Muckenhoupt Matrix Weights

We study matrix weights defined on the multivariate torus Td\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {T}}^d$$\end{document}. Sufficient conditions for a matrix weight to be in the Muckenhoupt A2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf {A}}_2$$\end{document}-class are studied, and two such sufficiency results obtained by S. Bloom for d=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d=1$$\end{document} are generalized to the multivariate setting. As an application, an A2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_2$$\end{document}-decomposition property is introduced for matrix weights and a BMO distance theorem for matrix weights is considered.

boundedness of the Hardy-Littlewood maximal operator introduces a notion of the A p -weight. Its importance is emphasized further in the papers by Hunt, Muckenhoupt and Wheeden [10], and by Coifman and Fefferman [3]. Among others, these papers deal with the boundedness of the conjugation operator. As it turned out, there is yet another interesting approach to this problem; using the Fefferman duality theorem and the class of BMO functions, it deals with the boundedness of the commutator operator (see the fundamental paper by Coifman, Rochberg and Weiss [5] for details).
The above-mentioned papers influenced a rather pioneering work of S. Bloom, whose PhD dissertation was defended in August of 1981 at Washington University in St Louis. In his dissertation [1], as well as in several publications (for us here the most important being [2]), Bloom develops a theory of matrix-valued weights, which he successfully applies to the weighted norm inequalities for vector-valued functions. It took almost twenty years for the full realization of the significance of Bloom's approach, but the theory was pushed forward through the seminal work of Treȋl and Volberg [24], Nazarov and Treȋl [14], Volberg [25], and Goldberg [7]. It is fairly obvious to extend the notion of the A p -weight from the scalar-valued case to the case of diagonal matrices, i.e., to the finite spectrum of the matrix. However, the extension to a wider class of matrices is far from trivial. For matrix-valued weights defined on a one-dimensional domain, Bloom develops the notion of a log-preserving unitary matrix (which preserve the A 2 -property) and shows that a wide range of unitary matrices, having Lipschitz coefficients, belong to the class LP of log-preserving matrices. Bloom's approach naturally leads toward the class of functions in BMO weighted spaces. The interest in Bloom's work and related subjects increased even more in the last several years with dozens of papers devoted to commutators, matrix weights, etc.
Our interest in the subject developed initially from the studies of shift invariant spaces in the theory of wavelets and other reproducing function systems (see [17] and [26] ). The relationship between Schauder bases and Riesz bases in this context guided us naturally toward the stability of Schauder basis property (see [18]) and the Garnett-Jones distance in BMO spaces (see [19]). In particular, we developed the notion of the A 2 -decomposition property for a large class of the so-called Calderón-Zygmund coverings, and we show that this decomposition is essential for the Garnett-Jones distance theorem (for the original work on the subject consider the paper by Garnett and Jones [6], but also the related work by Coifman and Rochberg [4] and by Soria [22]).
In this paper we address the question of the A 2 -decomposition property for matrix weights. The question presents us immediately with a technical problem of the extension of the Bloom method of the LP matrices to the case of matrix weights defined on a d-dimensional domain. The problem proved to be more involved than one may expect. We leave its full implementation (i.e., for functions with any Lipschitz-type condition) for future research. However, we were able to prove the desired result for functions in Lip 1 . The argument is somewhat complex and we present it in detail in Sect. 2. In Sect. 3 we establish the A 2 -decomposition property for matrix weights defined on a d-dimensional torus. The consequences of these results, together with the discussion about the intricacies of the BMO distance theorem for matrix weights, are presented in Sect. 4.

Muckenhoupt Weights
Let Q denote the family of cubes in the multivariate torus T d , d ∈ N. A (measurable) scalar weight w : where for any measurable subset E ⊂ T d of positive measure, we define and we use the notation We denote the class of such Muckenhoupt weights by A p (T d ). Even though the scalar A p -conditions are quite involved, they are still very much operational since quite large classes of, e.g., polynomial weights are known to satisfy the respective conditions, see, e.g. [20]. Now consider a measurable matrix-valued weight W : T d → C N ×N , taking values in the positive definite N × N -matrices. The matrix A p condition was introduced and studied in [14,24,25] and it is considerably more complicated than the scalar condition, and there are no known straightforward sufficient conditions on a matrix weight to ensure membership in the A p class except in very special cases (e.g., for diagonal weights and for weights with strong pointwise bounds on its spectrum).
S. Roudenko introduced an equivalent matrix A p condition in [21] which is often more straightforward to verify. Let 1 < p < ∞ and let q be the dual exponent, 1/ p + 1/q = 1. The matrix A p condition holds if and only if W : T d → C N ×N is measurable and positive definite a.e. and satisfies The norm · appearing in the integral is any matrix norm on the N × N matrices.
The family of such matrix weights is denoted by A p (T d ). In the special case N = 1 (1 × 1-matrices), one can verify that We will also need weighted vector-valued L p -spaces, 1 < p < ∞. For W : T d → C N ×N a matrix-valued function, which is measurable and positive definite a.e., let L p (W ) denote the family of measurable functions f : In order to turn L p (W ) into a Banach space, one has to factorize over { f : T d → C N ; f L p (W ) = 0}.

A Sufficient Condition Based on Commutators
We now turn to our first sufficient condition for membership in A p (T d ). We will primarily focus on A 2 (T d )-weights. Weighted BMO spaces and commutators will play an essential part in the sufficient condition, together with the fact that boundedness of the (discrete) Riesz transforms on L 2 (W ) is equivalent to W ∈ A 2 (T d ).
For later use, we notice that the more conventional (unweighted) BMO is given by We can now state the result. The univariate case, d = 1, is due to Bloom [1]. The new contribution is the multivariate case.

Theorem 2.2 Suppose the matrix weight W
Proof The case d = 1 has been proved by Bloom [1]. For d ≥ 2, we let R j denote the discrete Riesz transform, i.e., the periodized version of Notice that Moreover, where [·, ·] denotes the commutator. The fact that U is unitary implies that |u lk | ≤ 1, so using Eq. (2.3) we obtain Hence, there exists a constant C such that where we used that the 1 and 2 norms are equivalent on R N 2 +1 . Now, since we have the scalar condition λ k ∈ A 2 (T d ), and R j is a standard singular integral operator, By the generalized Bloom commutator result, see [9], using thatū rm ∈ B M O It now follows from [15,Corollary 4.2] that boundedness of the discrete Riesz trans- , which completes the proof.

Lipschitz Continuous Matrix Weights
The in Theorem 2.2 may be somewhat challenging to verify in specific cases. We shall now derive a more straightforward sufficient condition to be in Recall that a function f : We can now state the result. The univariate case, d = 1, is due to Bloom [1]. The multivariate case is, as far as we know, new. We will prove the proposition by analyzing certain averaging operators. As before, we let Q denote the collection of cubes in T d . For Q ∈ Q, and a locally integrable vector-valued function f : T d → C N , we consider the vector-valued operator where the integral is applied coordinate-wise.
The following fundamental property of the operators {A Q } Q∈Q was proved by Nazarov and Treȋl in [14], see also [7,  A well-known useful observation is that for any linear operator T , we have . We now turn to the proof of Theorem 2.3.
Proof We first assume that the elements of D have been normalized such that τ λ 1 , . . . , e τ λ N ).
For use later, we notice that for any linear operator T on L p (W τ ), with f ∈ L p (W τ ), where K 1/ p is any upper bound on the A p -norms of e τ λ 1 , . . . , e τ λ N . Notice, in particular, that K does not depend on Q.
We now write where the matrix U (x)U * (t) is continuous in t, and equals the identity at t = x. With δ ik the Kronecker delta, and recalling that |u i j | ≤ 1 since U is unitary, with M any upper bound for the Lipschitz constants of the collection {ū i j } i j . Consequently, With a view towards (2.4) with T = A Q , we now employ the estimate (2.6) to deduce the following where we used that |Q| diam(Q) d ≥ |x − t| d for x, t ∈ Q. If d = 1, we may skip the following step and continue the estimate (2.7) directly as indicated below in (2.8) with any p > 2. For d ≥ 2, however, we need an additional estimate. We fix q > 1 such that q(d − 1) < d and we define p by 1/ p + 1/q = 1, where we notice that p > 2. Pick s > 1 such that sq(d −1) < d and let 1/s +1/s = 1. ThenˆT We now adjust the value of τ so τ s = 1, which ensures that e τ s λ k ∈ A p , k = 1, . . . , N , and implies that exp(−s τ qλ k / p) ∈ L 1 (T d ), k = 1, . . . , N , with norms that depend only on the A 2 -constants of {e λ k }. Consequently, with these choices, We now continue from the estimate (2.7), where we used the estimate (2.4). Finally, using (2.8) once more, Now we use Hölder's inequality with parameter p/2 > 1 in the finite sum over r , where we used (2.5) in the last step. We use the estimate (2.4) to conclude that independent of Q. Hence, A Q extends to a uniformly bounded family of bounded operators on L p (W τ ). We now follow Bloom [1] and use Stein's interpolation theorem, see [23],to cover the case L 2 (W η ) for a suitable value of η > 0. Define the analytic class and notice that (2.9) implies that T 1 is bounded on the (unweighted) vector-valued L p (C N ). Also notice that T 0 = A Q is bounded (uniformly in Q) on L q (C N ), which follows from Theorem 2.4 in the scalar case. Using the observation that for z = x +iy, . The following lemma, which may be of independent interest, was used in the proof of Theorem 2.3. We let L ∞ N ×N denote the family of N × N -matrices defined on T d with entries in L ∞ (T d ).
Proof By the result of Roudenko [21], V ∈ A 2 (T d ) is equivalent to the uniform estimate over cubes Q, Now, we notice that by the uniform bound on the operator norm of B, using that V 1/2 and B 1/2 commute, However, using the fact that (BV ) 1/2 and V −1/2 are self-adjoint, and that for selfadjoint matrices of the same size C and D, C D = (C D) * = DC , Now we use the uniform bound on the operator norm of B −1 , and that V −1/2 and B −1/2 commute,ˆQˆQ By combining the above estimates, we obtain and we conclude that (V B) −1 ∈ A 2 (T d ) so V B ∈ A 2 (T d ) and the lemma follows.

The Matrix A 2 -Decomposition Property
In the recent paper [19], the present authors studied the BMO distance theorem of Garnett and Jones and certain decomposition properties of scalar Muckenhoupt weights related to the Jones factorization theorem, see [12]. In particular, it was proven that the BMO distance theorem holds if and only if a certain A 2 -decomposition property holds.
Here we consider a corresponding A 2 -decomposition property for matrix weights.
Definition 3.1 Let F be a family of A 2 (T d ) matrix weights. We say that F has the matrix A 2 -decomposition property provided there exist constants K = K (F), and δ := δ(F) > 0, such that for W ∈ F, there exist commuting matrix weights V , B, satisfying and B is a matrix with spectrum that is (essentially) bounded and bounded from below on T d .

Remark 3.2
The A 2 -decomposition property is trivially satisfied for any finite family of matrix weights in A 2 (T d ). In the scalar case it is known, see [19], that the full A 2 class has the A 2 -decomposition property.
It is not known at present whether the full class A 2 (T d ) has the matrix A 2decomposition property. We therefore restrict our attention to certain subsets of A 2 (T d ). We need the following definition of matrix BMO. Definition 3. 3 We say that a matrix weight W : Bloom proves in [1], in the case d = 1, that W ∈ A 2 (T d ) implies that log(W ) ∈ B M O. One can easily verify that Bloom's proof generalizes verbatim to the multivariate case. However, Bloom also constructs a symmetric B ∈ B M O for which exp(t B) / ∈ A 2 (T d ) for any t > 0. To avoid such ill-conditioned matrices, we follow Bloom [1] and define so-called log-preserving unitary matrices. We have the following immediate corollary to Theorem 2.3. We now prove that certain families of matrix weights associated with log-preserving matrices satisfy the A 2 -decomposition property.
with the corresponding A 2 -constant depending only on U and K . Put and notice that the spectrum of B γ is bounded, and bounded from below. The decomposition now provided the desired A 2 -decomposition.

The Distance to a Matrix in BMO
We conclude the paper by a study of the BMO distance to L ∞ for matrices related to weights in A 2 (T d ). For a matrix weight M : where we notice that ε(M) ≤ 1. In the scalar case, using a reverse Hölder estimate, one can deduce that ε is always strictly greater than one, but this self-improving type result is known to fail for matrix weights [1]. We introduce two notions of BMO distance to L ∞ for a matrices.     4 Bloom's result [1,Theorem 4.9] is proved for d = 1, but one can verify that the proof can be adapted to the case d > 1. For the sake of brevity, we leave the details for the reader.
We now turn to the main result of this section.