JMATH Journal of Mathematics 2314-4785 2314-4629 Hindawi Publishing Corporation 135245 10.1155/2013/135245 135245 Research Article Summation of Multiple Fourier Series in Matrix Weighted Lp-Spaces 0000-0002-9078-0594 Nielsen Morten Liu Baoding Department of Mathematical Sciences Aalborg University Fredrik Bajers Vej 7G 9220 Aalborg East Denmark aau.dk 2013 20 3 2013 2013 28 01 2013 16 02 2013 2013 Copyright © 2013 Morten Nielsen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper is concerned with rectangular summation of multiple Fourier series in matrix weighted Lp-spaces. We introduce a product Muckenhoupt Ap condition for matrix weights W and prove that rectangular Fourier partial sums converge in the corresponding matrix weighted space Lp(𝕋d;W), 1<p<, if and only if the weight satisfies the product Muckenhoupt Ap 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 be the family of nonnegative-definite m×m complex-valued matrices. A (periodic) matrix weight is by definition an integrable map W:𝕋d. For a measurable vector-valued function f=(f1,,fm)T:𝕋dm, let (1)fLp(𝕋d;W):=(𝕋d|W1/p(t)f(t)|pdt)1/p,1p<, where |·| denotes the usual norm on m. We let Lp(𝕋d;W) denote the family {f:𝕋dm:fLp(𝕋d;W)<}, and Lp(𝕋d;W) becomes a Banach space when we factorize over 𝒩={f:fLp(𝕋d;W)=0}.

In this paper we are interested in convergence properties of multiple trigonometric series in Lp(𝕋d;W) and how specific convergence properties of trigonometric series can be related to properties of the weight W. To be more specific, let Dn(t)=|k|ne-2πikt denote the univariate Dirichlet kernel, and for Nd we define the rectangular kernel DN(t):=j=1dDNj(tj). Then (2)SNf:=f*DN:=𝕋df(t)DN(·-t)dt,fL1(𝕋d),

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

It is well known (see, e.g., [1, Theorem 3.5.7]) that f-SN(f)Lp(𝕋d)0, as minjNj+, for 1<p<. An immediate corollary is that we have convergence of the partial sums SN(f), in Lp(𝕋d;Id), for 1<p< in the vector-valued case. However, it is not obvious what can be said about convergence of SN(f) in Lp(𝕋d;W) for a general matrix weight W. The main result of the preset paper completely characterizes the special class of weights that allow convergence; SN(f) converges if and only if the weight W satisfies a certain matrix Muckenhoupt Ap 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 Lp(𝕋d;W) if and only if the weight W satisfies the mentioned matrix Muckenhoupt Ap product condition.

At a first glance, the study of vector-valued operators like SN on Lp(𝕋d;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 {fk}k=1m in L2(d), the associated shift invariant space S is given by (4)S:=Span{fk(·-l):ld,k=1,2,,m}¯L2(d).

A natural question to pose is whether B:=Span{fk(·-l):ld,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:={eje2πik·t:kd,j=1,,m}, where {ej}j=1m is the standard basis for m, and let G the m×m Gram matrix be given by (5)Gi,j:=kdfi^(·-k)fj^(·-k)¯, where the Fourier transform is given by f^(ξ):=df(t)e-2πit·ξ  dt. One can show (see ) that the map U:L2(𝕋d;G)S, given by (6)U(τ):=(=1Nτ(ξ)ψ^),

is an isometric isomorphism between L2(𝕋d;G) and S, satisfying U(eje-2πik·t)=fj(·-k). Hence, the metric properties of B in S are equivalent to the properties of 𝒯 in L2(𝕋d;G). For example, U will map {SN} 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 Lp(𝕋d;W) that will be discussed below.

It was proved by the present author  that the rectangular partial sums {USN} converge in S precisely when {SN} 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 Ap condition for matrix weights. Then necessary and sufficient conditions for a convolution operator of product type (such as SN) to be bounded on Lp(𝕋d;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 <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M83"><mml:msub><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub><mml:mo mathvariant="bold">(</mml:mo><mml:msup><mml:mrow><mml:mi>𝕋</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msup><mml:mo mathvariant="bold">;</mml:mo><mml:mi>W</mml:mi><mml:mo mathvariant="bold">)</mml:mo></mml:math></inline-formula>

In this section we introduce a matrix Muckenhoupt Ap product condition suitable for dealing with convolution operators of product type such as the partial sum operator SN defined by (3). A sufficient condition for convolution operators of product type bounded on Lp(𝕋d;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 Lp(𝕋d;W) when W satisfies the product Ap condition.

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

More recently, Hunt-Muckenhoupt-Wheeden type results for matrix weights have been considered. The matrix Ap condition was introduced by Nazarov et al.  and they showed that it is the right condition for “standard” singular integral operators to be bounded on Lp(𝕋d;W). The Ap condition (1<p<) for weights W:d was originally stated in terms of dual matrices and average, but it was shown by Roudenko  to be equivalent to (7)supB(B(BW1/p(x)W-1/p(y)pdy|B|)p/pdx|B|)p/p, where the sup is taken over all open balls in 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 (d) denote the family of all rectangles in d of the form R=I1×I2××Id, with Ij being a bounded open interval in . Then we consider the following more restrictive subclass of matrix weights.

Definition 1.

Let W:𝕋d 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 PAp provided there exists a uniform constant cW such that (8)AW(R,p):=R(RW1/p(x)W-1/p(y)pdy|R|)p/pdx|R|cW,

for any R(d).

Remark 2.

For d=1, Definition 1 reduces to the standard matrix Ap condition on 𝕋, which we denote by Ap(𝕋). In the scalar case (i.e., m=1), Definition 1 reduces to the known product Ap condition for scalar weights, which has a long history; see  and references therein.

The similarity of conditions (8) and (7) implies that many results for matrix Ap 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:𝕋d be a matrix weight. Then the following statements are equivalent for 1<p<:

WPAp;

W-p/pPAp;

R(RW1/p(x)W-1/p(y)p(dx/|R|))p/p(dy/|R|)cW, for all  R(d).

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

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

Lemma 4.

Let W:𝕋d be a matrix weight, and let 1<p<. Then the following holds.

For any rectangles RR~(d), (9)AW(R,p)(|R~||R|)pAW(R~,p).

Suppose WPAp; then the univariate weight ξjW(ξ), obtained by fixing the variables ξk, kj, is uniformly in 𝔸p(𝕋) for a.e. (ξ1,,ξj-1,ξj+1,,ξd)𝕋d-1.

Proof.

For (a), we notice that whenever RR~(d), (10)AW(R,p)=R(RW1/p(x)W-1/p(t)pdt|R|)p/pdx|R|(|R~||R|)pR~(R~W1/p(x)W-1/p(t)pdt|R~|)p/pdx|R~|=(|R~||R|)pAW(R~,p).

Now we turn to the proof of (b). It suffices to consider W~(t):=W(t,ξ2,,ξd) for (ξ2,,ξd)𝕋d-1 fixed. Given an interval I, we form Rε=Iε(ξ2)××Iε(ξd), where Iε(ξj) is an interval of length 2ε centered at ξj. First suppose pp. Since WPAp, there exists a constant cW independent of I×Rε such that (11)1|Rε|2RεRε[×(w,v)W1/ppdw|I|)p/pdt|I|]I(dw|I|IW1/p(t,u)W-1/p×(w,v)W1/ppdw|I|IW1/p(t,u)W-1/p)p/pdt|I|]dudvRεI(dwdv|I|·|Rε|RεIW1/p(t,u)W-1/p(w,v)p×dwdv|I|·|Rε|)p/pdtdu|I|·|Rε|=AW(R,p)cW,where we have used the continuous embedding L1(Rε;dv/|Rε|)Lp/p(Rε;dv/|Rε|). Hence, by Lebesgue's differentiation theorem, for almost every (ξ2,,ξd)𝕋d-1, (12)cWlimε0+AW(I×Rε,p)=I(IW~1/p(t)W~-1/p(w)pdw|I|)p/pdt|I|=AW~(I,p),

where the constant is independent of I and (ξ2,,ξd). It follows that W~ is uniformly in 𝔸p(𝕋) for a.e. (ξ2,,ξd)𝕋d-1. In the case p<p, we use Lemma 3 to conclude that W-p/pPAp, which implies the following estimate:(13)RεI(dtdu|I|·|Rε|RεIW1/p(t,u)W-1/p(w,v)p×dtdu|I|·|Rε|)p/pdwdv|I|·|Rε|cW.

By repeating the argument from the pp case, we conclude that W~-p/p is uniformly in Ap(𝕋) which again by Lemma 3 implies that W~ is uniformly in Ap(𝕋).

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

Proposition 5.

Suppose that {KN}N0 is a sequence of convolution kernels defined on 𝕋 for which the corresponding operators (14)TNf:=𝕋f(t)KN(·-t)dt

are uniformly bounded on Lp(𝕋;W) whenever WAp(𝕋). Then the associated product convolution kernels (15)KN(ξ)=j=1dKNj(ξj),N=(N1,,Nd)0d,ξ𝕋d,

induce a uniformly bounded family of operators on Lp(𝕋d;W) for WPAp.

Proof.

Suppose that WPAp. 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 d3.

According to Lemma 4(b), Wξ1:=W(ξ1,·) and Wξ2:=W(·,ξ2) satisfy uniform Muckenhoupt Ap-conditions a.e. on 𝕋. Pick any fLp(𝕋2,W). By Fubini's theorem, fξ1:=f(ξ1,·)Lp(𝕋,Wξ1) and fξ2:=f(·,ξ2)Lp(𝕋,Wξ2) for a.e. [ξ1] and [ξ2], respectively.

We define (16)TN1f:=KN*fξ2:=𝕋fξ2(t)KN(·-t)dt,TM2f:=KM*fξ1:=𝕋fξ1(t)KM(·-t)dt.

Notice that TN,Mf=TN1TM2f. By assumption, (17)𝕋|Wξ11/p(ξ2)TM2fξ1(ξ2)|pdξ2C𝕋|Wξ11/p(ξ2)fξ1(ξ2)|pdξ2,a.e.[ξ1].

An integration yields (18)𝕋𝕋|W1/p(ξ1,ξ2)TM2f(ξ1,ξ2)|pdξ2dξ1C𝕋𝕋|W1/p(ξ1,ξ2)f(ξ1,ξ2)|pdξ2dξ1.

Similarly, (19)TN,MfLp(𝕋2,W)p=𝕋𝕋|W1/p(ξ1,ξ2)TN1TM2f(ξ1,ξ2)|pdξ1dξ2C𝕋𝕋|W1/p(ξ1,ξ2)TM2f(ξ1,ξ2)|pdξ1dξ2C2𝕋𝕋|W1/p(ξ1,ξ2)f|pdξ1dξ2.

It follows that the family {TN}N02 is uniformly bounded on Lp(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 Lp(𝕋d;W) when WAPp.

We need the following Lemma which gives an estimate of the norm of integral operators on L2(𝕋d;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|-1χR×R for some bounded rectangle Rd. For 1<p<, there exists a constant Cd independent of the particular choice of S such that the norm of S on Lp(𝕋d;W) is at most Cd·α·AW(R,p), with AW(R,p) given by (8). Moreover, the kernel α|R|-1χR×χR induces an operator with norm at least Cd-1·α·AW(R,p) on Lp(𝕋d;W).

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

We can now give a proof of Proposition 7. For K, we let 𝒫K=span{e2πik·:k;|k|K}.

Proposition 7.

Let W:𝕋d be a periodic matrix weight, and let {Kn}n1 be a sequence of real-valued trigonometric convolution kernels defined on 𝕋. Assume there exist constants c, C such that Kn𝒫c·n, with C-1nKn=Kn(0)Cn,  for n. Suppose that the corresponding product kernels (20)KN(ξ)=j=1dKNj(ξj),N0d,

induce a uniformly bounded family {TKN} of convolution operators on Lp(𝕋d;W). Then WPAp.

Proof.

We have to estimate AW(R,p) for an arbitrary rectangle R(d). The idea is to form a suitable product kernel KN 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 Kn𝒫c·n is real and Kn=Kn(0)Cn, so by Bernstein's inequality, KncCn2. We can thus find an integer M (independent of n) such that for t[-1/Mn,1/Mn] we have Kn(t)(1-(1/2Cd2))1/dKn, where Cd is the constant from Lemma 6.

Let a rectangle R=I1×I2××Id be given. For j=1,2,,d, with |Ij|>1/2M, we define Nj=0 and replace Ij with [-1/2,1/2) and obtain a possibly larger rectangle R~. By Lemma 4(b), there exists a universal constant b such that AW(R,p)bAW(R~,p) since |R~|(2M)d|R|. Next, for each j=1,2,,d with |Ij|1/2M, we choose an integer Nj1 such that (21)14M·1Nj|Ij|12M·1Nj.

Notice that for t,uIj, we have t-uIj-Ij[-1/MNj,1/MNj] so (22)KNj(t-u)(1-12Cd2)1/dKNj.

For notational convenience we put K0:=1 and form the product kernel (23)KN(ξ)=j=1dKNj(ξj).

The plan of attack is to use the simple fact that fχR~TKN(χR~f) is uniformly bounded in both R~ and N0d. We notice that fχR~TKN(χR~f) has integral kernel (24)S2(ξ,η):=χR~(η)χR~(ξ)KN(η-ξ).

We wish to estimate the operator norm of S2 from below. For that purpose we first consider the operator with kernel (25)S(ξ,η):=S1(ξ,η)-S2(ξ,η)S(ξ,η):=KNχR~(ξ)χR~(η)-χR~(ξ)χR~(η)KN(ξ-η).

Notice that estimate (22), together with the fact that KN=j=1dKNj(0), implies the following size estimate (26)|S(ξ,η)|=|KNχR~(ξ)χR~(η)-χR~(ξ)χR~(η)KN(ξ-η)|KN2Cd2χR~(ξ)χR~(η)=|R~|·KN2Cd2|R~|-1χR~(ξ)χR~(η).

According to Lemma 6, the kernel S induces an operator of norm at most (1/2)Cd-1|R~|·KNAW(R~,p) on Lp(𝕋d;W). At the same time, Lemma 6 shows that the operator with kernel S1(ξ,η)=|R~|KN·|R~|-1χR~(ξ)χR~(η) has norm at least Cd-1|R~|·KNAW(R~,p) on Lp(𝕋d;W). The triangle inequality for operator norms now implies that (27)12Cd|R~|·DNM(R~,W)S1-S2|S1-S2|Cd-1|R~|·DNM(R~,W)-S2,

so S2(1/2)Cd-1|R~|·DNAW(R~,p). Moreover, by (21), we see that |R~|·KNC(4M)-d, so we may conclude that (28)AW(R,p)bAW(R~,p)AW(R,p)2bCCd(4M)dS2AW(R,p)=CsupfLp(𝕋d;W)=1χR~TKN(χR~f)Lp(𝕋d;W)AW(R,p)CsupfLp(𝕋d;W)=1TKNfLp(𝕋d;W)C′′

with constant C′′ independent of R. We may finally conclude that WPAp(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 Lp(𝕋), 1<p<, by (29)H(f)(x):=p.v.𝕋f(t)cot(π(x-t))dt.

We lift H to a linear operator on Lp(𝕋;W), for any matrix weight W:𝕋, 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 . Later, Nazarov and Treĭl′ introduced in a new “Bellman function” method  to extend the theory to 1<p<. Volberg presented a different solution to the matrix weighted Lp boundedness of the Hilbert transform via Littlewood-Paley theory . The fundamental result is the following.

Theorem 8 (see [<xref ref-type="bibr" rid="B7">6</xref>, <xref ref-type="bibr" rid="B12">7</xref>, <xref ref-type="bibr" rid="B11">11</xref>]).

Let W:𝕋 be a matrix weight. Suppose 1<p<. Then the Hilbert transform is bounded on Lp(𝕋;W) if and only if WAp(𝕋).

We recall that the univariate Dirichlet kernel DN is given by (30)DN(t)=sin2π(N+1/2)tsinπt,N1,

and for fLp(𝕋) we define the associated partial sum operators, (31)SN(f):=k=-NNf^(k)e2πik·SN(f):=f*DN:=𝕋f(t)DN(·-t)dt.

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

Lemma 9.

Let W:𝕋 be a matrix weight in Ap(𝕋). Then the partial sum operators ff*DN are uniformly bounded on Lp(𝕋;W).

Proof.

We let P+=(1/2)(I+iH+S0) denote the Riesz projection onto Hp for fLp(𝕋;W), where S0f:=𝕋f(y)dy is the 0-order partial sum operator. It follows that P+ is bounded on Lp(𝕋;W) since H is bounded according to Theorem 8, and S0 is bounded according to [12, Lemma 1.5]. Notice that ffe2πiM· is a norm preserving operator on Lp(𝕋;W), just as in the scalar case. Then we observe that (32)f*DN=e-2πiN·P+(e2πiN·f)-e2πi(N+1)·P+(e-2πi(N+1)·f),

and the result follows.

For N=(N1,,Nd)0d, we form the product kernel DN(ξ):=j=1dDNj(ξj). One has (33)SN(f):=𝕋df(t)DN(·-t)dt.

We notice that DN𝒫N and DN(0)=DN=2N+1, so the following corollary follows directly from Propositions 5 and 7 and Lemma 9.

Corollary 10.

Let W:𝕋d be a matrix weight. For 1<p<, the operators {SN:Nd} are uniformly bounded on Lp(𝕋d;W) if and only if WAPp.

Remark 11.

It is easy to verify that vectors of trigonometric polynomials are dense in Lp(𝕋d;W), 1<p<, whenever W is a matrix weight (since WL1 so each entry in W is in L1(𝕋)). It therefore follows by standard techniques that the family {SN:Nd} is uniformly bounded on Lp(𝕋d;W) if and only if f-SN(f)Lp(𝕋;W)0, as minjNj+, for all fLp(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 (34)σN(f)=1(N1+1)(Nd+1)k1=0,,kd=0N1,,NdS(k1,k2,,kd)(f)σN(f)=𝕋df(t)FN(·-t)dt

with the product Féjer kernel given by FN(ξ)=j=1dFNj(ξj), where the scalar Féjer kernel is defined by (35)Fn(t)=1n(sin(nπt)sin(πt))2.

Notice that Fb𝒫n and Fn(0)=Fn=n. The scalar Jackson kernel is the normalized square of the Féjer kernel and given by (36)Jn(t)=3n(2n2+1)(sin(nπt)sin(πt))4.

The corresponding product kernel is JN(ξ)=j=1dJNj(ξj), N0d, and the rectangular Jackson summation operator is given by 𝒥N(f):=f*JN. Notice that Jn𝒫2n and Jn(0)=Jn=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 SN, σN, and 𝒥N on Lp(𝕋d;w) is equivalent to the condition WAPp.

Theorem 12.

Let W:𝕋d be a matrix weight. For 1<p<, the following conditions are equivalent:

WAPp;

the operators {SN:N0d} are uniformly bounded on Lp(𝕋d;W);

the operators {σN:N0d} are uniformly bounded on Lp(𝕋d;W);

the operators {𝒥N:N0d} are uniformly bounded on Lp(𝕋d;W).

Proof.

We first notice that each of the univariate kernels Dn,Fn, and Jn satisfies the hypothesis of Proposition 7, so (ii), (iii), and (iv) each implies that WAPp. Now, suppose that WAPp. 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).

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