PARAEXPONENTIALS, MUCKENHOUPT WEIGHTS, AND RESOLVENTS OF PARAPRODUCTS

CRISTINA PEREYRA AND LESLEY WARD

Abstract. We analyze the stability of Muckenhoupt’s RH_{p}^{d} and A_{p}^{d} classes of weights under a nonlinear operation, the λ-operation. We prove that the dyadic doubling reverse Hölder classes RH_{p}^{d} are not preserved under the λ-operation, but the dyadic doubling A_{p} classes A_{p}^{d} are preserved for 0 < λ < 1. We give an application to the structure of resolvent sets of dyadic paraproduct operators.

1. Introduction

The Muckenhoupt classes of weights consist of positive locally integrable functions satisfying certain integrability conditions on intervals; see Section 2. They arise, for instance, in connection with the boundedness of the Hilbert transform and maximal function operators on L^{p} spaces.

Weights in the Muckenhoupt classes are often represented as exponentials of functions in BMO; see [GC-RF]. For weights ω(x) defined on an interval J, with mean value one on J, a different representation, sometimes called the paraexponential, was introduced in [FKP]. The correspondence is realized by an infinite product, namely:

ω(x) = ∏_{I∈D(J)} (1 + b_{I}h_{I}(x)), b_{I} = \frac{⟨ω, h_{I}⟩}{m_{I}ω}; (1.1)

where D(J) denotes the intervals in the dyadic decomposition of the interval J, {h_{I}}_{I∈D(J)} are Haar functions, ⟨·, ·⟩ is the scalar product in L^{2}, and m_{I}ω is the mean value of ω on I.

It was proved in [FKP] that if there is an ε > 0 such that |b_{I}h_{I}| < 1 − ε for all I ∈ D(J), then the partial products converge to a weight in dyadic doubling A_{∞}^{d} if and only if b = ∑_{I∈D(J)} b_{I}h_{I} is a function of dyadic bounded mean oscillation, BMO^{d}.

1991 Mathematics Subject Classification. 42c, 47b; Secondary 41.

Key words and phrases. Muckenhoupt weights, reverse Hölder RH_{p}, A_{p}, doubling weights, dyadic paraproducts, paraexponentials.

Research supported in part by (CP) NSF grant #DMS-93-04580 and (LW) at MSRI by NSF grant #DMS-90-22140

A larger dictionary relating properties of ω and b can be found in [B]. In particular the dyadic doubling Muckenhoupt RH_{p}^{d} and A_{p}^{d} classes can be characterized by summation conditions on b.

We will consider only weights ω defined on an interval J, with mean value one on J. For such a weight we have, at least formally, the product representation (1.1). We are interested in the effect of multiplying the coefficient b_{I} in each factor of the product by the same number λ. Define

ω_{λ}(x) = ∏_{I∈D(J)} (1 + λb_{I}h_{I}(x)). (1.2)

We call the mapping that sends ω ↦ ω_{λ} the λ-operation.

To guarantee the convergence of the products we will only consider the case −1 ≤ λ ≤ 1. If ω is a weight in A_{∞}^{d} and −1 ≤ λ ≤ 1, then ω_{λ} is also in A_{∞}^{d}; see [FKP].

The λ-operation seems, at first sight, very similar to taking a λ power of the weight, ω^{λ}. The first two terms in the Taylor expansion of ω^{λ} = e^{λb} coincide with the first two terms obtained by expanding the infinite product in (1.2), but the third terms differ. Both the RH_{p}^{d} and the A_{p}^{d} classes are preserved under taking powers ω ↦ ω^{λ}, for 0 ≤ λ ≤ 1.

Question: Are the RH_{p}^{d} and A_{p}^{d} classes preserved under the λ-operation?

The answer is negative for the RH_{p}^{d} classes, and positive for the A_{p}^{d} classes.

Theorem 1.1. For each p > 1, there exist a weight ω in RH_{p}^{d} and a number λ ∈ (0, 1) such that ω_{λ} is not in RH_{p}^{d}.

For p > (1 − log 2)^{−1} > 1, examples with positive λ’s are given in [P2]. In this paper we give examples with positive λ’s for all p > 1. Examples for all p > 1, with −1 ≤ λ < 0, are given in [P1].

A related result in [P2] is that if ω ∈ RH_{p}^{d}, then there is a q > 1 such that ω_{λ} ∈ RH_{q}^{d} for all −1 ≤ λ ≤ 1.

Theorem 1.2. Given a weight ω ∈ A_{p}^{d}, then ω_{λ} ∈ A_{p}^{d} for all 1 ≤ p ≤ ∞ and all 0 ≤ λ ≤ 1.

However, the A_{p}^{d} spaces for 1 ≤ p < ∞ are not preserved by the λ-operation for negative λ; this can be shown by examples like those in [P1].

The λ-operation appears naturally in the study of the resolvents of the dyadic paraproduct [P2]. The dyadic paraproduct is a bilinear operator that appears in different guises in harmonic analysis, often replacing the ordinary product [M, Ch, D]. The dyadic paraproduct π_{b} associated to a function b is defined by

π_{b}f(x) = ∑_{I∈D(J)} m_{I}f b_{I}h_{I}(x), (1.3)

where m_{I}f is the mean value of f on the interval I, and b_{I} = ⟨b, h_{I}⟩. The paraproduct π_{b} is bounded on L^{p}(J) if and only if b ∈ BMO^{d}. The conditions on b that guarantee the existence of a bounded inverse of (I − λπ_{b}) on L_{o}^{p}(J) = {f ∈ L^{p}(J) : ∫_{J} f = 0} are described in terms of properties of the weight ω_{λ}. For doubling weights, the conditions reduce to ω_{λ} ∈ RH_{p}^{d}; see [P1]. More precisely:

Theorem 1.3 (P). Let b be a function in BMO^{d}, take ε > 0, and let λ be a real number such that |λb_{I}h_{I}| ≤ (1 − ε) for all I ∈ D(J), where b_{I} = ⟨b, h_{I}⟩. Then (I − λπ_{b})^{−1} exists and is bounded on L_{o}^{p}(J) if and only if the weight ω_{λ}(x) = ∏_{I∈D(J)} (1 +

λb_{I}h_{I}(x)) is in RH_{p}^{d}.

Question: If (I − λπ_{b})^{−1} exists and is bounded on L^{p} for λ = 1, is the same true for all λ ∈ [0, 1]?

Since the resolvent set of π_{b} is open, there are neighborhoods of λ = 0 and λ = 1 for which this holds. By the discussion above, the question is equivalent to asking whether the RH_{p}^{d} classes are preserved under the λ-operation. Theorem 1.1 shows that the answer is negative. Rephrasing the theorem we have:

Theorem 1.4. For each p > 1, there exist a number λ ∈ (0, 1), a function b ∈ BMO^{d}, and a number ε > 0 such that |b_{I}h_{I}| ≤ 1 − ε for all I ∈ D(J), the operator (I − π_{b})^{−1} exists and is a bounded operator on L_{o}^{p}(J), but the operator (I − λπ_{b}) is not invertible as an operator from L_{o}^{p}(J) into L_{o}^{p}(J).

In Section 2 we give notation, definitions, and some preparatory lemmas. In Section 3 we prove Theorem 1.2, and in Section 4 we construct the examples which establish Theorem 1.1.

As customary, C denotes a constant that may change from line to line.

2. Preliminaries

In this section we introduce the dyadic intervals and the Haar basis; and we define dyadic BMO^{d} in terms of a Carleson condition. Next we define the dyadic doubling Muckenhoupt classes A_{∞}^{d}, RH_{p}^{d}, and A_{p}^{d}; and we recall some of their properties. We state the Fefferman-Kenig-Pipher Product Representation Theorem for A_{∞}^{d} weights and Buckley’s Theorem characterizing A_{p}^{d} weights via summation conditions. Finally we define the λ-operation, and give a convexity lemma used in the proof of Theorem 1.2.

Let D denote the family of all dyadic subintervals of [0, 1], in other words all intervals of the form (j2^{−k}, (j + 1)2^{−k}], j, k integers, 0 ≤ k, 0 ≤ j ≤ 2^{k} − 1. Given any interval J, D(J) denotes the family of dyadic subintervals of J. Given an interval J we denote its left and right halves respectively by J_{l} and J_{r}. An interval Ĩ is the parent of an interval I if I is Ĩ_{l} or Ĩ_{r}.

The Haar function associated to an interval I is given by h_{I}(x) = |I|^{−1/2}(χ_{I_{r}}(x) −

χ_{I_{l}}(x)); here χ_{I} denotes the characteristic function of the interval I. The set of Haar functions indexed by D(J) forms a basis of L_{o}^{2}(J) = {f ∈ L^{2}(J) : ∫_{J} f = 0}; see [H].

A locally integrable function b on [0, 1] is in the space of dyadic bounded mean oscillation BMO^{d} if there is a constant C such that ∫_{J} |b(x) − m_{J}b|^{2} dx ≤ C |J| for all J ∈ D, where m_{J}b = \frac{1}{|J|} ∫_{J} b. The function b is in BMO^{d} if and only if the Carleson condition on the Haar coefficients b_{I} = ⟨b, h_{I}⟩ of b holds: ∑_{I∈D(J)} b_{I}^{2} ≤ C |J| for all J ∈ D, with a constant C independent of J; see [Ch, M].

2.1. Dyadic weights. We consider weights defined on the interval J_{0} = [0, 1].

A dyadic doubling weight ω is a positive locally integrable function such that ∫_{I}ω ≤ C ∫_{I} ω for all intervals I ∈ D, where Ĩ is the parent of I and C is a constant independent of I.

A weight ω is in the dyadic doubling A_{∞} class A_{∞}^{d} if ω is dyadic doubling, and there is a constant C such that \frac{1}{|I|} ∫_{I} ω ≤ C exp (\frac{1}{|I|} ∫_{I} log ω), for all I ∈ D.

A weight ω is in the dyadic doubling reverse Hölder class RH_{p}^{d}, for 1 < p < ∞, if

ω is dyadic doubling, and there is a constant C such that (\frac{1}{|I|} ∫_{I} ω^{p})^{1/p} ≤ C \frac{1}{|I|} ∫_{I} ω, for all I ∈ D.

A weight ω is in the dyadic doubling A_{p} class A_{p}^{d}, for 1 < p < ∞, if ω is dyadic

doubling, and there is a constant C such that (\frac{1}{|I|} ∫_{I} ω) (\frac{1}{|I|} ∫_{I} ω^{−1/(p−1)})^{p−1} ≤ C for all I ∈ D.

A weight ω is in the dyadic doubling A_{1} class A_{1}^{d} if ω is dyadic doubling, and there is a constant C such that \frac{1}{|I|} ∫_{I} ω ≤ C ω(x) for a.e. x ∈ I, for all I ∈ D.

The canonical examples are ω(x) = |x|^{α}. In this case ω ∈ A_{∞}^{d} if and only if α > −1, ω ∈ RH_{p}^{d} if and only if α > −1/p, for 1 < p < ∞ ω ∈ A_{p}^{d} if and only if −1 < α < p − 1, and w ∈ A_{1}^{d} if and only if −1 < α ≤ 0.

The class A_{∞}^{d} is the union of the RH_{p}^{d} classes, and also of the A_{p}^{d} classes. The class A_{1}^{d} is strictly contained in the intersection of the A_{p}^{d} classes. More precisely:

A_{∞}^{d} = ⋃_{p>1} RH_{p}^{d} = ⋃_{p>1} A_{p}^{d}, A_{1}^{d} ⊂ ⋂_{p>1} A_{p}^{d}. (2.1)

These properties are known for weights that satisfy the conditions above for all intervals I; see [GC-RF]. In our case we consider only dyadic intervals, but we also assume that the weights are dyadic doubling, which yields (2.1); see [B]. An explicit example of a function in ⋂_{p>1} A_{p}^{d} but not in A_{1}^{d} can be found in [JN].

To each weight ω ∈ A_{∞}^{d}, with mean value one on J_{0}, we associate a function b = b_{ω} so that

ω(x) = ∏_{I∈D} (1 + b_{I}h_{I}(x)), b_{I} = \frac{⟨ω, h_{I}⟩}{m_{I}ω}, (2.2)

and b(x) = ∑_{I∈D} b_{I}h_{I}(x).

Theorem 2.1 (R. Fefferman, Kenig, Pipher). Let {b_{I}}_{I∈D(J_{0})} and ε > 0 be given, with |b_{I}h_{I}| ≤ (1 − ε) for all I ∈ D. Then the product (2.2) belongs to A_{∞}^{d} if and only if b ∈ BMO^{d}, that is, if and only if there is a constant C such that

∑_{I∈D(J)} b_{I}^{2} ≤ C |J|, ∀ J ∈ D. (2.3)

If so, the weight ω is a dyadic doubling weight.

This is proved in [FKP].

The dyadic RH_{p}^{d} and A_{p}^{d} classes can also be characterized by summation conditions. Part (a) of the next theorem, and related results, appear in [B].

Theorem 2.2. Let ω be a dyadic doubling weight ω.

(a) (Buckley) ω ∈ A_{p}^{d}, 1 < p < ∞, if and only if there is a constant C such that

∑_{I∈D(J)} (\frac{m_{I}ω}{m_{J}ω} )^{\frac{−1}{p−1}} b_{I}^{2} ≤ C |J|, ∀ J ∈ D.

(b) w ∈ A_{1}^{d} if and only if there is a constant C such that

\frac{m_{J}ω}{m_{I}ω} ≤ C, ∀ I ∈ D(J), ∀ J ∈ D.

Proof of (b): (⇒) Integrate over I ∈ D(J) in the definition of A_{1}^{d}. (⇐) By the Lebesgue differentiation theorem, the limit of m_{I}ω as a sequence of intervals I ∈ D(J) shrinks to a point x ∈ J, is ω(x) for almost every x.

Let Ĩ be the parent of I. Denote by s_{I} the proportion of the mass of Ĩ that is carried by I. Then

2s_{I} = \frac{m_{I}ω}{m_{I}ω} = 1 + b_{I}h_{I}(x), ∀ x ∈ I. (2.4)

2.2. The λ-operation. The nonlinear operation that sends the weight ω given by ω = ∏_{I∈D}(1 + b_{I}h_{I}) into the weight ω_{λ},

ω ↦ w_{λ} = ∏_{I∈D}(1 + λb_{I}h_{I}), (2.5)

is called the λ-operation.

Remark 2.3. Dyadic doubling and A_{∞}^{d} weights are preserved under the λ-operation for −1 ≤ λ ≤ 1. This is a consequence of Theorem 2.1, and the observation that since |λb_{I}| ≤ |b_{I}|, b ∈ BMO^{d} implies λb ∈ BMO^{d}; see [P2].

To understand how the λ-operation affects RH_{p}^{d} and A_{p}^{d} weights, let us first consider how it affects quotients of mean values of ω over consecutive nested intervals. Accordingly define

s_{I}(λ) = \frac{m_{I}ω_{λ}}{2 m_{I}ω_{λ}} . (2.6)

The λ-operation is a nonlinear operation on the weight, but it is linear at the level of the s_{I}’s:

Lemma 2.4. s_{I}(λ) = \frac{1}{2} + λ(s_{I} − \frac{1}{2}).

Proof: This is an immediate consequence of (2.4).

Lemma 2.5. Let a_{1}, …, a_{n} be positive numbers. Let a_{j}(λ) = 1 + λ(a_{j} − 1), and A_{n}(λ) = ∏_{j=1}^{n} a_{j}(λ). Then

A_{n}(λ) ≥ min {1, A_{n}(1)} . (2.7)

Proof: On expanding the product we get

A_{n}(λ) = ∏_{k=1}^{n} ((1 − λ) + λa_{k}) = ∑_{i=0}^{n} (1 − λ)^{n−i}λ^{i} ∑_{k_{1}≠...≠k_{i}} ∏_{j=1}^{i} a_{k_{j}}, (2.8)

where the index notation k_{1} ≠ … ≠ k_{i} means that the k_{j}’s are pairwise different, and 1 ≤ k_{j} ≤ n. Now, using repeatedly the fact that for positive numbers the arithmetic mean is at least the geometric mean, we find that

∑_{k_{1}≠...≠k_{i}} ∏_{j=1}^{i} a_{k_{j}} ≥ N_{n,i} ( ∏_{k_{1}≠...≠k_{i}} ∏_{j=1}^{i} a_{k_{j}} )^{1/N_{n,i}}, (2.9)

where N_{n,i} is the number of ways one can choose i numbers from a collection of n numbers, that is, N_{n,i} = \frac{n!}{i!(n−i)!}. Observe that in the double product in (2.9), each a_{j} occurs as many times as possible ways one can choose i−1 numbers from a collection of n − 1 numbers, that is N_{n−1,i−1}. Since N_{n−1,i−1}/N_{n,i} = i/n, the right hand side of (2.9) is equal to:

N_{n,i}( ∏_{j=1}^{n} a_{j})^{i/n} = N_{n,i} (A_{n}(1))^{i/n}. (2.10)

We conclude that

A_{n}(λ) ≥ ∑_{i=0}^{n} (1 − λ)^{n−i}λ^{i}N_{n,i} (A_{n}(1))^{i/n} = (λ(A_{n}(1))^{1/n} + (1 − λ))^{n}.

Fix A ≥ 0. The function f(λ) = (λA + (1 − λ))^{n} is monotonic for λ ∈ [0, 1],

hence it is bounded below by min {f(0), f(1)}. Set A = (A_{n}(1))^{1/n}; then f(0) = 1, f(1) = A_{n}(1). This completes the proof of the lemma.

3. The theorem for A_{p}^{d} weights

Theorem 3.1. Given a weight ω ∈ A_{p}^{d}, then ω_{λ} ∈ A_{p}^{d} for all 1 ≤ p ≤ ∞ and all 0 ≤ λ ≤ 1.

Proof: By Remark 2.3, ω_{λ} is a dyadic doubling weight, and w ∈ A_{∞}^{d} implies ω_{λ} ∈ A_{∞}^{d}.

For dyadic intervals I ⊂ J, the ratios \frac{m_{I}ω}{m_{J} ω} and \frac{m_{I}ω_{λ}}{m_{J} ω_{λ}} can be explicitly computed as products of s_{K}’s and s_{K}(λ)’s respectively:

\frac{m_{I}ω}{m_{J}ω} = ∏_{I⊆K⊂J} 2s_{K}, and \frac{m_{I}ω_{λ}}{m_{J}ω_{λ}} = ∏_{I⊆K⊂J} 2s_{K}(λ), (3.1)

where K ∈ D(J), 2s_{K}(λ) = 1 + λ(2s_{K} − 1), and 2s_{K} = m_{K}ω/m_{K}ω.

These are the kind of products considered in Lemma 2.5. Since ω is dyadic doubling, (2.4) shows that ε ≤ 2s_{K} ≤ 2 − ε. In particular a_{K} = 2s_{K} > 0. We can apply Lemma 2.5 to each pair (I, J) of dyadic intervals, I ⊂ J, where n = log_{2}(|J|/|I|), A_{n}(λ) = m_{I}ω_{λ}/m_{J}ω_{λ}, and A_{n}(1) = m_{I}ω/m_{J}ω, to conclude that

(\frac{m_{I}ω_{λ}}{m_{J}ω_{λ}} )^{r} ≤ max \{1, (\frac{m_{I}ω}{m_{J}ω} )^{r}\} , (3.2)

for all negative r. Case 1 < p < ∞: By Theorem 2.2(a), it suffices to find a constant C such that

\frac{1}{|J|} ∑_{I∈D(J)} (\frac{m_{I}ω_{λ}}{m_{J}ω_{λ}} )^{\frac{−1}{p−1}} b_{I,λ}^{2} ≤ C ∀ J ∈ D, (3.3)

where b_{I,λ} = ⟨λb, h_{I}⟩ = λb_{I}.

We use (3.2), with r = −1/(p − 1), to estimate the left hand side of (3.3). Using the observation that b_{I,λ}^{2} ≤ b_{I}^{2} for 0 < λ < 1, we get

\frac{1}{|J|} ∑_{I∈D(J)} (\frac{m_{I}ω_{λ}}{m_{J}ω_{λ}} )^{\frac{−1}{p−1}} b_{I,λ}^{2} ≤ \frac{1}{|J|} ∑_{I∈D(J)} max \{1, (\frac{m_{I}ω}{m_{J}ω} )^{\frac{−1}{p−1}} \} b_{I}^{2}

≤ \frac{1}{|J|} ∑_{I∈D(J)} b_{I}^{2} + \frac{1}{|J|} ∑_{I∈D(J)} (\frac{m_{I}ω}{m_{J}ω} )^{\frac{−1}{p−1}} b_{I}^{2} ≤ C;

where C is a constant independent of the interval J. The last inequality holds since ω ∈ A_{p}^{d} ⊂ A_{∞}^{d}, by Theorems 2.1 and 2.2(a).

Case p = 1: By Theorem 2.2(b), it suffices to find a constant C such that

(\frac{m_{I}ω_{λ}}{m_{J}ω_{λ}} )^{−1} ≤ C, ∀ I ∈ D(J), ∀ J ∈ D. (3.4)

We use (3.2) again, with r = −1, to conclude that the left hand side of (3.4) is bounded by max \{1, (\frac{m_{I}w}{m_{J} w} )^{−1}\}, which in turn is bounded because ω ∈ A_{1}^{d}.

4. Examples in RH_{p}^{d}

Theorem 4.1. For each p > 1, there exist a dyadic doubling weight ω in RH_{p}^{d} and a number λ ∈ (0, 1) such that ω_{λ} is not in RH_{p}^{d}.

Proof: Our examples are of the following form. Let I_{i} = [2^{−i}, 2^{−i+1}] for i ≥ 1, and J_{i} = [0, 2^{−i}] for i ≥ 0. Fix numbers s_{i} ∈ (0, 1) for i ≥ 1. For x ∈ [0, 1] let

ω(x) = ∑_{i=1}^{∞} c_{i} χ_{I_{i}}(x), where c_{i} = 2^{i} s_{1} … s_{i−1}(1 − s_{i}). (4.1)

One can think of ω as the weight of mass one obtained by assigning the fractions s_{i} and 1 − s_{i} of the mass of J_{i−1} to the left half J_{i} and the right half I_{i} respectively of J_{i−1}, for each i ≥ 1. In the notation of Section 2,

s_{i} = s_{J_{i}} = \frac{m_{J_{i}}ω}{2 m_{J_{i−1}}ω} , and m_{J_{i}}ω = 2^{i} s_{1} … s_{i−1}s_{i}. (4.2)

We further assume that the sequence {s_{i}} is n-periodic: there is a smallest positive integer n such that s_{i+n} = s_{i} for all i ≥ 1. Such a ω is a dyadic doubling weight. For n-periodic weights of this form,

c_{kn+j} = (2^{n} s_{1} … s_{n})^{k} c_{j}, ∀ k ≥ 0, j ≥ 1. (4.3)

Claim: The RH_{p}^{d} condition \frac{1}{|I|} ∫_{I} ω^{p} ≤ C (\frac{1}{|I|} ∫_{I} ω)^{p}, for all I ∈ D, for an n-periodic weight of the form ( 4.1) reduces to the condition

2^{n}s_{1} … s_{n} < 2^{n/p}. (4.4)

To see this, first note that ω is constant on each dyadic interval I whose left endpoint is not 0, so the reverse Hölder p condition holds with C = 1 on these intervals. The only other dyadic intervals are J_{l} = [0, 2^{−l}], l ≥ 0.

Write l = qn + m where q and m are non-negative integers and 0 ≤ m ≤ n − 1. We use (4.3) to compute the mean of ω^{p} on J_{l} = ⋃_{i=l+1}^{∞} I_{i} :

\frac{1}{|J_{l}|} ∫_{J_{l}} ω^{p} = 2^{l} ∑_{i=l+1}^{∞} ∫_{I_{i}} ω^{p} = 2^{l} ∑_{i=qn+m+1}^{∞} c_{i}^{p} 2^{−i}

= 2^{l} ∑_{k=q}^{∞} ∑_{j=1}^{n} (c_{kn+j+m})^{p} 2^{−(kn+j+m)}

= 2^{l} ∑_{k=q}^{∞} ∑_{j=1}^{n} (2^{n}s_{1} … s_{n})^{kp} (c_{j+m})^{p} 2^{−(kn+j+m)}

(4.5)  = 2^{l} d_{p,m} ∑_{k=q}^{∞} [(2^{n} s_{1} … s_{n})^{p} 2^{−n}]^{k} ,

where d_{p,m} = ∑_{j=1}^{n} (c_{j+m})^{p} 2^{−(j+m)}.

The series (4.5) converges if and only if 2^{n} s_{1} … s_{n} < 2^{n/p}; this is condition (4.4). The mean of ω on J_{l} is

\frac{1}{|J_{l}|} ∫_{J_{l}} ω = 2^{l} d_{1,m} ∑_{k=q}^{∞} (s_{1} … s_{n})^{k}; (4.6)

this series converges since each s_{i} is less than one.

If 2^{n} s_{1} … s_{n} < 2^{n/p}, then on summing the series in (4.5) and (4.6) we find

\frac{\frac{1}{|J_{l}|} ∫_{J_{l}} ω^{p}}{(\frac{1}{|J_{l}|} ∫_{J_{l}} ω)^{p}} = 2^{l(1−p)} \frac{d_{p,m}}{(d_{1,m})^{p}} 2^{n(p−1)q} \frac{[1 − (s_{1} … s_{n})]^{p}}{1 − (2^{n}s_{1} … s_{n})^{p} 2^{−n}} , (4.7)

for all l = qn + m ≥ 0 with q ≥ 0 and 0 ≤ m ≤ n − 1. This expression depends on p, n, and m but is independent of q, since d_{p,m} and d_{1,m} are independent of q and the exponent of 2 in the right hand side of (4.7) reduces to m(1−p). Taking the maximum over m ∈ {0, …, n − 1}, we see that there is a uniform upper bound, depending only on p and n, for \frac{1}{|J_{l}|} ∫_{J_{l}} ω^{p}/(\frac{1}{|J_{l}|} ∫_{J_{l}} ω)^{p}, for all l ≥ 0. Therefore ω ∈ RH_{p}^{d}, which establishes the claim.

Let f(x_{1}, …, x_{n}) = 2^{n} x_{1} … x_{n}. Then by the claim above, the n-periodic weight ω associated to the point (s_{1}, …, s_{n}) in the open unit n-cube (0, 1)^{n} is in RH_{p}^{d} if and only if f(s_{1}, …, s_{n}) < 2^{n/p}.

By Lemma 2.4, the λ-operation takes the weight ω with sequence {s_{i}} to a new weight ω_{λ} with sequence {s_{i}(λ)}, where

s_{i}(λ) = \frac{1}{2} + λ (s_{i} − \frac{1}{2}) . (4.8)

In the unit n-cube, the λ-operation moves the point P : (s_{1}, …, s_{n}) towards the centre Q : (1/2, …, 1/2) of the cube along the line segment joining P to Q. For λ ∈ (0, 1), the point P_{λ} : (s_{1}(λ), …, s_{n}(λ)) associated to ω_{λ} lies on L between P and Q.

We wish to find a weight ω and a number λ ∈ (0, 1) such that ω ∈ RH_{p}^{d} but ω_{λ} ∉ RH_{p}^{d}. To do this, we will exhibit a line L through Q, and points P and P_{λ} on L, such that: (i) P and P_{λ} are inside the open unit cube (0, 1)^{n}, and P_{λ} is between P and Q; (ii) the value of f at P is strictly less than 2^{n/p}; and (iii) the value of f at P_{λ} is at least 2^{n/p}. Then: (i)′ there is a unique λ ∈ (0, 1) such that the λ-operation takes the weight ω associated to P to the weight ω_{λ} associated to P_{λ}; (ii)′ the weight ω associated to P is in RH_{p}^{d}; and (iii)′ the weight ω_{λ} associated to P_{λ} is not in RH_{p}^{d}.

Fix a ∈ (0, 1/2). Let L be the line through (a, 1, …, 1) and Q; it has equation (a + (\frac{1}{2} − a) t, 1 − \frac{t}{2}, …, 1 − \frac{t}{2}), and values of t ∈ [0, 2] correspond to points in the closed unit cube [0, 1]^{n}. Let g(t) be the restriction of f to L, for t ∈ [0, 2]:

g(t) = (2a + (1 − 2a)t)(2 − t)^{n−1}. (4.9)

Then g′(t) = (2 − t)^{n−2} [−n (1 − 2a) t + 2 (1 − (n + 1) a)].

If n > 2, g has two critical points; if n = 2, g has a single critical point. Since g(2) = f(1 − 2a, 0, …, 0) = 0, and f is positive in the open unit cube, a critical point of g at t = 2 is a minimum. The critical point which occurs at

t_{m} = \frac{2 (1 − (n + 1) a)}{n(1 − 2a)} (4.10)

is a maximum for g. Denote this point by P_{m}. The maximum value is

g(t_{m}) = \frac{2^{n} (n − 1)^{n−1} (1 − a)^{n}}{n^{n} (1 − 2a)^{n−1}} . (4.11)

This maximum is taken inside the open unit cube if and only if t_{m} > 0; in other words, if and only if a < (n + 1)^{−1}. In this case, P_{m} lies between (a, 1, …, 1) and the centre Q of the cube.

The maximum value of g is a continuous, increasing function of a, for a ∈ [0, (n + 1)^{−1}]. We denote this function by h(a) = g(t_{m}) (see (4.11) above). Then h′(a) > 0 for a ∈ [0, (n + 1)^{−1}]. Also, h(0) = 2^{n} (n − 1)^{n−1}n^{−n}, and h(\frac{1}{n+1}) = \frac{2^{n}}{n+1}. Therefore, if p > 1 is large enough that

2^{n/p} < \frac{2^{n}}{n + 1}, (4.12)

then we can choose a_{p} ∈ (0, (n+1)^{−1}) so that h(a_{p}) ≥ 2^{n/p} and so that the maximum value h(a_{p}) of f along the line segment from (a_{p}, 1, …, 1) to Q is taken on at a point inside the open unit cube.

Notice that if p is too large, we may not be able to find a line segment on which the maximum value h(a_{p}) of f is actually equal to 2^{n/p}, since the smallest h(a_{p}) is h(0) which is strictly larger than one (if n > 2).

Define P_{λ} as follows: if h(0) < 2^{n/p} < \frac{2^{n}}{n+1}, then choose the unique a_{p} ∈ (0, (n+1)^{−1}) such that h(a_{p}) = 2^{n/p}, and let P_{λ} = P_{m} for that a_{p}. If 2^{n/p} ≤ h(0), choose a_{p} small enough that f(a_{p}, 1, …, 1) = 2^{n} a_{p} < 2^{n/p}, and let P_{λ} = P_{m} for that a_{p}.

In both cases, f(a_{p}, 1, …, 1) < 2^{n/p} while the value of f at P_{m} is at least 2^{n/p}. Since g is increasing for 0 < t < t_{m}, we may choose a point P between (a_{p}, 1, …, 1) and P_{λ} such that the value of f at P is strictly less than 2^{n/p}. Let λ be the unique number in (0, 1) such that the λ-operation takes this point P to P_{λ}.

The weights ω and ω_{λ} associated to P and P_{λ} respectively, together with the number λ, furnish our example: ω is in RH_{p}^{d} but ω_{λ} is not in RH_{p}^{d}.

The critical value of p, from (4.12), is

(4.13) p_{c} = \frac{n log 2}{n log 2 − log(n + 1)}.

Examples for all p > p_{c} can be found among the n-periodic weights of the form described above. Letting n tend to infinity, we see that p_{c} tends to 1, and so we have found examples for all p > 1.

References

[B] S. BUCKLEY, Summation conditions on weights. Michigan Math. J. 40 #1, 153-170 (1993).

[Ch] M. CHRIST, Lectures on singular integral operators. Regional Conferences Series in Math, AMS # 77 (1990).

[D] G. DAVID, Wavelets and singular integrals on curves and surfaces. Springer Verlag # 1465 (1991).

[FKP] R. FEFFERMAN, C. KENIG, J. PIPHER, The theory of weights and the Dirichlet problem for elliptic equations. Annals of Math. # 134, 65-124 (1991).

[GC-RF] J. GARCIA-CUERVA, J.L. RUBIO de FRANCIA, Weighted norm inequalities and related

topics. North Holland (1985).

[H] A. HAAR, Zur Theorie der orthogonalen Funktionensysteme. Math. Ann. # 69, 331-371 (1910).

[JN] R. JOHNSON, C. J. NEUGEBAUER, Homeomorphisms preserving A_{p}^{d}. Revista Matemática Iberoamericana Vol. 3, No. 2, 249-273 (1987).

[M] Y. MEYER, Ondelettes et Opérateurs II. Herman (1990).

[P1] M.C. PEREYRA, On the resolvents of dyadic paraproducts. Revista Matemática Iberoamericana Vol. 10 No. 3, 627-664 (1994).

[P2] M.C. PEREYRA, On the resolvent of the dyadic paraproduct, and a nonlinear operation on RH_{p} weights. Cont. Math. of AMS. Vol 189, 461-471 (1995).

Cristina Pereyra, Department of Mathematics, Princeton University, Princeton NJ 08544-0001

E-mail address: crisp@math.princeton.edu

Lesley Ward, Department of Mathematics, Rice University, Houston TX 77251-1892; and MSRI, 1000 Centennial Drive, Berkeley CA 94720-5070

E-mail address: lesley@math.rice.edu