VECTOR A_{2} WEIGHTS AND A HARDY-LITTLEWOOD MAXIMAL FUNCTION

MICHAEL CHRIST AND MICHAEL GOLDBERG

A basic chapter of the subject of singular integral operators is the weighted norm theory, which provides a necessary and sufficient condition on a nonnegative function w for such operators, and for the Hardy-Littlewood maximal function M, to be bounded in L^{p} with respect to the measure w(x) dx. See [3], [6], [12]. More recently, aspects of this theory have been extended first by Treil and Volberg [17, 18], then by other authors, to functions taking values in finite-dimensional Hilbert spaces, with weights taking values in the corresponding spaces of Hermitian forms. These extensions have relied on new ideas, quite different from those employed in the scalar case by earlier authors, and have shed new light on the scalar theory.

Nonetheless, some aspects of the scalar theory had apparently not been successfully generalized, including M itself. Indeed, doubts have been expressed [20, 17] as to whether there could exist any useful analogue of M.

In this note we introduce a vector analogue of the Hardy-Littlewood maximal function, and prove its boundedness, in the simplest case, p = 2. In doing so, we seek firstly, to clarify the relationships between the new vector theory, and the more familiar scalar case, and secondly, to open up an alternative approach to the subject, which might lead to vector analogues of other features of the scalar theory. The second author has demonstrated [5] that the analogue of our theorem holds for all 1 < p < ∞, and that the boundedness of singular integral operators on L^{p}(ℝ^{n}, H, w) can be deduced from our theorem, in the spirit of [3].

When specialized to the scalar case, our analysis differs from that in [3] in that we study the operator f ↦ w^{1/2}M(w^{−1/2}f) on unweighted L^{2}, rather than M on w–weighted L^{2}. Although the direct analysis in this conjugated framework is slightly more intricate than [3], its generalization from the scalar to the vector setting is transparent.

1. A Maximal Operator

Let H be a separable Hilbert space, possibly of infinite dimension. Denote by M the space of all nonnegative Hermitian quadratic forms on H that are bounded with respect to the norm of H. The same symbol ∥ · ∥ will denote the norm either of an element of H, or of an element of M, considered as a linear operator from H to H. If w : ℝ^{n} ↦ M is measurable and locally integrable in the sense that the function x ↦ ∥w(x)∥ is locally integrable, define L^{p}(ℝ^{n}, H, w(x) dx) to be the Banach space of all equivalence classes of measurable functions f : ℝ^{n} ↦ H for which ∫_{ℝ^{n}}⟨w(x)f(x), f(x)⟩^{p/2} dx is finite.

Date: March 6, 2000. Updated April 26, 2000.

The first author was supported in part by NSF grant DMS-9970660. He thanks the staff of the Bamboo Garden hotel in Shenzhen, PRC for the hospitable atmosphere in which a portion of this work was done.

The second author was supported by an NSF graduate fellowship.

To simplify notation we will systematically write g_{E} = \frac{1}{|E|} ∫_{E} g, where E ⊂ ℝ^{n}, |E| denotes the Lebesgue measure of E, g takes values in ℝ, H, or M, and the integral is taken with respect to Lebesgue measure. For v ∈ M or ℝ^{+} and r ∈ ℝ, the notation v_{E}^{r} means (v_{E})^{r}.

A_{2} = A_{2}(ℝ^{n}, H) is by definition the class of all w : ℝ^{n} ↦ M such that w, w^{−1} are locally integrable and there exists C < ∞ such that

(1.1)  ∥w_{Q}^{1/2}(w^{−1})_{Q}^{1/2}∥ ≤ C  for every cube Q ⊂ ℝ^{n}.

The smallest constant C satisfying this inequality will be called the A_{2} “norm” of w. Treil and Volberg [17] have proved that if H has finite dimension, then the Hilbert transform H maps L^{2}(ℝ, H, w(x) dx) boundedly to itself, if and only if w ∈ A_{2}.

When H has finite dimension, (1.1) is equivalent^{1} to

(1.2)  \frac{1}{|Q|} ∫_{Q} ∥w^{1/2}(y)(w^{−1})_{Q}^{1/2}∥^{2} dy ≤ C;

another equivalent formulation is that \frac{1}{|Q|} ∫_{Q} ∥w_{Q}^{1/2}w^{−1/2}(y)∥^{2} dy ≤ C.

Now H preserves L^{2}(ℝ, H, w(x) dx), if and only if the operator f ↦ w^{1/2}(x)H(w^{−1/2}f)(x) maps L^{2}(ℝ, H, dx) boundedly to itself. Our analogue of the Hardy-Littlewood maximal function likewise acts on L^{2}(ℝ^{n}, H, dx) rather than on a weighted space, but M_{w}f(x) is scalar-valued, rather than H-valued. Definition.

(1.3)  M_{w}f(x) = sup_{Q∋x} \frac{1}{|Q|} ∫_{Q} ∥w^{1/2}(x)w^{−1/2}(y)f(y)∥ dy.

In the scalar case, the unweighted L^{2} boundedness of M_{w} is equivalent to the weighted boundedness of the Hardy-Littlewood maximal function. Maximal Theorem. Let H be a finite-dimensional Hilbert space. Let w ∈ A_{2}(ℝ^{n}, H). Then there exists δ > 0, depending only on the A_{2} constant of w and the dimension of H, such that the maximal operator M_{w} is bounded from L^{p}(ℝ^{n}, H, dx) to L^{p}(ℝ^{n}, ℝ, dx) whenever |p − 2| < δ. Conversely, if M_{w} is bounded for p = 2, then w ∈ A_{2}.

2. Preparations

In this section we outline several preliminary results, deferring the proofs to a later section. The linchpin of the scalar theory, as developed in [3], is the reverse Hölder inequality. Our first ingredient is an analogue for vector A_{2} weights, when H has finite dimension; this is the only step in our analysis that requires finite-dimensionality.

1To see this, use the equivalence of norm with trace in finite dimensions.

VECTOR A_{2} WEIGHTS 3

Proposition 2.1. Let H have finite dimension, and let w ∈ A_{2}(H). Then there exist r > 2 and C < ∞ such that for every Q,

(2.1)  ∫_{Q} ∥w^{1/2}(y)(w^{−1})_{Q}^{1/2}∥^{r} dy ≤ C|Q|.

The same holds for ∥w_{Q}^{1/2}w^{−1/2}(y)∥. The constants r, C can be taken to depend only on an upper bound for the A_{2} norm of w, and on the dimension of H.

While we are unaware of any explicit statement of this result in the literature, it is a rather direct consequence of the corresponding result from the scalar theory. However, one should note that whereas this inequality for r ≤ 2 is trivial for finitedimensional H, it is not automatic in infinite dimensions. We will refer to it as a “reverse Hölder” inequality, even when r ≤ 2.

By standard reasoning, involving the utilization of two incompatible dyadic grids, it suffices to consider the supremum only over dyadic cubes Q in defining M_{w}. Throughout the remainder of the paper, all cubes will hence be assumed to be dyadic.

We introduce the auxiliary maximal function

(2.2)  M′_{w}f(x) = sup_{x∈Q} \frac{1}{|Q|} ∫_{Q} ∥w_{Q}^{1/2}w^{−1/2}(y)f(y)∥ dy.

Denote also by m the Hardy-Littlewood maximal function in ℝ^{n}, acting on scalarvalued functions. Lemma 2.2. Let w ∈ A_{2}(H). Suppose that w satisfies a reverse Hölder inequality (2.1) for some exponent r > 2. Then there exist C < ∞ and δ > 0, depending only on the constants in (2.1), such that M′_{w} is bounded from L^{p}(ℝ^{n}, H, dx) to L^{p}(ℝ^{n}, ℝ, dx), for every p > 2 − δ. In particular, the conclusion holds whenever H has finite dimension.

Proof. A direct consequence of the reverse Hölder inequality (2.1) is that M′_{w}f(x) ≤ C(m(∥f∥^{r′}))^{1/r′} for all f, x, where r′ < 2 is conjugate to the exponent r of (2.1). Consequently, M′_{w} is bounded from L^{p}(H) to L^{p}, for all p > r′. □

The new twist in our analysis is the presence of the variable factor w^{1/2}(x), rather than the constant w_{Q}^{1/2}, in the definition of M_{w}. We will proceed by comparing M_{w} to M′_{w} via a sort of conditional expectation argument.

The following quite simple result will be proved below. Lemma 2.3. For any measurable M–valued function v and any cube Q ⊂ ℝ^{n},

(2.3)  ∥(v^{−1})_{Q}^{−1/2}v_{Q}^{−1/2}∥ ≤ 1.

We assume here that ∥v∥, ∥v^{−1}∥ belong to L^{1}(Q), and that v_{Q}, (v^{−1})_{Q} are invertible elements of M; both these hypotheses hold for all v ∈ A_{2}.

To any cube Q associate the quantity

(2.4)  N(x, Q) = sup_{x∈S⊂Q} ∥w^{1/2}(x)(w^{−1})_{S}^{1/2}∥.

The function x ↦ N(x, Q) will be denoted by N_{Q}.

The next lemma is a refinement of the reverse Hölder inequality, whose proof will be given later.

Lemma 2.4. Let w ∈ A_{2} satisfy a reverse Hölder inequality (2.1) with exponent r ≥ 1. Then there exists C < ∞ such that for every dyadic cube Q ⊂ ℝ^{n}, with the same exponent r,

(2.5)  ∫_{Q} N_{Q}(x)^{r} dx ≤ C|Q|.

In particular, if H has finite dimension, then this conclusion holds for some r > 2.

3. Proof of Theorem

Let w ∈ A_{2}, and let p ≥ 1 be any finite exponent. To prove the main theorem, it suffices to show that if M′_{w} is bounded from L^{p}(ℝ^{n}, H, dx) to L^{p}(ℝ^{n}, ℝ, dx), and if the functions N_{Q} satisfy the conclusion of Lemma 2.4 with exponent r = p, then M_{w} is likewise bounded.

Consider an arbitrary function f ∈ L^{p}(ℝ^{n}, H), not identically zero. Let {Q_{ı}^{ȷ}} be the collection of all maximal dyadic cubes Q satisfying

(3.1)  2^{j} ≤ \frac{1}{|Q|} ∫_{Q} ∥w_{Q}^{1/2}w^{−1/2}(y)f(y)∥ dy < 2^{j+1}.

Define

(3.2)  E(ȷ) = ∪_{ı}Q_{ı}^{ȷ} \ ( ∪_{k} Q_{k}^{ȷ+1}).

The sets Q_{ı}^{ȷ}∩E(ȷ) are pairwise disjoint, as ȷ, ı range over all possible values. Moreover, modulo Lebesgue null sets, E_{ȷ} = {x : 2^{ȷ} ≤ M′_{w}(x) < 2^{ȷ+1}}.

If M_{w}f(x) = 0, then x may be disregarded in our analysis. To each x ∈ ℝ^{n} for which M_{w}f(x) > 0, associate a dyadic cube R_{x} satisfying

(3.3)  \frac{1}{|R_{x}|} ∫_{R_{x}} ∥w^{1/2}(x)w^{−1/2}(y)f(y)∥ dy ≥ \frac{1}{2}M_{w}f(x).

Fix ȷ, ı and consider any x ∈ E(ȷ) ∩ Q_{ı}^{ȷ}.

(3.4) M_{w}f(x) ≤ 2N(x, Q_{ı}^{ȷ}) · ∥(w^{−1})_{R_{x}}^{−1/2}w_{R_{x}}^{−1/2}∥ · \frac{1}{|R_{x}|} ∫_{R_{x}} ∥(w_{R_{x}})^{1/2}w^{−1/2}(y)f(y)∥ dy

≤ 2N(x, Q_{ı}^{ȷ})\frac{1}{|R_{x}|} ∫_{R_{x}} ∥(w_{R_{x}})^{1/2}w^{−1/2}(y)f(y)∥ dy ≤ 4N(x, Q_{ı}^{ȷ})2^{ȷ},

by first Lemma 2.3, then the definitions of N(x, Q_{ı}^{ȷ}) and of E(ȷ). Thus

(3.5)  M_{w}f ≤ 4 ∑_{j=−∞}^{∞} 2^{ȷ} ∑_{ı} N(x, Q_{ı}^{ȷ})χ_{Q_{ı}^{ȷ}∩E(ȷ)},

where χ_{S} denotes the characteristic function of a set S. Thus by Lemma 2.4,

∥M_{w}f∥_{p}^{p} ≤ 4^{p} ∑_{ȷ∈ℤ}2^{pȷ} ∑_{ı} ∫_{Q_{ı}^{ȷ}} N(·, Q_{ı}^{ȷ})^{p}

≤ C ∑_{ȷ} 2^{pȷ} ∑_{ı} |Q_{ı}^{ȷ}| ≤ C ∑_{ȷ} 2^{pȷ}|{x : M′_{w}f(x) ≥ 2^{ȷ}}| ≤ C∥M′_{w}f∥_{p}^{p}.

Remark. We have implicitly proved certain implications for the case of infinitedimensional Hilbert spaces. Assume that w ∈ A_{2}.

(1) If w satisfies the reverse Hölder inequality for some exponent r > 2, then M_{w} is bounded on L^{p} for all |p − 2| < δ, and M′_{w} is bounded for all p > 2 − δ, for some δ > 0.

(2) If M′_{w} is bounded on L^{2}, and if w satisfies (2.1) with r = 2, then M_{w} is bounded on L^{2}.

4. Proofs of Lemmas

Proof of reverse Hölder inequality. It suffices to show \frac{1}{|Q|} ∫_{Q} ∥w∥^{r} ≤ C(\frac{1}{|Q|} ∫_{Q} ∥w∥)^{r}, where r > 1, C < ∞ depend only on the dimension n of the ambient space, the dimension of H, and the A_{2} norm of w. For then the conclusion stated follows by applying this inequality to BwB, where B = (w^{−1})_{Q}^{1/2}. The A_{2} norm of BwB equals that of w; this holds for any B ∈ M, by Lemma 3.5 of [17]. And \frac{1}{|Q|} ∫_{Q} ∥BwB∥ is bounded by the A_{2} norm squared of w.

It is known that for any nonnegative, locally integrable function u defined on ℝ^{n}, the set of all exponents q > 1 for which there exists C < ∞ for which

(4.1)  (\frac{1}{|Q|} ∫_{Q} u^{q})^{1/q} ≤ C \frac{1}{|Q|} ∫_{Q} u for all cubes Q

is open. Volberg and Treil [17], Lemma 3.6, have proved that

\frac{1}{|Q|} ∫_{Q} ∥w∥ ≤ C(\frac{1}{|Q|} ∫_{Q} ∥w∥^{1/2})^{2},

that is, u(x) = ∥w(x)∥^{1/2} satisfies (4.1) with q = 2, and the lemma follows. □

An alternative proof uses an easy consequence of the A_{2} condition: For any nonzero v ∈ H, for any w ∈ A_{2}(H), the scalar function ⟨w(x)v, v⟩ is an A_{2} weight, with A_{2} constant bounded uniformly in v. Hence these scalar weights satisfy uniform reverse Hölder inequalities. Therefore the scalar case implies the finite-dimensional vector case, by taking the trace of w_{Q}^{−1/2}w(x)w_{Q}^{−1/2}.

Proof of Lemma 2.3. Let ξ ∈ H be arbitrary. For any M–valued function u,

∥ξ∥^{2} = \frac{1}{|Q|} ∫_{Q}⟨u^{1/2}(x)ξ, u^{−1/2}(x)ξ⟩ dx ≤ ⟨ξ, u_{Q}ξ⟩^{1/2} · ⟨ξ, (u^{−1})_{Q}ξ⟩^{1/2}.

Set u = (v^{−1})_{Q}^{1/2} ∘ v ∘ (v^{−1})_{Q}^{1/2}, apply Cauchy-Schwarz, and eliminate a common factor of ∥ξ∥^{3/2} from both sides of the inequality, then square, to deduce that ∥ξ∥ ≤ ∥v_{Q}^{1/2}(v^{−1})_{Q}^{1/2}ξ∥. Inverting the matrix on the right yields the lemma. □

Proof of Lemma 2.4. We argue informally, assuming that ∫_{Q} N_{Q}^{r} ≤ B|Q| for all Q for some finite B, and deriving an a priori upper bound on B. To turn this argument into a legitimate proof is a routine exercise.

Let A < ∞ be a large constant to be specified later. Decompose w^{1/2}(x)(w^{−1})_{S}^{1/2} = [w^{1/2}(x)(w^{−1})_{Q}^{1/2}] ∘ [(w^{−1})_{Q}^{−1/2}(w^{−1})_{S}^{1/2}]. Consider all maximal dyadic cubes S_{ȷ} ⊂ Q satisfying ∥(w^{−1})_{Q}^{−1/2}(w^{−1})_{S}^{1/2}∥ > A.

For any x ∉ ∪_{ȷ}S_{ȷ}, N(x, Q) ≤ A∥[w^{1/2}(x)(w^{−1})_{Q}^{1/2}∥. Thus ∫_{Q\∪_{ȷ}S_{ȷ}} N(x, Q)^{r} dx ≤ C|Q|, by virtue of the reverse Hölder inequality (2.1).

We claim that | ∪_{ȷ} S_{ȷ}| ≤ |Q|/2, provided that A is sufficiently large, relative to the A_{2} constant of w. Granting this, we conclude that

(4.2)  ∫_{∪_{ȷ}S_{ȷ}} N_{Q}^{r} = ∑_{ȷ} ∫_{S_{ȷ}} N_{Q}^{r} ≤ B ∑_{ȷ} |S_{ȷ}| ≤ \frac{1}{2}B|Q|.

All together, we would find that B ≤ C + (B/2), where C < ∞ depends only on the A_{2} norm of w.

To prove the claim note first that for any index ȷ,

A^{2} < ∥(w^{−1})_{Q}^{−1/2}(w^{−1})_{S_{ȷ}}^{1/2}∥^{2} = ∥(w^{−1})_{Q}^{−1/2}(w^{−1})_{S_{ȷ}}(w^{−1})_{Q}^{−1/2}∥

≤ \frac{1}{|S_{ȷ}|} ∫_{S_{ȷ}} ∥w_{Q}^{1/2}w^{−1}(y)w_{Q}^{1/2}∥ dy = \frac{1}{|S_{ȷ}|} ∫_{S_{ȷ}} ∥w_{Q}^{1/2}w^{−1/2}∥^{2}.

Therefore by the pairwise disjointness of the cubes S_{ȷ} and the A_{2} condition, summing the preceding inequality over ȷ yields

(4.3)  A^{2} ∑_{ȷ} |S_{ȷ}| ≤ ∫_{Q} ∥w_{Q}^{1/2}w^{−1/2}(y)∥^{2} dy ≤ C|Q|,

justifying the claim. □

Denote by A_{Q} the averaging operators

A_{Q}f(x) = χ_{Q}(x)\frac{1}{|Q|} ∫_{Q} w^{1/2}(x)w^{−1/2}(y)f(y) dy.

Lemma 4.1. Let H be any Hilbert space. Then the operators A_{Q} are bounded from L^{2}(ℝ^{n}, H, dx) to itself uniformly for all cubes Q ⊂ ℝ^{n}, if and only if w ∈ A_{2}.

Of course, boundedness of M′_{w} , or of M_{w}, directly implies uniform boundedness of {A_{Q}}.

For the proof, let f ∈ L^{2}(ℝ^{n}, H) be supported in Q. Then

\frac{1}{|Q|}∥A_{Q}f∥_{2}^{2} = \frac{1}{|Q|} ∫_{Q} ∥\frac{1}{|Q|} ∫_{Q} w^{1/2}(x)w^{−1/2}(y)f(y) dy∥^{2} dx

= ∥\frac{1}{|Q|} ∫_{Q} w_{Q}^{1/2}w^{−1/2}(y)f(y) dy∥^{2}.

By duality and Cauchy-Schwarz, the supremum of this last expression over all f ∈ L^{2} having norm ≤ 1 equals

sup_{ξ∈H, ∥ξ∥≤1} \frac{1}{|Q|^{2}} ∫_{Q} ∥w^{−1/2}(y)w_{Q}^{1/2}ξ∥^{2} dy

= \frac{1}{|Q|} sup_{ξ∈H, ∥ξ∥≤1} \frac{1}{|Q|} ∫_{Q}⟨ξ, w_{Q}^{1/2}w^{−1}(y)w_{Q}^{1/2}ξ⟩ dy

= \frac{1}{|Q|} sup_{ξ∈H, ∥ξ∥≤1} ∥(w^{−1})_{Q}^{1/2}w_{Q}^{1/2}ξ∥^{2} = \frac{1}{|Q|}∥w_{Q}^{1/2}(w^{−1})_{Q}^{1/2}∥^{2}.

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

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