MATRIX A_{p} WEIGHTS VIA MAXIMAL FUNCTIONS

MICHAEL GOLDBERG

Abstract. The matrix A_{p} condition extends several results in weighted norm theory to functions taking values in a finite-dimensional vector space. Here we show that the matrix A_{p} condition leads to L^{p}-boundedness of a Hardy-Littlewood maximal function, then use this estimate to establish a bound for the weighted L^{p} norm of singular integral operators.

1. Preliminaries

Weighted Norm theory forms a basic component of the study of singular integrals. Here one attempts to characterize those measure spaces over which a broad class of singular integral operators remain bounded. For the case of singular integral operators on C-valued functions in Euclidean space, the answer is given by the Hunt-Muckenhoupt-Wheeden theorem [10]. It states that the necessary and sufficient condition for boundedness in L^{p}(dμ) is that dμ = W (x) dx and the function W

satisfies the A_{p} condition, namely: (\frac{1}{|B|} ∫_{B} W dx)^{1/p}(\frac{1}{|B|} ∫_{B} W^{−p′/p}dx)^{1/p′} ≤ C for

all balls B ⊂ R^{n}.

The A_{p} condition requires considerable interpretation in order to apply it to weighted measures of C^{d}-valued functions. First, the weight W (x) should take values in the space of positive d×d Hermitian forms. This raises concerns about the order in which products are taken, since matrices need not commute, and also what it means for the quantity on the left-hand side to be uniformly bounded. Treil [21] conjectured that the correct statement of the matrix A_{2} condition should be

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

where exponents 1/2 indicate operator powers of a nonnegative matrix. This was subsequently proven in [23] and again in [24].

If p is different from 2, the matrix A_{p} condition cannot be written in terms of averages of operator powers of weight W. Averages still play a crucial role, however it is more accurate to regard W (x) as a Banach space norm on C^{d} rather than a matrix. A correct formulation of the matrix A_{p} condition, which is also the subject of this note, first appeared in [12] and [25]. Because their statements do not appear similar, it is especially important to understand what properties matrix A_{p} weights share with their scalar counterparts. This is discussed further in the next section.

Date: June 20, 2001.

The author was supported in part by an NSF graduate research fellowship.

Boundedness estimates on singular integral operators were originally obtained by way of the Hardy-Littlewood maximal function M. If a scalar weight W possesses the A_{∞} property (several equivalent definitions are given in [18]), then the L^{p} norm of any singular integral is dominated by the L^{p} norm of M via a distributional argument commonly known as the good-λ inequality. The A_{p} condition is specifically required to ensure that ∥Mf∥_{L^{p}(W)} ≤ C∥f∥_{L^{p}(W)}.

Some of these techniques fail to generalize to the case of vector-valued functions with matrix weights. There is no known analogue of the A_{∞} property to create simultaneous estimates for every exponent p. The weak-L^{p}(W) spaces used to prove boundedness of the Hardy-Littlewood maximal function are not well defined in this setting. In general, much of the ability to compare objects and dominate one by another is lost when the objects are vectors rather than scalars. The theory of matrix weights has consequently evolved along much different lines. One fundamental technique employed in both [23] and [25] is to choose a good basis (often inspired by Haar functions) in L^{p}(W) and consider the integral operator as a matrix acting on the coefficient space. Estimates may then be made separately on the matrix and on the coefficient embedding operator. Even in the scalar case these ideas have yielded new results and new ways of approaching weighted norm problems.

In this note we attempt to tackle the difficulties of extending the classical theory, or else circumvent them. Some arguments may be borrowed nearly word for word, some remain intact only if they are presented in a specific manner. Our hope is to discover which properties of scalar A_{p} weights admit some generalization to the case of vectorvalued functions and matrix weights, leading to a more complete understanding of the matrix A_{p} class.

Let T be a singular integral operator associated to kernel K(x) in the sense that T f(x) = ∫_{R^{n}} K(x − y)f(y) dy for almost every x outside the support of f. The following regularity hypotheses are to be assumed for K:

(1)  |K(x)| ≤ C|x|^{−n} and |∇K(x)| ≤ C|x|^{−n−1}

and additionally we suppose that for some choice of p, 1 < p < ∞, the bound ∥T f∥_{L^{p}} ≤ A∥f∥_{L^{p}} holds for all f ∈ L^{p}. One may then apply T to functions taking values in C^{d} by allowing it to act separately on each coordinate function, that is: (T f)_{j} = T f_{j}. This new operator, also denoted by T, is a singular integral operator whose associated convolution kernel is K times the identity matrix.

In a similar manner, define the truncated operators T_{ϵ} to be convolution with K_{ϵ}(x) = χ_{{|x|>ϵ}}K(x) for all ϵ > 0. Note that T and the T_{ϵ} all commute with pointwise multiplication by any constant matrix Λ, in other words ΛT f = T (Λf).

A matrix weight W is a function on R^{n} taking values in d × d positive-definite matrices, with weighted norm space L^{p}(W) defined by

(2)  ∥f∥_{L^{p}(W)}^{p} = ∫_{R^{n}} |W^{1/p}f|^{p}dx

One is often concerned with the relationship between a weight and its average over arbitrary balls. The most straightforward notion of an average, W_{B} = \frac{1}{|B|} ∫_{B} W dx, turns out to be useful only in the study of L^{2}(W). With any exponent p ≠ 2, this does not properly respect the structure of the underlying L^{p}-space. The following definitions are needed instead:

A metric ρ = ρ_{x}( ·) denotes a family of Banach space norms on C^{d}, indexed by x ∈ R^{n}. The weighted norm space L^{p}(ρ) is given by

∥f∥_{L^{p}(ρ)}^{p} = ∫_{R^{n}} [ρ_{x}(f(x))]^{p} dx

Note that for any matrix weight W, L^{p}(W) is isometrically equivalent to L^{p}(ρ) with the metric ρ_{x}(e) = |W^{1/p}(x)e|. Given a ball B ⊂ R^{n} and an exponent p > 1, let ρ_{p,B} be defined by the formula

ρ_{p,B}(e) = (\frac{1}{|B|} ∫_{B} [ρ_{x}(e)]^{p} dx)^{1/p}

This will be our method for averaging the metric ρ over a ball B.

The dual metric ρ^{∗} is defined pointwise in x to be

ρ_{x}^{∗}(e) = sup_{f∈C^{d}} \frac{|(e, f)|}{ρ_{x}(f)}

One immediate consequence is that (e, f) ≤ ρ_{x}^{∗}(e)ρ_{x}(f). Proposition 1.1. For any e ∈ C^{d} and any ball B ⊂ R^{n}, ρ_{p′,B}^{∗}(e) ≥ (ρ_{p,B})^{∗}(e).

Proof. Given two vectors e, f ∈ C^{d},

(e, f) ≤ \frac{1}{|B|} ∫_{B} ρ_{x}^{∗}(e)ρ_{x}(f) dx

≤ (\frac{1}{|B|} ∫_{B} [ρ_{x}^{∗}(e)]^{p′} dx)^{1/p′} · (\frac{1}{|B|} ∫_{B} [ρ_{x}(f)]^{p} dx)^{1/p}

= ρ_{p′,B}^{∗}(e)ρ_{p,B}(f)

In other words, ρ_{p′,B}^{∗}(e) ≥ \frac{(e, f)}{ρ_{p,B}(f)}. The proof is completed by taking the supremum

over all f ∈ C^{d}. □

A metric ρ is called an A_{p} metric if there exists some constant C < ∞ so that the opposite statement

(3)  ρ_{p′,B}^{∗}(e) ≤ C(ρ_{p,B})^{∗}(e) for all balls B ⊂ R^{n}

is also true. Since the averages over cubes and balls in R^{n} differ by no more that a fixed constant, A_{p} metrics satisfy an analogous condition for cubes, and vice versa. Stated either way, the A_{p} condition characterizes an important class of weighted measures. Theorem 1. (Nazarov,Treil [12], Volberg [25]) Let d < ∞. The following statements are equivalent:

1) The Hilbert Transform is bounded on L^{p}(ρ).

2) ρ is an A_{p} metric.

We will prove this theorem again for metrics which are induced by some matrix weight W. There is no loss of generality because for fixed dimension d < ∞ every metric can be uniformly approximated by matrix weights. Proposition 1.2. Let d < ∞. Given a Banach space norm ρ_{x} on C^{d}, there exists a positive selfadjoint matrix W_{x} such that

(4)  ρ_{x}(e) ≤ |W_{x}(e)| ≤ \sqrt{d} · ρ_{x}(e) for all e ∈ C^{d}.

Proof. Let O represent the unit ball of ρ_{x}, and E the ellipsoid of maximal volume contained in O. There exists a positive selfadjoint matrix W_{x} such that W_{x}(E) is the standard unit ball in C^{d}. The image W_{x}(O) is a convex balanced set containing the unit ball, and containing no ellipsoid of greater volume.

If there exists a point v ∈ W_{x}(O) with |v| > \sqrt{d}, then by convexity the boundary of W_{x}(O) can only be tangent to the unit sphere at points w such that

(w, v) ≤ \frac{1}{|v|} < \frac{1}{\sqrt{d}}

For some δ > 0 the ellipsoid with major axis length e^{δ} in the direction of v and minor axes length e^{−δ/(|v|^{2}−1)} in every direction perpendicular to v is also contained in W_{x}(O). This has strictly greater volume than the unit ball, contradicting the property of W_{x}(O) stated above. □

It is now possible to state the A_{p} condition in terms of matrix weights, though some precision is lost in the process. Given a matrix weight W and a ball B ⊂ R^{n}, define a Banach space norm X_{B} on C^{d} by considering the L^{p}(W) norm of characteristic functions on B.

∥v∥_{X_{B}} = |B|^{−1/p}∥χ_{B}v∥_{L^{p}(W)}

By proposition 1.2 there exists a positive-definite d × d matrix V_{B} such that ∥v∥_{X_{B}} ≤ |V_{B}v| ≤ d^{1/2}∥v∥_{X_{B}}. From a heuristic standpoint, V_{B} might be considered an “L^{p} average” of W^{1/p} over ball B. With p′ = \frac{p}{p−1} the dual exponent to p, let V′_{B} be an L^{p′} average of W^{−1/p}. In summary, matrices V_{B}, V′_{B} enjoy the following properties:

\begin{aligned}|V_{B}v| & ∼ |B|^{−1/p}∥χ_{B}W^{1/p}v∥_{L^{p}} \\ |V′_{B}v| & ∼ |B|^{−1/p′}∥χ_{B}W^{−1/p}v∥_{L^{p′}} \end{aligned} ; (5)

Remark. The definition of V_{B} and V′_{B} depends implicitly on the method used to approximate Banach space norms by matrices. For the purposes of our discussion, V_{B} and V′_{B} may be any two matrices satisfying (5).

The statement about weights taking the place of proposition 1.1 is

|V_{B}V′_{B}e| ≥ |e| for all vectors e ∈ C^{d} and balls B ⊂ R^{n}.

A matrix weight W satisfies the matrix A_{p} condition if V_{B}V′_{B} are uniformly bounded as operators on C^{d}; that is

(6)  ∥V_{B}V′_{B}∥ ≤ C < ∞ for all balls B ⊂ R^{n}

The exact value of C depends on the choice of V_{B} and V′_{B}, and is therefore determined here only up to a factor of d.

Our approach to Theorem 1 is styled after Coifman and Fefferman’s proof [5] in the scalar (d = 1) case. Two technical problems arise immediately: first that general d × d matrices do not commute with one another, and second the matter of defining a maximal operator for vector-valued functions. To choose pointwise a vector with the largest ℓ^{2}(C^{d}) magnitude is clearly wrong because the effect of weight W (x) may depend strongly on the direction. In the special case where W is uniformly nonsingular (i.e. ∥W (x)∥ · ∥W^{−1}(x)∥ ≤ C for all x) this can be controlled by a constant factor, but we have no such a priori assumptions about W.

For this reason our analysis will take place primarily in unweighted L^{q} spaces, following [4]. Rather than deal with T directly, we consider the action of W^{1/p}T W^{−1/p} on functions in L^{q}(dx). With the family of truncated operators W^{1/p}T_{ϵ}W^{−1/p} in mind, we define the maximal truncated operator (W^{1/p}T)_{∗} to be

(7)  (W^{1/p}T)_{∗}f(x) = sup_{ϵ>0} |W^{1/p}T_{ϵ}f(x)|

with the convention that f = W^{−1/p}g and g is a function in L^{q}(dx). One estimate from the scalar theory that remains wholly intact is the bound

(8)  |W^{1/p}T W^{−1/p}g|(x) ≤ |(W^{1/p}T)_{∗}W^{−1/p}g|(x) + C|g(x)|

The constant C depends only on our choice of operator T but not on the function g. This will allow us to infer the boundedness of T by controlling the behavior of its truncations. Our primary results are the following four theorems, numbered according to the section in which they appear: Four Theorems. (3.2) If W is a matrix A_{p} weight, there exists δ > 0 such that the vector Hardy-Littlewood maximal function M_{w} (defined in section 3) is a bounded operator from L^{q}(R^{n}; C^{d}) to L^{q}(R; R) whenever |p − q| < δ. (4.2): Given a singular integral operator T as above, and a weight W ∈ A_{p}, there exists δ > 0 such that (W^{1/p}T)_{∗}W^{−1/p} is a bounded operator from L^{q}(R^{n}; C^{d}) to L^{q}(R; R) whenever |p − q| < δ. (5.1): Consequently W^{1/p}T W^{−1/p} is bounded on L^{q}(R^{n}; C^{d}) for these exponents q. (5.2): In particular, T is bounded on L^{p}(W) if W ∈ A_{p}. With one additional hypothesis on the structure of T, the converse statement is also true. Remark. The exponent W^{1/p} is used throughout, even when we are considering functions under an L^{q} norm with q ≠ p. This places us squarely in the setting of [25], where the A_{p} metric W^{1/p} is the basic object of study. Theorem 5.1 then asserts that any A_{p} metric is also an A_{q} metric for all q in some open interval containing p.

2. Properties of A_{p} Weights

We would like first to characterize the matrix A_{p} class in a more transparent manner by borrowing a lemma from [12]:

Proposition 2.1. A metric ρ_{x} satisfies the A_{p} condition if and only if the operators f → χ_{B} \frac{1}{|B|} ∫_{B} f dx are uniformly bounded on L^{p}(ρ). In fact, the uniform bound is equal to the A_{p} constant of ρ.

Proof. The L^{p}(ρ) norm of χ_{B} \frac{1}{|B|} ∫_{B} f dx is given by \frac{1}{|B|}( ∫_{B} [ρ_{y}( ∫_{B} f dx)]^{p}dy)^{1/p}, which in turn is equal to |B|^{−1/p′}ρ_{p,B}( ∫_{B} f dx). Therefore

sup_{∥f∥_{L^{p}(ρ)}=1} ∥χ_{B} \frac{1}{|B|} ∫_{B} f dx∥_{L^{p}(ρ)} = sup_{f} sup_{e∈C^{d}} |B|^{−1/p′} \frac{∫_{B}(e, f(x)) dx}{(ρ_{p,B})^{∗}(e)}

= sup_{e∈C^{d}} |B|^{−1/p′} \frac{∥χ_{B}e∥_{L^{p′}(ρ^{∗})}}{(ρ_{p,B})^{∗}(e)} = sup_{e∈C^{d}} \frac{ρ_{p′,B}^{∗}(e)}{(ρ_{p,B})^{∗}(e)}

Equality between the first and second lines takes place because L^{p}(ρ) is the dual space of L^{p′}(ρ^{∗}). □

Corollary 2.2. Let ρ be an A_{p} metric. For any vector v ∈ C^{d}, ρ_{x}(v)^{p} is a scalar A_{p} weight with constant less than or equal to that of ρ.

Proof. Let ϕ be any scalar function and consider f = ϕv. The weighted norm of f is ∥f∥_{L^{p}(ρ)} = (∫_{B} ϕ^{p} [ρ_{x}(v)]^{p}dx)^{1/p}. Proposition 2.1 applied to f states that all maps ϕ → χ_{B} \frac{1}{|B|} ∫_{B} ϕ dx are uniformly bounded on the L^{p} space with measure [ρ_{x}(v)]^{p}dx, with norms less than the A_{p} constant of ρ. We now apply Proposition 2.1 again, this time in the scalar setting, to conclude that [ρ_{x}(v)]^{p} is a scalar A_{p} weight whose constant is also less than the A_{p} constant of ρ. □

Corollary 2.3. If W is a matrix A_{p} weight, then ∥W ∥ is a scalar A_{p}weight.

Proof. Let e_{i} be the standard unit basis for C^{d}. Since W (x) is a nonnegative and selfadjoint operator at each point x,

\begin{aligned}∥W (x)∥ = ∥W^{2/p}(x)∥^{p/2} & ∼ [tr(W^{2/p}(x))]^{p/2} \\ & = ( ∑_{i=1}^{d} |W^{1/p}(x)e_{i}|^{2})^{p/2} ∼ ∑_{i=1}^{d} |W^{1/p}(x)e_{i}|^{p} \end{aligned} ; (9)

pointwise in x. By corollary 2.2, each individual function |W^{1/p}(x)e_{i}|^{p} is a scalar A_{p} weight, therefore their sum is as well. □

Remarks. Both of these corollaries are proven in [23] for the case p = 2, and are adapted here with minimal alteration.

From this point forward we will work exclusively in the language of matrix weights. While our primary definition of A_{p} weights (6) is decidedly less elegant than that of A_{p} metrics (3), the ability to use notaion and theorems from linear algebra makes it a worthwhile sacrifice.

One crucial feature in the theory of scalar A_{p} weights is the presence of “Reverse Hölder” inequalities estimating the average of W^{1+ϵ} in terms of the average of W. We will employ inequalities of a similar character as the centerpiece of our analysis.

Proposition 2.4. Let W be an A_{p} weight. Then there exist δ > 0 and constants C_{q} such that for all balls B ⊂ R^{n},

(10)  \frac{1}{|B|} ∫_{B} ∥W^{1/p}(y)V′_{B}∥^{q}dy ≤ C_{q}, all q < p + δ

(11)  \frac{1}{|B|} ∫_{B} ∥V_{B}W^{−1/p}(y)∥^{q}dy ≤ C_{q}, all q < p′ + δ

Proof. We will verify only the first of these statements. The second one is proven in an identical manner with the starting point that W^{−p′/p} is an A_{p′} weight.

By Corollary 2.2, all functions of the form |W^{1/p}(y)V′_{B}e|^{p} are scalar A_{p} weights with A_{p} norms bounded uniformly in e. It is therefore possible to choose q and C_{q} so that the Reverse Hölder inequality

\frac{1}{|B|} ∫_{B} |W^{1/p}(y)V′_{B}e|^{q}dy ≤ C_{q}(\frac{1}{|B|} ∫_{B} |W^{1/p}(y)V′_{B}e|^{p}dy)^{q/p}

is satisfied for all e ∈ C^{d}.

Let e_{i} once again be the standard unit basis for C^{d}. It is useful to remember that the norm of any d×d matrix M (not necessarily Hermitian) is controlled by its action on the vectors e_{i} via the formula

∥M∥ ≤ d^{1/2} sup_{i} |Me_{i}|

We may now estimate the desired integral:

\frac{1}{|B|} ∫_{B} ∥W^{1/p}(y)V′_{B}∥^{q}dy ≤ \frac{1}{|B|} ∫_{B} (d^{1/2} sup_{i} |W^{1/p}(y)V′_{B}e_{i}|)^{q}dy

≤ d^{q/2} ∑_{i=1}^{d} \frac{1}{|B|} ∫_{B} |W^{1/p}(y)V′_{B}e_{i}|^{q}dy ≤ C_{q} ∑_{i=1}^{d} (\frac{1}{|B|} ∫_{B} |W^{1/p}(y)V′_{B}e_{i}|^{p}dy)^{q/p}

∼ C_{q} ∑_{i=1}^{d} |V_{B}V′_{B}e_{i}|^{q} ≤ d · C_{q}∥V_{B}V′_{B}∥^{q} ≤ C_{q}.

□

Note. In later sections we will also use the slightly weaker inequality

(12)  |B|^{−1} ∫_{B} ∥W^{1/p}(y)V_{B}^{−1}∥^{q}dy ≤ C_{q}, all q < p + δ

whose proof follows the above calculations almost word for word.

3. The Hardy-Littlewood Maximal Function

There is a wide variety of possible maximal functions to choose from, each of which has its own advantages and limitations. In [4] we first considered an auxiliary maximal function M′_{w}, given by

(13)  M′_{w}g(x) = sup_{x∈B} \frac{1}{|B|} ∫_{B} |V_{B}W^{−1/p}(y)g(y)| dy

Although the intuitive meaning of M′_{w} is unclear, one may approach it with the classical tools of weak-type inequalities and interpolation. A direct application of the second reverse Hölder inequality (11) proves the following lemma. Lemma 3.1. Let W be an A_{p} weight. Then there exists δ > 0 such that

∥M′_{w}g∥_{L^{q}} ≤ C_{q}∥g∥_{L^{q}(R^{n};C^{d})}, all g ∈ L^{q}, all q > p − δ.

Sketch of Proof. The reverse Hölder inequality allows us to extend Proposition 2.1 to exponents p − δ < q < ∞. For this maximal function one may use the Vitali Covering Lemma to obtain a weak-type (q, q) estimate. The result then follows from the Marcinkiewicz Interpolation Theorem.

The vector Hardy-Littlewood maximal function M_{w} is defined as

(14)  M_{w}g(x) = sup_{x∈B} \frac{1}{|B|} ∫_{B} |W^{1/p}(x)W^{−1/p}(y)g(y)|dy

The following equivalent definition of M_{w} is often quite useful:

(15)  M_{w}g(x) = M(|W^{1/p}(x)W^{−1/p}(·)g(·)|)(x)

Here M denotes the classical Hardy-Littlewood maximal operator acting on scalarvalued functions. The only difference between M_{w} and M′_{w} is the presence of a weight W^{1/p}(x) rather than an average weight V_{B} over a ball containing x. The reverse Hölder inequalities suggest that A_{p} weights are often pointwise comparable to their averages, in which case ∥M_{w}g∥ would be controlled by ∥M′_{w}g∥. For a range of exponents near p, this line of reasoning can be made precise. Theorem 3.2. Let W be an A_{p} weight. Then there exists δ > 0 such that

∥M_{w}g∥_{L^{q}} ≤ C_{q}∥g∥_{L^{q}(R^{n};C^{d})}, all g ∈ L^{q}, all |p − q| < δ.

Proof. Let us suppose for a moment that the suprema defining M_{w}g and M′_{w}g are taken over cubes in some dyadic grid. The entire preceding discussion holds for maximal functions over cubes, so in particular we can still estimate ∥M′_{w}g∥ via Lemma 3.1. For each point x, choose a (dyadic) cube R_{x} such that

M_{w}g(x) ≤ 2|R_{x}|^{−1} ∫_{R_{x}} |W^{1/p}(x)W^{−1/p}(y)g(y)|dy

≤ 2∥W^{1/p}(x)V_{R_{x}}^{−1}∥ · (|R_{x}|^{−1} ∫_{R_{x}} |V_{R_{x}}W^{−1/p}(y)g(y)|dy).

For each integer j, define {S_{j}} to be the collection of dyadic cubes R = R_{x} that are maximal with respect to the property 2^{j} ≤ |R|^{−1} ∫_{R} |V_{R}W^{−1/p}(y)g(y)|dy < 2^{j+1}. Maximality insures that whenever M_{w}g(x) ≠ 0 the cube R_{x} is contained in some S_{j} with

|R_{x}|^{−1} ∫_{R_{x}} |V_{R_{x}}W^{−1/p}(y)g(y)|dy ≤ 2|S_{j}|^{−1} ∫_{S_{j}} |V_{S_{j}}W^{−1/p}(y)g(y)|dy.

When j is fixed, the disjoint union ∪_{j}S_{j} is contained in the set where M′_{w}g(x) ≥ 2^{j}.

Consider the functions N_{Q}(x) = sup_{x∈R⊂Q} ∥W^{1/p}(x)V_{R}^{−1}∥, defined for x ∈ Q. By virtue

of the preceding two statements, the inequality M_{w}g(x) ≤ 4 · 2^{j+1}N_{S_{j}}(x) must hold for some number j (this is trivial at the points where M_{w}g(x) = 0). It follows that

(16)  ∥M_{w}g∥_{L^{q}}^{q} ≤ C ∑_{j=−∞}^{∞} 2^{jq} ∑_{S_{j}} ∫_{S_{j}} (N_{S_{j}}(x))^{q}dx

By Lemma 3.3 below, we can continue the estimate as follows:

∥M_{w}g∥_{L^{q}}^{q} ≤ C ∑_{j=−∞}^{∞} 2^{jq} ∑_{S_{j}} |S_{j}| ≤ C ∑_{j=−∞}^{∞} 2^{jq} |{M′_{w}g ≥ 2^{j}}| ≤ C∥M′_{w}g∥_{L^{q}}^{q}

The proof is then complete by Lemma 3.1.

□

Lemma 3.3. Let W be a matrix A_{p} weight and functions N_{Q}(x) be defined as above. Then there exist δ > 0 and C_{q} < ∞ such that for all dyadic Q,

∫_{Q} (N_{Q}(x))^{q}dx ≤ C_{q}|Q| for all q < p + δ

Proof. We present an informal argument here, assuming that ∫_{Q} N_{Q}^{q} ≤ B|Q| with some finite B then deriving an a priori bound for B. This may be readily adapted into a rigorous proof.

Let A < ∞ be a large constant to be specified later. Denote by {R_{j}} the set of maximal cubes satisfying ∥V_{Q}V_{R_{j}}^{−1}∥ > A. Outside of ∪_{j}R_{j}, N_{Q}(x) ≤ A∥W^{1/p}(x)V_{Q}^{−1}∥.

Thus ∫_{Q\∪_{j}R_{j}} (N_{Q}(x))^{q}dx ≤ C|Q|, seen by applying reverse Hölder inequality (12).

We claim that ∑_{j} |R_{j}| < \frac{1}{2}|Q| if A is sufficiently large. Remember first that ∥V_{Q}V_{R_{j}}^{−1}∥ = ∥V_{R_{j}}^{−1}V_{Q}∥ ≤ C∥V′_{R_{j}}V_{Q}∥, by Proposition 1.1. It follows that

(17) |R_{j}| · ∥V′_{R_{j}}V_{Q}∥^{p′} = sup_{|e|=1} |R_{j}| · |V′_{R_{j}}V_{Q}e|^{p′}

∼ sup_{|e|=1} ∫_{R_{j}} |W^{−1/p}(y)V_{Q}e|^{p′}dy ≤ ∫_{R_{j}} ∥W^{−1/p}(y)V_{Q}∥^{p′}dy

The cubes R_{j} are disjoint from one another, so

A^{p′} ∑_{j} |R_{j}| < C ∫_{∪_{j}R_{j}} ∥W^{−1/p}(y)V_{Q}∥^{p′}dy ≤ C ∫_{Q} ∥W^{−1/p}(y)V_{Q}∥^{p′} ≤ C|Q|

This estimate shows that for A large enough, ∑_{j} |R_{j}| < \frac{1}{2}|Q|, and the value of A may be chosen independently of Q.

Inside each cube R_{j}, we may assume that N_{Q}(x) = N_{R_{j}}(x), otherwise the bound N_{Q}(x) ≤ A∥W^{1/p}(x)V_{Q}^{−1}∥ still holds. Then

(18)  ∫_{∪_{j}R_{j}} (N_{Q}(x))^{q} = ∑_{j} ∫_{R_{j}} (N_{R_{j}}(x))^{q} ≤ B ∑_{j} |R_{j}| < \frac{1}{2}B|Q|.

Putting these pieces together, we would discover that B ≤ C + \frac{1}{2}B, where C < ∞ is determined by the constants in the reverse Hölder inequality. □

This concludes the proof that matrix A_{p} weights enjoy L^{q}-boundedness of the dyadic Hardy-Littlewood maximal function for a range of exponents |q−p| < δ. There is a standard argument employing two incompatible dyadic grids [7] for extending results of this kind to the general setting. Thus the Hardy-Littlewood maximal function as we originally defined it (as a supremum over balls containing x) is bounded in L^{q} for the same range of exponents q.

4. A Distributional Inequality

Proposition 4.1. Let W be a matrix A_{p} weight and fix q < 2 + δ. Then there exist positive constants 0 < b < 1, c > 0 depending only on q, the A_{p} “norm” of W, and the dimensions d, n such that

|{x ∈ R^{n} : (W^{1/p}T)_{∗}f(x) > α; max(M′_{w}(W^{1/p}f)(x), M_{w}(W^{1/p}f)(x)) < cα}|

(19)  < \frac{1}{2}b^{q}|\{x ∈ R^{n} : (W^{1/p}T)_{∗}f(x) > bα\}|

for all f ∈ C_{c}^{∞}(R^{n}; C^{d})

From this point onward we follow as closely as possible in the footsteps of Coifman and Fefferman [5], decomposing the set where (W^{1/p}T)_{∗}f > bα into a union of cubes and proving the desired inequality on each cube separately. Our decomposition uses a slightly modified version of the Whitney covering lemma, stated below. Covering Lemma. Given a set E ⊂ R^{n} of finite (Lebesgue) measure, there exists a collection {Q_{j}} of pairwise disjoint cubes such that:

i) E ⊂ ∪_{j}Q_{j} up to sets of measure zero

ii) |Q_{j} ∩ E| ≥ \frac{1}{2}|Q_{j}|
iii) |3Q_{j} ∩ E^{c}| ≥ C_{n}|3Q_{j}|

A simple consequence of statements i) and ii) is that ∑_{j} |Q_{j}| ≤ 2|E|.

Proof. Let {Q_{j}} be the collection of dyadic cubes maximal under the property that |Q ∩ E| ≥ \frac{1}{2}|Q|. Then conditions ii) and iii) hold with constant C_{n} = \frac{1}{2} · (\frac{2}{3})^{n}. The first condition also holds because as ϵ → 0, the ratio |B(x, ϵ) ∩ E|/|B(x, ϵ)| → 1 at almost every x ∈ E. □

Proof of Proposition 4.1. Write f = W^{−1/p}g and let

E = {x ∈ R^{n} : (W^{1/p}T)_{∗}f(x) > bα}

Apply the covering lemma to obtain cubes {Q_{j}} with the specified properties. It suffices to verify that in each cube Q = Q_{j} there is a distributional inequality

(20) |\{x ∈ Q : (W^{1/p}T)_{∗}f(x) > α; max(M′_{w}g(x), M_{w}g(x)) < cα\}| < \frac{1}{4}b^{q}|Q|.

For this we use a construction similar to the one in [5]. Let O be the ball with the same center as Q_{j} and radius 5 diam (Q). By the covering lemma and inequality (11), there exists a point \overline{x} ∈ 3Q such that

(W^{1/p}T)_{∗}f(\overline{x}) < bα and ∥V_{O}W^{−1/p}(\overline{x})∥ < C

Let B = B(\overline{x}, 3 diam(Q_{j})). Since B ⊂ O and is of comparable size, ∥V_{B}V_{O}^{−1}∥ is bounded by a constant and hence ∥V_{B}W^{−1/p}(\overline{x})∥ < C.

Assume |\{x ∈ Q : M′_{w}g(x) < cα\}| ≥ \frac{1}{4}b^{q}|Q|, otherwise the proposition is trivially satisfied. Then there exists a point \overline{y} ∈ Q such that

M′_{w}g(\overline{y}) < cα and ∥V_{B}W^{−1/p}(\overline{y})∥ ≤ Cb^{−1}

Write f_{1} = χ_{B}f and f_{2} = χ_{B^{c}}f. By the sublinearity of (W^{1/p}T)_{∗}, the set where (W^{1/p}T)_{∗}f(x) > α is contained in the union of sets (W^{1/p}T)_{∗}f_{i}(x) > α/2, i = 1, 2.

The operator T_{∗} is weak-type (1, 1). This fact is easily obtained from the scalar case when d is finite, but is also true in general [17]. Consequently,

|\{(V_{B}T)_{∗}f_{1}(x) > \frac{α}{2R}\}| ≤ \frac{AR}{α}∥V_{B}f_{1}∥_{L_{1}(R^{n};C^{d})}

Here we are using the property that operator T_{∗} commutes with multiplication by any constant matrix, in this case V_{B}. Furthermore,

∥V_{B}f_{1}∥_{L^{1}} = ∫_{B} |V_{B}f(y)|dy ≤ |B|M′_{w}g(\overline{y}) ≤ Ccα|Q|

with the end result that |\{x ∈ Q : (V_{B}T)_{∗}f_{1}(x) > \frac{α}{2R}\}| ≤ CcR|Q|.

It follows that |\{x ∈ Q : (W^{1/p}T)_{∗}f_{1}(x) > \frac{α}{2}\}| ≤ (CcR + C′R^{−p})|Q| for all R > 0, because the Reverse Hölder inequality (10) guarantees that ∥W^{1/p}(x)V_{B}^{−1}∥ < R except on a set of measure less than C′R^{−p}. Taking the infimum over R,

(21)  |{x ∈ Q : (W^{1/p}T)_{∗}f_{1}(x) > α/2}| ≤ C_{0}c^{p/(p+1)}|Q|

For the second estimate, we begin by noting that the point \overline{x} is chosen so that (W^{1/p}T)_{∗}f(\overline{x}) < bα and ∥V_{B}W^{−1/p}(\overline{x})∥ < C. Then (V_{B}T)_{∗}f(\overline{x}) < Cbα. Our estimate for |{(W^{1/p}T)_{∗}f_{2}(x) > α/2}| relies on the following inequality which holds for all x ∈ Q.

(22) (V_{B}T)_{∗}f_{2}(x) ≤ (V_{B}T)_{∗}f(\overline{x}) + C′M(|V_{B}f|)(\overline{y})

≤ Cbα + C′∥V_{B}W^{−1/p}(\overline{y})∥ · M(|W^{1/p}f|)(\overline{y})

≤ Cbα + C′∥V_{B}W^{−1/p}(\overline{y})∥ · M_{w}g(\overline{y}) ≤ (Cb + C′b^{−1}c)α

In the preceding expressions M(·) denotes the scalar Hardy-Littlewood maximal function.

Imitating the method for the |(W^{1/p}T)_{∗}f_{1}| estimate, we see that

|{x ∈ Q : (W^{1/p}T)_{∗}f_{2}(x) > R(Cb + C′b^{−1}c)α}| ≤ AR^{−r}|Q|

where r may be chosen so that q < r < p + δ. Once again (10) has been invoked, this time to guarantee that ∥W^{1/p}V_{B}^{−1}∥ > R only on a set of measure less than CR^{−r}|B|. Set R equal to (4bC)^{−1}. Then

(23)  |{x ∈ Q : (W^{1/p}T)_{∗}f_{2}(x) > (1/4 + C_{1}b^{−2}c)α}| ≤ C_{2}b^{r}|Q|

Statement (20) is then verified by choosing b < (8C_{2})^{1/q−r} and c sufficiently small. Summing over all cubes Q_{j} proves the proposition. □

Corollary 4.2. With c as in Proposition (4.1),

∥(W^{1/p}T)_{∗}f∥_{L^{q}}^{q} ≤ 2c^{−q}∥ max(M′_{w}(W^{1/p}f), M_{w}(W^{1/p}f))∥_{L^{q}}^{q}

for all f ∈ C_{c}^{∞}(R^{n}; C^{d})

Proof. If both sides of (19) are multiplied by qα^{q−1} and integrated over the the interval 0 ≤ α < ∞, the resulting inequality is

∫_{R^{n}} ([(W^{1/p}T)_{∗}f]^{q} − c^{−q} max ([M′_{w}(W^{1/p}f)]^{q}, [M_{w}(W^{1/p}f)]^{q}))_{+}dx

≤ \frac{1}{2} ∫_{R^{n}}[(W^{1/p}T)_{∗}f]^{q} dx

from which it follows that

∥(W^{1/p}T)_{∗}f∥_{L^{q}}^{q} − \frac{1}{c^{q}} ∥ max(M′_{w}(W^{1/p}f), M_{w}(W^{1/p}f))∥_{L^{q}}^{q} ≤ \frac{1}{2}∥(W^{1/p}T)_{∗}f∥_{L^{q}}^{q}

The remaining task is to verify that the L^{q} norm of (W^{1/p}T)_{∗}f is finite. A key estimate is the fact that T_{∗}f(x) ≤ C_{f}(1 + |x|)^{−n} for all f ∈ C_{c}^{∞}, where C_{f} depends on f. Then

(W^{1/p}T)_{∗}f(x) ≤ C∥W ∥^{1/p}(1 + |x|)^{−n}

There are many ways to show that the expression on the right-hand side is in L^{q}, all exploiting the fact that ∥W ∥ is a scalar A_{p} weight. One possibility is to choose any nontrivial (scalar) function ϕ ≥ 0 ∈ C_{c}^{∞}. We have shown in Theorem 3.2 that ∥W ∥^{1/p}M(∥W ∥^{−1/p}ϕ) ∈ L^{q} whenever |p − q| < δ.

On the other hand, C(1 + |x|)^{−n} ≤ M(∥W ∥^{−1/p}ϕ), which completes the proof. □

5. The Main Theorem

Theorem 5.1. Let T be a linear operator whose associated convolution kernel K(x) satisfies the hypotheses in (1), and which acts separately on each coordinate function of f (in other words, (T f)_{j} = T f_{j}). Let W be a matrix A_{p} weight.

There exists δ > 0 such that W^{1/p}T W^{−1/p} is a bounded operator on L^{q}(R^{n}; C^{d}) whenever |q − p| < δ.

Proof. As in the scalar case, the truncated operators T_{ϵ} possess a weak limit T_{0}, and T = T_{0} + A, where A is a bounded pointwise multiplier. In dimensions d > 1, A = A(x) is a matrix-valued function, but the hypothesis ΛT Λ^{−1} = T requires A(x) to be a scalar L^{∞} function multiplied by the identity matrix.

The function W^{1/p}T W^{−1/p}g is dominated pointwise by g and (W^{1/p}T)_{∗}(W^{−1/p}g), as in equation (8):

|W^{1/p}T W^{−1/p}g(x)| = |W^{1/p}T_{0}W^{−1/p}g(x) + A(x)g(x)|

≤ |(W^{1/p}T)_{∗}(W^{−1/p}g)(x)| + C|g(x)|.

The triangle inequality for L^{q}-norms immediately yields the result

(24)  ∥W^{1/p}T W^{−1/p}g∥_{L^{q}} ≤ ∥(W^{1/p}T)_{∗}W^{−1/p}g∥_{L^{q}} + C∥g∥_{L^{q}}

For all g such that W^{−1/p}g ∈ C_{c}^{∞}, the right-hand side is controlled by ∥g∥_{L^{q}}. Observe that W^{q/p} is a locally integrable matrix-valued function. Then C_{c}^{∞}(R^{n}; C^{d}) is a dense subset of L^{q}(W^{q/p}). The map f ∈ L^{q}(W^{q/p}) → g = W^{1/p}f ∈ L^{q}(dx) is an invertible isometry, so its image W^{1/p}(C_{c}^{∞}) is dense in L^{q}. Thus the boundedness of W^{1/p}T W^{−1/p} may then be extended to all functions g ∈ L^{q}(R^{n}; C^{d}), |p − q| < δ. □

A converse statement, with some minor modifications, is also true.

Theorem 5.2. Suppose that T is a convolution operator as above, with the additional nondegeneracy hypothesis that there exists some unit vector u ∈ R^{n} such that |K(ru)| ≥ a|r|^{−n}, all r ∈ R \ {0}. If T is a bounded operator on L^{p}(W), then W is an A_{p} weight.

In order to prove this theorem we first need a result about integral operators with bounded and compactly supported kernels:

Proposition 5.3. Let S be an integral operator Sf(x) = ∫_{R^{n}} S(x, y)f(y) whose (scalar) kernel S(x, y) is supported in B ×B and satisfies the bound |S(x, y)| ≤ |B|^{−1} for all (x, y) ∈ B × B.

The norm of S as an operator on L^{p}(W) is less than C_{d}∥V_{B}V′_{B}∥, where C_{d} is a dimensional constant independent of the particular choice of S. In the special case S_{0}(x, y) = |B|^{−1}χ_{B×B}, the operator norm of S_{0} is also greater than C_{d}^{−1}∥V_{B}V′_{B}∥.

Proof. This is a straightforward calculation similar to those found in Section 2. Let f be any function in L^{p}(W). We first estimate the size of W^{1/p}(x)Sf(x) pointwise for each x.

|W^{1/p}(x)Sf(x)| = |W^{1/p}(x) ∫_{B} S(x, y)f(y) dy|

= | ∫_{B} S(x, y)W^{1/p}(x)f(y) dy| ≤ |B|^{−1} ∫_{B} |W^{1/p}(x)f(y)|dy

≤ |B|^{−1}( ∫_{B} ∥W^{1/p}(x)W^{−1/p}(y)∥^{p′}dy)^{1/p′} · ∥f∥_{L^{p}(W)}.

As in Section 2, we now introduce an orthonormal basis of vectors e_{i} spanning C^{d}.

( ∫_{B} ∥W^{1/p}(x)W^{−1/p}(y)∥^{p′}dy)^{1/p′}

≤ ( ∫_{B} (d^{1/2} sup_{i} |W^{−1/p}(y)W^{1/p}(x)e_{i}|)^{p′}dy)^{1/p′}

≤ d^{1/2}( ∑_{i=1}^{d} ∫_{B} |W^{−1/p}(y)W^{1/p}(x)e_{i}|^{p′}dy)^{1/p′}

≤ C_{d}( ∑_{i=1}^{d} |B| · |V′_{B}W^{1/p}(x)e_{i}|^{p′})^{1/p′} ≤ C_{d}|B|^{1/p′}∥V′_{B}W^{1/p}(x)∥

which leads to the estimate |W^{1/p}(x)Sf(x)| ≤ C_{d}|B|^{−1/p}∥V′_{B}W^{1/p}(x)∥ · ∥f∥_{L^{p}(W)}.

Then for all ∥f∥_{L^{p}(W)} ≤ 1, it follows that

(25) ∥Sf∥_{L^{p}(W)} ≤ C(|B|^{−1} ∫_{B} ∥V′_{B}W^{1/p}(x)∥^{p}dx)^{1/p}

≤ C_{d}(|B|^{−1} ∫_{B} (d^{1/2} sup_{i} |W^{1/p}(x)V′_{B}e_{i}|)^{p})^{1/p}

≤ C_{d}( ∑_{i} |B|^{−1} ∫_{B} |W^{1/p}(x)V′_{B}e_{i}|^{p})^{1/p}

∼ C_{d}( ∑_{i} |V_{B}V′_{B}e_{i}|^{p})^{1/p} ≤ C_{d}∥V_{B}V′_{B}∥

The second assertion is a restatement of Proposition 2.1. □

Proof of Theorem 5.2. First, let ϵ > 0 be small enough so that 2ϵ+ϵ^{2} < \frac{1}{2}C_{d}^{−2}. There exists a number t_{0} < ∞ such that

(26) |K(v) − K(rt_{0}u)| ≤ ϵ|K(rt_{0}u)| whenever v ∈ B(rt_{0}u, 2r), all r ∈ R \ {0}.

This is seen to be true because |K(rt_{0}u)| ≥ \frac{a}{t_{0}^{n}|r|^{n}} but |∇K(x)| ≤ \frac{C}{t_{0}^{n+1}r^{n+1}} for all

x ∈ B(rt_{0}u, r). It suffices to choose t_{0} > \frac{2C}{ϵa}.

Let B denote the ball B(y, r) in R^{n}, and B′ the translated ball B′ = B(y+rt_{0}u, r). We wish to consider the operator S_{B} defined by

S_{B}f = χ_{B}T (χ_{B′}T (χ_{B}f))

This is an integral operator whose kernel S_{B}(x, y) = χ_{B×B} ∫_{B′} K(x − z)K(z − y) dz is supported in B × B. If T acts boundedly on L^{p}(W), so too does S_{B} with operator norm less than or equal to ∥T ∥^{2}.

The restrictions {x, y ∈ B, z ∈ B′} guarantee that z − y ∈ B(rt_{0}u, 2r) and x − z ∈ B(−rt_{o}u, 2r). Thus the values of K(z − y) and K(x − z) do not vary much over the region of integration. Using the bounds established in (26), we rewrite S_{B}(x, y) as the sum of a characteristic function and a small remainder:

(27)  S_{B}(x, y) = |B|K(rt_{0}u)K(−rt_{0}u)χ_{B×B} + S_{1}(x, y),

where |S_{1}(x, y)| ≤ \frac{1}{2}C_{d}^{−2}|B| · |K(rt_{0}u)K(−rt_{0}u)|

According to Proposition 5.3, the first term corresponds to an operator with norm at least C∥V_{B}V′_{B}∥. In terms of other constants, C is proportional to a^{2}t_{0}^{−2n}C_{d}^{−1}. The operator corresponding to the second term has norm no more than half as great. It follows that ∥S_{B}∥ ≥ \frac{1}{2}C∥V_{B}V′_{B}∥. Then

(28)  ∥V_{B}V′_{B}∥ ≤ 2C^{−1}∥S_{B}∥ ≤ 2C^{−1}∥T ∥^{2} < ∞

for all balls B ⊂ R^{n}, and W is an A_{p} weight. □

Corollary 5.4. If W is a matrix A_{p} weight, there exists δ > 0 such that W^{q/p} is an A_{q} weight whenever |q − p| < δ. In other words, an A_{p} metric is also an A_{q} metric for all |q − p| < δ. Remarks. We could have proven this statement directly in section 2, using the reverse Hölder inequality to show that operators f → χ_{B} \frac{1}{|B|} ∫_{B} f dx are uniformly bounded on L^{q}(W^{q/p}). To do so would have added another computation without simplifying the subsequent discussion in any way.

Recall that a matrix weight W ∈ A_{p} if and only if the averaging operators A_{B} defined by

A_{B}f = χ_{B} \frac{1}{|B|} ∫_{B} f dx

are uniformly bounded on L^{p}(W). An equivalent statement is that the conjugated operators W^{1/p}A_{B}W^{−1/p} are uniformly bounded on the unweighted space L^{p}(C^{d}). It is trivial to observe that A_{B} are uniformly bounded on L^{∞}(C^{d}) with norm 1. By interpolation on the analytic family of operators^{1}

{W^{(1−z)/p}A_{B}W^{(z−1)/p}, 0 ≤ Re(z) ≤ 1}

we find that W^{1/r}A_{B}W^{−1/r} are uniformly bounded on L^{r}(dx) for all r > p, leading to another result well known in the scalar case: Proposition 5.5. If W is a matrix A_{p} weight, then W is also a matrix A_{r} weight for all r > p.

One crucial difference must be noted. We cannot use the reverse Hölder inequality in this setting to extend the range of exponents to r > p − δ. If we could, then by corollary 5.4 and proposition 5.5 for each weight W ∈ A_{p} there would exist numbers r < q < p such that W^{q/r} ∈ A_{q} ⊂ A_{p}. Instead, counterexamples are known; in [1] a matrix A_{2} weight W is constructed for which W^{s} ∉ A_{2} for any s > 1.

On a speculative note, perhaps this (suspected) lack of self-improvement is related to the absence of a unifying matrix A_{∞} class whose elements are all contained in some A_{p} with p finite. We do not claim to have proven anything here, nor have we investigated thoroughly the union of the A_{p}-weight classes in search of a common A_{∞} property. It has been suggested [25] that the scalar A_{∞} condition generalizes instead to an entire spectrum of A_{p,∞} conditions, one for each exponent p, in the matrix setting.

1Following [16], with the slight modification F_{z}(ψ) = |ψ|^{\frac{α(z)}{α} −1}ψ

6. The Case d = ∞

Most of the estimates in the preceding discussion fail when the dimension d is infinite. Banach space norms may not be representable by matrices, and traces (when defined at all) are no longer comparable to operator norms. Most importantly, the main theorem is false. Gillespie et al. [9] have constructed operator A_{2} weights W for which the Hilbert Transform is unbounded on L^{2}(W).

The test function f in their counterexample is constructed out of Haar functions on different length-scales, with the signs chosen so that each new piece contributes positively to the overall L^{2}(W) norm of T f. Linearity of T is needed to ensure that the whole of T f will be equal to the sum of the various parts, and also to bound from below an expectation over choices of signs. When applied to merely sublinear operators such as a maximal function, the argument is less successful. We do not presently know if the Hardy-Littlewood maximal operator M_{w} is bounded or not.

References

[1] M. Bownik, Inverse volume inequalities for matrix weights, Indiana Univ. Math. J. 50 (2001), no. 1, 383–410.

[2] M. Christ, Weighted norm inequalities and Schur’s lemma, Studia Math. 78 (1984), 309-319.

[3] M. Christ and R. Fefferman, A note on weighted norm inequalities, Proc. Amer. Math. Soc. 87 (1983), 447-448.

[4] M. Christ and M. Goldberg, Vector A_{2} Weights and a Hardy-Littlewood Maximal Function, Trans. Amer. Math. Soc. 353 (2001), 1995-2002.

[5] R. R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241-250.

[6] R. Fefferman, C. E. Kenig, and J. Pipher, The theory of weights and the Dirichlet problem for elliptic equations, Ann. Math. 134 (1991), 65-124.

[7] J. Garnett and P. W. Jones, BMO from dyadic BMO, Pacific J. Math. 99 (1982), 351-371.

[8] A. Gillespie, F. Nazarov, S. Pott, S. Treil, A. Volberg, Logarithmic Growth for Matrix Martingale Transform, preprint, Michigan State Univ., 1998.

[9] , Logarithmic Growth for Weighted Hilbert Transform and Vector Hankel Operators, preprint, Michigan State Univ., 1998.

[10] R. Hunt, B. Muckenhoupt, and R. Wheeden, Weighted norm inequalities for conjugate function and Hilbert transform, Trans. Amer. Math. Soc. 176 (1973), 227-251.

[11] B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207-226.

[12] F. L. Nazarov and S. R Treil, The hunt for a Bellman function: applications to estimates for singular integral operators and to other classical problems of harmonic analysis, (Russian) Algebra i Analiz 8 (1996), no. 5, 32–162; translation in St. Petersburg Math. J. 8 (1997), no. 5, 721–824.

[13] F. L. Nazarov and S. R. Treil, The weighted norm inequalities for Hilbert transform are now trivial, C. R. Acad. Sci. Paris Sér. I Math. 323 (1996), no. 7, 717–722.

[14] F. Nazarov, S. R Treil, and A. Volberg, Counterexample to the infinite-dimensional Carleson embedding theorem, C. R. Acad. Sci. Paris Sér. I Math. 325 (1997), no. 4, 383–388.

[15] F. Nazarov, S. R. Treil, and A. Volberg, The Bellman functions and two-weight inequalities for Haar multipliers, J. Amer. Math. Soc. 12 (1999), no. 4, 909–928.

[16] E. M. Stein, Interpolation of linear operators, Trans. Amer. Math. Soc. 83 (1956), 482–492.

[17] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, NJ, 1970.

[18] E. M. Stein, Harmonic Analysis, Princeton University Press, Princeton, NJ, 1993.

[19] S. R. Treil, The geometric approach to weight estimates of the Hilbert transform, (Russian) Funktsional. Anal. i Prilozhen. 17 (1983), no. 4, 90–91.

[20] S. R. Treil, An operator approach to weighted estimates of singular integrals, (Russian) Investigations on linear operators and the theory of functions, XIII. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 135 (1984), 150–174.

[21] S. R. Treil, Geometric methods in spectral theory of vector valued functions: Some recent results, Operator Theory: Adv. Appl. 42 (1989), 209-280.

[22] S. R. Treil and A. L. Volberg, Weighted embeddings and weighted norm inequalities for the Hilbert transform and the maximal operator, Algebra i Analiz 7 (1995), no. 6, 205–226; translation in St. Petersburg Math. J. 7 (1996), no. 6, 1017–1032.

[23] S. Treil and A. Volberg, Wavelets and the angle between past and future, J. Funct. Anal. 143 (1997), no. 2, 269–308.

[24] S. Treil and A. Volberg, Continuous frame decomposition and a vector Hunt-Muckenhoupt-Wheeden theorem, Ark. Mat. 35 (1997), no. 2, 363–386.

[25] A. Volberg, Matrix A_{p} weights via S-functions, J. Amer. Math. Soc. 10 (1997), no. 2, 445–466.

Department of Mathematics, University of California, Berkeley, CA 94720-3840

  E-mail address: mikeg@math.berkeley.edu