Lemma 6 for nonproduct 𝐴_𝑝-weights can be found in Goldberg [10]. We leave the straightforward adaptation of the proof in [10] to the product case for the reader.

We can now give a proof of Proposition 7. For 𝐾∈ℕ, we let 𝒫_𝐾=span{𝑒^{2𝜋𝑖𝑘⋅}∶𝑘∈ℤ;|𝑘|≤𝐾}.

Proposition 7. Let 𝑊∶𝕋^𝑑→ℳ be a periodic matrix weight, and let {𝐾_𝑛}_𝑛≥1 be a sequence of real-valued trigonometric convolution kernels defined on 𝕋. Assume there exist constants 𝑐, 𝐶 such that 𝐾_𝑛∈𝒫_𝑐⋅𝑛, with 𝐶⁻¹𝑛≤‖𝐾_𝑛‖_∞=𝐾_𝑛(0)≤𝐶𝑛,for 𝑛∈ℕ. Suppose that the corresponding product kernels 𝐾_𝐍(𝜉)=^𝑑_𝑗=1𝐾_𝑁𝑗𝜉_𝑗,𝐍∈ℕ^𝑑₀, (20)
induce a uniformly bounded family {𝑇_𝐾𝐍} of convolution operators on 𝐿_𝑝(𝕋^𝑑;𝑊). Then 𝑊∈𝑃𝐴_𝑝.

Proof. We have to estimate 𝐴_𝑊(𝑅,𝑝) for an arbitrary rectangle 𝑅∈ℛ(𝑑). The idea is to form a suitable product kernel 𝐾_𝐍 that is “large” on 𝑅 in the sense that the corresponding operator can be well approximated by an integral operator of the type considered in Lemma 6.
By assumption, the kernel 𝐾_𝑛∈𝒫_𝑐⋅𝑛 is real and ‖𝐾_𝑛‖_∞=𝐾_𝑛(0)≤𝐶𝑛, so by Bernstein's inequality, ‖𝐾^′_𝑛‖_∞≤𝑐𝐶𝑛². We can thus find an integer 𝑀 (independent of 𝑛) such that for 𝑡∈[−1/𝑀𝑛,1/𝑀𝑛] we have 𝐾_𝑛(𝑡)≥(1−(1/2𝐶²_𝑑))^1/𝑑‖𝐾_𝑛‖_∞, where 𝐶_𝑑 is the constant from Lemma 6.
Let a rectangle 𝑅=𝐼₁×𝐼₂×⋯×𝐼_𝑑 be given. For 𝑗=1,2,…,𝑑, with |𝐼_𝑗|>1/2𝑀, we define 𝑁_𝑗=0 and replace 𝐼_𝑗 with [−1/2,1/2) and obtain a possibly larger rectangle 𝑅. By Lemma 4(b), there exists a universal constant 𝑏 such that 𝐴_𝑊(𝑅,𝑝)≤𝑏𝐴_𝑊(𝑅,𝑝) since |𝑅|≤(2𝑀)^𝑑|𝑅|. Next, for each 𝑗=1,2,…,𝑑 with |𝐼_𝑗|≤1/2𝑀, we choose an integer 𝑁_𝑗≥1 such that 1⋅14𝑀𝑁_𝑗≤𝐼_𝑗≤1⋅12𝑀𝑁_𝑗. (21)
Notice that for 𝑡,𝑢∈𝐼_𝑗, we have 𝑡−𝑢∈𝐼_𝑗−𝐼_𝑗⊂[−1/𝑀𝑁_𝑗,1/𝑀𝑁_𝑗] so 𝐾_𝑁𝑗()≥1𝑡−𝑢1−2𝐶²_𝑑^1/𝑑𝐾_𝑁𝑗∞. (22)
For notational convenience we put 𝐾₀∶=1 and form the product kernel 𝐾_𝐍(𝜉)=^𝑑_𝑗=1𝐾_𝑁𝑗𝜉_𝑗. (23)
The plan of attack is to use the simple fact that 𝑓→𝜒_𝑅𝑇_𝐾𝐍(𝜒_𝑅𝑓) is uniformly bounded in both 𝑅 and 𝐍∈ℕ^𝑑₀. We notice that 𝑓→𝜒_𝑅𝑇_𝐾𝐍(𝜒_𝑅𝑓) has integral kernel 𝑆₂𝜉,𝜂∶=𝜒_𝑅𝜂𝜒_𝑅(𝜉)𝐾_𝐍𝜂−𝜉. (24)
We wish to estimate the operator norm of 𝑆₂ from below. For that purpose we first consider the operator with kernel 𝑆𝜉,𝜂∶=𝑆₁𝜉,𝜂−𝑆₂𝐾𝜉,𝜂∶=_𝐍∞𝜒_𝑅(𝜉)𝜒_𝑅𝜂−𝜒_𝑅(𝜉)𝜒_𝑅𝜂𝐾_𝐍𝜉−𝜂. (25)
Notice that estimate (22), together with the fact that ‖𝐾_𝐍‖_∞=∏^𝑑_𝑗=1𝐾_𝑁𝑗(0), implies the following size estimate 𝑆=𝐾𝜉,𝜂_𝐍∞𝜒_𝑅(𝜉)𝜒_𝑅𝜂−𝜒_𝑅(𝜉)𝜒_𝑅𝜂𝐾_𝐍≤𝐾𝜉−𝜂_𝐍∞2𝐶²_𝑑𝜒_𝑅(𝜉)𝜒_𝑅𝜂=𝑅⋅𝐾_𝐍∞2𝐶²_𝑑𝑅⁻¹𝜒_𝑅(𝜉)𝜒_𝑅𝜂. (26)
According to Lemma 6, the kernel 𝑆 induces an operator of norm at most (1/2)𝐶_𝑑⁻¹|𝑅|⋅‖𝐾_𝐍‖_∞𝐴_𝑊(𝑅,𝑝) on 𝐿_𝑝(𝕋^𝑑;𝑊). At the same time, Lemma 6 shows that the operator with kernel 𝑆₁(𝜉,𝜂)=|𝑅|‖𝐾_𝐍‖_∞⋅|𝑅|⁻¹𝜒_𝑅(𝜉)𝜒_𝑅(𝜂) has norm at least 𝐶_𝑑⁻¹|𝑅|⋅‖𝐾_𝐍‖_∞𝐴_𝑊(𝑅,𝑝) on 𝐿_𝑝(𝕋^𝑑;𝑊). The triangle inequality for operator norms now implies that 12𝐶_𝑑𝑅⋅𝐷_𝐍∞𝑀≥𝑆𝑅,𝑊₁−𝑆₂≥𝑆₁−𝑆₂≥𝐶_𝑑⁻¹𝑅⋅𝐷_𝐍∞𝑀𝑅,𝑊−𝑆₂, (27)
so ‖𝑆₂‖≥(1/2)𝐶_𝑑⁻¹|𝑅|⋅‖𝐷_𝐍‖_∞𝐴_𝑊(𝑅,𝑝). Moreover, by (21), we see that |𝑅|⋅‖𝐾_𝐍‖_∞≥𝐶(4𝑀)^−𝑑, so we may conclude that 𝐴_𝑊𝑅,𝑝≤𝑏𝐴_𝑊𝑅,𝑝≤2𝑏𝐶𝐶_𝑑(4𝑀)^𝑑𝑆₂=𝐶^′sup_{𝑓𝐿𝑝𝑑(𝕋;𝑊)=1}𝜒_𝑅𝑇_𝐾𝐍𝜒_𝑅𝑓_{𝐿𝑝(𝕋𝑑;𝑊)}≤𝐶^′sup_{𝑓𝐿𝑝𝑑(𝕋;𝑊)=1}𝑇_𝐾𝐍𝑓_{𝐿𝑝(𝕋𝑑;𝑊)}≤𝐶^′′ (28)
with constant 𝐶^′′ independent of 𝑅. We may finally conclude that 𝑊∈𝑃𝐴_𝑝(𝑑).

3. Summation of Multiple Trigonometric Series

This section contains applications of the results of Section 2 to convolution operators induced by rectangular Dirichlet, Fejér, and Jackson kernels. The Dirichlet kernels correspond to standard rectangular trigonometric summation while the Fejér kernels generate the corresponding Cesàro means. The Jackson kernels are (normalized) squares of the Fejér kernels, and they induce the well-known Jackson approximation by trigonometric polynomials.

We begin by studying the univariate Dirichlet kernel. The Hilbert transform 𝐻 is defined on 𝐿_𝑝(𝕋), 1<𝑝<∞, by 𝐻𝑓(𝑥)∶=p.v._𝕋𝑓(𝑡)cot(𝜋(𝑥−𝑡))𝑑𝑡. (29)

We lift 𝐻 to a linear operator on 𝐿_𝑝(𝕋;𝑊), for any matrix weight 𝑊∶𝕋→ℳ, by letting it act coordinatewise.

Treil and Volberg completely characterized when the Hilbert transform 𝐻 is bounded in the matrix case on 𝕋 when 𝑝=2; see [11]. Later, Nazarov and Treĭl′ introduced in a new “Bellman function” method [6] to extend the theory to 1<𝑝<∞. Volberg presented a different solution to the matrix weighted 𝐿_𝑝 boundedness of the Hilbert transform via Littlewood-Paley theory [7]. The fundamental result is the following.

Theorem 8 (see [6, 7, 11]). Let 𝑊∶𝕋→ℳ be a matrix weight. Suppose 1<𝑝<∞. Then the Hilbert transform is bounded on 𝐿_𝑝(𝕋;𝑊) if and only if 𝑊∈𝐴_𝑝(𝕋).

We recall that the univariate Dirichlet kernel 𝐷_𝑁 is given by 𝐷_𝑁(𝑡)=sin2𝜋(𝑁+1/2)𝑡sin𝜋𝑡,𝑁≥1, (30)

and for 𝑓∈𝐿_𝑝(𝕋) we define the associated partial sum operators, 𝑆_𝑁𝑓∶=^𝑁_𝑘=−𝑁𝑓(𝑘)𝑒^{2𝜋𝑖𝑘⋅}=𝑓∗𝐷_𝑁∶=_𝕋𝑓(𝑡)𝐷_𝑁(⋅−𝑡)𝑑𝑡. (31)

We have the following lemma which follows easily from Theorem 8.

Lemma 9. Let 𝑊∶𝕋→ℳ be a matrix weight in 𝐴_𝑝(𝕋). Then the partial sum operators 𝑓→𝑓∗𝐷_𝑁 are uniformly bounded on 𝐿_𝑝(𝕋;𝑊).

Proof. We let 𝑃₊=(1/2)(𝐼+𝑖𝐻+𝑆₀) denote the Riesz projection onto 𝐻^𝑝 for 𝑓∈𝐿_𝑝(𝕋;𝑊), where 𝑆₀𝑓∶=∫_𝕋𝑓(𝑦)𝑑𝑦 is the 0-order partial sum operator. It follows that 𝑃₊ is bounded on 𝐿_𝑝(𝕋;𝑊) since 𝐻 is bounded according to Theorem 8, and 𝑆₀ is bounded according to [12, Lemma 1.5]. Notice that 𝑓→𝑓𝑒^{2𝜋𝑖𝑀⋅} is a norm preserving operator on 𝐿_𝑝(𝕋;𝑊), just as in the scalar case. Then we observe that 𝑓∗𝐷_𝑁=𝑒^{−2𝜋𝑖𝑁⋅}𝑃₊𝑒^{2𝜋𝑖𝑁⋅}𝑓−𝑒^{2𝜋𝑖(𝑁+1)⋅}𝑃₊𝑒^{−2𝜋𝑖(𝑁+1)⋅}𝑓, (32)
and the result follows.

For 𝐍=(𝑁₁,…,𝑁_𝑑)∈ℕ^𝑑₀, we form the product kernel 𝐷_𝐍(𝜉)∶=∏^𝑑_𝑗=1𝐷_𝑁𝑗(𝜉_𝑗). One has 𝑆_𝐍(𝐟)∶=_𝕋𝑑𝐟(𝑡)𝐷_𝐍(⋅−𝑡)𝑑𝑡. (33)

We notice that 𝐷_𝑁∈𝒫_𝑁 and 𝐷_𝑁(0)=‖𝐷_𝑁‖_∞=2𝑁+1, so the following corollary follows directly from Propositions 5 and 7 and Lemma 9.

Corollary 10. Let 𝑊∶𝕋^𝑑→ℳ be a matrix weight. For 1<𝑝<∞, the operators {𝑆_𝐍∶𝐍∈ℕ^𝑑} are uniformly bounded on 𝐿_𝑝(𝕋^𝑑;𝑊) if and only if 𝑊∈𝐴𝑃_𝑝.

Remark 11. It is easy to verify that vectors of trigonometric polynomials are dense in 𝐿_𝑝(𝕋^𝑑;𝑊), 1<𝑝<∞, whenever 𝑊 is a matrix weight (since 𝑊∈𝐿₁ so each entry in 𝑊 is in 𝐿₁(𝕋)). It therefore follows by standard techniques that the family {𝑆_𝐍∶𝐍∈ℕ^𝑑} is uniformly bounded on 𝐿_𝑝(𝕋^𝑑;𝑊) if and only if ‖𝐟−𝑆_𝐍(𝐟)‖_{𝐿𝑝(𝕋;𝑊)}→0, as min_𝑗𝑁_𝑗→+∞, for all 𝐟∈𝐿_𝑝(𝕋;𝑊).
Corollary 10 relies on basic localization properties of the Dirichlet kernel. However, many well-known summation kernels share the necessary properties needed to apply Propositions 5 and 7. Let us illustrate this fact by considering two specific examples.
The rectangular Cesàro summation is given by 𝜎_𝐍(𝐟)=1𝑁₁+1⋯𝑁_𝑑+1^{𝑁1,…,𝑁𝑑}_{𝑘1=0,…,𝑘𝑑=0}𝑆_{(𝑘1,𝑘2,…,𝑘𝑑)}(𝐟)=_𝕋𝑑𝐟(𝑡)𝐹_𝐍(⋅−𝑡)𝑑𝑡 (34)
with the product Féjer kernel given by 𝐹_𝐍(𝜉)=∏^𝑑_𝑗=1𝐹_𝑁𝑗(𝜉_𝑗), where the scalar Féjer kernel is defined by 𝐹_𝑛(𝑡)=1𝑛sin(𝑛𝜋𝑡)sin(𝜋𝑡)². (35)
Notice that 𝐹_𝑏∈𝒫_𝑛 and 𝐹_𝑛(0)=‖𝐹_𝑛‖_∞=𝑛. The scalar Jackson kernel is the normalized square of the Féjer kernel and given by 𝐽_𝑛(𝑡)=3𝑛2𝑛²()+1sin𝑛𝜋𝑡sin(𝜋𝑡)⁴. (36)
The corresponding product kernel is 𝐽_𝐍(𝜉)=∏^𝑑_𝑗=1𝐽_𝑁𝑗(𝜉_𝑗), 𝐍∈ℕ^𝑑₀, and the rectangular Jackson summation operator is given by 𝒥_𝐍(𝑓)∶=𝑓∗𝐽_𝐍. Notice that 𝐽_𝑛∈𝒫_2𝑛 and 𝐽_𝑛(0)=‖𝐽_𝑛‖_∞=𝑛.
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 𝑆_𝐍, 𝜎_𝐍, and 𝒥_𝐍 on 𝐿_𝑝(𝕋^𝑑;𝑤) is equivalent to the condition 𝑊∈𝐴𝑃_𝑝.

Theorem 12. Let 𝑊∶𝕋^𝑑→ℳ be a matrix weight. For 1<𝑝<∞, the following conditions are equivalent: (i)𝑊∈𝐴𝑃_𝑝; (ii)the operators {𝑆_𝐍∶𝐍∈ℕ^𝑑₀} are uniformly bounded on 𝐿_𝑝(𝕋^𝑑;𝑊); (iii)the operators {𝜎_𝐍∶𝐍∈ℕ^𝑑₀} are uniformly bounded on 𝐿_𝑝(𝕋^𝑑;𝑊); (iv)the operators {𝒥_𝐍∶𝐍∈ℕ^𝑑₀} are uniformly bounded on 𝐿_𝑝(𝕋^𝑑;𝑊).

Proof. We first notice that each of the univariate kernels 𝐷_𝑛,𝐹_𝑛, and 𝐽_𝑛 satisfies the hypothesis of Proposition 7, so (ii), (iii), and (iv) each implies that 𝑊∈𝐴𝑃_𝑝. Now, suppose that 𝑊∈𝐴𝑃_𝑝. Then (ii) holds by Corollary 10. To conclude, we just need to recall that Cesàro and Jackson summations are both regular summation methods, so (ii) implies both (iii) and (iv).