New York Journal of Mathematics

New York J. Math. 12 (2006) 19–37.

On certain weighted moving averages and their differentiation analogues

Parthena Avramidou

Abstract. Let (X, Σ, μ, T) be a measure-preserving dynamical system, and {I_{n}} a sequence of intervals of nonnegative integers moving to infinity with increasing cardinality. Rosenblatt and Wierdl constructed optimal weights w_{n} for the averages of the form

\frac{1}{w_{n}} ∑_{k∈I_{n}} f ∘ T^{k}

to converge a.e. in L_{1}. In this paper, we provide modified versions of those weights to address the question of optimality for more general weighted averages and their differentiation analogues.

Contents

1. Introduction 19 2. Weighted moving averages and optimal weights 21 3. Weighted differentiation averages and optimal weights 29 Acknowledgments 36 References 37

1. Introduction

Let (X, Σ, μ, T) be a measure-preserving dynamical system. Pointwise convergence of averages of the form \frac{1}{#I_{n}} ∑_{k∈I_{n}} f ∘ T^{k} (called moving averages) depends on existence of maximal inequalities, where I_{n}’s are intervals of integers and #I_{n} denotes the cardinality of I_{n}. Bellow et al. [4] provided a geometric criterion, the so-called cone condition, which controls boundedness of the associated maximal operators. Whenever this criterion fails, it is natural to investigate the rate of divergence. For the diverging moving averages \frac{1}{n} ∑_{n^{2}<k≤n^{2}+n} f ∘ T^{k} (studied in [6]), multiplying by the weights 1/n remedies a.e. convergence. Rosenblatt and Wierdl

19^{ISSN 1076-9803/06}

Received November 29, 2005.

Mathematics Subject Classification. Primary 28D99,37A45; Secondary 47B38.

Key words and phrases. ergodic theory, differentiation.

[8] constructed appropriate weights, so-called correct factors, which impose maximal inequalities to the weighted averages in L_{1}. The correct factors are optimal, that is smallest possible, whenever the intervals I_{n} have increasing lengths.

Let A and B be sets of integers. We define the difference set

A − B = {a − b : a ∈ A, b ∈ B}.

For a sequence of nonempty finite sets of nonnegative integers {U_{n}}_{n∈ℕ}, the correct factors are defined by

Q_{n} = # ⋃_{i=1}^{n}(U_{n} − U_{i}).

Consider the set of indices

𝒰_{n} = {i : #U_{i} ≤ #U_{n}}.

We define the modified correct factors by

Q̃_{n} = # ⋃_{i∈𝒰_{n}} (U_{n} − U_{i}).

In this paper we show that the modified correct factors not only preserve the good qualities of the correct factors, that is the weighted operators satisfy maximal inequalities, but they further extend them in the sense that optimality holds for all moving averages. For example, consider the diverging moving averages over the intervals I_{2n−1} = [n^{2}, n^{2} +n), I_{2n} = [n^{3}, n^{3} +n^{2}). In this case, the modified correct factors are optimal while the correct factors are not. Another significant advantage of the modified correct factors is that they can be used as a geometric criterion for the a.e. convergence of moving averages. This is presented in the following theorem.

Theorem 1.1. Let {I_{n}}_{n∈ℕ} be a sequence of bounded intervals of nonnegative integers given by I_{n} = [v_{n}, v_{n} + #I_{n}) where v_{n}, #I_{n} tend to infinity. Then, for every measure-preserving system (X, Σ, μ, T), there is a constant C so that for every f ∈ L_{1}(X) and λ > 0,

μ \{x ∈ X : sup_{n} |\frac{1}{#I_{n}} ∑_{k∈I_{n}} f(T^{k}(x))| > λ\} ≤ C \frac{∥f∥_{1}}{λ}

if and only if

sup_{n} \frac{Q̃_{n}}{#I_{n}} < +∞.

Consider the moving averages over the intervals I_{2n−1} = [0, (n + 1)!), I_{2n} = [(n + 1)!, (n + 1)! + n!). Notice that sup_{n} \frac{Q_{n}}{#I_{n}} = +∞, when sup_{n} \frac{Q̃_{n}}{#I_{n}} < +∞. By the latter and Theorem 1.1 we conclude that the associated moving averages converge pointwise (see also Remark 2.6 in [8]).

The interplay between ergodic theory and real variable harmonic analysis was established by Calderón’s Transfer Principle in [5]. Consequently, parallel results appeared for example in [4] and [7], and in [2] and [3]. We modify the correct factors for differentiation operators analogous to moving averages, and we show that they display similar behavior. Furthermore, we investigate the question of optimality of these weights, which was not addressed in [8].

In Section 2, we provide in detail the properties of the modified correct factors and the proof of Theorem 1.1. Many examples and propositions are included to illustrate the differences between the new weights and the ones introduced in [8]. In Section 3, we discuss the differentiation analogue of the modified correct factor and analyze the similarities and differences in behavior relative to its ergodic version. Sufficient conditions are provided for the modified correct factors to be optimal.

2. Weighted moving averages and optimal weights

Throughout this section we stay in the ergodic framework of a measure-preserving system (X, Σ, μ, T). Let f be a μ-almost everywhere finite Σ-measurable function. Let I be a nonempty, finite set of nonnegative integers, and n, w be positive integers. We make use of the following types of operators

M_{(I,w)}f = \frac{1}{w} ∑_{n∈I} f ∘ T^{n},

M_{I}f = M_{(I,#I)}f = \frac{1}{#I} ∑_{n∈I} f ∘ T^{n}.

Let Ω be an infinite collection of lattice points in ℤ^{2} with positive second coordinate. We denote by Ω_{α} the lattice points in the union of solid cones with aperture α > 0 and vertex in Ω, and for any integer height λ > 0, we let

Ω_{α}(λ) = {k ∈ ℤ : (k, λ) ∈ Ω_{α}}.

The maximal function associated with Ω is given by

M_{Ω}^{∗}f(x) = sup_{(k,n)∈Ω} \frac{1}{n} ∑_{j=0}^{n−1} |f(T^{k+j}x)|.

Definition 2.1. We say that M_{Ω}^{∗} is weak type (p, p) for p < ∞ if

μ\{x ∈ X : M_{Ω}^{∗}f(x) > λ\} ≤ (C \frac{∥f∥_{p}}{λ} )^{p} .

We say that M_{Ω}^{∗} is strong type (p, p) for 1 < p ≤ ∞ if it bounded from L_{p} to itself.

The following geometric criterion on the approach regions which characterizes the existence of maximal inequalities for ergodic moving averages is essential in understanding the statement of Theorem 1.1.

Theorem 2.2 ([4]). (i) Assume that there exist constants A < +∞ and α > 0 such that #Ω_{α}(λ) ≤ Aλ for all integers λ > 0; then M_{Ω}^{∗} is weak type (1, 1) and strong type (p, p) for 1 < p ≤ ∞.

(ii) If M_{Ω}^{∗} is weak type (p, p) for some p > 0 then for every α > 0 there exists A_{α} < +∞ such that for all integers λ > 0 we have #Ω_{α}(λ) ≤ A_{α}λ.

Remark 2.3. For I_{n} = [v_{n}, v_{n} + l_{n}) with v_{n}, l_{n} → ∞ we set Ω = {(v_{n}, l_{n})}_{n∈ℕ}. Then Theorem 2.2 characterizes the pointwise convergence of the moving averages M_{I_{n}} in all L_{p} spaces, p ≥ 1. This is a consequence of the Banach Principle, which asserts that the set of functions where a.e. convergence holds is a closed set, and the fact that {f ∈ L_{p}(X) : f ∘ T = f} ⊕ cl_{∥·∥_{p}}{f − f ∘ T : f ∈ L_{∞}} is a dense subset of that set.

Question 2.4. Suppose that sup_{k} M_{I_{k}}|f| is not weak type (1, 1). What are the optimal (i.e., smallest) weights w_{k} so that sup_{k} M_{(I_{k},w_{k})}|f| becomes weak type (1, 1)?

We use the modified correct factors as weights. The next two theorems show the sufficiency of the weights Q̃_{n} to impose pointwise convergence, and their optimality. Their proofs are variations of the corresponding theorems in [8], and therefore omitted. Interested readers may refer to [1] for details.

Theorem 2.5. Let {I_{n}}_{n∈ℕ} be a sequence of finite sets of positive integers. For every measure-preserving system (X, Σ, μ, T), f ∈ L_{1}(X) and λ > 0 we have

μ \{sup_{n} |M_{(I_{n},Q̃_{n})}f| > λ\} ≤ \frac{∥f∥_{1}}{λ} .

Let α_{n} = #ℐ_{n}. It is easy to see that whenever I_{n} are bounded intervals one has the estimate

Q̃_{n} ≤ ∑_{i∈ℐ_{n}}(#I_{n} + #I_{i} − 1) ≤ 2α_{n} · #I_{n}. (2.1)

As a consequence of Equation (2.1) and Theorem 2.5 we have the next corollary.

Corollary 2.6. Let {I_{n}}_{n∈ℕ} be a sequence of bounded intervals of nonnegative integers. For any measure-preserving system (X, Σ, μ, T), f ∈ L_{1}(X) and λ > 0, we have

μ \{sup_{n} |M_{(I_{n},α_{n}·#I_{n})}f| > λ\} ≤ \frac{2}{λ}∥f∥_{1}.

Thus, for f ∈ L_{1}(X),

M_{(I_{n},α_{n}·#I_{n})}f(x) → 0 a.e.

For sequences {I_{n}}_{n∈ℕ} of intervals with lengths tending to infinity, Q̃_{n} are optimal weights.

Theorem 2.7. Let {I_{n}}_{n∈ℕ} be a sequence of bounded intervals of nonnegative integers with {#I_{n}} tending to infinity. Let c(n) be a sequence of positive numbers satisfying

lim_{n→∞} c(n) = +∞.

For every nonatomic ergodic probability measure-preserving system (X, Σ, μ, T) there is a residual set in L_{1} of functions so that

sup_{n} |M_{(I_{n},Q̃_{n}/c(n))}f(x)| = +∞

for almost every x ∈ X.

The largeness of Q̃_{n} in terms of the size of the intervals is a measure of whether these weights are trivial or not. An upper bound is given in (2.1).

Remark 2.8. If

∑_{n} \frac{#I_{n}}{w_{n}} < +∞,

then w_{n} are trivial sufficient factors for M_{(I_{n},w_{n})} to satisfy weak type (1,1) maximal inequalities. It is enough to notice that

μ {sup_{n} |M_{(I_{n},w_{n})}f| > λ} ≤ ∑_{n} μ{|M_{(I_{n},w_{n})}f| > λ}

≤ ∑_{n} \frac{#I_{n}}{w_{n}} · \frac{∥f∥_{1}}{λ}

≤ C \frac{∥f∥_{1}}{λ} .

As a consequence of Theorem 2.7 we conclude that Q̃_{n} are nontrivial sufficient weights.

Corollary 2.9. If {I_{n}}_{n∈ℕ} is a sequence of bounded intervals of nonnegative integers with {#I_{n}} tending to infinity, then

∑_{n} \frac{#I_{n}}{Q̃_{n}} = +∞.

A new characterization of pointwise convergence for moving averages is given in terms of the relative size of the length of the intervals to the corresponding correct factor. It indicates the connection between the size of the cross sections involved in Theorem 2.2 and the correct factors.

Proof of Theorem 1.1. Suppose that Q̃_{n} ≤ C · #I_{n} for some constant C and every n ∈ ℕ. Then for every f ∈ L_{1},

|M_{I_{n}}f(x)| ≤ M_{I_{n}}|f|(x) ≤ CM_{(I_{n},Q̃_{n})}|f|(x),

which implies that for every λ > 0

μ {sup_{n} |M_{I_{n}}f| > λC} ≤ μ {sup_{n} M_{(I_{n},Q̃_{n})}|f| > λ} ,

and Theorem 2.5 finishes the proof in this direction.

Conversely, assume that M_{I_{n}} satisfy weak type (1,1) maximal inequalities. By Theorem 2.2, there exist constants A < ∞ and α > 0 such that #Ω_{α}(s) ≤ As for every positive integer height s. Without loss of generality we may assume that 2α ≥ π/2.

Fix n_{0}. We prove the existence of a constant C, independent of n_{0}, for which Q̃_{n_{0}} ≤ Cl_{n_{0}}.

Choose an integer constant C_{1} such that (C_{1} − 1) tan α ≥ 1. Then

C_{1} ≥ 1 + \frac{1}{tan α} > 1.

Therefore, at height s = C_{1}l_{n_{0}} we have

#Ω_{α}(s) ≤ AC_{1}l_{n_{0}}. (2.2)

Notice that

Ω_{α}(s) = ⋃_{{n: l_{n}<s}} J_{n}^{s}

where J_{n}^{s} are the intervals of nonnegative integers given by

J_{n}^{s} = v_{n} + tan α(−(s − l_{n}), (s − l_{n})).

We reflect about the origin and appropriately translate J_{n}^{s} to obtain U_{n}^{s}, given by

U_{n}^{s} = v_{n_{0}} − v_{n} + tan α(−(s − l_{n}), (s − l_{n})).

Then, by Equation (2.2),

(2.3)  # ⋃_{{n: l_{n}<s}} U_{n}^{s} = # ⋃_{{n: l_{n}<s}} J_{n}^{s}
≤ AC_{1}l_{n_{0}}.

For all n with l_{n} ≤ l_{n_{0}} < s we have

I_{n_{0}} − I_{n} = v_{n_{0}} − v_{n} + (−l_{n}, l_{n_{0}})
⊆ v_{n_{0}} − v_{n} + (−l_{n_{0}}, l_{n_{0}})
⊆ v_{n_{0}} − v_{n} + tan α(−l_{n_{0}}(C_{1} − 1), l_{n_{0}}(C_{1} − 1))
⊆ v_{n_{0}} − v_{n} + tan α(−(C_{1}l_{n_{0}} − l_{n}), C_{1}l_{n_{0}} − l_{n}))
= U_{n}^{s}.

Therefore, using also Equation (2.3),

\begin{aligned}Q̃_{n_{0}} & = # ⋃_{n∈𝒜_{n_{0}}} (I_{n_{0}} − I_{n}) \\ & ≤ # ⋃_{n∈𝒜_{n_{0}}} U_{n}^{s} \\ & ≤ # ⋃_{\{n: l_{n}<s\}} U_{n}^{s} \\ & ≤ Cl_{n_{0}}. \end{aligned}  □

Definition 2.10. Let {M_{n}}_{n∈ℕ} be a sequence of linear operators from L_{1} to L_{1}. We say that the strong sweeping out property holds for the operators M_{n} if for every ε > 0 there is a set A ∈ Σ such that:

(a) μ(A) < ε,

(b) lim sup_{n→∞} M_{n}χ_{A}(x) = 1 a.e., and

(c) lim inf_{n→∞} M_{n}χ_{A}(x) = 0 a.e.

One may think of strong sweeping out property as the extreme of divergence.

Example 2.11. Let I_{k} = [v_{k}, v_{k} + l_{k}) with v_{k} = 2^{k} and

l_{k} = \begin{cases}2s, if k = 2s − 1 \\ s, if k = 2s. \end{cases}

Then

Q̃_{2s−1} = # ( ⋃_{i=1}^{2s−1}(I_{2s−1} − I_{i}) ∪ ⋃_{j=s}^{2s}(I_{2s−1} − I_{2j}))
≥ s^{2}.

It follows that

sup_{s} \frac{Q̃_{2s−1}}{#I_{2s−1}} = +∞,

and therefore a weak type (1, 1) maximal inequality is not possible for M_{I_{n}}. In particular, the corresponding moving averages satisfy the strong sweeping out property (see Corollary 5 in [4]).

In the previous example, the strong sweeping out property is not a coincidence, as is pointed out below.

Theorem 2.12. Let {I_{n}}_{n∈ℕ} be a sequence of finite intervals of nonnegative integers with lengths tending to infinity. If

sup_{n} \frac{Q̃_{n}}{#I_{n}} = +∞

then in every nonatomic ergodic probability measure-preserving system (X, Σ, μ, T) the operators M_{I_{n}} satisfy the strong sweeping out property.

Proof. Similar to the proof of Theorem 2.5b in [8]. □

Our next task is to compare the two versions of correct factors. We assume nondecreasing lengths for the intervals I_{n} and we investigate how much larger is the modified correct factor.

First, consider intervals with not strictly increasing lengths, for which the correct factors are not necessarily optimal.

Proposition 2.13. Let {I_{n}}_{n∈ℕ} be a sequence of disjoint bounded intervals of nonnegative integers with lengths satisfying:

(1) #I_{i} < #I_{i+1} for every 1 ≤ i < k_{1} and every m_{n} ≤ i < k_{n+1} for n ∈ ℕ,

(2) #I_{i} = #I_{k_{n}} for every k_{n} ≤ i ≤ m_{n} for n ∈ ℕ,

where k_{n} and m_{n} are two increasing sequences of positive integers with the property sup_{n}(m_{n} − k_{n}) = +∞. Then sup_{n} \frac{Q̃_{n}}{#I_{n}} = +∞. Moreover, if

sup_{n} \frac{(m_{n} − k_{n} + 1) · #I_{k_{n}}}{∑_{i=1}^{n−1}(m_{i} − k_{i} + 1) · #I_{k_{i}}} < +∞

then

sup_{n} \frac{Q̃_{n}}{Q_{n}} < +∞.

Proof. Let I_{n} = [v_{n}, v_{n} + l_{n}). Fix n = k_{s} + j for some s and 0 ≤ j ≤ m_{s} − k_{s}. Let

A_{i} = ⋃_{r=0}^{m_{i}−k_{i}}(I_{k_{s}+j} − I_{k_{i}+r}).

By the definition of Q_{n} and Q̃_{n},

Q_{n} = # \underbrace{⋃_{i=1}^{n}(I_{n} − I_{i})}_{B}

= ⋃_{i=1}^{k_{1}−1}(I_{n} − I_{i}) ∪ ⋃_{i=1}^{s−1} ⋃_{r=1}^{k_{i+1}−m_{i}−1}(I_{n} − I_{m_{i}+r}) ∪ ⋃_{i=1}^{s−1} A_{i} ∪ ⋃_{p=0}^{j}(I_{n} − I_{k_{s}+p}),

Q̃_{n} = #B ∪ ⋃_{t=j+1}^{m_{s}−k_{s}}(I_{k_{s}+j} − I_{k_{s}+t}).

Since I_{n} are disjoint, we obtain the following bounds for Q̃_{n} and ∪A_{i}:

Q_{n} + (m_{s} − k_{s} − j)l_{k_{s}} ≤ Q̃_{n} ≤ Q_{n} + 2(m_{s} − k_{s} + 1)l_{k_{s}} (2.4)

# ⋃_{i=1}^{s−1} A_{i} ≥ ∑_{i=1}^{s−1}(m_{i} − k_{i} + 1)l_{k_{i}}. (2.5)

From the left-hand side of (2.4) we conclude that sup_{n} \frac{Q̃_{n}}{#I_{n}} = +∞.

Combining (2.4) and (2.5),

\frac{Q̃_{n}}{Q_{n}} ≤ 1 + 2\frac{(m_{s} − k_{s} + 1)l_{k_{s}}}{∑_{i=1}^{s−1}(m_{i} − k_{i} + 1)l_{k_{i}}} < C.  □

Remark 2.14. The converse of the previous proposition does not hold. For example, the intervals {I_{n}}_{n∈ℕ} given by I_{n+i} = [n! + i(n − 1)!, n! + (i + 1)(n − 1)!) for every 0 ≤ i ≤ n − 1, satisfy

sup_{n} \frac{(m_{n} − k_{n} + 1) · #I_{k_{n}}}{∑_{i=1}^{n−1}(m_{i} − k_{i} + 1) · #I_{k_{i}}} = +∞,

and simultaneously

\frac{Q̃_{n}}{Q_{n}} < +∞.

The fact that the A_{i}’s defined in the proof of the previous proposition are all disjoint plays a crucial role.

Next we consider intervals with increasing lengths for which the correct factors are optimal.

Lemma 2.15. Let {I_{n}}_{n∈ℕ} be a sequence of disjoint bounded intervals of nonnegative integers with lengths increasing to infinity. Then Q_{n} is an increasing sequence of positive integers.

Proof. Let I_{n} = [v_{n}, v_{n} + l_{n}). Fix n ∈ ℕ, and let k ∈ {1, ⋯, n − 1}. We denote

D_{n,k} = I_{n} − I_{k}.

We relate the sets D_{n,k} ∪ D_{n,k+1} and D_{n+1,k} ∪ D_{n+1,k+1}.

There are two possibilities for D_{n,k} ∩ D_{n,k+1}:

• D_{n,k} ∩ D_{n,k+1} ≠ ∅: this is equivalent to v_{k+1} − v_{k} − l_{k} < l_{n} which implies that

v_{k+1} − v_{k} − l_{k} < l_{n+1} or D_{n+1,k} ∩ D_{n+1,k+1} ≠ ∅.

Then

D_{n,k} ∪ D_{n,k+1} = v_{n} + (−v_{k+1} − l_{k+1}, −v_{k} + l_{n}),

and

D_{n+1,k} ∪ D_{n+1,k+1} = v_{n+1} + (−v_{k+1} − l_{k+1}, −v_{k} + l_{n+1}).

• D_{n,k} ∩ D_{n,k+1} = ∅: this is equivalent to v_{k+1} − v_{k} ≥ l_{k} + l_{n} and

D_{n,k} ∪ D_{n,k+1} = v_{n} + (−v_{k} − l_{k}, −v_{k} + l_{n}) ∪ (−v_{k+1} − l_{k+1}, −v_{k+1} + l_{n}).

Then we have two possible cases for v_{k+1} − v_{k} − l_{k} − l_{n+1}:

– if v_{k+1} − v_{k} − l_{k} < l_{n+1} then D_{n+1,k} ∩ D_{n+1,k+1} ≠ ∅, which implies

D_{n+1,k} ∪ D_{n+1,k+1} = v_{n+1} + (−v_{k+1} − l_{k+1}, −v_{k} + l_{n+1}).

– if v_{k+1} − v_{k} − l_{k} ≥ l_{n+1} then D_{n+1,k} ∩ D_{n+1,k+1} = ∅, which implies

D_{n+1,k} ∪ D_{n+1,k+1} = v_{n+1} + (−v_{k} − l_{k}, −v_{k} + l_{n+1}) ∪ (−v_{k+1} − l_{k+1}, −v_{k+1} + l_{n+1}).

Hence,

v_{n+1} − v_{n} + D_{n,k} ∪ D_{n,k+1} ⊂ D_{n+1,k} ∪ D_{n+1,k+1},

which produces

Q_{n} = # ⋃_{k=1}^{n} D_{n,k} = # (v_{n+1} − v_{n} + ⋃_{k=1}^{n} D_{n,k}) ≤ # ⋃_{k=1}^{n+1} D_{n+1,k} = Q_{n+1}.  □

Proposition 2.16. Let {I_{n}}_{n∈ℕ} be a sequence of finite intervals of nonnegative integers with lengths increasing to infinity. Assume that:

(1) #I_{i} < #I_{i+1} for every 1 ≤ i < k_{1} and every m_{n} ≤ i < k_{n+1} for n ∈ ℕ,

(2) I_{i} = [v_{k_{n}} + (i − k_{n})l_{k_{n}}, v_{k_{n}} + (i − k_{n} + 1)l_{k_{n}}) for every k_{n} ≤ i ≤ m_{n} for

n ∈ ℕ, where k_{n} and m_{n} are two increasing sequences of positive integers with the property sup_{n}(m_{n} − k_{n}) = +∞. Then

sup_{n} \frac{Q̃_{n}}{Q_{n}} < +∞ if and only if sup_{n} \frac{(m_{n} − k_{n})l_{k_{n}}}{Q_{n}} < +∞.

Proof. Notice that for m_{n} ≤ j < k_{n+1} we have Q̃_{j} = Q_{j}.

Fix k_{n} ≤ j < m_{n}. A basic calculation shows that

⋃_{i=j+1}^{m_{n}}(I_{j} − I_{i}) = (−(m_{n} − j + 1)l_{k_{n}}, 0).

Hence,

Q̃_{j} = Q_{j} + #(−(m_{n} − j + 1)l_{k_{n}}, −l_{k_{n}})
= Q_{j} + (m_{n} − j)l_{k_{n}}

which gives

max_{k_{n}≤j<m_{n}} \frac{Q̃_{j}}{Q_{j}} = 1 + max_{k_{n}≤j<m_{n}} \frac{(m_{n} − j)l_{k_{n}}}{Q_{j}} .

Since

max_{k_{n}≤j<m_{n}}(m_{n} − j) = m_{n} − k_{n}

and, by Lemma 2.15,

inf_{k_{n}≤j<m_{n}} Q_{j} = Q_{k_{n}},

we obtain

max_{k_{n}≤j<m_{n}} \frac{Q̃_{j}}{Q_{j}} = 1 + \frac{(m_{n} − k_{n})l_{k_{n}}}{Q_{k_{n}}} ,

and finally

sup_{n} \frac{Q̃_{j}}{Q_{j}} = 1 + sup_{n} \frac{(m_{n} − k_{n})l_{k_{n}}}{Q_{k_{n}}} .  □

Remarks 2.17. Let {I_{n}}_{n∈ℕ} be a sequence of finite intervals of nonnegative integers with lengths increasing to infinity (not necessarily strictly). Q_{n} are optimal weights. We examine how much Q̃_{n} differ from Q_{n}. Notice that lim_{n} \frac{Q̃_{n}}{Q_{n}} = +∞ is not possible since Q_{n} and Q̃_{n} agree for infinitely many indices n.

(1) If the lengths of {I_{n}} strictly increase to infinity, then Q_{n} = Q̃_{n} for every n.

(2) If the lengths of {I_{n}} increase to infinity and the number of intervals with the same length is uniformly bounded, then Q_{n} and Q̃_{n} are equivalent up to a constant.

(3) If the lengths of {I_{n}} increase to infinity and the number of intervals with the same length is unbounded, then it is possible to have Q_{n} and Q̃_{n} to be equivalent up to a constant or not. Consider the intervals given for each n by I_{n+i} = [2^{n} + in, 2^{n} + (i + 1)n) for every 0 ≤ i ≤ n − 1. Then we have sup_{n} \frac{Q̃_{n}}{Q_{n}} < +∞, by Proposition 2.13. To the other end, consider the sequence of intervals {I_{n}} given for each n by I_{n+i} = [n! + in, n! + (i + 1)n) for every 0 ≤ i ≤ (n − 1)! − 1. Then, by Proposition 2.16 we have sup_{n} \frac{Q̃_{n}}{Q_{n}} = +∞. Notice that in the last example consecutive A_{i}’s are not disjoint, which is in contrast to the example in Remark 2.14.

Remarks 2.18. (1) Consider a sequence of finite intervals of nonnegative integers {I_{n}}_{n∈ℕ} with lengths tending not monotonically to infinity. Depending both on the number of intervals that spoil the increasing property, and the number of those with the same length, the modified correct factor can be much smaller than the original one. There is a unique permutation π such that {I_{π(n)}}_{n∈ℕ} has increasing lengths and is “moving” to infinity. For intervals of the same cardinality, the smaller the left endpoint of I_{n}, the smaller the index after π.

(2) The question of optimal weights for general moving averages with the number of intervals at each step not uniformly bounded remains open. It may involve a generalized cone condition which is still missing.

3. Weighted differentiation averages and optimal weights

We consider the space (ℝ^{n}, ℒ, m) where ℒ is the σ-algebra of the Lebesgue measurable sets and m is the Lebesgue measure. Let I be a set of positive measure in ℝ and Q be a positive real number. We make use of the following operators

N_{(I,w)}f(·) = \frac{1}{w} ∫_{I} f(t + ·) dt,

N_{I}f(·) = N_{(I,m(I))}f(·) = \frac{1}{m(I)} ∫_{I} f(t + ·) dt.

Let Ω be an infinite collection of points in ℝ_{+}^{n+1} and Ω_{α} be the union of solid cones with aperture α > 0 and vertex in Ω. For any positive height λ, Ω_{α}(λ) is defined by

Ω_{α}(λ) = {x ∈ ℝ^{n} : (x, λ) ∈ Ω_{α}}.

The maximal operator associated with Ω_{α} is denoted by

N_{Ω_{α}}^{∗}f(·) = sup_{(x,y)∈Ω_{α}} \frac{1}{m(B(x, y))} ∫_{B(x,y)} |f(t + ·)| dt.

Theorem 3.1 ([7]). (i) Assume that there exist constants A < +∞ and α > 0 such that m(Ω_{α}(λ)) ≤ Aλ for every λ > 0; then N_{Ω_{α}}^{∗} is weak type (1, 1) and strong type (p, p) for 1 < p ≤ ∞.

(ii) If N_{Ω_{α}}^{∗} is weak type (p, p) for some p > 0 then for every α > 0 there exists A_{α} < +∞ such that for all λ > 0 we have m(Ω_{α}(λ)) ≤ A_{α}λ.

Remark 3.2. Let I_{n} = [v_{n}, v_{n} + l_{n}) be on the real line with v_{n}, l_{n} → 0. For Ω = {(v_{n}, l_{n})}_{n∈ℕ} the previous theorem characterizes the pointwise behavior of the operators N_{I_{n}}.

Question 3.3. Suppose that sup_{k} N_{I_{k}}|f| is not weak type (1, 1). What are the optimal weights w_{k} so that sup_{k} N_{(I_{k},w_{k})}|f| becomes weak type (1, 1)?

Let {I_{n}}_{n∈ℕ} be a sequence of sets of real numbers with nonzero Lebesgue measure. Let

ℐ_{n} = {i : m(I_{i}) ≤ m(I_{n})}.

We define Q̃_{n} by

Q̃_{n} = m ( ⋃_{i∈ℐ_{n}}(I_{n} − I_{i})) .

Theorem 3.4. Let {I_{n}}_{n∈ℕ} be a sequence of sets of real numbers of finite, nonzero Lebesgue measure. There exists constant C > 0 so that for every f ∈ L_{1}(ℝ) and λ > 0,

m \{sup_{n} |N_{(I_{n},Q̃_{n})}f| > λ\} ≤ \frac{C}{λ} ∥f∥_{1}.

Remarks 3.5. (1) If

∑_{n} \frac{m(I_{n})}{w_{n}} < +∞

then w_{n} are trivial sufficient factors for N_{(I_{n},w_{n})} to satisfy weak type (1,1) maximal inequalities (see also Remark 2.8).

(2) Addressing the question of optimality for the weights Q̃_{n}, in the sense of Theorem 2.7, it is natural to first investigate the convergence of ∑_{n} \frac{m(I_{n})}{Q̃_{n}} (see also Corollary 2.9).

Let I_{n} = [v_{n}, v_{n} + l_{n}) and let d_{n} denote the distance between I_{n} and I_{n+1}. Suppose that l_{n}, v_{n} decrease strictly to zero and d_{n} decrease to zero but not strictly. Fix n and let k ≥ n.

m((I_{n} − I_{k}) ∩ (I_{n} − I_{k+1})) ≠ 0 ⇔
v_{n} − v_{k} + l_{n} > v_{n} − v_{k+1} − l_{k+1} ⇔
l_{n} > v_{k} − (v_{k+1} + l_{k+1}) ⇔
l_{n} > d_{k}.

Let k(n) be the smallest integer i ≥ n so that d_{i} < l_{n}. It follows immediately that from k(n) onwards the consecutive I_{n} − I_{k} have nontrivial intersection. Therefore

(3.1)  Q̃_{n} = Q_{n} = ∑_{i=n}^{k(n)−1}(l_{i} + l_{n}) + (v_{n} + l_{n}) − (v_{n} − v_{k(n)} − l_{k(n)})
≥ (k(n) − n)l_{n}.

As a consequence of (3.1) we have the following lemma.

Lemma 3.6. For a sequence of intervals {I_{n}}_{n∈ℕ} as given above:

(i) If sup_{n}(k(n) − n) = +∞ then sup_{n} \frac{Q_{n}}{l_{n}} = +∞.

(ii) If ∑_{n} \frac{1}{k(n)−n} < +∞ then ∑_{n} \frac{l_{n}}{Q_{n}} < +∞.

Proposition 3.7. There exists a sequence of intervals {I_{n}}_{n∈ℕ} for which

∑_{n} \frac{m(I_{n})}{Q̃_{n}} < +∞.

Proof. Choose l_{n} = \frac{1}{2^{n}} ↘ 0. Fix n_{0}, to be determined later. Let

w(n) = (n_{0} + n) + (n_{0} + n)^{2}.

Let {d_{n}} be given by

d_{n} = \begin{cases}l_{n_{0}} & if n_{0} ≤ n < n_{0} + n_{0}^{2} \\ l_{n_{0}+s+1} & if w(s) ≤ n < w(s + 1) for some s ≥ 0. \end{cases}

Then d_{n} decreases to zero but not strictly. Such a choice of l_{n} and d_{n} makes {I_{n}} well-defined, since

∑_{n=n_{0}}^{∞} l_{n} < +∞,

and

∑_{n=n_{0}}^{∞} d_{n} = n_{0}^{2}l_{n_{0}} + ∑_{s=0}^{∞}[w(s + 1) − w(s)]l_{n_{0}+s+1}

= \frac{n_{0}^{2}}{2^{n_{0}}} + 2 ∑_{s=0}^{∞} \frac{s + n_{0} + 1}{2^{s+n_{0}+1}}

= \frac{n_{0}^{2}}{2^{n_{0}}} + 2 ∑_{s=n_{0}+1}^{∞} \frac{s}{2^{s}}

which is finite, by the integral test.

The integer n_{0} is chosen so that the intervals move to the origin.

Next, we compute v_{n}: w(s) ≤ n < w(s + 1) for some s ≥ 0, so

v_{n} = ∑_{i=n}^{∞} d_{i} + ∑_{i=n+1}^{∞} l_{i}

= [w(s + 1) − n]l_{n_{0}+s+1} + 2 ∑_{i=n_{0}+s+1}^{∞} \frac{i}{2^{i}} + ∑_{i=n+1}^{∞} l_{i}

= \frac{(s + 1)^{2} + (2n_{0} + 1)(s + 1) + (n_{0}^{2} + n_{0}) − n}{2^{n_{0}+s+1}} + 2 ∑_{i=n_{0}+s+1}^{∞} \frac{i}{2^{i}} + ∑_{i=n+1}^{∞} \frac{i}{2^{i}}

= (I)_{n} + (II)_{n} + (III)_{n} < +∞.

As n → ∞ all three terms in the last equation tend to zero.

Moreover, v_{n} ↘ 0: If n+1 is in the same block then (I)_{n} > (I)_{n+1}, (II)_{n} = (II)_{n+1} and (III)_{n} > (III)_{n+1}. If n + 1 is in the next block, namely n = w(s + 1) − 1, then

v_{n} = \frac{1}{2^{n_{0}+s+1}} + 2 ∑_{k=n_{0}+s+1}^{∞} \frac{k}{2^{k}} + ∑_{i=w(s+1)}^{∞} \frac{1}{2^{i}} ,

v_{n+1} = \frac{s + n_{0} + 2}{2^{n_{0}+s+1}} + 2 ∑_{k=n_{0}+s+2}^{∞} \frac{k}{2^{k}} + ∑_{i=w(s+1)+1}^{∞} \frac{1}{2^{i}} ,

and v_{n} > v_{n+1} holds, since

\frac{s + n_{0} + 2}{2^{n_{0}+s+1}} < \frac{1}{2^{n_{0}+s+1}} + 2\frac{n_{0} + s + 1}{2^{n_{0}+s+1}}.

Fix n. For n ≤ i ≤ n + n^{2} − 1 we have d_{i} ≥ l_{n}, when d_{n+n^{2}} = l_{n+1} < l_{n}. Therefore k(n) = n + n^{2}, and Lemma 3.6 finishes the proof. □

The previous proposition already indicates the existence of delicate differences between moving averages and their differentiation analogues. There exists a characterization of weak boundedness of the maximal operator of N_{I_{n}} in terms of the relative size of Q̃_{n} to m(I_{n}). This characterization is analogous to the one for ergodic moving averages.

Theorem 3.8. Let {I_{n}}_{n∈ℕ} be a sequence of intervals of real numbers given by I_{n} = [v_{n}, v_{n} + l_{n}), where l_{n}, v_{n} tend to zero. Then, there exists a constant C > 0 so that for every f ∈ L_{1}(ℝ) and λ > 0,

m \{sup_{n} |N_{I_{n}}f| > λ\} ≤ \frac{C}{λ} ∥f∥_{1}

if and only if

sup_{n} \frac{Q̃_{n}}{l_{n}} < +∞.

Proof. Similar to the proof of Theorem 1.1. □

Remark 3.9. The proof of the above theorem includes that at any height l_{n} there exists a height C_{1}l_{n} for which we have

Q̃_{n} ≤ m(Ω_{α}(C_{1}l_{n})).

Therefore,

\frac{l_{n}}{Q̃_{n}} ≥ \frac{1}{C_{1}} · \frac{C_{1}l_{n}}{m(Ω_{α}(C_{1}l_{n}))} ≥ \frac{1}{C_{1}} · inf_{u≥l_{n}} \frac{u}{m(Ω_{α}(u))}.

Let inf_{u≥l_{n}} \frac{u}{m(Ω_{α}(u))} = η(l_{n}). Choosing suitably the angle α allows us to have C_{1} as close to one as desired. Hence, we may assume that C_{1} is almost one, and then

\frac{l_{n}}{Q̃_{n}} N_{I_{n}}f(x) ≥ η(l_{n})N_{I_{n}}f(x).

That relates the modified correct factor Q̃_{n} with the weights used by Nagel and Stein [7] in their Section 3.

The following proposition shows that whenever ∑_{n} m(I_{n})/Q̃_{n} < +∞, it is always possible to have a better factor than Q̃_{n}.

Proposition 3.10. Suppose that sup_{n} \frac{Q̃_{n}}{m(I_{n})} = +∞ and ∑_{n} \frac{m(I_{n})}{Q̃_{n}} < +∞. Let

F_{n} = ρ_{n}m(I_{n}) where ρ_{n} ≪ Q̃_{n} and ∑_{n} \frac{1}{ρ_{n}} < +∞. Then, there exists a constant C such that for every λ > 0 and f ∈ L_{1}(ℝ),

m \{sup_{n} |N_{(I_{n},F_{n})}f| > λ\} ≤ \frac{C}{λ} ∥f∥_{1}.

Proof. We may assume that f ≥ 0, and λ = 1. We replace the sup with max by using the interval 1 ≤ n ≤ U where U is a large number.

Let

A = {x : N_{(I_{n},F_{n})}f(x) > 1 for some n, 1 ≤ n ≤ U}.

We show that m(A) ≤ C∥f∥_{1}.

Notice that

A = ⋃_{n=1}^{U} A_{n}

where

A_{n} = {x : N_{(I_{n},F_{n})}f(x) > 1}
= {x : N_{I_{n}}f(x) > ρ_{n}}.

By Chebyshev’s inequality,

m(A_{n}) ≤ \frac{1}{ρ_{n}} ∥N_{I_{n}}f∥_{1} ≤ \frac{1}{ρ_{n}} ∥f∥_{1}.

Then

m(A) ≤ ∑_{n=1}^{U} m(A_{n}) ≤ ∥f∥_{1} ∑_{n=1}^{U} \frac{1}{ρ_{n}} < ∥f∥_{1} ∑_{n=1}^{∞} \frac{1}{ρ_{n}} = C∥f∥_{1}.  □

Proposition 3.7 shows the possibility of convergence of the series ∑_{n} m(I_{n})/Q̃_{n}, in which case Q̃_{n} are not the optimal weights. Now we show that divergence is also possible. We use the same notation as in Proposition 3.10.

Let

ε_{n} = \frac{l_{n}}{∑_{k>n} l_{k}} .

Lemma 3.11. If ∑_{n} l_{n} < +∞ then ∑_{n} ε_{n} = +∞.

Proof. Let σ_{n} = ∑_{k=n}^{∞} l_{k}. Then σ_{n} → 0. Notice that

ε_{n} = (σ_{n} − σ_{n+1})/σ_{n+1}.

It follows that

∑_{n} ε_{n} ≥ ∫_{0}^{c} \frac{1}{x} dx = +∞.  □

Lemma 3.12. Suppose l_{n}, v_{n} ↘ 0, ε_{n} → 0, and for d_{n} one of the following conditions holds:

(1) ∑_{k≥n} d_{k} ≤ Cl_{n}.

(2) ∑_{k≥n} d_{k} = R_{n}l_{n} with sup_{n} R_{n} = +∞ and sup_{n} R_{n}ε_{n} < +∞.

(3) ∑_{k≥n} d_{k} = R_{n}l_{n} with sup_{n} R_{n} = +∞, sup_{n} R_{n}ε_{n} = +∞ and ∑_{n} \frac{1}{R_{n}} = ∞. Then

sup_{n} \frac{Q_{n}}{l_{n}} = +∞ and ∑_{n} \frac{l_{n}}{Q_{n}} = +∞.

Proof. Suppose that condition (1) holds. Since l_{n}, v_{n} ↘ 0 there exists an upper bound for the weight Q_{n},

Q_{n} ≤ v_{n} + 2l_{n},

which gives an upper bound for Q_{n}/l_{n},

\frac{Q_{n}}{l_{n}} ≤ 2 + \frac{v_{n}}{l_{n}} . (3.2)

Decomposing v_{n} in terms of l_{n} and d_{n} we have

(3.3)  v_{n} = ∑_{k>n} l_{k} + ∑_{k≥n} d_{k}

≤ \frac{l_{n}}{ε_{n}} + Cl_{n}

≤ C \frac{l_{n}}{ε_{n}}

where we use the definition of ε_{n}, condition (1), and ε_{n} → 0.

On the other hand,

v_{n} ≥ ∑_{k>n} l_{k}

= \frac{l_{n}}{ε_{n}} ,

which implies

\frac{v_{n}}{l_{n}} ≥ \frac{1}{ε_{n}} → +∞.

Moreover,

sup_{n} \frac{v_{n}}{l_{n}} = +∞

produces, by (3.2),

sup_{n} \frac{Q_{n}}{l_{n}} = +∞.

Finally, Equation (3.3) yields

\frac{Q_{n}}{l_{n}} ≤ \frac{C}{ε_{n}} ⇒
∑_{n} \frac{l_{n}}{Q_{n}} ≥ C ∑_{n} ε_{n}.

By Lemma 3.11, we conclude that

∑_{n} \frac{l_{n}}{Q_{n}} = +∞.

The cases when conditions (2) or (3) are satisfied follow similarly. □

Example 3.13. I_{n} = [\frac{1}{n}, \frac{1}{n} + \frac{1}{n^{2}} ) satisfy the hypothesis of Lemma 3.12 with condition (1).

The next theorem answers the question of optimality of the correct factor for a large class of differentiation operators.

Theorem 3.14. Let {I_{n}}_{n∈ℕ} be a sequence of disjoint intervals of real numbers denoted by I_{n} = [v_{n}, v_{n} + l_{n}) with l_{n}, v_{n} tending to zero. Suppose that

inf_{n} \frac{1}{Q̃_{n}} ∑_{k>n} l_{k} > 0.

Let {c_{n}} be a sequence of real numbers such that lim_{n} c_{n} = +∞. Then, for every C > 0 there is λ > 0 and f ∈ L_{1}^{+}(ℝ) so that

m \{sup_{n} N_{(I_{n}, \frac{Q̃n}{cn})}f > λ\} > \frac{C}{λ} ∥f∥_{1}.

Proof. Fix C > 0. Let

α_{n} = \frac{c_{n}}{Q̃_{n}} ∑_{k>n} l_{k}.

Then

α_{n} ≥ c_{n} inf_{n} \frac{1}{Q̃_{n}} ∑_{k>n} l_{k}

and since lim_{n} c_{n} = +∞ we conclude that lim_{n} α_{n} = +∞. Consequently, there exists n_{0} ∈ ℕ so that for every n ≥ n_{0} we have α_{n} > C.

Choose

λ ∈ (\frac{c_{n_{0}}}{Q̃_{n_{0}}} , \frac{c_{n_{0}+1}}{Q̃_{n_{0}+1}} ) .

For μ = δ_{0}, and for all n > n_{0} and all x ∈ U_{n} = −I_{n},

∫_{I_{n}} dμ(x + t) = 1.

Let

k_{n_{0}} = max \{n ≥ n_{0} : \frac{c_{n}}{Q̃_{n}} ≤ λ and \frac{c_{m}}{Q̃_{m}} for every m > n\} .

Then for all n > k_{n_{0}} we have

\frac{c_{n}}{Q̃_{n}} ∫_{I_{n}} dμ(x + t) > λ

or

\{x ∈ ℝ : N_{(I_{n}, \frac{Q̃n}{cn})}μ(x) > λ\} ⊇ U_{n}.

Moreover, since U_{n} are disjoint,

m \{sup_{n} N_{(I_{n}, \frac{Q̃n}{cn})}μ(x) > λ\} ≥ m \{ sup_{n>k_{n_{0}}} N_{(I_{n}, \frac{Q̃n}{cn})}μ(x) > λ\}

≥ m ( ⋃_{n>k_{n_{0}}} U_{n})

= ∑_{n>k_{n_{0}}} l_{n}

= α_{k_{n_{0}}} · \frac{Q̃_{k_{n_{0}}}}{c_{k_{n_{0}}}}

> C · \frac{1}{λ}

= \frac{C}{λ} ∥μ∥_{1}.

Since ∑_{n} l_{n} < +∞, a significant portion of this sum occurs in the first N terms, where N is sufficiently large. Therefore, for fixed C > 0 there are λ > 0 and N sufficiently large positive integer so that

m \{x ∈ ℝ : sup_{n≤N} N_{(I_{n}, \frac{Q̃n}{cn})}μ(x) > λ\} > \frac{C}{λ} ∥μ∥_{1}.

Consider an integrable function ϕ on ℝ such that ∫ ϕ = 1, and for t > 0 define ϕ_{t}(x) = 1/tϕ(x/t). Then it can be shown that

lim_{t→0} ϕ_{t} = μ.

It follows that for every ε > 0 there is t_{0} > 0 such that for all t ≥ t_{0} and n ≤ N,

∫_{I_{n}} ϕ_{t}(x + u) du > ∫_{I_{n}} dμ(x + u) − \frac{ε}{C}

where

C = max_{n≤N} \frac{c_{n}}{Q̃_{n}} .

Therefore,

sup_{n≤N} N_{(I_{n}, \frac{cn}{Q̃n})}μ(x) > λ

implies

sup_{n≤N} N_{(I_{n}, \frac{cn}{Q̃n})}ϕ_{t}(x) > λ − ε.

Hence, for every C > 0 there exist λ > 0, N ∈ ℕ and t > 0 such that

m \{x ∈ ℝ : sup_{n≤N} N_{(I_{n}, \frac{Q̃n}{cn})}ϕ_{t}(x) > λ − ε\} > \frac{C}{λ} ∥ϕ_{t}∥_{1}.

Taking ε → 0 finishes the proof. □

Example 3.15. Lemma 3.12 with conditions (1) or (2) implies that

Q_{n} ≤ v_{n} + 2l_{n} ≤ \frac{C}{ε_{n}} l_{n},

therefore

\frac{1}{Q_{n}} ∑_{k>n} l_{k} ≥ C \frac{ε_{n}}{l_{n}} ∑_{k>n} l_{k} = C > 0,

which gives

inf_{n} \frac{1}{Q_{n}} ∑_{k>n} l_{k} > 0.

For such sequences of intervals, Q̃_{n} are indeed the optimal factors for a maximal inequality to hold for N_{(I_{n},Q̃_{n})}.

Remark 3.16. Since it is not clear whether the condition in Theorem 3.14 is also necessary for the modified correct factor to be optimal, further investigation is required.

Acknowledgments

The author would like to acknowledge the invaluable discussions with Professor Joseph Rosenblatt.

weighted moving averages and differentiation analogues 37

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Department of Mathematics, Ohio State University, Columbus, Ohio 43210 avramidou@math.ohio-state.edu

This paper is available via http://nyjm.albany.edu/j/2006/12-2.html.