Wavelets and the angle between past and future

S. Treil^{∗} and A. Volberg^{†}

1. Introduction.

In this paper we consider the weighted norm inequalities with matrix weight. Namely, let W be a d × d matrix weight, i.e. a L^{1} function whose values are selfadjoint nonnegative d × d matrices. We suppose that the weight W is defined on a set R where R is either the unit circle 𝕋 = {z ∈ ℂ : |z| = 1} or the real line ℝ.

Let L^{2} = L^{2}(ℂ^{d}) be the space of square summable functions on R with values in ℂ^{d}, H^{2} = H^{2}(ℂ^{d}) be the corresponding Hardy space of analytic functions and P_{+} be the orthogonal projection in L^{2} onto H^{2}. Let T denote the Hilbert Transform, T = −iP_{+} + i(I − P_{+}). The Hilbert Transform admits well known singular integral representation, for example on the real line ℝ

T f(s) = \frac{1}{π} p.v. ∫_{ℝ} \frac{f(t)}{t − s}dt .

The question we are interested in is under what conditions on W the following weighted norm inequality for T holds (say for all f ∈ L^{2} ∩ L^{∞}):

∫_{R}(W (t)T f(t), T f(t))dt ≤ C ∫_{R}(W (t)f(t), f(t))dt

Clearly this inequality is equivalent to the same inequality for P_{+} (with another constant) .

If we define a weighted space L^{2}(W) as the space of all measurable ℂ^{d}-valued functions on R satisfying

∥f∥_{L^{2}(W)}^{2} =^{def} ∫_{R}(W (t)f(t), f(t))dt < ∞ ,

(of course we should factorize it over the subspace of functions of norm 0), then the last inequality means that T (or, equivalently P_{+}) is a bounded operator in L^{2}(W).

If the dimension d = 1 and everything is scalar-valued the answer is given by the famous Hunt — Muckenhoupt — Wheeden theorem, that says that the Muckenhoupt condition (A_{2})

sup_{I} \frac{1}{|I|} ∫_{I} W · \frac{1}{|I|} ∫_{I} W^{−1} < ∞

∗Partially supported by NSF grant DMS 9304011

†Partially supported by NSF grant DMS 9302728

(supremum is taken over all intervals) is necessary and sufficient in this case.

The main result of the article is that the vector Muckenhoupt condition (A_{2}),

sup_{I} ∥(\frac{1}{|I|} ∫_{I} W dm)^{1/2} (\frac{1}{|I|} ∫_{I} W^{−1}dm)^{1/2}∥ < ∞

is necessary and sufficient for the boundedness of Hilbert Transform in L^{2}(W) with matrix weight. This was conjectured by the first author in [4], and the conjecture turns out to be true.

As a byproduct of our techniques we obtain some results on unconditional basis of wavelets in the weighted space L^{2}(W). We believe that results in such generality are new even in scalar-valued case, as for matrix-valued case we have not seen anything about it in the literature.

In this introduction we shall try to answer the following 3 questions the reader could have already asked him(her)self.

1. Why is this problem interesting ?

2. Why is this problem difficult? Why cannot it be easily reduced to the scalar-valued case?

3. What are connections with other areas?

1.1. Origin of the problem.

We think nobody needs to be convinced of the importance of the scalar case (Hunt — Muckenhoupt – Wheeden theorem). One can probably think of many ways to generalize their result to vector case, and we want to convince the reader that our generalization is a “right” one. And we think the simplest way to do that is to explain the origin of the problem.

The problem has two origins (at least for us). The first one is the theory of stationary processes.

Let us consider a multivariate random stationary process. For the simplicity we consider the case of discrete time. Let W be the spectral measure of the process; in our case it is a measure whose values are d × d nonnegative selfadjoint matrices. The reader can think of it as of a matrix whose entries are complex measures μ_{i,j} such that for any Borel set E the matrix {μ_{i,j}(E)}_{i,j=1}^{d} is nonnegative.

The geometry of the process is described by the geometry of the sequence of subspaces z^{n}ℂ^{d}, n ∈ ℤ in a weighted space L^{2}(W) = L^{2}(W, ℂ^{d}). The space L^{2}(W) consists of all functions on 𝕋 with values in ℂ^{d} such that

∥f∥_{L^{2}(W)}^{2} =^{def} ∫_{R}(dW(t)f(t), f(t)) < ∞ .

In this representation the past of the process is span {z^{n}ℂ^{d} : n < 0} and the future is span {z^{n}ℂ^{d} : n ≥ 0}; the angle between past and future is nonzero means simply that the Riesz projection P_{+} is bounded in the weighted space L^{2}(W). This property for stationary Gaussian processes (the angle between past and future is nonzero) is called in probability literature uniform mixing of past and future.

If the angle between past and future is nonzero, then for any vector e ∈ ℂ^{d} the angle between subspaces span {z^{n}e : n < 0} and span {z^{n}e : n ≥ 0} in L^{2}(W) is nonzero. If a measure μ is defined on Borel sets E by μ(E) = (W(E)e, e) the last condition means that the angle between antianalytic polynomials span {z^{n} : n < 0} and analytic polynomials span {z^{n} : n ≥ 0} in the (scalar) weighted space L^{2}(μ) is positive. Equivalently, we can say that the Riesz Projection P_{+} (or Hilbert Transform T) is bounded in the weighted space L^{2}(μ). It is well known that this is possible if and only if the measure μ is absolutely continuous and its density w satisfies the (scalar) Muckenhoupt condition (A_{2}).

Therefore if the angle between past and future is positive, the spectral measure W of the process is necessarily absolutely continuous, and the question about the angle gives rise to our problem.

Similarly, if we consider stationary processes with continuous time, we arrive at the problem about Hilbert Transform on ℝ.

Another motivation for the problem comes from the theory of Toeplitz operators. Let us consider a Toeplitz operator T_{F} with symbol F, T_{F} : H^{2}(ℂ^{d}) → H^{2}(ℂ^{d}),

T_{F} f = P_{+}F f , f ∈ H^{2}(ℂ^{d}) .

Here F is a bounded function whose values are d × d matrices.

It is well known, see [6] that the Toeplitz operator T_{F} is invertible if and only if its symbol F can be factorized as F = G_{1}^{∗}G_{2}, where G_{1}, G_{2} are H^{2} d × d matrix-functions (H^{2} means with entries in H^{2}), such that G_{1}^{−1}, G_{2}^{−1} ∈ H^{2} and such that the operator

G_{2}^{−1}P_{+}(G_{1}^{−1})^{∗}  (1.1)

is bounded on H^{2}. We would like to point out that the operator (1.1) is the (formal) inverse to T_{F}, so the above result is very natural.

Since for any f ∈ (H^{2})^{⊥} we have G_{2}^{−1}P_{+}(G_{1}^{−1})^{∗}f = 0, we can conclude that the operator (1.1) is bounded on H^{2} if and only if it is bounded on L^{2}. If we denote W = G_{1}G_{1}^{∗}, V = (G_{2}^{−1})^{∗}G_{2}^{−1} then the boundedness of the operator (1.1) on L^{2} is equivalent to the following weighted norm inequality

∫_{R}(V (t)P_{+}f(t), P_{+}f(t))dt ≤ C ∫_{R}(W (t)f(t), f(t))dt  (1.2)

It is trivial and well known that given an invertible Toeplitz operator T_{F}, its symbol F has to be invertible in L^{∞}. Therefore the weights V and W are equivalent in a sense that there exists a constant C such that for all e ∈ ℂ^{d} and for almost all t ∈ R

\frac{1}{C} (V (t)e, e) ≤ (W (t)e, e) ≤ C(V (t)e, e) .

That means that we can replace in (1.2) V by W or vice versa, so we again arrive at our problem.

1.2. Specifics of the matrix-valued case.

We would like to mention that the problem can not be reduced to the scalar case. The proof for the scalar case can not be reproduced to matrix valued weights. The main reason is that the original proof by Hunt — Muckenhoupt — Wheeden [1], and all its modifications (see [2, 3]) heavily use maximal functions. And it is very difficult for us to imagine how can one introduce working maximal functions for matrix-valued weights. Indeed, for scalar weighted spaces we have a very simple but wonderful fact that a function f belongs L^{p}(μ) if and only if |f| ∈ L^{p}(μ). And we do not have an analogue of this for matrix-valued weights, even for p = 2.

As an illustration what kind of difficulties one can encounter in the matrix-valued case let us present a very simple example. It is trivial in the scalar case that if we have an integral operator in L^{2}(μ) with positive (scalar) kernel, and we know that an operator with a bigger kernel is bounded, then the original operator is bounded too. This statement (even for scalar kernels) does not hold for weighted L^{2}-spaces with matrix weights (at least we do not know a proof of this fact).

1.3. Main idea.

The main idea of the proof is very simple. We choose an appropriate basis and show that the matrix of the operator generates a bounded operator in ℓ^{2}. This idea itself is nothing new — it is a standard idea in the wavelet technique.

We would like to point out to two details that distinguish our approach. The first is that the basis system we use is just usual (normalized) Haar system. We believe that for our problem the Haar system is the most appropriate one. Let us mention that at first glance the weighted Haar system like the one used in the solution of Calderon problem [9], or smooth wavelets would be more appropriate. However it is not the case, and the reader can easily see this after he works some time with systems and matrices. In particular, it was shown in [9] that the weighted Haar system forms a Riesz basis in the non-weighted L^{2}, but this is not true for our weights (even in scalar case). In our case the weighted Haar system is not even minimal.

The other distinction is that we do not use the Schur test to prove the boundedness of the operator in the sequence space ℓ^{2}. We certainly would prefer to use this test (because it is well known, simple and actually gives necessary and sufficient condition to positive kernels) but we encounter very serious difficulties here. The Schur test multipliers that cancel out in scalar case do not cancel in the matrix case, because of a certain matrix between them. So we have to use another and less known test, the so called Senichkin — Vinogradov test ^{1} and some of its modifications, see [7] (see also Theorems 6.3, 7.3, 7.4 in the text).

1After the paper was submitted it was pointed to us by E. Stein that such “dohbling kernel” trick was used long before by different authors. The paper by A. N. Kolmogorov and G. Silvestrov [11] is the earliest known to us reference

The basic idea of the proof lies entirely in the vein of wavelets theory, but the essential technicalities seem to be new in this context, and we never met them in the literature.

We present the proof only for the real line ℝ, but one can easily see that (almost) the same proof works just fine for the circle 𝕋.

Note that we consider only matrix-valued case. We do not know whether the results are true for operator-valued weights (d = ∞).

1.4. Acknowledgments.

We are grateful to Michael Frazier and Igor Verbitsky for encouraging and valuable discussions.

2. Matrix Muckenhoupt condition.

Let W be a matrix-valued weight. We say that the weight W satisfies matrix Muckenhoupt condition (A_{2}) (or simply Muckenhoupt condition (A_{2})) if

sup_{I} ∥(\frac{1}{|I|} ∫_{I} W dm)^{1/2} (\frac{1}{|I|} ∫_{I} W^{−1}dm)^{1/2}∥ < ∞

where the supremum is taken over all intervals I ⊂ ℝ. If W is a usual scalar function, this condition is simply the well known Muckenhoupt condition (A_{2}).

In this section we show that this condition is necessary for the Hilbert transform T (or, equivalently, the Riesz Projection P_{+} to be bounded in the weighted space L^{2}(W).

Let W be an d × d matrix weight. Throughout the paper we always assume that the weight W is non-degenerate in the sense that there is no vector e ∈ ℂ^{d} such that W (t)e = 𝕆 a.e., because otherwise we can always restrict ourselves to the orthogonal complement of such e.

The following lemma holds for an arbitrary measure space X, although we shall apply it for the real line ℝ or its subsets endowed with Lebesgue measure. For this reason we still use the notation dt for the measure.

Let k be a scalar-valued function in L^{2} ∪ L^{∞}, ∥k∥_{2} = 1, K be a subspace of L^{2}(ℂ^{d}).

K =^{def} kℂ^{d} = {ke : e ∈ ℂ^{d}} ,

and let P denotes the orthogonal projection (in the non-weighted space L^{2} = L^{2}(ℂ^{d})) onto K. It is easy to see that P is given by the formula

P f = k · (∫_{X} f(t)\overline{k(t)}dt) .

Lemma 2.1. Given a d × d matrix weight W, the projection P introduced above extends (from a dense set L^{2} ∩ L^{2}(W)) to a bounded operator on L^{2}(W) if and only if both weights W |k|^{2} and W^{−1}|k|^{2} belong L^{1}. In this case the norm of P in L^{2}(W) is exactly

∥[∫ W |k|^{2}]^{1/2} [∫ W^{−1}|k|^{2}]^{1/2}∥ .

Proof. First note that if P extends continuously to a bounded operator in L^{2}(W) then W |k|^{2} ∈ L^{1}. Indeed, if W |k|^{2} ∉ L^{1}, one can find a vector e ∈ ℂ^{d} such that

∫_{X} (W (t)e, e)|k(t)|^{2}dt = ∞ .

The latter means that ke ∉ L^{2}(W). It is a trivial exercise to find a scalar function f ∈ L^{2} such that fe ∈ L^{2}(W) and (f, k)_{L^{2}} = α ≠ 0. Then P fe = αke ∉ L^{2}(W), and so P is not bounded in L^{2}(W).

Let now W |k|^{2} ∈ L^{1}. Denote by L the orthogonal complement of K in L^{2}, and let L_{0} =^{def} L ∩ L^{2}(W). For any vector e ∈ ℂ^{d}

dist_{L^{2}(W)}{ke, L_{0}} = dist_{L^{2}}{W^{1/2}ke, W^{1/2}L_{0}} =
= sup { (W^{1/2}ke, g) : g ∈ (W^{1/2}L_{0})^{⊥} , ∥g∥ = 1} .

Suppose for a moment that the weight W^{−1} exists and is uniformly bounded. Then it is easy to see that (W^{1/2}L_{0})^{⊥} = W^{−1/2}K So we can continue the above chain of equalities

dist_{L^{2}(W)}(ke, L_{0}) = sup {|(W^{1/2}ke, W^{−1/2}kf )_{L^{2}} | : f ∈ ℂ^{d} , ∥W^{−1/2}kf∥ ≤ 1 } =

= sup {|(ke, kf)_{L^{2}}| : f ∈ ℂ^{d} , ∫_{ℝ}(W^{−1}(t)f, f) |k(t)|^{2} dt ≤ 1 } =
= sup {|(e, f)_{L^{2}}| : f ∈ ℂ^{d} , ([∫ W^{−1}|k|^{2}] f, f) ≤ 1 } =

= sup {|(e, [∫ W^{−1}|k|^{2}]^{−1/2} g)_{L^{2}} | : g ∈ ℂ^{d} , ∥g∥ ≤ 1 } = ∥[∫ W^{−1}|k|^{2}]^{−1/2} e∥

So we have proved that under the assumption W^{−1} ∈ L^{∞}

dist_{L^{2}(W)}(ke, L_{0}) = ∥[∫ W^{−1}|k|^{2}]^{−1/2} e∥ .  (2.1)

Let now W be a general weight. Define W_{ε} =^{def} W +εI. As it was shown above (2.1) holds for W_{ε}. Since for any f ∈ K + L_{0} its weighted norm ∥f∥_{L^{2}(W_{ε})} decreases as ε decreases, and has limit ∥f∥_{L^{2}(W)} as ε → 0, it follows that

dist_{L^{2}(W)}(ke, L_{0}) = lim_{ε→0+} dist_{L^{2}(W_{ε})}(ke, L_{0}) = lim_{ε→0+} [∫_{ℝ}(W_{ε}^{−1}(t)e, e)|k|^{2} dt]^{1/2}

This implies that if dist_{L^{2}(W)}(ke, L_{0}) is nonzero for all nonzero e, then W^{−1}|k|^{2} ∈ L^{1} and formula (2.1) holds. But, as it will be shown below, that is exactly the case when P admits continuous extension to L^{2}(W).

It follows from elementary geometry of Hilbert spaces that

∥P ∥_{L^{2}(W)} = sup \{\frac{∥ke∥_{L^{2}(W)}}{dist_{L^{2}(W)}(ke, L_{0})} : e ∈ ℂ^{d}, e ≠ 0 \} =

= sup \begin{cases}∥ & [∫ W |k|^{2}] & 1/2 & e & ∥ \\ ∥ & [∫ W^{−1}|k|^{2}] & −1/2 & e & ∥ \end{cases} : e ∈ ℂ^{d}, e ≠ 0 \} =

= sup {∥[∫ W |k|^{2}]^{1/2} [∫ W^{−1}|k|^{2}]^{1/2} f∥ : f ∈ ℂ^{d}, ∥f∥ = 1 } =

= ∥[∫ W |k|^{2}]^{1/2} [∫ W^{−1}|k|^{2}]^{1/2}∥ .

and therefore the part “only if” of the lemma is proved.

To prove the “if” part of the lemma one has to reverse the reasoning above. □

Let k_{λ} be the normalized reproducing kernel

k_{λ} =^{def} (\frac{2 Im λ}{π})^{1/2} \frac{1}{(z − \overline{λ})} , λ ∈ ℂ_{+} .

Note that |k_{λ}(t)|^{2} is the Poisson kernel for ℂ_{+}, i.e. the harmonic extension W (λ) of a function W at the point λ ∈ ℂ_{+} is given by the formula

W (λ) = ∫_{ℝ} W (t)|k_{λ}(t)|^{2}dt .

Lemma 2.2. Assume the Riesz projection P_{+} is bounded in the weighted space L^{2}(W). Then

sup_{λ∈ℂ_{+}} ∥[W (λ)]^{1/2} [W^{−1}(λ)]^{1/2}∥ < ∞  (2.2)

where W (λ), W^{−1}(λ) denote harmonic extensions at the point λ of W and W^{−1} respectively. The latter condition in its turn implies the matrix Muckenhoupt condition

sup_{I} ∥(\frac{1}{|I|} ∫_{I} W dm)^{1/2} (\frac{1}{|I|} ∫_{I} W^{−1}dm)^{1/2}∥ < ∞  (2.3)

where supremum is taken over all intervals I ⊂ ℝ.

Proof. Define a subspace K_{λ},

K_{λ} =^{def} k_{λ}ℂ^{d} = {k_{λ}e : e ∈ ℂ^{d}} ,

and let P_{λ} denotes the orthogonal projection (in the non-weighted space L^{2} = L^{2}(ℂ^{d})) onto K_{λ}. The operator P_{λ} can be represented as

P_{λ}f = P_{+}f − b_{λ}P_{+}\overline{b}_{λ}f , f ∈ L^{2}(ℂ^{d}) ,

where b_{λ} stands for an elementary Blaschke factor with zero at λ ∈ ℂ_{+}, b_{λ} =^{def} (z −λ)/(z −\overline{λ}).

Since P_{+} is bounded in L^{2}(W) and the multiplication by a unimodular function b_{λ} is a unitary operator there, operators P_{λ}, λ ∈ ℂ_{+} are uniformly bounded. Applying Lemma 2.1 and taking into account that

∫ W |k_{λ}|^{2} = W (λ) ,  and  ∫ W^{−1}|k_{λ}|^{2} = W^{−1}(λ) ,

we can obtain (2.2).

Condition (2.2) can be rewritten as

sup_{λ∈ℂ_{+}} sup \begin{cases}∥ & [W (λ)] & 1/2 & e & ∥ \\ ∥ & [W^{−1}(λ)] & −1/2 & e & ∥ \end{cases} : e ∈ ℂ^{d}, e ≠ 0 \} < ∞ .  (2.4)

Fix a point λ ∈ ℂ_{+} and denote by I = I_{λ} the interval [Re λ − Im λ, Re λ + Im λ] ⊂ ℝ. Since for the Poisson P_{λ}(t) kernel we have on the interval I

P_{λ}(t) = \frac{Im λ}{π|t − \overline{λ}|^{2}} ≥ \frac{1}{π Im λ} = \frac{2}{π|I|} ,

the following inequality holds for any matrix weight W:

W_{I} =^{def} \frac{1}{|I|} ∫_{I} W ≤ \frac{π}{2} W (λ) , I = I_{λ};

the inequality means usual inequality for quadratic forms. Therefore for I = I_{λ}

\frac{(W_{I}e, e)}{([(W^{−1})_{I}]^{−1} e, e)} ≤ (\frac{π}{2} )^{2} \frac{(W (λ)e, e)}{([W^{−1}(λ)]^{−1} e, e)}  (2.5)

Since

∥(W_{I})^{1/2} ((W^{−1})_{I})^{1/2}∥ = sup \begin{cases}∥ & [W_{I}] & 1/2 & e & ∥ \\ ∥ & [(W^{−1})_{I}] & −1/2 & e & ∥ \end{cases} : e ∈ ℂ^{d}, e ≠ 0 \} < ∞ ,

inequalities (2.4) and (2.5) imply (2.3). □

3. Properties of matrix Muckenhoupt weights.

First we will prove some lemmas about general (not necessary Muckenhoupt) matrix weights.

Lemma 3.1. Let W be a nonnegative, measurable, d × d matrix function on a measure space X , μ. Then

∫_{X} ∥W (t)∥dμ(t) ≤ d ∥∫_{X} W (t)dμ(t)∥

Proof. Since for a nonnegative d × d matrix A we have ∥A∥ ≤ trace A ≤ d∥A∥, we can conclude that

∫_{X} ∥W (t)∥dμ(t) ≤ ∫_{X} trace (W (t))dμ(t) = trace (∫_{X} W (t)dμ(t)) ≤ d · ∥∫_{X} W (t)dμ(t)∥ .

□

Given a matrix weight W and a set I ⊂ ℝ let us denote by W (I) the integral W (I) =^{def} ∫_{I} W (t)dt and by W_{I} the average value W_{I} =^{def} W (I)/|I|.

Lemma 3.2. Let W be a d×d matrix-valued weight such that W and W^{−1} are summable on a measurable set I. Then for any vector e ∈ ℂ^{d}

\frac{(W_{I}e, e)}{([(W^{−1})_{I}]^{−1} e, e)} ≥ 1 .

Proof. The statement of the lemma has very simple geometrical interpretation. The numerator (W_{I}e, e) is exactly the norm of a function identically equal e in the weighted space L^{2}(I, W), and the denominator ([(W^{−1})_{I}]^{−1} e, e) is the distance (in the same space) from this function to the space of all (vector)-functions satisfying ∫_{I} f(t)dt = 0. The formal proof that we preset below, is, in essence, contained in the proof of Lemma 2.1. The vector f that we use here is the same vector f from the proof of Lemma 2.1 (up to the notation).

Define a vector f = [W^{−1}(I)]^{−1} e. Then

|I| · ([W^{−1}(I)]^{−1} e, e) = |I|(e, f) = ∫_{I} (W^{1/2}(t)e, W^{−1/2}(t)f ) dt ≤
≤ (∫_{I} (W (t)e, e) dt))^{1/2} (∫_{I} (W^{−1}(t)f, f ) dt)^{1/2} =
= (W (I)e, e)^{1/2} (W^{−1}(I)f, f)^{1/2} = (W (I)e, e)^{1/2} ([W^{−1}(I)]^{−1} e, e)^{1/2}

Therefore

|I|^{2} ≤ \frac{(W (I)e, e)}{([W^{−1}(I)]^{−1} e, e)} .

The latter inequality is equivalent to the statement of the lemma. □

Corollary 3.3. For the matrix weight W from Lemma 3.2 above the operator A = [W_{I}]^{1/2}[(W^{−1})_{I}]^{1/2} is an expanding operator, i.e. ∥Ae∥ ≥ ∥e∥ for all vectors e.

Proof. Follows immediately from Lemma 3.2. □

And now some properties of (A_{2}) weights. First of all note that (A_{2}) condition is symmetric, that means a matrix weight W satisfies (A_{2}) if and only if W^{−1} does.

It is well-known that usual scalar A_{2}-weights satisfy the doubling property. A similar property holds for matrix A_{2}-weights.

Lemma 3.4. Let W satisfy the vector condition (A_{2}). There exists a constant C = C(W) such that for any e ∈ ℂ^{d} and any intervals I ⊂ J, |J| ≤ 2|I|, the following inequality

∫_{J}(W (t)e, e)dt ≤ C ∫_{I}(W (t)e, e)dt.

holds.

Proof. Lemma 3.2 implies the first inequality in the following chain:

|I|^{2} ≤ \frac{(W (I)e, e)}{([W^{−1}(I)]^{−1} e, e)} ≤ \frac{(W (J)e, e)}{([W^{−1}(I)]^{−1} e, e)} ≤ \frac{(W (J)e, e)}{([W^{−1}(J)]^{−1} e, e)} ≤ A|J|^{2} ≤ 4A|I|^{2}

(the fourth inequality in the chain is simply a Muckenhoupt condition). So

|I|^{2} ≤ \frac{(W (I)e, e)}{([W^{−1}(I)]^{−1} e, e)} ≤ \frac{(W (J)e, e)}{([W^{−1}(I)]^{−1} e, e)} ≤ 4A|I|^{2}

and hence

\frac{(W (J)e, e)}{(W (I)e, e)} ≤ 4A .

□

Lemma 3.5. Let W be a d × d matrix Muckenhoupt weight, and A be a positive nonsingular d × d (constant) matrix. Then the weight W =^{def} AW A also satisfies the Muckenhoupt condition (A_{2}) with the same constant.

Proof. Clearly, W_{I} = AW_{I}A and (W^{−1})_{I} = A^{−1}(W^{−1})_{I}A^{−1}. Operator B = (W_{I})^{1/2} A is a Hermitian square root of W_{I}, that means B^{∗}B = W_{I}. Therefore we can write a polar decomposition for the operator B = (W_{I})^{1/2} A = U (W_{I})^{1/2}, where U is a unitary matrix. Similarly, for an operator C = ((W^{−1})_{I})^{1/2} A^{−1} one can write C = V ((W^{−1})_{I})^{1/2}, where V again is a unitary matrix.

Therefore

BC^{∗} = (W_{I})^{1/2} ((W^{−1})_{I})^{1/2} = U (W_{I})^{1/2} ((W^{−1})_{I})^{1/2} V^{∗} ,

and so

∥(W_{I})^{1/2} ((W^{−1})_{I})^{1/2}∥ = ∥(W_{I})^{1/2} ((W^{−1})_{I})^{1/2}∥ .
□

Lemma 3.6. For a d × d matrix Muckenhoupt weight W the following inequality holds;

\frac{1}{|I|} ∫_{I} ∥W ∥ ≤ C (\frac{1}{|I|} ∫_{I} ∥W ∥^{1/2})^{2}

(Clearly, since d < ∞ the norm here can be replaced by trace, Hilbert–Schmidt norm, or any S_{p} norm.)

Proof. One can easily see from Lemma 2.1 (with k = |I|^{−1/2}χ_{I}) that the Muckenhoupt condition is equivalent to the uniform boundedness of all averaging operators

f ↦ χ_{I} \frac{1}{|I|} ∫_{I} f, f ∈ L^{2}(ℂ^{d}) .

Fix any direction (unit vector) e ∈ ℂ^{d}, and consider the restrictions of these operators on the subspace {fe : f ∈ L^{2}}. It follows that for any e ∈ ℂ^{d}, ∥e∥ = 1 the scalar weight w(t) = (W (t)e, e) is a Muckenhoupt weight with the same or better constant as W. So any diagonal element of W is a Muckenhoupt weight (with the control over the constant).

It is very easy to see, that if d scalar weights satisfy the Muckenhoupt condition with the constant at most C, then their sum also satisfy the Muckenhoupt condition with the constant at most dC. Therefore trace W is a Muckenhoupt weight.

The conclusion of the lemma is well known (and now almost trivial) for scalar weights, so

\frac{1}{|I|} ∫_{I} trace W ≤ C (\frac{1}{|I|} ∫_{I}(trace W)^{1/2})^{2}

Since d < ∞ we can replace trace by the norm. □

4. A Carleson measure.

The main result of this section is the following theorem about matrix Muckenhoupt weights. Given a dyadic interval I let Q_{I} denote the corresponding “Carleson square”, Q_{I} =^{def} {z ∈ ℂ_{+} : Re z ∈ I, 0 ≤ Im z ≤ |I|}, and let λ(I) be the middle point of the upper side of Q_{I}, i.e. Im λ(I) = |I| and Re λ(I) be the center of I. Let δ_{λ} denotes the unit mass at the point λ. By I_{1} and I_{2} we denote the left and the right halves of I respectively.

Theorem 4.1. Let W be a d × d Muckenhoupt weight. Then the measure

M = ∑_{I∈D} ∥(W_{I})^{−1/2}(W_{I_{1}} − W_{I_{2}})(W_{I})^{−1/2}∥^{2} |I|δ_{λ(I)}

is Carleson. That means there exists a constant C such that

∑_{I∈D, I⊂J} ∥(W_{I})^{−1/2}(W_{I_{1}} − W_{I_{2}})(W_{I})^{−1/2}∥^{2} |I| ≤ C|J|  (4.1)

for all dyadic intervals J

Remark. Formally speaking, the theorem claims that the measure M is a dyadic Carleson measure, since in the definition of Carleson measure the interval J in (4.1) is an arbitrary, not necessarily dyadic, interval. But any interval can be covered by at most 2 dyadic intervals of lengths comparable with the length of the interval itself, so a dyadic Carleson measure is a usual Carleson measure.

For (A_{∞}) scalar weights this was proved in [10]. Our proof is formally different. We do not touch (A_{∞}) matrix weights, but for scalar (A_{∞}) weights our proof also works.

To prove the theorem we need some lemmas.

Lemma 4.2. Let A and B be nonsingular positive d × d matrices. Then

\sqrt{det A det B} ≤ det (\frac{A + B}{2} )

Proof. If the lemma holds for some matrices A and B, it holds for all matrices of form C^{∗}AC, C^{∗}BC — this is because determinant is multiplicative. Taking C = [(A + B)/2]^{−1/2} one can see that it is sufficient to prove the lemma only in the case (A + B)/2 = I.

Let now (A + B)/2 = I. Then A = I + Δ, B = I − Δ, Δ = Δ^{∗}. Let α_{1}, α_{2}, …, α_{d} be eigenvalues of Δ. Then the eigenvalues of A and B are 1 + α_{1}, 1 + α_{2}, …, 1 + α_{d} and 1 − α_{1}, 1 − α_{2}, …, 1 − α_{d} respectively. Since (1 + α_{k})(1 − α_{k}) = 1 − α_{k}^{2} ≤ 1,

det(AB) = det A det B = ∏_{k=1}^{d}(1 + α_{k})(1 − α_{k}) ≤ 1 = [det (\frac{A + B}{2} )]^{1/2}

□

Proof of Theorem 4.1. For a dyadic interval I let us denote

μ(I) =^{def} det W_{I} , ν(I) =^{def} det(W^{−1})_{I} , m(I) =^{def} μ(I)ν(I) = det (W_{I})^{1/2}(W_{I}^{−1})(W_{I})^{1/2}) .

Let I_{1} and I_{2} be the right and left halves of the interval I. Since W_{I} = (W_{I_{1}} + W_{I_{2}})/2, Lemma 4.2 implies μ(I)^{2} ≥ μ(I_{1})μ(I_{2}) and similarly, ν(I)^{2} ≥ ν(I_{1})ν(I_{2}). Let us estimate the ratio

\frac{μ(I_{1})μ(I_{2})}{μ(I)^{2}} = \frac{det W_{I_{1}} det W_{I_{2}}}{(det W_{I})^{2}} = det ((W_{I})^{−1/2}W_{I_{1}}(W_{I})^{−1/2}) det ((W_{I})^{−1/2}W_{I_{2}}(W_{I})^{−1/2})

Denote

A =^{def} (W_{I})^{−1/2}W_{I_{1}}(W_{I})^{−1/2} , B =^{def} (W_{I})^{−1/2}W_{I_{2}}(W_{I})^{−1/2} .

Since (A + B)/2 = I, we can write A = I + Δ, B = I − Δ. Let α_{1}, α_{2}, …, α_{d} be the eigenvalues of Δ, then

det A det B = ∏_{k=1}^{d}(1 + α_{k}) · ∏_{k=1}^{d}(1 − α_{k}) = ∏_{k=1}^{d}(1 − α_{k}^{2}) = exp (∑_{k=1}^{d} log(1 − α_{k}^{2})) ≤

≤ exp (− ∑_{k=1}^{d} α_{k}^{2}) = exp (−trace (Δ^{2})) = exp (−\frac{1}{4}trace ((A − B)^{2}))

So we have proved that

μ(I) ≥ (μ(I_{1})μ(I_{2}))^{1/2} exp \{\frac{1}{2} · \frac{1}{4}trace (((W_{I})^{−1/2}(W_{I_{1}} − W_{I_{2}})(W_{I})^{−1/2})^{2})\}

Taking into account that ν(I)^{2} ≥ ν(I_{1})ν(I_{2}) we get

m(I) ≥ (m(I_{1})m(I_{2}))^{1/2} exp \{\frac{1}{2} · \frac{1}{4}trace ([(W_{I})^{−1/2}(W_{I_{1}} − W_{I_{2}})(W_{I})^{−1/2}]^{2})\}  (4.2)

(let us remind that m(I) =^{def} μ(I)ν(I) ).

Applying (4.2) to I_{1}, I_{2}, then to halves of these intervals we get on the nth step

m(I) ≥ [ ∏_{|J|=|I|2^{−n}^{J∈D: J⊂I,}} m(J)]^{1/2^{n}} exp (\frac{1}{8} ∑_{|J|>|I|2^{−n}^{J∈D: J⊂I,}} \frac{|J|}{|I|}trace ([(W_{J})^{−1/2}(W_{J_{1}} − W_{J_{2}})(W_{J})^{−1/2}]^{2}))

Since by Corollary 3.3 operators (W_{J})^{1/2}((W^{−1})_{J})^{1/2} are expanding, it follows that m(J) ≥ 1. On the other hand m(I) ≤ C (Muckenhoupt condition), so

∑_{|J|>|I|2^{−n}^{J∈D: J⊂I,}} \frac{|J|}{|I|}trace ([(W_{J})^{−1/2}(W_{J_{1}} − W_{J_{2}})(W_{J})^{−1/2}]^{2}) ≤ C

for all n (C does not depend on n and I). The latter is exactly the conclusion of the lemma. □

5. Construction of a vector Haar system.

5.1. Preliminaries about bases of subspaces.

Consider a system of subspaces E_{n} in a Hilbert H space. For simplicity let us suppose that the system is complete, i.e. the only vector orthogonal to all E_{n} is 0.

Let us recall that the system is called minimal if there exists a family of bounded (not necessary orthogonal) projections E^{n}

E^{n}f = \{ \begin{matrix}f, & f ∈ E_{n} \\ 0, & f ∈ E_{k} , k ≠ n . \end{matrix}

and is called uniformly minimal if projections E^{n} are uniformly bounded.

For a minimal system of subspaces E_{n} one can define the so-called biorthogonal or dual system E′_{n} =^{def} (E^{n})^{∗}H. Clearly E′_{n} = span {E_{k} : k ≠ n}^{⊥}.

A complete system of subspaces E_{n} is called a Riesz basis or unconditional basis if there exists an isomorphism from H onto another Hilbert space H′ that maps the system E_{n} into an orthogonal system. Such an isomorphism is called the orthogonalizer of the system. Another equivalent definition: a system of subspaces E_{n} is a Riesz basis if there exists a constant C such that for any (finite) collection of vectors f_{n} ∈ E_{n}

\frac{1}{C} ∑ ∥f_{k}∥^{2} ≤ ∥∑ f_{k}∥^{2} ≤ C ∑ ∥f_{k}∥^{2}

A system of vectors f_{n} is said to be an unconditional basis if the corresponding system of one-dimensional spaces is an unconditional basis, and we call the system of vectors a Riesz basis if, in addition, it is almost normalized, i.e.

0 < inf_{n} ∥f_{n}∥ ≤ sup_{n} ∥f_{n}∥ < ∞ .

In this case the norm ∥ ∑ c_{k}f_{k}∥ is equivalent to ℓ^{2}-norm of the sequence {c_{n}}_{n}. We need the following well known fact about Riesz bases, see [7].

Theorem 5.1 A complete system of subspaces E_{n} is a Riesz basis if and only if it is uniformly minimal and the following two “imbedding theorems” hold:

1. ∑ ∥P_{E_{n}}f∥^{2} ≤ C∥f∥^{2} for all f ∈ H;

2. ∑ ∥P_{E′_{n}}f∥^{2} ≤ C∥f∥^{2} for all f ∈ H.

5.2. Construction of a vector Haar system.

Let D denote the set of all dyadic intervals on ℝ. Given a dyadic interval I ∈ D let I_{1} and I_{2} denote its left and right halves respectively, and let h_{I} be a scalar Haar function, h_{I} = |I|^{−1/2}(χ_{I_{2}} − χ_{I_{1}}). Denote by e_{I}^{k}, k = 1, 2, ..., d and orthonormal basis of eigenvalues of W (I) (or, equivalently of W_{I}) in ℂ^{d}. Let

w_{I}^{k} = (\frac{1}{|I|} ∫_{I}(W (t)e_{I}^{k}, e_{I}^{k})dt)^{−1/2} = [(W_{I}e_{I}^{k}, e_{I}^{k})]^{−1/2} = [([W_{I}]^{−1} e_{I}^{k}, e_{I}^{k})]^{1/2} = ∥[W_{I}]^{−1/2} e_{I}^{k}∥ .

Then the system

f_{I}^{k} = w_{I}^{k}h_{I}e_{I}^{k} , I ∈ D, k = 1, 2, ..., d

is a normalized (∥f_{I}^{k}∥_{L^{2}(W)} = 1) complete system in L^{2}(W).

Let us recall that a complete system of vectors f_{n} in a Hilbert space is called a Riesz basis if there exists a constant C > 0 such that

\frac{1}{C} ∑_{n} |c_{n}|^{2} ≤ ∥ ∑_{n} c_{n}f_{n}∥^{2} ≤ C ∑_{n} |c_{n}|^{2}

for all finitely supported sequences c_{n}.

Theorem 5.2. Let the weight W satisfy vector Muckenhoupt condition (A_{2}). Then the system

f_{I}^{k} = w_{I}^{k}h_{I}e_{I}^{k} , I ∈ D, k = 1, 2, ..., d

is a Riesz basis in L^{2}(W).

Note that if the weight W satisfies the Muckenhoupt condition (A_{2}), the the system E_{I} is uniformly minimal. Indeed, the subspaces E_{I} are orthogonal in the non-weighted space L^{2}, so the corresponding projections P^{I} are simply orthogonal (in the non-weighted L^{2}) projections onto E_{I}. By Lemma 2.1 the norm of P^{I} is

∥(\frac{1}{|I|} ∫_{I} W )^{1/2} (\frac{1}{|I|} ∫_{I} W^{−1})^{1/2}∥ ≤ C < ∞

hence the system is uniformly minimal.

Proof of Theorem 5.2. Vectors f_{I}^{k}, k = 1, 2, …, d form an orthonormal (in L^{2}(W)) basis in E_{I}, and hence we know that to prove the theorem it is sufficient to show that the system of subspaces E_{I} forms a Riesz basis. Since we already now that the system of subspaces is uniformly minimal, it is sufficient to show that two imbedding theorems hold.

The system of functions E_{I} in the weighted space L^{2}(W) has the same geometry as the system W^{1/2}E_{I} in the non-weighted space L^{2}. It is easy to see that the biorthogonal to the latter system is W^{−1/2}E_{I}. So, due to the symmetry of the Muckenhoupt condition we only need to check one imbedding theorem.

To show that the imbedding theorem holds we need a modified Haar system which is orthogonal in the weighted space L^{2}(W). Namely, there exist vectors ẽ_{I}^{k} ∈ ℂ^{d} such that the system

g_{I}^{k} =^{def} w_{I}^{k}h_{I}e_{I}^{k} + χ_{I} ẽ_{I}^{k}

is orthogonal in L^{2}(W). Clearly, one only has to check that for all I and k

∫_{I}(W (t)g_{I}^{k}(t), f)dt = 0 ∀f ∈ ℂ^{d}

One can rewrite the above condition as

0 = |I|^{−1/2}w_{I}^{k} [(W (I_{2})e_{I}^{k}, f) − (W (I_{1})e_{I}^{k}, f)] − (W (I)ẽ_{I}^{k}, f) ,

and since this should hold for all f ∈ ℂ^{d} it follows that

ẽ_{I}^{k} = |I|^{−1/2}w_{I}^{k}W (I)^{−1}(W (I_{1}) − W (I_{2}))e_{I}^{k} .

Taking into account that w_{I}^{k} = ∥[W_{I}]^{−1/2} e_{I}^{k}∥ and e_{I}^{k} is an eigenvector of W_{I} we can rewrite

ẽ_{I}^{k} = |I|^{−1/2}W (I)^{−1}(W (I_{1}) − W (I_{2}))(W_{I})^{−1/2}e_{I}^{k} =

= \frac{1}{2}|I|^{−1/2}W_{I}^{−1}(W_{I_{1}} − W_{I_{2}})(W_{I})^{−1/2}e_{I}^{k} =^{def} A_{I}e_{I}^{k} ,

where

A_{I} =^{def} \frac{1}{2}|I|^{−1/2}W_{I}^{−1}(W_{I_{1}} − W_{I_{2}})(W_{I})^{−1/2}  (5.1)

The constructed functions g_{I}^{k} = f_{I}^{k}+χ_{I} ẽ_{I}^{k} are orthogonal to each other, but not necessarily normalized. It is possible to show that the system is almost normalized, that means

0 < inf_{I,k} ∥g_{I}^{k}∥_{W} ≤ sup_{I,k} ∥g_{I}^{k}∥_{W} < ∞

We shall prove only that sup_{I,k} ∥g_{I}^{k}∥_{W} < ∞, since we need only this inequality for our purposes. It is sufficient to show that the vectors ẽ_{I}^{k} we added can not be too big, namely that

∫_{I}(W (t)ẽ_{I}^{k}, ẽ_{I}^{k})dt = (W (I)A_{I}e_{I}^{k}, A_{I}e_{I}^{k}) ≤ C < ∞ .

or, recalling the formula for A_{I},

\frac{1}{2} ∥(W_{I})^{−1/2} (W_{I_{1}} − W_{I_{2}}) (W_{I})^{−1/2}e_{I}^{k}∥ ≤ C

But operators (W_{I})^{−1/2} (W_{I_{1}} − W_{I_{2}}) (W_{I})^{−1/2} are uniformly bounded. It follows, for example, from Theorem 4.1, because the norms of these operators forms a Carleson measure and therefore they have to be uniformly bounded.

There is also a simple direct proof of this fact. First of all W_{I_{1}} ≤ 2W_{I} since clearly W (I_{1}) ≤ W (I). Therefore

(W_{I})^{−1/2}W_{I_{1}}(W_{I})^{−1/2} ≤ 2(W_{I})^{−1/2}W_{I}(W_{I})^{−1/2} = 2I

and therefore ∥(W_{I})^{−1/2}W_{I_{1}}(W_{I})^{−1/2}∥ ≤ 2. Repeat the same for I_{2} and we are done.

Now we are going to reduce the theorem to some imbedding theorem, kind of weighted Carleson imbedding theorem. First of all, since g_{I}^{k} are orthogonal in L^{2}(W)

∑_{k=1,2,...,d^{I∈D,}} \frac{1}{∥g_{I}^{k}∥_{W}^{2}} |(f, g_{I}^{k})|^{2} ≤ ∥f∥_{W}^{2} , ∀f ∈ L^{2}(W) ,

and because ∥g_{I}^{k}∥_{W} are uniformly bounded

∑_{k=1,2,...,d^{I∈D,}} |(f, g_{I}^{k})|^{2} ≤ C∥f∥_{W}^{2} ,  ∀f ∈ L^{2}(W) .  (5.2)

We want to prove the first assertion of Theorem 5.1 for the system of subspaces E_{I}, or equivalently, since the vectors f_{I}^{k}, k = 1, 2, …, d form an orthonormal (in L^{2}(W)) basis in E_{I},

∑_{k=1,2,...,d^{I∈D,}} |(f, f_{I}^{k})_{W} |^{2} ≤ C∥f∥_{W}^{2} ,  ∀f ∈ L^{2}(W) .  (5.3)

Since g_{I}^{k} = f_{I}^{k} + A_{I}e_{I}^{k} and (5.2) holds, it is sufficient to prove that

∑_{k=1,2,...,d^{I∈D,}} |(f, χ_{I}A_{I}e_{I}^{k})_{W} |^{2} ≤ C∥f∥_{W}^{2} ,  ∀f ∈ L^{2}(W) .  (5.4)

Recalling that the system e_{I}^{k} k = 1, 2, …, d is an orthonormal basis in ℂ^{d} (eigenvectors of a selfadjoint matrix W_{I}) and using the notation f_{I} = |I|^{−1} ∫_{I} f(t)dt we can rewrite the above inequality as

∑_{I∈D} |I|^{2} ∥A_{I}^{∗}(W f)_{I}∥^{2} ≤ C∥f∥_{W}^{2} , ∀f ∈ L^{2}(W) ,

or equivalently

∑_{I∈D} |I|^{2} ∥A_{I}^{∗}(W^{1/2}f)_{I} ∥^{2} ≤ C∥f∥^{2} ,  ∀f ∈ L^{2} ;  (5.5)

here we have used the fact the multiplication by W^{−1/2} is a unitary operator from L^{2} to L^{2}(W).

Let μ_{I} =^{def} |I| · ∥(W_{I})^{−1/2}(W_{I_{1}} − W_{I_{2}})(W_{I})^{−1/2}∥^{2} (note that μ_{I} here are different from μ_{I} in the proof of Theorem 4.1). Since for any vector f ∈ ℂ^{d} the inequality ∥A_{I}^{∗}f∥ ≤ \frac{1}{2}(μ_{I})^{1/2}|I|^{−1} ∥(W_{I})^{−1/2}f∥ holds, the inequality (5.5) follows immediately from the following one

∑_{I∈D} μ_{I} ∥(W_{I})^{−1/2}(W^{1/2}f)_{I} ∥^{2} ≤ C∥f∥^{2} ,  ∀f ∈ L^{2} .  (5.6)

So we have reduced the theorem to the imbedding (5.6). □

6. Proof of the imbedding theorem.

Theorem 6.1. Let a d × d matrix weight W satisfy the Muckenhoupt condition (A_{2}),

and let μ_{I} =^{def} |I| · ∥(W_{I})^{−1/2}(W_{I_{1}} − W_{I_{2}})(W_{I})^{−1/2}∥^{2}. Then

∑_{I∈D} μ_{I} ∥(W_{I})^{−1/2}(W^{1/2}f)_{I} ∥^{2} ≤ C∥f∥^{2} ,  ∀f ∈ L^{2} .  (6.1)

where the constant C depends only on the Muckenhoupt constant of W.

Let us remind the following “dyadic” version of a Carleson imbedding theorem (as it was shown above in Theorem 4.1 the measure ∑_{I∈D} μ_{I}δ_{λ(I)} is Carleson). Let us remind that λ(I) is a point in ℂ_{+} such that Im λ(I) = |I| and Re λ(I) is the center of I.

Theorem 6.2. Let numbers μ(I), I ∈ D be such that the measure M = ∑_{I∈D} μ_{I}δ_{λ(I)} is Carleson. That means there exists a constant C such that that for any dyadic interval J

∑_{I⊂J, I∈D} μ(I) ≤ C|J| .

Then for any (scalar) function f ∈ L^{2}

∑_{I∈D} |f_{I}|^{2}μ(I) ≤ K(C) · ∥f∥_{2}^{2} .

This theorem follows immediately from the classical Carleson imbedding theorem, because for a positive function f the average f_{I} is estimated as f_{I} ≤ Cf(λ(I)), where f(λ(I)) is a harmonic extension of f at the point λ(I) and C is an absolute constant.

Proof of Theorem 6.1. The proof resembles the proof of the Carleson imbedding theorem that can be found in [7].

Denote by J the imbedding operator, J : L^{2}(ℂ^{d}) → ℓ^{2}({μ(I)}) = ℓ^{2}({μ(I)}, ℂ^{d}),

J f = {(W_{I})^{−1/2}(W^{1/2}f)_{I}}_{I∈D} ∈ ℓ^{2}({μ(I)}_{I∈D}) .

The formal adjoint J^{∗} to J is given by the formula

J^{∗} \{α_{I}\}_{I∈D} = ∑_{I∈D} \frac{1}{|I|}χ_{I}W^{1/2}(W_{I})^{−1/2}α_{I} , \{α_{I}\}_{I∈D} ∈ ℓ^{2}(\{μ(I)\}, ℂ^{d}) .

A block (J J^{∗})_{J,I} of the operator (J J^{∗}) is nonzero only if J ∩I ≠ ∅. Clearly (J J^{∗})_{I,J} = ((J J^{∗})_{J,I})^{∗} and if I ⊂ J

(J J^{∗})_{J,I} = \frac{1}{|J|}(W_{J})^{−1/2}(W_{I})^{1/2} .

One can estimate for I ⊂ J

∥(J J^{∗})_{I,J}∥ = ∥(J J^{∗})_{J,I}∥ = \frac{1}{|J|} ∥(W_{J})^{−1/2}(W_{I})^{1/2}∥ ≤ \frac{1}{|J|} ∥((W^{−1})_{J})^{1/2} (W_{I})^{1/2}∥ =^{def} t_{J,I} ,

and put t_{I,J} = t_{J,I}. Put t_{I,J} = 0 for I ∩ J = ∅. Now to prove the theorem it is sufficient to prove that the matrix {t_{I,J}}_{I,J∈D} generates a bounded operator on ℓ^{2}({μ(I)}).

We shall use the following well known test of boundedness of an integral operator, see [7].

Theorem 6.3 (Senichkin – Vinogradov Test). Let Z be a measure space, k be a locally summable nonnegative measurable function on Z × Z. Let

∫_{Z} k(s, t)k(s, x)ds ≤ C [k(x, t) + k(t, x)] , a.e. on Z × Z .

Then for any positive g ∈ L^{2}(Z)

∫∫ _{Z×Z}k(s, t)g(s)g(t)dsdt ≤ 2C∥g∥_{L^{2}(Z)}^{2} ,

and the integral operator with kernel k(s, t) is bounded in L^{2}(Z).

Since the matrix {t_{I,J}}_{I,J∈D} is symmetric, it is sufficient to show that the matrix

{T_{I,J}}_{I,J∈D}

T_{I,J} = \{ \begin{matrix}t_{I,J}, & I ⊂ J \\ 0 & I ⊄ J. \end{matrix}

generates a bounded operator in ℓ^{2}({μ(I)}).

Since T_{I,J} ≠ 0 only if I ⊂ J we need to estimate the sum

∑_{I∈D} T_{I,J}T_{I,J′}μ(I)

only when J ∩ J′ ≠ ∅. Without loss of generality we can assume J′ ⊂ J, then

∑_{I∈D} T_{I,J}T_{I,J′}μ(I) = ∑_{I∈D, I⊂J′} T_{I,J}T_{I,J′}μ(I) =

= ∑_{I∈D, I⊂J′} \frac{1}{|J||J′|} ∥(W_{J})^{−1/2}(W_{I})^{1/2}∥ ∥(W_{J′})^{−1/2}(W_{I})^{1/2}∥ μ(I)

We can estimate

∥(W_{J})^{−1/2}(W_{I})^{1/2}∥ = ∥(W_{J})^{−1/2}(W′_{J})^{1/2}(W′_{J})^{−1/2}(W_{I})^{1/2}∥ ≤
≤ ∥(W_{J})^{−1/2}(W′_{J})^{1/2}∥ ∥(W′_{J})^{−1/2}(W_{I})^{1/2}∥ .

Hence

∑_{I∈D} T_{I,J}T_{I,J′}μ(I) ≤ ∑_{I∈D, I⊂J′} \frac{1}{|J||J′|} ∥(W_{J})^{−1/2}(W_{J′})^{1/2}∥ ∥(W_{J′})^{−1/2}(W_{I})^{1/2}∥^{2} μ(I)

By Lemma 3.5 the weight (W_{J′})^{−1/2}W (W_{J′})^{−1/2} is Muckenhoupt with the same constant as W, so we can use Lemma 3.6 to estimate ∥(W_{J′})^{−1/2}(W_{I})^{1/2}∥^{2}:

∥(W_{J′})^{−1/2}(W_{I})^{1/2}∥^{2} = ∥(W_{J′})^{−1/2}W_{I}(W_{J′})^{−1/2}∥ ≤ \frac{1}{|I|} ∫_{I} ∥(W_{J′})^{−1/2}W (W_{J′})^{−1/2}∥ ≤

≤ C (\frac{1}{|I|} ∫_{I} ∥(W_{J′})^{−1/2}W^{1/2}∥)^{2} = C (\frac{1}{|I|} ∫_{I} ∥(W_{J′})^{−1/2}W^{1/2}∥ χ_{J′})^{2}

Since the measure ∑_{I∈D} μ(I)δ_{λ(I)} is Carleson (see Theorem 4.1), one can use Theorem 6.2

\frac{|J||J′|}{∥(W_{J})^{−1/2}(W_{J′})^{1/2}∥} ∑_{I∈D, I⊂J′} T_{I,J}T_{I,J′}μ(I) ≤ C ∑_{I∈D, I⊂J′} (\frac{1}{|I|} ∫_{I} ∥(W_{J′})^{−1/2}W^{1/2}∥ χ_{J′})^{2} μ(I) ≤

≤ C′ ∥ ∥(W_{J′})^{−1/2}W^{1/2}∥ χ_{J′} ∥_{L^{2}}^{2} = C′ ∫_{J′} ∥(W_{J′})^{−1/2}W (W_{J′})^{−1/2}∥ ≤

≤ C′′ ∥(W_{J′})^{−1/2} [∫_{J′} W ] (W_{J′})^{−1/2}∥ = C′′|J′|;

the last inequality here holds by Lemma 3.1. Therefore

∑_{I∈D} T_{I,J}T_{I,J′}μ(I) ≤ \frac{C′′}{|J|} ∥(W_{J})^{−1/2}(W_{J′})^{1/2}∥ = C′′(T_{J′,J} + T_{J,J′})

and we are done. □

7. Boundedness of the Hilbert transform.

Let T denote Hilbert transform

T f(x) = p.v. \frac{1}{π} ∫_{ℝ} \frac{f(t)}{x − t}dt .

In this section we are going to prove the following theorem.

Theorem 7.1. The operator T is bounded in the weighted space L^{2}(W) (or, equivalently, that W^{1/2}T W^{−1/2} is bounded in the regular vector-valued space L^{2}) if and only if the matrix weight W satisfies the Muckenhoupt condition (A_{2}).

Remark. The “only if” part of the theorem is already proved, see Lemma 2.2. As one can see from the proof below, the “if” part holds not only for Hilbert Transform, but for more general singular integral operators. In fact it is proved below that any operator whose matrix in the Haar system is estimated by the sum of matrices from Lemmas 7.7, 7.8 is bounded in the weighted space L^{2}(W) if the weight W satisfies the Muckenhoupt (A_{2}) condition.

Proof of Theorem 7.1: The beginning.

To prove the theorem consider the matrix of the operator W^{1/2}T W^{−1/2} with respect to the systems of vectors W^{1/2}f_{I}^{k} and W^{−1/2}g_{I}^{k}, where f_{I}^{k} is the vector Haar system constructed in Section 5,

f_{I}^{k} = w_{I}^{k}h_{I}e_{I}^{k} , w_{I}^{k} = ∥[W_{I}]^{−1/2}e_{I}^{k}∥;

here h_{I} is usual (scalar) Haar function on a dyadic interval I, and vectors e_{I}^{1}, e_{I}^{2}, …, e_{I}^{d} ∈ ℂ^{d} are eigenvectors of W_{I}. The system g_{I}^{k} is the corresponding system for W^{−1} namely

g_{I}^{k} = w̃_{I}^{k}h_{I} ẽ_{I}^{k} , w̃_{I}^{k} = ∥[(W^{−1})_{I}]^{−1/2}ẽ_{I}^{k}∥ ,

where ẽ_{I}^{1}, ẽ_{I}^{2}, …, ẽ_{I}^{d} ∈ ℂ^{d} are eigenvectors of (W^{−1})_{I}.

By Theorem 5.2 systems f_{I}^{k} and g_{I}^{k} form Riesz bases in L^{2}(W) and L^{2}(W^{−1}) respectively, so the systems W^{1/2}f_{I}^{k} and W^{−1/2}g_{I}^{k} are two Riesz bases in L^{2}. Therefore it is sufficient to show that the matrix

{(W^{1/2}T W^{−1/2} · W^{1/2}f_{J}^{l}, W^{−1/2}g_{I}^{k})}_{I,J∈D, k,l=1,2,...,d}

is a matrix of a bounded operator in ℓ^{2}. One can rewrite this matrix as

{w_{J}^{l}w̃_{I}^{k}(T h_{J}, h_{I})(e_{J}^{l}, ẽ_{I}^{k})}_{I,J∈D, k,l=1,2,...,d} ,

or, gathering d × d blocks and recalling the definitions of w_{I}^{k} and w̃_{J}^{l},

{(T h_{J}, h_{I})[(W^{−1})_{I}]^{−1/2}[W_{J}]^{−1/2}}_{I,J∈D} .

If we estimate the norm of each block and the resulting matrix of positive number turns out to be the matrix of a bounded operator in ℓ^{2}, we are done.

By Corollary 3.3

∥[(W^{−1})_{I}]^{−1/2}[W_{J}]^{−1/2}∥ ≤ ∥[W_{I}]^{1/2}[(W^{−1})_{J}]^{1/2}∥ .

The number (T h_{J}, h_{I}) is also easy to estimate. For a dyadic interval J let Z(J) denote the interval 3J (the interval J itself and the two neighbors of the same size), and s(J) denotes three-point set formed of the center point and endpoints of J (T h_{J} has logarithmic singularities at points of s(J), that explains the notation). Recall that λ(I) denotes the point in ℂ_{+} such that Re λ(I) is the center of I and Im λ(I) = |I|. If |I| ≤ |J| we denote by ρ_{I,J} the distance from λ(I) to s(J).

Lemma 7.2. If |I| ≤ |J| then

|(T h_{I}, h_{J})| = |(T h_{J}, h_{I})| ≤ \{ \begin{matrix}C \frac{|I|^{3/2}|J|^{3/2}}{ρ_{I,J}^{3}} , & if I ⊄ Z(J) \\ C \frac{|I|^{3/2}}{ρ_{I,J}} \frac{1}{|J|^{1/2}} , & if I ⊂ Z(J) \end{matrix}

This is a standard and explicit computation and we omit the proof.

7.1. Some auxiliary matrices.

We are going to estimate our matrix by the sum of two matrices: one deals with the singularity at ∞, another one with the singularity that occurs when I is close to s(J).

We need the following two tests of the boundedness of an integral operator.

Theorem 7.3 (Modified Senichkin — Vinogradov test, 1). Let Z be a measure space, k_{1}, k_{2} be locally summable nonnegative measurable functions on Z × Z, and let K = k_{1} + k_{2}. Suppose that

∫_{Z} k_{1,2}(s, t)k_{1,2}(s, x)ds ≤ C [K(x, t) + K(t, x)] , a.e. on Z × Z .

Then the integral operators with kernels k_{1} and k_{2} are bounded on L^{2}

Proof. Let f ∈ L^{2}, ∥f∥_{2} = 1 be a nonnegative function such that

P =^{def} ∫∫_{Z×Z} k_{1}(s, t)f(s)f(t)dsdt < ∞ , Q =^{def} ∫∫_{Z×Z} k_{2}(s, t)f(s)f(t)dsdt < ∞ .

Then

P^{2} = (∫_{Z} f(s) (∫_{Z} k_{1}(s, t)f(t)dt) ds)^{2} ≤ ∥f∥_{L^{2}(Z)}^{2} · ∥∫_{Z} k_{1}(., t)f(t)dt∥^{2} ≤
≤ ∫_{Z} ds (∫∫_{Z×Z} k_{1}(s, t)k_{1}(s, x)f(t)f(x)dtdx) =
= ∫∫_{Z×Z} f(t)f(x) (∫_{Z} k_{1}(s, t)k_{1}(s, x)ds) ≤
≤ 2C ∫_{Z}[k_{1}(t, x) + k_{2}(t, x)]f(t)f(x)dtdx = 2C(P + Q) ,

and similarly,

Q^{2} ≤ 2C(P + Q) .

Therefore

(P + Q)^{2} ≤ 2(P^{2} + Q^{2}) ≤ 8C(P + Q) .

This implies (P + Q) ≤ 8C, so P ≤ 8C, Q ≤ 8C. From here one can conclude that the corresponding integral operators are bounded and their norms can be estimated above as 2\sqrt{2C}. □

Theorem 7.4 (Modified Senichkin — Vinogradov test, 2). Let Z be a measure space, k, b be locally summable nonnegative measurable functions on Z × Z, and the integral operator with kernel b is bounded (with the norm B). Suppose that

∫_{Z} k(s, t)k(s, x)ds ≤ C [k(x, t) + k(t, x)] + b(x, t) , a.e. on Z × Z .

Then the integral operator with kernel k is bounded on L^{2}(Z) (with norm estimated by a constant depending on C and B).

Proof. As in the proof of Theorem 7.3 take f ∈ L^{2}, ∥f∥_{2} = 1 such that

Q =^{def} ∫∫_{Z×Z} k(s, t)f(s)f(t)dsdt < ∞ .

Applying the same reasoning as in Theorem 7.3 one can conclude that Q^{2} ≤ 2CQ + B. □

Let w be a (scalar) Muckenhoupt weight, I be an interval and λ(I) ∈ ℂ_{+} be a point such that Re λ(I) is a center of I and Im λ(I) = |I|. Let w(λ) be the Poisson extension of w at the point λ ∈ ℂ_{+}.

Lemma 7.5. Let w be a scalar Muckenhoupt weight. There exists a constant C depending only on Muckenhoupt constant of w such that w(λ(I)) ≤ Cw_{I}.

Remark. The opposite inequality (with some absolute constant) is trivial.

Corollary 7.6. For a d × d matrix Muckenhoupt weight W there exists a constant C depending only on the Muckenhoupt constant of W such that W (λ(I)) ≤ CW_{I}.

Proof. Take a vector e ∈ ℂ^{d}. The scalar weight w, w(t) = (W (t)e, e) is clearly a Muckenhoupt weight with the same or better constant as W. So by Lemma 7.5 (W (λ(I))e, e) ≤ C(W_{I}e, e). □

Let us recall that D denotes the set of all dyadic intervals. Denote by {t_{I,J}}_{I,J∈D} a symmetric matrix defined by

t_{I,J} = \frac{|I|^{3/2}}{|J|^{3/2}} · \frac{|J|^{2}}{|λ(I) − \overline{λ(J)}|^{2}} ,  |I| ≤ |J|  (7.2)

Lemma 7.7. Given a d × d matrix Muckenhoupt weight W the matrix

{t_{I,J} ∥[(W^{−1})_{J}]^{1/2}[W_{I}]^{1/2}∥}_{I,J∈D}

generates a bounded operator in ℓ^{2} with the norm at most M where M depends only on d and the Muckenhoupt constant of W.

Proof. Since the matrix t_{I,J} is symmetric, it is sufficient to show that matrices T_{I,J}^{1}, T_{I,J}^{2} defined by T_{I,J}^{1} = T_{I,J}^{2} = 0 for |J| < |I| and

T_{I,J}^{1} = t_{I,J} ∥[(W^{−1})_{J}]^{1/2}[W_{I}]^{1/2}∥ ,  T_{I,J}^{2} = t_{I,J} ∥[W_{J}]^{1/2}[(W^{−1})_{I}]^{1/2}∥ ,  |I| ≤ |J|

generates bounded operators. Let us apply the modified Senichkin — Vinogradov Test (Theorem 7.3). Take some interval J′. Since in Theorem 7.3 we need an estimate by a symmetric sum K(x, t) + K(t, x), we can assume without loss of generality that |J′| ≤ |J|. We want to estimate

∑_{I: |I|≤|J′|} T_{I,J}^{1,2}T_{I,J′}^{1,2}

First of all note that for all s, x, t ∈ ℂ_{+}

|x − \overline{t}| = |x − \overline{s} + \overline{s} − \overline{t}| ≤ |x − \overline{s}| + |s − t| ≤ |x − \overline{s}| + |s − \overline{t}|

and therefore

|x − \overline{s}|^{−1}|s − \overline{t}|^{−1} ≤ |x − \overline{t}|^{−1} (|x − \overline{s}|^{−1} + |s − \overline{t}|^{−1}) .

Hence, if |I| ≤ |J′| ≤ |J| then

t_{I,J}t_{I,J′} = \frac{|I|^{3/2}}{|J|^{3/2}} · \frac{|I|^{3/2}}{|J′|^{3/2}} · \frac{|J|^{2}}{|λ(I) − \overline{λ(J)}|^{2}} · \frac{|J′|^{2}}{|λ(I) − \overline{λ(J′)}|^{2}} ≤

≤ \frac{|J′|^{1/2}}{|J|^{3/2}} · |J|^{2}|I|^{3} \frac{1}{|λ(J′) − \overline{λ(J)}|^{2}} (\frac{1}{|λ(I) − \overline{λ(J)}|^{2}} + \frac{1}{|λ(I) − \overline{λ(J′)}|^{2}} ) =

= t_{J′,J}|I|^{3}|J′|^{−1} (\frac{1}{|λ(I) − \overline{λ(J)}|^{2}} + \frac{1}{|λ(I) − \overline{λ(J′)}|^{2}} )

Let us note that

∥[(W^{−1})_{J}]^{1/2}[W_{I}]^{1/2}∥ = ∥[(W^{−1})_{J}]^{1/2}[W_{J′}]^{1/2}[W_{J′}]^{−1/2}[W_{I}]^{1/2}∥ ≤
≤ ∥[(W^{−1})_{J}]^{1/2}[W_{J′}]^{1/2}∥ ∥[W_{J′}]^{−1/2}[W_{I}]^{1/2}∥ ≤
≤ ∥[(W^{−1})_{J}]^{1/2}[W_{J′}]^{1/2}∥ ∥[(W^{−1})_{J′}]^{1/2}[W_{I}]^{1/2}∥

(the last inequality holds by Corollary 3.3). Therefore

∑_{I : |I|≤|J′|} ∥[(W^{−1})_{J}]^{1/2}[W_{I}]^{1/2}∥ ∥[(W^{−1})_{J′}]^{1/2}[W_{I}]^{1/2}∥ \frac{|I|^{3}}{|λ(I) − \overline{λ(J′)}|^{2}} ≤

≤ ∥[(W^{−1})_{J}]^{1/2}[W_{J′}]^{1/2}∥ ∑_{I : |I|≤|J′|} ∥[(W^{−1})_{J′}]^{1/2}[W_{I}]^{1/2}∥^{2} |I|^{2} \frac{|J′|}{|λ(I) − \overline{λ(J′)}|^{2}} ≤

C ∥[(W^{−1})_{J}]^{1/2}[W_{J′}]^{1/2}∥ ∑_{k=0}^{∞} |J′|2^{−k} ∫_{ℝ} \frac{|J′|}{|t − \overline{λ(J′)}|^{2}} ∥[(W^{−1})_{J′}]^{1/2}W (t)[(W^{−1})_{J′}]^{1/2}∥ dt ≤

≤ C_{1} ∥[(W^{−1})_{J}]^{1/2}[W_{J′}]^{1/2}∥ |J′| ∥∫_{ℝ} \frac{|J′|}{|t − \overline{λ(J′)}|^{2}} [(W^{−1})_{J′}]^{1/2}W (t)[(W^{−1})_{J′}]^{1/2}dt∥

(the last inequality holds by Lemma 3.1.) By Lemma 3.5 the weight

[(W^{−1})_{J′}]^{1/2}W (.)[(W^{−1})_{J′}]^{1/2}

is a Muckenhoupt weight with the same or better constant than W, therefore by Corollary 7.6

∫_{ℝ} \frac{|J′|}{|t − \overline{λ(J′)}|^{2}} [(W^{−1})_{J′}]^{1/2}W (t)[(W^{−1})_{J′}]^{1/2}dt ≤ A[(W^{−1})_{J′}]^{1/2}W (λ(J′))[(W^{−1})_{J′}]^{1/2} ≤

≤ C[(W^{−1})_{J′}]^{1/2}W_{J′}[(W^{−1})_{J′}]^{1/2} ≤ C′I

(the Muckenhoupt condition). Therefore we can continue the chain of inequalities:

∑_{I : |I|≤|J′|} ∥[(W^{−1})_{J}]^{1/2}[W_{I}]^{1/2}∥ ∥[(W^{−1})_{J′}]^{1/2}[W_{I}]^{1/2}∥ \frac{|I|^{3}}{|λ(I) − \overline{λ(J′)}|^{2}} ≤

≤ C ∥[(W^{−1})_{J}]^{1/2}[W_{J′}]^{1/2}∥ |J′| .

Similarly,

∥[(W^{−1})_{J′}]^{1/2}[W_{I}]^{1/2}∥ = ∥[(W^{−1})_{J′}]^{1/2}[W_{J}]^{1/2}[W_{J}]^{−1/2}[W_{I}]^{1/2}∥ ≤
≤ ∥[(W^{−1})_{J′}]^{1/2}[W_{J}]^{1/2}∥ ∥[W_{J}]^{−1/2}[W_{I}]^{1/2}∥ ≤
≤ ∥[(W^{−1})_{J′}]^{1/2}[W_{J}]^{1/2}∥ ∥[(W^{−1})_{J}]^{1/2}[W_{I}]^{1/2}∥ ,

and therefore

∑_{I : |I|≤|J′|} ∥[(W^{−1})_{J}]^{1/2}[W_{I}]^{1/2}∥ ∥[(W^{−1})_{J′}]^{1/2}[W_{I}]^{1/2}∥ \frac{|I|^{3}}{|λ(I) − \overline{λ(J)}|^{2}} ≤

≤ ∥[(W^{−1})_{J′}]^{1/2}[W_{J}]^{1/2}∥ ∑_{I : |I|≤|J′|} ∥[(W^{−1})_{J}]^{1/2}[W_{I}]^{1/2}∥^{2} |I|^{2} \frac{|J|}{|λ(I) − \overline{λ(J)}|^{2}} ≤

≤ C ∥[(W^{−1})_{J′}]^{1/2}[W_{J}]^{1/2}∥ ∑_{k=0}^{∞} |J′|2^{−k} ∫_{ℝ} \frac{|J|}{|t − \overline{λ(J)}|^{2}} ∥[(W^{−1})_{J}]^{1/2}W (t)[(W^{−1})_{J}]^{1/2}∥ dt ≤

≤ C_{1} ∥[(W^{−1})_{J′}]^{1/2}[W_{J}]^{1/2}∥ |J′| ∥∫_{ℝ} \frac{|J|}{|t − \overline{λ(J)}|^{2}} [(W^{−1})_{J}]^{1/2}W (t)[(W^{−1})_{J}]^{1/2}dt∥ ≤

≤ C ∥[(W^{−1})_{J′}]^{1/2}[W_{J}]^{1/2}∥ |J′| .

Gathering all the calculations together we obtain (assuming |J′| ≤ |J| and therefore |I| ≤ |J′|)

∑_{I} T_{I,J}^{1}T_{I,J′}^{1} ≤ Ct_{J′,J} (∥[(W^{−1})_{J}]^{1/2}[W_{J′}]^{1/2}∥ + ∥[(W^{−1})_{J′}]^{1/2}[W_{J}]^{1/2}∥) = C(T_{J′,J}^{1} + T_{J′,J}^{2}) .

The weight W^{−1} is a Muckenhoupt weight, and the similar reasoning implies

∑_{I} T_{I,J}^{2}T_{I,J′}^{2} ≤ C(T_{J′,J}^{1} + T_{J′,J}^{2}) .

Therefore by Theorem 7.3 both matrices generate bounded operators. □ Denote by {k_{I,J}}_{I,J∈D} a symmetric matrix defined for |I| ≤ |J| by

k_{I,J} = \{ \begin{matrix}\frac{|I|^{3/2}}{ρ_{I,J}} \frac{1}{|J|^{1/2}} , & I ⊂ Z(J) \\ 0, & I ⊄ Z(J) \end{matrix}  (7.3)

Lemma 7.8. Given a matrix Muckenhoupt weight W the matrix

{k_{I,J}∥[W_{I}]^{1/2}[(W^{−1})_{J}]^{1/2}∥}_{I,J∈D}

is the matrix of a bounded in ℓ^{2} operator (with norm depending on the Muckenhoupt constant of W).

Proof of Theorem 7.1: the completion. Lemmas 7.7 and 7.8 imply immediately that the Hilbert transform T is bounded in L^{2}(W). Indeed, one can see from Lemma 7.2 that |(T h_{I}, h_{J})|∥[W_{I}]^{1/2}[(W^{−1})_{J}]^{1/2}∥ is estimated above by

C · t_{I,J}∥[W_{I}]^{1/2}[(W^{−1})_{J}]^{1/2}∥

if I ⊄ Z(J) and J ⊄ Z(I), and by

C · k_{I,J}∥[W_{I}]^{1/2}[(W^{−1})_{J}]^{1/2}∥

if I ⊂ Z(J) or J ⊂ Z(I). So it is estimated by the sum of these matrices, and each matrix is bounded in ℓ^{2} by Lemmas 7.7 and 7.8 respectively. □

Proof of Lemma 7.8. Since everything is symmetric with respect to change of W to W^{−1} it is sufficient to prove that only one “half” of the matrix is a bounded operator in ℓ^{2}; the reasoning for the other “half” is absolutely the same. Namely we shall prove that the matrix K_{I,I}, K_{I,J} = 0 if |J| < |I| and

K_{I,J} = k_{I,J}∥[W_{I}]^{1/2}[(W^{−1})_{J}]^{1/2}∥ if |I| ≤ |J|

generates a bounded operator in ℓ^{2}.

We apply Theorem 7.4 to the matrix K_{I,J}. Suppose that J′ is a dyadic interval, |J′| ≤ |J|. First, acting like in the proof of Lemma 7.7 notice that

∥[(W^{−1})_{J}]^{1/2}[W_{I}]^{1/2}∥ = ∥[(W^{−1})_{J}]^{1/2}[W_{J′}]^{1/2}[W_{J′}]^{−1/2}[W_{I}]^{1/2}∥ ≤
≤ ∥[(W^{−1})_{J}]^{1/2}[W_{J′}]^{1/2}∥ ∥[W_{J′}]^{−1/2}[W_{I}]^{1/2}∥ ≤
≤ ∥[(W^{−1})_{J}]^{1/2}[W_{J′}]^{1/2}∥ ∥[(W^{−1})_{J′}]^{1/2}[W_{I}]^{1/2}∥

(the last inequality holds by Corollary 3.3). Therefore

∑_{I∈D, |I|≤|J′|} K_{I,J}K_{I,J′} ≤ ∥[(W^{−1})_{J}]^{1/2}[W_{J′}]^{1/2}∥ ∑_{I∈D, I⊂Z(J′)} k_{I,J}k_{I,J′} ∥[(W^{−1})_{J′}]^{1/2}[W_{I}]^{1/2}∥^{2}

For a set A denote

ω(A) = trace {[(W^{−1})_{J′}]^{1/2}[W (A)][(W^{−1})_{J′}]^{1/2}}

and let ω_{A} =^{def} \frac{1}{|A|}ω(A). Clearly

∥[(W^{−1})_{J′}]^{1/2}[W (A)]^{1/2}∥^{2} = ∥[(W^{−1})_{J′}]^{1/2}[W (A)][(W^{−1})_{J′}]^{1/2}∥ ≍ ω(A) ;

the inequality ≤ is trivial, and the inequality ≥ follows from Lemma 3.1 We consider 3 cases: Case 1: J′ ⊂ Z(J) and ρ_{J′,J} ≥ 10|J′|. In this case consider dyadic Whitney intervals of

Z(J′) \ s(J′), and let A_{J} denote the union of all such intervals of length |J′|2^{−j} (there are at most 6 such intervals). If a dyadic interval I ⊂ A_{j} (I can be of arbitrary length), then ρ_{I,J′} ≍ |A_{j}| ≍ |J′|2^{−j}. Taking into account that under the above assumptions ρ_{I,J} ≍ ρ_{J′,J}, one can conclude

∥[(W^{−1})_{J}]^{1/2}[W_{J′}]^{1/2}∥^{−1} ∑_{I∈D, |I|≤|J′|} K_{I,J}K_{I,J′} ≤ C ∑_{I∈D, I⊂Z(J′)} k_{I,J}k_{I,J′}ω_{I} ≤

≤ \frac{C}{ρ_{J′,J}|J|^{1/2}|J′|^{1/2}} ∑_{I∈D, I⊂Z(J′)} \frac{|I|^{2}ω(I)}{ρ_{I,J′}}

Estimating the last sum we have

∑_{I∈D, I⊂Z(J′)} \frac{|I|^{2}ω(I)}{ρ_{I,J′}} ≤ C ∑_{j=0}^{∞} ∑_{I⊂A_{j}} \frac{|I|^{2}ω(I)}{ρ_{I,J′}} ≤

≤ C ∑_{j=0}^{∞} \frac{∑_{t=0}^{∞} |J′|^{2}2^{−2j}2^{−2t}ω(A_{j})}{|A_{j}|} ≤ C ∑_{j=0}^{∞} |J′|2^{−j}ω(A_{j}) ≤ C|J′|ω(J′)

The Muckenhoupt condition implies that ω(J′) ≤ C|J′| and therefore

∑_{I∈D, |I|≤|J′|} K_{I,J}K_{I,J′} ≤ C ∥[(W^{−1})_{J}]^{1/2}[W_{J′}]^{1/2}∥ \frac{|J′|^{3/2}}{ρ_{J′,J}|J|^{1/2}} = C(K_{J′,J} + K_{J,J′}) ,

and we are done in this case.

Case 2: J′ ⊄ Z(J). In this case ρ_{J′,J} ≥ |J|, and in most situations K_{I,J}K_{I,J′} = 0. If K_{I,J}K_{I,J′} ≠ 0 for some I, then ρ_{J′,J} ≤ 3|J|, so ρ_{J′,J} ≍ |J|. We can apply the same reasoning as in Case 1 to get the estimate

∑_{I∈D, |I|≤|J′|} K_{I,J}K_{I,J′} ≤ C ∥[(W^{−1})_{J}]^{1/2}[W_{J′}]^{1/2}∥ \frac{|J′|^{3/2}}{ρ_{J′,J}|J|^{1/2}} .

The problem is that for our choice of J and J′ we have K_{J′,J} + K_{J,J′} = 0, so we can not estimate the right side by this sum. But then \frac{|J′|^{3/2}}{ρ_{J′,J} |J|^{1/2}} ≍ \frac{|J′|^{3/2}}{|J|^{3/2}} ≍ t_{J′,J}, so

∑_{I∈D, |I|≤|J′|} K_{I,J}K_{I,J′} ≤ CT_{J′,J} ,

where {T_{I,J}}_{I,J∈D} is the matrix from Lemma 7.7 which we know generates a bounded operator in ℓ^{2}.

Case 3: J′ ⊂ J, ρ_{J′,J} < 10|J′|. Note that ρ_{J′,J} ≥ |J′| always, so ρ_{J′,J} ≍ |J′|. We sum over all dyadic I ⊂ Z(J′).

Consider dyadic Whitney intervals for Z(J′) \ (s(J′) ∪ s(J)), and denote by A_{j} the union of all intervals of length |J′|2^{−j}. Note that there exists at most 12 such intervals (2 for each point of s(J′) ∪ s(J); of course it is not a sharp estimate).

Let ρ_{I} =^{def} min{ρ_{I,J}, ρ_{I,J′}}, then ρ_{I} ≍ |J′|2^{−j} if I ⊂ A_{j}. Again

∑_{I∈D, |I|≤|J′|} K_{I,J}K_{I,J′} ≤ ∥[(W^{−1})_{J}]^{1/2}[W_{J′}]^{1/2}∥ ∑_{I∈D, I⊂Z(J′)} k_{I,J}k_{I,J′}ω_{I} ≤

≤ C \frac{∥[(W^{−1})_{J}]^{1/2}[W_{J′}]^{1/2}∥}{|J|^{1/2}|J′|^{1/2}} ∑_{I∈D, I⊂Z(J′)} \frac{|I|^{2}ω(I)}{ρ_{I}^{2}} .

Estimating the last sum we obtain

∑_{I∈D, I⊂Z(J′)} \frac{|I|^{2}ω(I)}{ρ_{I}^{2}} ≤ C ∑_{j=0}^{∞} \frac{∑_{t=0}^{∞} |J′|^{2}2^{−2j}2^{−2t}ω(A_{j})}{|A_{j}|^{2}} ≤ Cω(J′) = C|J′|ω_{J′} .

Since ρ_{J′,J} ≍ |J′| and by the Muckenhoupt condition ω_{J′} ≤ C,

\frac{|J′|^{1/2}}{|J|^{1/2}} ω_{J′} ≤ C \frac{|J′|^{3/2}}{ρ_{J′,J}|J|^{1/2}}

and therefore

∑_{I∈D, |I|≤|J′|} K_{I,J}K_{I,J′} ≤ C ∥[(W^{−1})_{J}]^{1/2}[W_{J′}]^{1/2}∥ \frac{|J′|^{3/2}}{ρ_{J′,J}|J|^{1/2}} = C(K_{J′,J} + K_{J,J′}) .

□

8. Unconditional bases of wavelets.

8.1. Construction of a wavelet basis.

Let we have so-called biorthogonal wavelets, i.e. functions φ, ψ such that the system of dilations and shifts of φ is an unconditional basis in L^{2}, and dilations and shifts of ψ forms a dual system to it. More precisely, given a dyadic interval I with left endpoint x_{I} and length |I| = 2^{−ν} define

φ_{I}(x) = |I|^{−1/2}φ(2^{ν}(x − x_{I})) , ψ_{I}(x) = |I|^{−1/2}ψ(2^{ν}(x − x_{I})) .

Note that for I = [0, 1], φ_{I} = φ, ψ_{I} = ψ, and ∥φ_{I}∥_{2} = ∥φ∥_{2}, ∥ψ_{I}∥_{2} = ∥ψ∥_{2}.

Formally we only assume that the system of functions φ_{I}, I ∈ D is a complete minimal (and therefore, uniformly minimal) system in the (non-weighted) scalar L^{2}, and the system ψ_{I}, I ∈ D is a dual system to it (of course we can replace φ and ψ: everything is absolutely symmetric here).

We assume that functions φ, ψ satisfy some very minimal conditions, namely

1. φ, ψ are bounded, and moreover |φ(x)| ≤ C/|x|^{2}, |ψ(x)| ≤ C/|x|^{2} at ∞;

2. φ, ψ, are absolutely continuous and φ′ and ψ′ are bounded and satisfy the estimates |φ′(x)| ≤ C/|x|^{2}, |ψ′(x)| ≤ C/|x|^{2} at ∞;

Condition 1 implies

3. φ, ψ ∈ L^{1}(ℝ), and, moreover ∫_{R}^{∞} |φ|, ∫_{−∞}^{−R} |φ|, ∫_{R}^{∞} |ψ|, ∫_{−∞}^{−R} |ψ| ≤ C/|R|

Conditions 1 — 3 imply

4. ∫_{ℝ} φ(t)dt = ∫_{ℝ} ψ(t)dt = 0.

Indeed, if, for example, ∫_{ℝ} φ_{I}(t)dt ≠ 0, it is an easy exercise to show that φ is not orthogonal to ψ_{I} for some big enough dyadic interval I.

Denote W^{I} =^{def} ∫ W |φ|^{2}, and let e_{I}^{k}, k = 1, 2, …, d be an orthonormal basis of eigenvalues of W^{I}. Put

w_{I}^{k} = [(W^{I}e_{I}^{k}, e_{I}^{k})]^{−1/2} = [([W^{I}]^{−1} e_{I}^{k}, e_{I}^{k})]^{1/2} = ∥[W^{I}]^{−1/2} e_{I}^{k}∥ .

(note that numbers w_{I}^{k} here are different from those in section 5). The system

f_{I}^{k} = w_{I}^{k}φ_{I}e_{I}^{k} , I ∈ D, k = 1, 2, ..., d

is a normalized (∥f_{I}^{k}∥_{L^{2}(W)} = 1) system in L^{2}(W). Note that we can not apriori say that it is complete in L^{2}(W).

Define a system of subspaces E_{I} = E_{I}^{φ} ⊂ L^{2}(W)

E_{I} = L(f_{I}^{k} : k = 1, 2, …, d) , I ∈ D

Note, that the subspaces E_{I} do not depend on the weight W.

We can easily write the spectral projections P^{I} defined in section 5.1

E^{I}f = \{ \begin{matrix}f, & f ∈ E_{I} \\ 0, & f ∈ E_{J}, J ≠ I \end{matrix}

Namely, E^{I}f = φ · (∫_{ℝ} \overline{ψ(t)}f(t)dt)

Acting as in the proof of Lemma 2.1 one can show that the norm ∥P^{I}∥ is exactly

∥[∫ W |φ_{I}|^{2}]^{1/2} [∫ W^{−1}|ψ_{I}|^{2}]^{1/2}∥

Property 1 of φ, ψ implies that |φ_{I}|^{2} and |ψ_{I}|^{2} is majorized by Poisson kernel P_{λ(I)}

|φ_{I}(t)|^{2} , |ψ_{I}(t)|^{2} ≤ \frac{C|I|}{|t − λ(I)|^{2}} = CP_{λ(I)} t ∈ ℝ ,

therefore by Corollary 7.6 ∫ W |φ_{I}|^{2} ≤ CW (λ(I)) ≤ C′W_{I} and ∫ W^{−1}|ψ_{I}|^{2} ≤ CW^{−1}(λ(I)) ≤ C′(W^{−1})_{I}. Hence the Muckenhoupt condition (A_{2}) implies that the system of subspaces E_{I} is uniformly minimal.

We want again to bring the reader’s attention to the fact that we do not know apriori whether the system of subspaces E_{I} is complete. However it does not matter for the proof of uniform minimality. To be able to apply all definition we need to restrict ourself to the closed linear span of E_{I}, and the norms of spectral projections E^{I} can only decrease.

The following theorem is the main result of this chapter

Theorem 8.1. Let φ, ψ satisfy conditions 1 — 4 above, W be a matrix d × d Muckenhoupt weight. Then the system of subspaces E_{I} constructed above forms an unconditional basis in L^{2}(W).

The proof of the theorem consists of several steps. The following theorem is one of the main steps.

Theorem 8.2. Let W be a Muckenhoupt weight. Then the operator J : ℓ^{2}(ℂ^{d}) → L^{2},

J {f_{I}} = ∑_{I∈D} φ_{I}W^{1/2}(W_{I})^{−1/2}f_{I}

is bounded (with norm estimated by a constant depending on φ, d and the Muckenhoupt constant of W)

Proof. Let us recall that in Section 5 we have proved that the system of Haar subspaces E_{I}^{h} form a Riesz basis in L^{2}(W) if W is a Muckenhoupt weight. Since W and W^{−1} are Muckenhoupt weight with the same Muckenhoupt constant, the system E_{I}^{h}, I ∈ D is also a Riesz basis in L^{2}(W^{−1}), or, equivalently, the system W^{−1/2}E_{I}^{h}, I ∈ D is a Riesz basis in L^{2}

Any vector in the subspace E_{I}[χ] can be represented as h_{I}e, where h_{I} is the normalized Haar function, and e ∈ ℂ^{d}. It is more convenient to represent an arbitrary function in E_{I}[χ] as f = h_{I} ([W^{−1}]_{I})^{−1/2} e; in this case ∥f∥_{L^{2}(W)} = ∥e∥_{ℂ^{d}}.

Let ẽ_{I}^{k}, k = 1, 2, …, d be an orthonormal basis of eigenvectors of ([W^{−1}]_{I})^{−1/2}. Then the system h_{I}W^{−1/2} ([W^{−1}]_{I})^{−1/2} ẽ_{I}^{k}, k = 1, 2, …, d is an orthonormal basis in W^{−1/2}E_{I}[χ]. So, the system {g_{I}^{k}}

g_{I}^{k} = h_{I}W^{−1/2} ([W^{−1}]_{I})^{−1/2} ẽ_{I}^{k}, I ∈ D , k = 1, 2, …, d

is a Riesz basis in L^{2}.

To prove that the operator J is bounded it is sufficient to take an orthogonal basis f_{I}^{k} in ℓ^{2}(ℂ^{d}) and show that the matrix of J

(J f_{J}^{l}, g_{I}^{k}), , I, J ∈ D , k, l = 1, 2, …, d

generates a bounded operator in ℓ^{2}. Let vectors f_{I}^{k} ∈ ℓ^{2}(ℂ^{d}) be a natural (for our purposes) orthonormal basis in ℓ^{2}(ℂ^{d}), namely, Ith component of f_{I}^{k} is e_{I}^{k}, and all other components are 0, then the matrix can be rewritten as

(φ_{J}, h_{I})_{L^{2}}(W_{J}^{−1/2}e_{J}^{l}, ([W^{−1}]_{I})^{−1/2} ẽ_{I}^{k}) , I, J ∈ D , k, l = 1, 2, …, d .

Grouping d × d blocks we get

(φ_{J}, h_{I})_{L^{2}} [(W^{−1})_{I}]^{−1/2} (W_{J})^{−1/2} , I, J ∈ D ,

and we have to prove that this matrix generates a bounded operator in ℓ^{2}(ℂ^{d}). By Corollary 3.3 (W_{I})^{−1} ≤ (W^{−1})_{I}, and similarly [(W^{−1})_{I}]^{−1} ≤ W_{I}, therefore

∥[(W^{−1})_{I}]^{−1/2} (W_{J})^{−1/2}∥ ≤ ∥(W_{I})^{1/2}([W^{−1}]_{J})^{1/2}∥ .

We want to prove that the matrix

|(φ_{J}, h_{I})_{L^{2}}| ∥(W_{I})^{1/2}([W^{−1}]_{J})^{1/2}∥ ,  I, J ∈ D  (8.1)

generates a bounded operator in ℓ^{2} (of course it is sufficient).

The scalar product |(φ_{J}, h_{I})_{L^{2}}| can be estimated as

|(φ_{J}, h_{I})_{L^{2}}| ≤ C \frac{|I|^{3/2}}{|J|^{3/2}} \frac{|J|^{2}}{|λ(I) − \overline{λ(J)}|^{2}}  for |I| ≤ |J|  (8.2)

If |I| ≥ |J| one can write another estimate. Let as in Section 7 Z(I) denotes the interval 3I (the interval I itself an the two neighbors of the same size), s(I) stands for the three point set formed of the center point and endpoints of I, and ρ_{J,I} be the distance from λ(J) to s(I). Then for |I| ≥ |J|

|(φ_{J}, h_{I})_{L^{2}}| ≤ C \frac{|J|^{3/2}}{|I|^{3/2}} \frac{|I|^{2}}{|λ(I) − \overline{λ(J)}|^{2}} ,  if J ⊄ Z(I)  (8.3)

and

|(φ_{J}, h_{I})_{L^{2}}| ≤ C \frac{|J|^{3/2}}{|I|^{1/2}} · \frac{1}{ρ_{J,I}} ,  if J ⊂ Z(I)  (8.4)

Then the matrix (8.1) is bounded above by a sum of matrices from Lemmas 7.7, 7.8 (modulo changing I to J and W to W^{−1} in some places), so the corresponding operator is bounded.

To complete the proof of the theorem we need to explain the estimates (8.2) — (8.4). The first one is easiest to explain:

|(φ_{J}, h_{I})_{L^{2}}| ≤ sup_{I} |φ′(t)| · |I|^{3/2} ,

and by property 2 of φ

sup_{I} |φ′(t)| ≤ C|J|^{−3/2} \frac{|J|^{2}}{|λ(I) − \overline{λ(J)}|^{2}} ,

that gives us the required estimate.

The estimate (8.3) is also fairly simple: first of all

|(φ_{J}, h_{I})_{L^{2}}| ≤ sup_{I} |φ(t)| · |I|^{1/2} .

Since J ⊄ Z(I), we have

sup_{I} |φ(t)| ≤ C sup_{I} \begin{matrix}|J| & 3/2 \\ |t − \overline{λ(J)}| & 2 \end{matrix} ≍ \begin{matrix}|J| & 3/2 \\ |λ(I) − \overline{λ(J)}| & 2 \end{matrix} ,

and above two estimates imply (8.3).

To prove (8.4) let first estimate ∫_{I} φ_{J}. Suppose J ⊂ I = [a, b]. Then, since the integral ∫_{R} φ_{J} = 0 we can apply property 3

|∫_{I} φ_{J} | = |∫_{−∞}^{a} φ_{J} + ∫_{b}^{∞} φ_{J} | ≤ C \frac{|J|^{3/2}}{min \{|λ(J) − a|, |λ(J) − b|\}}  (8.5)

If J ⊄ I (and |J| ≤ |I|) then I is strictly on the left or on the right side of J. Suppose (without loss of generality) the later. Then we can write ∫_{I} φ_{J} = ∫_{a}^{∞} φ_{J} − ∫_{b}^{∞} φ_{J}, and again the estimate (8.5) holds.

So

|(φ_{J}, h_{I})_{L^{2}}| ≤ \frac{|J|^{3/2}}{ρ_{J,I}} \frac{1}{|I|^{1/2}}

and we are done. □

Theorem 8.3. For an arbitrary numerical sequence α = {α_{I}}_{I∈D}, the operator E_{α},

E_{α}f = ∑_{I∈D} α_{I}E^{I}f

is a bounded operator in L^{2}(W) and its norm is bounded by ∥{α_{I}}∥_{∞}.

In particular this theorem implies that the system of subspaces E_{I}, I ∈ D forms a Riesz basis in the closure of its linear span in L^{2}(W).

Proof of Theorem 8.3. The operator E_{α} in L^{2}(W) is unitarily equivalent to the operator W^{1/2}E_{α}W^{−1/2} in L^{2}. If J_{W}^{φ} denotes the operator from Theorem 8.2 J_{W}^{φ} : ℓ^{2}(ℂ^{d}) → L^{2},

J_{W}^{φ} {f_{I}} = ∑_{I∈D} φ_{I}W^{1/2}(W_{I})^{−1/2}f_{I} ,

the operator E_{α} can be factorized as

E_{α} = J_{W}^{φ} M(J_{W^{−1}}^{ψ})^{∗} ,

where M is the multiplication by the sequence α_{I}(W_{I})^{1/2}([W^{−1}]_{I})^{1/2} in ℓ^{2}(ℂ^{d}),

M{f_{I}}_{I∈D} = {α_{I} · (W_{I})^{1/2}([W^{−1}]_{I})^{1/2}f_{I}}_{I∈D} .

Because of Muckenhoupt condition the operator M is bounded, and by Theorem 8.2 operators J_{W}^{φ} and J_{W^{−1}}^{ψ} are bounded too. □

The following theorem shows that the system of subspaces E_{I}, I ∈ D is a complete system in L^{2}(W). Note, that it does not follows automatically from the fact that the system is complete in L^{2} (completeness of polynomials in weighted L^{2}-spaces on the real line ℝ is a nontrivial question, for example).

Theorem 8.4. If W is a d × d Muckenhoupt weight, the system of subspaces E_{I}, I ∈ D is complete in L^{2}(W).

Proof. Since the system E_{I}, I ∈ D is a Riesz basis in L^{2}, then for any f ∈ L^{2} the series

∑_{I∈D} E^{I}f

converges unconditionally (partial sums converge independently of order) in the norm of L^{2} to f.

Theorem 8.3 implies that for any f ∈ L^{2}(W) partial sums of

∑_{I∈D} E^{I}f  (8.6)

are uniformly bounded. Since E_{I}, I ∈ D is a Riesz basis (an isomorphic image of an orthogonal basis) in the closure of its linear span in L^{2}(W), implies that the series (8.6) converges (in L^{2}(W)) for any f ∈ L^{2}(W).

Let H = L^{2} ∩ L^{2}(W). Clearly H is dense in both L^{2} and L^{2}(W). Then for any f ∈ H the series (8.6) converges in norms of both L^{2} and L^{2}(W). The imbedding operator from L^{2} to L^{2}(W) (with natural domain) is a closed operator — this is a well-known fact that follows, for example, from the fact that for any sequence that converges in L^{2} one can find a subsequence that converges almost everywhere. Therefore the series (8.6) converges to f in both norms, so H is in the L^{2}(W) closed linear span of E_{I}, I ∈ D. □

Remark. The results of this section can be extended to the more general case of wavelet transform, when we do not assume that the system φ_{I}, I ∈ D is a Riesz basis or even uniformly minimal. We only assume that the functions φ, ψ satisfy conditions 1 — 4 above, and that the series

∑_{I∈D} φ_{I}(f, ψ_{I})_{L^{2}} =^{def} ∑_{I∈D} E^{I}f

converges to f for all f in a dense set in L^{2}. Theorems 8.2, 8.3, 8.4 hold and all proofs work in this case. Of course in Theorem 8.3 E^{I} denote not spectral projections but simply operators given by the formula

E^{I}f = φ_{I}(f, ψ_{I})_{L^{2}} = φ_{I} · ∫_{ℝ} f\overline{ψ_{I}} .

Theorem 8.1 can be easily reformulated as follows: If W is a d × d matrix Muckenhoupt weight, then for any f ∈ L^{2}(W) the series

∑_{I∈D} φ_{I}(f, ψ_{I})_{L^{2}} =^{def} ∑_{I∈D} E^{I}f

converges unconditionally (independently of order) to f in the norm on L^{2}(W). All the proofs remain practically the same.

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