New York Journal of Mathematics

New York J. Math. 3A (1998) 135–148.

Convergence of Moving Averages of Multiparameter Superadditive Processes

Doğan Çömez

Abstract. It is shown that moving averages sequences are good in the mean for multiparameter strongly superadditive processes in L_{1}, and good in the pmean for multiparameter admissible superadditive processes in L_{p}, 1 ≤ p < ∞. Also, using a decomposition theorem in L_{p}-spaces, a.e. convergence of the moving averages of multiparameter superadditive processes with respect to positive L_{p}-contractions, 1 < p < ∞, is obtained.

Contents

1. Introduction 135 2. Convergence in the p-Mean 137 3. Almost Everywhere Convergence 143 4. Concluding Discussions 146 References 147

1. Introduction

Beginning with the moving averages theorem of Bellow, Jones and Rosenblatt [BJR_{1}], determining the conditions that ensure a.e. convergence (or divergence) of moving averages of various processes has been a subject of intensive study. Subsequently, a.e. convergence of moving averages has been obtained in several different settings [AD, Ç_{2}, ÇF, JO_{1}, JO_{2}]. In [JO_{1}, JO_{2}] the moving averages theorem has been extended to the operator setting. Generalization of this theorem to (multiparameter) superadditive processes relative to measure preserving transformations (MPTs) is due to Ferrando [F]. Recently, the a.e. convergence of the moving averages of superadditive processes relative to positive L_{p}-contractions, 1 < p < ∞, has been obtained [Ç_{2}].

In proving the a.e. convergence of moving averages, a condition on the sequence, called the cone condition, plays a vital role. It has been observed that the class of

\begin{matrix}c \\ ◯ \end{matrix}1998 State University of New York ISSN 1076-9803/98

Received January 27, 1998.

Mathematics Subject Classification. Primary 47A35, Secondary 28D99.

Key words and phrases. superadditive processes, admissible processes, moving averages, almost everywhere convergence, convergence in the mean.

This work was supported in part by ND-EPSCoR through NSF OSR-9452892.

sequences satisfying the cone condition is the same as the class of B-sequences, introduced by Akcoglu and Déniel [AD]. Multiparameter B-sequences are introduced in [F]. It turns out, however, that for the norm convergence of moving averages of the additive processes this cone condition is not necessary.

In this article we will investigate the a.e. and norm convergence of the moving averages of multiparameter superadditive processes relative to positive L_{p}- contractions. Our results will generalize some results in [Ç_{2}, F] to the multiparameter operator theory setting, and the result in [JO_{1}] and the norm convergence result in [BJR_{2}] to the superadditive setting.

Let (X, F, μ) be a σ-finite measure space, and let T and S be commuting positive linear L_{p}(X)-contractions, 1 ≤ p < ∞ fixed. A family of real-valued functions F = {F_{(m,n)}}_{m≥0,n≥0} ⊂ L_{p} with F_{(0,0)} = F_{(1,0)} = F_{(0,1)} is called a (two parameter) (T, S)-superadditive process if, for all m ≥ 0, n ≥ 0,

F_{(m+k,n)} ≥ F_{(m,n)} +T^{m}F_{(k,n)} if k > 0, and F_{(m,n+l)} ≥ F_{(m,n)} +S^{n}F_{(m,l)} if l > 0.

F is called a strongly (T, S)-superadditive process, if, for all 0 < k < m, 0 < l < n,

F_{(m+k,n+l)} ≥ F_{(m,n)} + T^{m}F_{(k,n+l)} + S^{n}F_{(m+k,l)} − T^{m}S^{n}F_{(k,l)}.

When both {F_{(m,n)}} and {−F_{(m,n)}} are (T, S)-superadditive, then {F_{(m,n)}} is called (T, S)-additive process. Clearly, (T, S)-additive processes are necessarily of the form {∑_{i,j=0}^{m−1,n−1} T^{i}S^{j}F_{(1,1)}}. If there exists g ∈ L_{p} such that, for all m, n > 0, F_{(m,n)} ≤ ∑_{i,j=0}^{m−1,n−1} T^{i}S^{j}g, then F is called dominated, and the function g is called a dominant for F . If sup_{m,n≥1} \frac{1}{mn}∥F_{(m,n)}∥_{p} < ∞, the process is called bounded. It is well known that, when p = 1, any bounded superadditive process relative to MPTs has a dominant g that satisfies ∫ g = γ_{F} := sup_{m,n≥1} \frac{1}{mn}∥F_{(m,n)}∥_{1} < ∞, called an exact dominant [AS_{3}, S].

Remark 1. Any strongly superadditive process satisfying F_{(m,0)} = F_{(0,n)} ≡ 0, for all m, n ≥ 0, is a superadditive process (which will be assumed throughout this article). In one parameter case, strong superadditivity and superadditivity coincide. Furthermore, if F is a superadditive process with F_{(1,1)} ≥ 0, then F_{(m,n)} ≥ 0, for all m > 0, n > 0.

Remark 2. If F is a (T, S)-superadditive process, then

G = {G_{(m,n)}} = ∑_{i,j=0}^{m−1,n−1} T^{i}S^{j}F_{(1,1)}

is a (T, S)-additive process, and hence F′_{(m,n)} = F_{(m,n)} − G_{(m,n)} is a positive superadditive process. Therefore, we can always assume that a superadditive process is positive. Also, F′ = {F′_{(m,n)}} is dominated if F is dominated.

Throughout this article, any sequence of the form w = {(a_{n}, r_{n})}, with a_{n} > 0, r_{n} > 0 for all n, and r_{n} → ∞, will be called a moving average sequence (MAS). A (two parameter) MAS is a sequence w = {(a_{n}, r_{n})}, where a_{n} = (a_{n}^{1}, a_{n}^{2}), r_{n} = (r_{n}^{1}, r_{n}^{2}), with a_{n}^{i} > 0, r_{n}^{i} > 0 for all n, and r_{n}^{i} → ∞. We will call the MASs w^{1} = {(a_{n}^{1}, r_{n}^{1})} and w^{2} = {(a_{n}^{2}, r_{n}^{2})} as the components of w, each of which are MASs themselves.

If F is a T -superadditive process and w = {(a_{n}, r_{n})} is a one parameter MAS, we define the averages of F along w by \frac{1}{r_{n}} T^{a_{n}}F_{r_{n}}. (Hence, if F is T -additive, the averages along w will be A_{w,n}(T)F_{1} := \frac{1}{r_{n}} T^{a_{n}} ∑_{i=0}^{r_{n}−1} T^{i}F_{1}.) Similarly, if F is a (T, S)-superadditive process and w = {(a_{n}, r_{n})} is a (two parameter) MAS, then we define the averages of F along w by \frac{1}{|r_{n}|}T^{a_{n}^{1}}S^{a_{n}^{2}}F_{(r_{n}^{1},r_{n}^{2})}, where |r_{n}| = r_{n}^{1}r_{n}^{2} . (So, for (T, S)-additive processes the averages along w will be A_{w,n}(T, S)F_{(1,1)} = \frac{1}{|r_{n}|}T^{a_{n}^{1}}S^{a_{n}^{2}} ∑_{i=0}^{r_{n}^{1}−1} ∑_{j=0}^{r_{n}^{2}−1} T^{i}S^{j}F_{(1,1)}.) It should be noted here that, in the superadditive setting, it is possible to give alternative definitions of moving averages, however, such averages may fail to converge a.e. and in the mean as shown in [ÇF, F].

Let T be a positive linear operator on L_{p}. A MAS w is called good in the p-mean for T if, for every f ∈ L_{p}, lim_{n} A_{w,n}(T)f exists in the L_{p}-norm. w is called good in the p-mean if it is good in the p-mean for all (operators induced by) MPTs. Similarly, a MAS w is called good a.e. for T if, for every f ∈ L_{p}, lim_{n} A_{w,n}(T)f exists a.e., and w is called good a.e. if it is good a.e. for all (operators induced by) MPTs. A (two parameter) MAS w is called good in the p-mean (good a.e.) for linear operators T and S if lim_{n} A_{w,n}(T, S)f exists in the L_{p}-norm (a.e.) for all f ∈ L_{p}. A MAS w is called good in the p-mean (good a.e.) for a (T, S)-superadditive process F if lim_{n→∞} \frac{1}{|r_{n}|}T^{a_{n}^{1}}S^{a_{n}^{2}}F_{(r_{n}^{1},r_{n}^{2})} exists in L_{p}-norm (a.e.).

In what follows, for practicality, all the propositions that require the cone condition will be stated in terms of B-sequences. Of course, one can easily restate them using the cone condition. Also, we will use the notation n = (n, n) , for any integer n ≥ 0. We will assume that ℤ_{+}^{2} is ordered in the usual manner, that is, (m, n) ≤ (u, v) if m ≤ u, n ≤ v, and (m, n) < (u, v) if (m, n) ≤ (u, v) and (m, n) ≠ (u, v).

2. Convergence in the p-Mean

Moving averages of additive processes converge a.e. if and only if the MAS satisfy the cone condition. An example of MAS for which a.e. convergence fails is given in [AdJ]. On the other hand, the situation is different for the norm convergence. Indeed, by a criterion of Bellow Jones and Rosenblatt [BJR_{2}, Corollary 1.8], if w = \{(a_{n}, r_{n})\} is a MAS and ν_{n}f(x) = \frac{1}{r_{n}} ∑_{i=0}^{r_{n}−1} T^{a_{n}+i}f(x), then

ν̂_{n}(γ) = \frac{1}{r_{n}} (γ^{a_{n}} \frac{1 − γ^{r_{n}}}{1 − γ}) → 0 for all |γ| = 1, γ ≠ 1.

Hence any MAS is good in the p-mean (with a T -invariant limit). Recently, in [ÇF] the following is proved:

Theorem A. Let T be a positive L_{p}-contraction, 1 < p < ∞, or a positive Dunford-Schwartz operator on L_{1}. If {n_{k}} is a sequence of positive integers which is good in the p-mean for a class of (super)additive processes relative to MPTs, then it is good in the p-mean for T -(super)additive processes of the same class.

Theorem A implies that a MAS is good in the p-mean for positive L_{p}-contractions, when 1 < p < ∞, or for L_{1} − L_{∞}-contractions (i.e. Dunford-Schwartz operators). Naturally, one asks if the same is valid for superadditive processes. Since the tool employed in [BJR_{2}] is inherently applicable to additive processes, one needs other techniques to answer this question. Also, there are sequences (not MAS) which are good in the mean for additive processes but not so for superadditive processess [ÇF]. In this section we will answer this question for both one-parameter and multiparameter (super)additive processes.

The following result, that can be proved in an arbitrary Banach space setting, tells us that for the mean convergence of additive processes, we can consider one parameter case only:

Proposition 2.1. Let T and S be commuting contractions on a Banach space X. If w = {(a_{n}, r_{n})}, r_{n}^{i} → ∞, is a MAS whose components are good in the p-mean for T and S, respectively, then w is good in the p-mean for T and S.

Proof. Observe first that, the norm limit is (T and S) invariant. Hence

lim_{n} A_{w^{1},n}(T)x = E_{1}x, and lim_{n} A_{w^{2},n}(S)x = E_{2}x,

for some projections E_{1} and E_{2}. Since T and S commute, so do the projections. Then

∥A_{w,n}(T, S)x−E_{1}E_{2}x∥_{p} ≤ ∥A_{w,n}(T, S)x−A_{w^{2},n}(S)E_{1}x∥_{p}+∥A_{w^{2},n}(S)E_{1}x−E_{1}E_{2}x∥_{p}

and, by assumption, both the terms on the right hand side tend to zero. □

Since MASs are good in the p-mean for additive processes relative to MPTs, Theorem A and Proposition 2.1 imply that MASs are good in the p-mean for additive processes relative to positive L_{p}-contractions, when 1 < p < ∞, or for Dunford-Schwartz operators on L_{1}. Hence, we obtain:

Corollary 2.2. Multiparameter MASs are good in the p-mean for positive L_{p}- contractions (when 1 < p < ∞) or Dunford-Schwartz operators on L_{1}.

For the rest of this section, unless stated otherwise, we will consider superadditive processes relative to MPTs only. The solution to the problem for superadditive processes will be studied in two cases.

Case 1. p = 1. The following is a two parameter version of a lemma of Akcoglu and Sucheston [AS_{3}], which is proved similarly. So, we omit the proof.

Lemma 2.3. Let F be a positive (T, S)-superadditive process, where T and S are positive L_{p}-contractions, 1 ≤ p < ∞. If h_{k} = \frac{1}{k^{2}} F_{k}, k > 1, then (with the convention that sums over void sets are zero) F_{n} ≥ ∑_{i=0}^{n−k−1} ∑_{j=0}^{n−k−1} T^{i}S^{j}h_{k}.

If F ⊂ L_{1} is a bounded positive strongly (T, S)-superadditive process with an exact dominant δ ∈ L_{1}, then, together with Lemma 2.3, this yields that, for all n > k ≥ 1,

H_{n−k}^{k} ≤ F_{n} ≤ G_{n},

where G_{(m,n)} = ∑_{i=0}^{m−1} ∑_{j=0}^{n−1} T^{i}S^{j}δ, and H_{(m,n)}^{k} = ∑_{i=0}^{m−1} ∑_{j=0}^{n−1} T^{i}S^{j}h_{k} . So, for a MAS w = {(a_{n}, r_{n})}, if n is large enough, we have

0 ≤ \frac{1}{|r_{n}|}(F_{r_{n}} − H_{r_{n}−k}^{k}) ≤ \frac{1}{|r_{n}|}(G_{r_{n}} − H_{r_{n}−k}^{k}).

Both the processes {G_{(m,n)}} and {H_{(m,n)}^{k}} are positive (T, S)-additive processes.

Thus, by Corollary 2.2, the averages \frac{1}{|r_{n}|}T^{a_{n}^{1}}S^{a_{n}^{2}}G_{r_{n}} and \frac{1}{|r_{n}|}T^{a_{n}^{1}}S^{a_{n}^{2}}H_{r_{n}}^{k} converge in the L_{1}-norm. Observe that

L_{1} − lim_{n} \frac{1}{|r_{n}|}T^{a_{n}^{1}}S^{a_{n}^{2}}H_{r_{n}}^{k} = L_{1} − lim_{n} \frac{1}{|r_{n}|}T^{a_{n}^{1}}S^{a_{n}^{2}}H_{r_{n}−k}^{k}.

Hence,

lim_{n} \frac{1}{|r_{n}|}T^{a_{n}^{1}}S^{a_{n}^{2}}(G_{r_{n}} − H_{r_{n}−k}^{k}) = lim_{n} A_{w,n}(T, S)(δ − h_{k}) = δ^{∗} − g_{k}

exists in the L_{1}-norm, where δ^{∗} = L_{1}−lim A_{w,n}(T, S)δ and g_{k} = L_{1}−lim_{n} A_{w,n}(T, S)h_{k}. Consequently

0 ≤ L_{1} − lim \frac{1}{|r_{n}|}T^{a_{n}^{1}}S^{a_{n}^{2}}(F_{r_{n}} − H_{r_{n}−k}^{k}) ≤ δ^{∗} − g_{k}. (1)

Lemma 2.4. If F is a positive (T, S)-superadditive process, where T and S are positive L_{p}-contractions, then for every k ≥ 1, g_{k} ≤ g_{2k} and ∥g_{k}∥_{1} ≤ γ_{F}.

Proof. By superadditivity, F_{2k} ≥ F_{k} + T^{k}F_{k} + S^{k}F_{k} + T^{k}S^{k}F_{k}, for all k ≥ 1. Therefore,

g_{2k} = L_{1} − lim_{n} \frac{1}{|r_{n}|}T^{a_{n}^{1}}S^{a_{n}^{2}}H_{r_{n}}^{2k}

≥ L_{1} − lim_{n} \frac{1}{4|r_{n}|}T^{a_{n}^{1}}S^{a_{n}^{2}}(H_{r_{n}}^{k} + T^{k}H_{r_{n}}^{k} + S^{k}H_{r_{n}}^{k} + T^{k}S^{k}H_{r_{n}}^{k})

= \frac{1}{4}[L_{1} − lim_{n} \frac{1}{|r_{n}|}(T^{a_{n}^{1}}S^{a_{n}^{2}}H_{r_{n}}^{k} + T^{a_{n}^{1}+k}H_{r_{n}}^{k} + S^{a_{n}^{2}+k}H_{r_{n}}^{k} + T^{a_{n}^{1}+k}S^{a_{n}^{2}+k}H_{r_{n}}^{k})] = g_{k}.

On the other hand, since ∥\frac{1}{|r_{n}|}T^{a_{n}^{1}}S^{a_{n}^{2}}H_{r_{n}}^{k}∥_{1} ≤ ∥\frac{1}{k^{2}} F_{k}∥_{1} ≤ γ_{F} , for every k, the second assertion also follows. □

Theorem 2.5. Let T and S be commuting MPTs and w = {(a_{n}, r_{n})} be a MAS. Then w is good in the 1-mean for bounded strongly (T, S)-superadditive processes and the limit is T -invariant.

Proof. Let F be a bounded strongly (T, S)-superadditive process. Then there exists an exact dominant δ for F. Since w is good in the 1-mean for additive processes, we can assume that F is positive. Given ϵ > 0, pick k such that ∥h_{k}∥_{1} = ∥\frac{1}{k^{2}} F_{k}∥_{1} > γ_{F} −ϵ/2. By (1), 0 ≤ L_{1} −lim_{n} \frac{1}{|r_{n}|}T^{a_{n}^{1}}S^{a_{n}^{2}}[F_{r_{n}} −H_{r_{n}}^{k}] ≤ δ^{∗} −g_{k} (where g_{k} and δ^{∗} are as defined above). It follows from the measure preserving property that ∫ g_{k}dμ > γ_{F} − \frac{ϵ}{2}. For the same reason, and since δ is an exact dominant, we also have γ_{F} = ∫ δ = ∫ δ^{∗}. By Lemma 2.4, {g_{2^{i}k}}_{i} is an increasing sequence of L_{1}-functions, hence there exists g ∈ L_{1} such that g_{k} ↑ g in L_{1}-norm. So, there exists a k (can be assumed equal to the previous one) such that ∥g_{k} − g∥_{1} < \frac{ϵ}{2}. Then

∥δ^{∗} − g∥_{1} ≤ ∥δ^{∗} − g_{k}∥_{1} + ∥g_{k} − g∥_{1} < [∫ δ^{∗} − ∫ g_{k}] + \frac{ϵ}{2} < (γ_{F} − γ_{F} + \frac{ϵ}{2}) + \frac{ϵ}{2} = ϵ,

since ∫ g_{k} = ∫ h_{k} > γ_{F} − \frac{ϵ}{2}. Arbitrariness of ϵ implies that L_{1} −lim_{n} \frac{1}{|r_{n}|}T^{a_{n}^{1}}S^{a_{n}^{2}}F_{r_{n}} exists (and is equal to g ). The invariance of the limit follows from the invariance of g_{k}’s and of δ^{∗}. □

Theorem A yields to the extension of one parameter version of Theorem 2.5 to the operator setting when T is a positive Dunford-Schwartz operator:

Corollary 2.6. Let T be a positive Dunford-Schwartz operator on L_{1} and w = {(a_{n}, r_{n})} be a MAS. Then w is good in the 1-mean for bounded T -superadditive processes and the limit is T -invariant.

Remark 3. Since the method of proof in [ÇF] is valid only in one parameter case, extending Theorem 2.5 to operator setting requires a different approach (see the discussion in Section 4). Also, in this approach one has to use a technique that does not depend on the existence of an (exact) dominant for F, since in the multiparameter operator setting no result on the existence of an exact dominant for superadditive processes is known.

Case 2. 1 < p < ∞. The solution to the problem in this case will be considered for one parameter processes first. In [DK], Derriennic and Krengel constructed an example of a positive superadditive process in L_{2}, satisfying sup_{n} ∥\frac{1}{n}F_{n}∥_{2} < ∞, whose averages do not converge in the L_{2}-norm. Hence for the convergence in the p-mean for superadditive processes one needs more than boundedness condition on the process. One possibility is the condition (∗) considered in the next section (for a.e. convergence). Although it leads to a.e. convergence for (multiparameter) superadditive processes, the tools available do not yield to a conclusive result for the norm convergence if (∗) is assumed. In [ÇF] it has been observed that for a more restrictive class of superadditive processes, namely Chacon admissible processes, one can obtain affirmative results both for a.e. and norm convergence. That is why, for the rest of this section we will work with such processes.

Definition 1. A family of functions {f_{n}} ⊂ L_{p} is called a Chacon T-admissible family (or simply admissible family) if T f_{i} ≤ f_{i+1} for all i ≥ 1.

If {f_{n}} is T -admissible, then the process F = {F_{n}}, where F_{n} = ∑_{i=0}^{n−1} f_{i}, is a T - superadditive process, called an admissible process. The following is an important property of admissible processes:

Proposition 2.7. Let F ⊂ L_{p}, 1 < p < ∞, be a positive T -admissible process, where T is a MPT. If F is bounded, then sup_{k} ∥f_{k}∥_{p} < ∞.

Proof. By the measure preserving property of T and by the admissibility,

∥f_{k}∥_{p}^{p} = ∥T^{j}f_{k}∥_{p}^{p} = \frac{1}{r} ∑_{j=0}^{r−1} ∥T^{j}f_{k}∥_{p}^{p} ≤ \frac{1}{r} ∑_{j=0}^{r−1} ∥f_{k+j}∥_{p}^{p} = \frac{1}{r} ∑_{j=0}^{r−1} ∥F_{k+j+1} − F_{k+j}∥_{p}^{p}.

First let 2 ≤ p < ∞. In that case, by Clarkson’s inequality for 2 ≤ p < ∞ and superadditivity, we have

\begin{aligned}∥F_{k+j+1} − F_{k+j}∥_{p}^{p} & ≤ 2^{p−1}(∥F_{k+j+1}∥_{p}^{p} + ∥F_{k+j}∥_{p}^{p}) − ∥F_{k+j+1} + F_{k+j}∥_{p}^{p} \\ & ≤ 2^{p}[∥F_{k+j+1}∥_{p}^{p} − ∥F_{k+j}∥_{p}^{p}]. \end{aligned}

Thus,

\begin{aligned}∥f_{k}∥_{p}^{p} & ≤ \frac{1}{r} ∑_{j=0}^{r−1} ∥F_{k+j+1} − F_{k+j}∥_{p}^{p} ≤ \frac{2^{p}}{r} ∑_{j=0}^{r−1} ∥F_{k+j+1}∥_{p}^{p} − ∥F_{k+j}∥_{p}^{p} \\ & = \frac{2^{p}}{r} [∥F_{k+r}∥_{p}^{p} − ∥F_{k}∥_{p}^{p}] ≤ \frac{2^{p}}{r} ∥F_{k+r}∥_{p}^{p} ≤ \frac{2^{p}(k + r)}{r} sup_{n≥1} \frac{1}{n}∥F_{n}∥_{p}^{p}. \end{aligned}

Therefore, taking the limit r → ∞, we have ∥f_{k}∥_{p}^{p} ≤ 2^{p} sup_{n} \frac{1}{n}∥F_{n}∥_{p}^{p} < ∞, proving the assertion. When 1 < p < 2, again applying Clarkson’s inequality for this case, we have

∥f_{k}∥_{p}^{q} ≤ \frac{1}{r} ∑_{j=0}^{r−1} ∥F_{k+j+1} − F_{k+j}∥_{p}^{q} ≤ \frac{1}{r} ∑_{j=0}^{r−1} 2^{q}[∥F_{k+j+1}∥_{p}^{q} − ∥F_{k+j}∥_{p}^{q}] = \frac{2^{q}}{r} ∥F_{k+r}∥_{p}^{q}.

Now, the assertion follows as before. □

Theorem 2.8. Let T be a MPT and F be a bounded T-admissible process. Then any MAS w is good in the p-mean for F, 1 < p < ∞, and the limit is T -invariant.

Proof. Since {f_{i}} ⊂ L_{p} is an admissible family, ∑_{j=0}^{n−1} T^{j}f_{0} ≤ F_{n}, and hence we can assume that f_{i} ≥ 0, i ≥ 0. For convenience, define P_{i} = f_{i} − T f_{i−1}, i ≥ 1, where we set P_{0} = f_{0}. Observe that, by Clarkson’s inequalities and Proposition 2.7,

∫ P_{r}^{p} = ∫ (f_{r} − T f_{r−1})^{p} ≤ C_{p}[∫ f_{r}^{p} − ∫ f_{r−1}^{p}] < ∞ (C_{p} = 2^{p−1} or 2^{q−1}). (2)

Now we will use a technique employed in [ÇF]: for a fixed positive integer k, define

g_{n}^{k}(x) = \begin{cases}f_{k}(T^{n−k}x) & for n > k \\ f_{n}(x) & for 0 ≤ n ≤ k. \end{cases}

Then, it follows that

(3) f_{n}(x) − g_{n}^{k}(x) = \begin{cases}0 & if 0 ≤ n ≤ k \\ ∑_{i=1}^{m} P_{k+i}(T^{m−i}x) & for n > k, where m = n − k. \end{cases}

Define D_{i}(x) = ∑_{n=0}^{r_{i}−1} f_{n}(x) − g_{n}^{k}(x). By making use of (3) we estimate that

D_{i}(x) ≤ ∑_{n=0}^{r_{i}−1} ∑_{r=k+1}^{n} P_{r}(T^{n−r}x).

Next, if we let

b_{k,t}(x) = ∑_{r=k+1}^{t} P_{r}(T^{r}x)  and  b_{k}(x) = lim_{t→∞} b_{k,t}(x),

then b_{k,t} ≥ 0, and b_{k} ≥ 0. Using the Lebesgue monotone convergence theorem, we obtain that

∫_{X} b_{k}^{p}dμ = lim_{t→∞} ∫_{X} b_{k,t}^{p}dμ ≤ ∑_{r=k+1}^{∞} ∫ P_{r}^{p} ≤ C_{p} lim_{r→∞} ∫ f_{r}^{p} < ∞, (4)

by Proposition 2.7 and (2). Because b_{k,t} ↑ b_{k} and b_{k} ∈ L_{p} we conclude that T^{j}b_{k,t} ↑ T^{j}b_{k} in L_{p}, for all j, since T is strongly continuous. Therefore, (4) implies that

T^{a_{i}}D_{i} ≤ ∑_{n=0}^{r_{i}−1} T^{a_{i}+n}b_{k}.

On the other hand, observe that, since T is measure preserving, as k → ∞,

∥\frac{1}{r_{i}} ∑_{n=0}^{r_{i}−1} T^{a_{i}+n}b_{k}∥_{p}^{p} ≤ ∥b_{k}∥_{p}^{p} ≤ ∑_{i=k+1}^{∞} ∫ P_{i}^{p} ↓ 0.

By assumption G_{k} := L_{p} − lim_{i→∞} \frac{1}{r_{i}} T^{a_{i}} ∑_{n=0}^{r_{i}−1} g_{n}^{k} exists and is T -invariant. Since, for all n ≥ 1, g_{n}^{k} ≤ g_{n}^{k+1}, we also have G^{k} ≤ G^{k+1}. Therefore, {G^{k}} is a monotone increasing sequence of functions in L_{p}, and consequently, G = lim_{k→∞} G^{k} exists in L_{p} and is T -invariant. Now, given ϵ > 0, find a positive integer K such that for k ≥ K, ∥b_{k}∥_{p}^{p} < ϵ/3, ∥\frac{1}{r_{i}} T^{a_{i}} ∑_{n=0}^{r_{i}−1} g_{n}^{k} − G^{k}∥_{p}^{p} < ϵ/3, and ∥G − G^{k}∥_{p}^{p} < ϵ/3. Then,

∥\frac{1}{r_{i}} T^{a_{i}} ∑_{n=0}^{r_{i}−1} f_{n}−G∥_{p}^{p} ≤ ∥\frac{1}{r_{i}} T^{a_{i}} ∑_{n=0}^{r_{i}−1}(f_{n}−g_{n}^{k})∥_{p}^{p}+∥\frac{1}{r_{i}} T^{a_{i}} ∑_{n=0}^{r_{i}−1} g_{n}^{k}−G^{k}∥_{p}^{p}+∥G−G^{k}∥_{p}^{p} < ϵ,

proving the assertion. □

Now, as an immediate consequence of Theorem 2.8 and Theorem A, we have:

Corollary 2.9. Let T be a positive L_{p}-contraction, 1 < p < ∞. If F is a bounded T -admissible process, then any MAS w is good in the p-mean for F, and the limit is T -invariant.

In order to obtain the two-parameter version of Theorem 2.8, we define twoparameter (T, S)-admissible processes as a family {f_{i,j}} ⊂ L_{p} such that T f_{i,j} ≤ f_{i+1,j}, and Sf_{i,j} ≤ f_{i,j+1}, for all i ≥ 0, j ≥ 0. As before, any such (T, S)- admissible family defines a (T, S)-superadditive processes {F_{(m,n)}}, where F_{(m,n)} = ∑_{i=0}^{m−1} ∑_{j=0}^{n−1} f_{i,j}.

Theorem 2.10. Let T and S be commuting MPTs and F = {F_{(m,n)}} be a bounded (T, S)-admissible process. Then any MAS w is good in the p-mean for F, 1 < p < ∞.

Proof. Let w^{1} = {(a_{n}^{1}, r_{n}^{1})} and w^{2} = {(a_{n}^{2}, r_{n}^{2})} be the components of w. Define, for i ≥ 0, g_{i} = L_{p} − lim_{n} \frac{1}{r_{n}^{2}} S^{a_{n}^{2}} ∑_{j=0}^{r_{n}^{2}−1} f_{i,j}. By Theorem 2.8, these functions g_{i} ∈ L_{p} are well defined. Observe that, by (T, S)-admissiblity of F and strong continuity of T,

T g_{i} = L_{p} − lim_{n} \frac{1}{r_{n}^{2}} T S^{a_{n}^{2}} ∑_{j=0}^{r_{n}^{2}−1} f_{i,j} ≤ L_{p} − lim_{n} \frac{1}{r_{n}^{2}} S^{a_{n}^{2}} ∑_{j=0}^{r_{n}^{2}−1} f_{i+1,j} = g_{i+1},

implying that {g_{i}} is a T -admissible process. Furthermore, boundedness of F implies that this process is bounded also. Hence g := L_{p} − lim \frac{1}{r_{n}^{1}} T^{a_{n}^{1}} ∑_{i=0}^{r_{n}^{1}−1} g_{i} exists by Theorem 2.8. Now, the inequality

∥\frac{1}{|r_{n}|}T^{a_{n}^{1}}S^{a_{n}^{2}}F_{(r_{n}^{1},r_{n}^{2})} − g∥_{p} ≤ ∥\frac{1}{|r_{n}|}T^{a_{n}^{1}}S^{a_{n}^{2}}F_{(r_{n}^{1},r_{n}^{2})} − \frac{1}{r_{n}^{1}} T^{a_{n}^{1}} ∑_{i=0}^{r_{n}^{1}−1} g_{i}∥_{p}

+ ∥\frac{1}{r_{n}^{1}} T^{a_{n}^{1}} ∑_{i=0}^{r_{n}^{1}−1} g_{i} − g∥_{p}

proves the theorem. □

Remark 4. In the proofs of Theorem 2.8 and Theorem 2.10 only superadditivity is needed as opposed to strong superadditivity (in Theorem 2.5.) Hence, the assertion of Theorem 2.10 is valid for any n-parameter bounded admissible process, n ≥ 1.

Remark 5. The arguments employed in the proofs of Theorem 2.8 and Theorem 2.10 can be repeated almost verbatim (in fact, more simply) when p = 1, with the same conclusions. Therefore: if F ⊂ L_{1} is an n-parameter bounded admissible (superadditive) process, n ≥ 1, then any MAS w is good in the 1-mean for F.

As remarked earlier, it is possible to define averages of a superadditive process along a MAS in an alternative fashion. More precisely, if w = {(a_{n}, r_{n})} is a MAS, the averages of a superadditive process F along w can be defined as

\frac{1}{r_{n}} (F_{a_{n}+r_{n}} − F_{r_{n}}). (†)

It is observed in [ÇF] that the averages (†) may fail to converge in the mean (and a.e.) There, when p = 1, it is shown that if the process is T -admissible and w is a B-sequence, then such averages do converge a.e. and in the L_{1}-norm. The argument used there is very similar to the proof of Theorem 2.8. Hence, the same proof works for the convergence in the p-mean if the averages of admissible processes along w is defined by (†).

3. Almost Everywhere Convergence

In this section we study the a.e. convergence of moving averages of multiparameter superadditive processes relative to positive L_{p}-contractions, 1 < p < ∞. The averages we will consider are those averages along sequence of B-cubes, which are multiparameter B-sequences w = {(a_{n}, r_{n})}, with r_{n} := r_{n}^{1} = r_{n}^{2}. (See [F, JO_{1}] for definitions.) The components of all the B-cubes form one parameter B-sequences. When p = 1, a.e. convergence of moving averages of bounded superadditive processes (relative to MPTs) along B-sequences is proved in [F].

We will assume that all the processes under study will satisfy the condition

lim inf_{v→∞} ∥\frac{1}{v^{2}} ∑_{i,j=0}^{v−1} F_{(i,j)} − T F_{(i−1,j)} − SF_{(i,j−1)} + T SF_{(i−1,j−1)}∥_{p} < ∞. (∗)

This condition was introduced and used in [EH] to obtain the a.e. convergence of multiparameter superadditive processes with respect to positive L_{p}-contractions. If a positive (T, S)-superadditive process F satisfies the condition (∗), then the sequence {ϕ_{v}} ⊂ L_{p} is a bounded set in L_{p}, where

ϕ_{v} = \frac{1}{v^{2}} ∑_{i,j=0}^{v−1} F_{(i,j)} − T F_{(i−1,j)} − SF_{(i,j−1)} + T SF_{(i−1,j−1)}.

Hence, {ϕ_{v}} is weakly sequentially compact. Consequently, along a subsequence {v_{i}}, the weak limit of {ϕ_{v}} exists, that is, there is a function ϕ ∈ L_{p} such that, ϕ = w − lim_{i→∞} ϕ_{v_{i}}.

It is known that, if F is a positive strongly (T, S)-superadditive process, then, for any v > 1 and for any 1 ≤ n ≤ v, (1 − \frac{n}{v})^{2}F_{n} ≤ ∑_{i,j=0}^{n−1} T^{i}S^{j}ϕ_{v} [Ç_{1}, EH]. Thus, for a positive strongly (T, S)-superadditive process F satisfying (∗), by the strong continuity of T and S, we have

F_{n} ≤ ∑_{i,j=0}^{n−1} T^{i}S^{j}ϕ,

where ϕ = w − lim_{i→∞} ϕ_{v_{i}}. This shows that:

Lemma 3.1. Let F be a positive strongly (T, S)-superadditive process satisfying (∗), then it has a dominant ϕ ∈ L_{p}.

The following result of Akcoglu and Sucheston will be very instrumental in proving the a.e. convergence. We state it here for easy reference.

Theorem B. [AS_{1}] Let U be a positive L_{p}-contraction, 1 < p < ∞ fixed. Then there is a unique decomposition of X into sets E and E^{c} such that

(i) E is the support of a U-invariant function h ∈ L_{p} , and the support of each U-invariant function is contained in E.

(ii) The subspaces L_{p}(E) and L_{p}(E^{c}) are both invariant under U.

Now, as in Section 2 (case p = 1), if H_{(m,n)}^{k} = ∑_{i=0}^{m−1} ∑_{j=0}^{n−1} T^{i}S^{j}h_{k}, for k ≥ 1, then Lemma 2.3, together with Lemma 3.1, imply that, for a positive strongly (T, S)-superadditive process F, if n > k, H_{n−k}^{k} ≤ F_{n} ≤ G_{n}, where in this case G_{(m,n)} = ∑_{i=0}^{m−1} ∑_{j=0}^{n−1} T^{i}S^{j}ϕ. Hence, if w = {(a_{n}, r_{n})} is a B-sequence, for n large enough, we have

0 ≤ \frac{1}{n^{2}} (F_{n} − H_{n−k}^{k}) ≤ \frac{1}{n^{2}} (G_{n} − H_{n−k}^{k}).

Both {G_{(m,n)}} and {H_{(m,n)}^{k})} are positive (T, S)-additive processes, so, by the

Theorem of Jones and Olsen [JO_{1}, Theorem 2.1], the averages \frac{1}{r_{n}^{2}} T^{a_{n}^{1}}S^{a_{n}^{2}}G_{r_{n}} and

\frac{1}{r_{n}^{2}} T^{a_{n}^{1}}S^{a_{n}^{2}}H_{r_{n}}^{k} converge a.e. (and also, lim_{n} \frac{1}{r_{n}^{2}} T^{a_{n}^{1}}S^{a_{n}^{2}}H_{r_{n}}^{k} = lim_{n} \frac{1}{r_{n}^{2}} T^{a_{n}^{1}}S^{a_{n}^{2}}H_{r_{n}−k}^{k} a.e.) Hence,

lim_{n} \frac{1}{r_{n}^{2}} T^{a_{n}^{1}}S^{a_{n}^{2}}(G_{r_{n}} − H_{r_{n}−k}^{k}) = lim_{n} A_{w,n}(T, S)(ϕ − h_{k}) exists a.e.

Since,

∥ ∑_{n=0}^{∞} \frac{1}{r_{n}^{2}} (T^{a_{n}^{1}+r_{n}} − T^{a_{n}^{1}}) ∑_{j=a_{n}^{2}}^{a_{n}^{2}+r_{n}−1} S^{j}(ϕ − h_{k})∥_{2}^{2} ≤ 2∥ϕ − h_{k}∥_{2}^{2} ∑_{n=0}^{∞} \frac{1}{r_{n}^{2}} < ∞,

it follows that this limit is T -invariant. Similarily, the limit is also S-invariant. Hence, if E is the maximal support of a non-negative (T, S)-invariant function h, then this limit is zero on the set E^{c} by Theorem B. Furthermore, the process 1_{E}F is strongly (T_{E}, S_{E})-superadditive process, where T_{E} and S_{E} are restrictions of T and S to L_{p}(E), respectively. (We will denote them with T and S for simplicity.)

Now, let m = h^{p}.μ be a new (finite) measure on X, and consider the operators T̂f = h^{−1}T (fh) and Ŝf = h^{−1}S(fh), f ∈ L_{p}(m). Then they are positive L_{p}(m)- contractions with T̂1 = 1 and Ŝ1 = 1, and ∫ T̂fdm = ∫ fdm = ∫ Ŝfdm [Ç_{2}, DK]. Therefore, they can be extended to Markovian operators on L_{1}(m), which will still be denoted by T̂ and Ŝ. Furthermore, A_{w,n}(T, S)g, g ∈ L_{p}(μ), converge μ-a.e. if and only if A_{w,n}(T̂, Ŝ)(h^{−1}g) converge m-a.e.

Theorem 3.2. Let T and S be commuting positive linear L_{p}-contractions and F be a strongly (T, S)-superadditive process satisfying (∗). If {(a_{n}, r_{n})} is a sequence of B-cubes, then it is good a.e. for F.

Proof. Since (T, S)-additive processes converge a.e. by the theorem of Jones and Olsen [JO_{1}], we can assume F be nonnegative. Then, by the observations above we can assume for the rest of the proof that X = E. Let m = h^{p}.μ, where h is a (T, S)- invariant function given by Theorem B. Consider the operators T̂f = h^{−1}T (fh) and Ŝf = h^{−1}S(fh), f ∈ L_{p}(m), which are positive L_{p}(m)-contractions with T̂1 = 1 and Ŝ1 = 1, and are Markovian on L_{1}(m). If F′ = {h^{−1}F_{(m,n)}}, then F′ ⊂ L_{1}(m) is a strongly (T̂, Ŝ)-superadditive bounded process with γ_{F′} = sup_{m,n≥1} ∫ \frac{1}{mn}h^{−1}F_{(m,n)}dm. Therefore, γ_{F′} = lim_{m,n→∞} \frac{1}{mn} ∫ h^{−1}F_{(m,n)}dm [Ç_{1}, EH]. Now given ϵ > 0, find k ≥ 1 such that \frac{1}{k^{2}} ∫ h^{−1}F_{k}dm > γ_{F′} − ϵ. Define H_{(m,n)} = H_{(m,n)}^{k} = ∑_{i=0}^{m−1} ∑_{j=0}^{n−1} T^{i}S^{j}h_{k}, where h_{k} = \frac{1}{k^{2}} F_{k}, and G_{(m,n)} = ∑_{i=0}^{m−1} ∑_{j=0}^{n−1} T^{i}S^{j}ϕ. Also, define H′_{(m,n)} = h^{−1}H_{(m,n)}, and G′_{(m,n)} = h^{−1}G_{(m,n)}. Then {H_{(m,n)}} is a positive (T, S)-additive process, and so {H′_{(m,n)}} ⊂ L_{1}(m) is a (T̂, Ŝ)-additive process. Hence,

0 ≤ ∫ lim sup_{n} \frac{1}{r_{n}^{2}} T̂^{a_{n}^{1}} Ŝ^{a_{n}^{2}}(F′_{r_{n}} − H′_{r_{n}})dm ≤ ∫ lim sup_{n} \frac{1}{r_{n}^{2}} T̂^{a_{n}^{1}} Ŝ^{a_{n}^{2}}(G′_{r_{n}} − H′_{r_{n}})dm.

Since lim_{n} \frac{1}{r_{n}^{2}} T^{a_{n}^{1}}S^{a_{n}^{2}}(G_{r_{n}} −H_{r_{n}}) exists μ-a.e., lim_{n} \frac{1}{r_{n}^{2}} T̂^{a_{n}^{1}} Ŝ^{a_{n}^{2}}(G′_{r_{n}} −H′_{r_{n}}) exists m-a.e. too. Therefore, using Fatou’s Lemma,

∫ lim sup_{n} \frac{1}{r_{n}^{2}} T̂^{a_{n}^{1}} Ŝ^{a_{n}^{2}}(G′_{r_{n}} − H′_{r_{n}})dm ≤ lim_{n} ∫ \frac{1}{r_{n}^{2}} T̂^{a_{n}^{1}} Ŝ^{a_{n}^{2}}(G′_{r_{n}} − H′_{r_{n}})dm.

Consequently,

0 ≤ ∫ lim sup_{n} \frac{1}{r_{n}^{2}} T̂^{a_{n}^{1}} Ŝ^{a_{n}^{2}}(F′_{r_{n}} − H′_{r_{n}})dm

≤ lim_{n} \frac{1}{r_{n}^{2}} ∫ G′_{r_{n}}dm − lim_{n} \frac{1}{r_{n}^{2}} ∫ H′_{r_{n}}dm ≤ ∫ h^{−1}ϕdm − lim_{n} \frac{u^{2}}{r_{n}^{2}} ∫ \frac{1}{k^{2}} h^{−1}F_{k}dm

= ∫ h^{−1}ϕdm − (γ_{F′} − ϵ),

where u = [r_{n}/k], integer part of r_{n}/k. On the other hand, observe that ∫ h^{−1}ϕdm = ∫ h^{p−1}ϕdμ and h^{p−1} ∈ L_{q}(μ). Since ϕ = w−lim ϕ_{v_{k}}, and T̂ and Ŝ are Markovian, (using v instead of v_{k}, for convenience) we have

∫ h^{p−1}ϕdμ = lim_{v} ∫ h^{p−1}ϕ_{v}dμ = lim_{v} ∫ h^{p−1}[\frac{1}{v^{2}} ∑_{i,j=0}^{v−1} F_{(i,j)} − T F_{(i−1,j)} − SF_{(i,j−1)}r + T SF_{(i−1,j−1)}]dμ = lim_{v} ∫ [\frac{1}{v^{2}} ∑_{i,j=0}^{v−1} F′_{(i,j)} − T̂F′_{(i−1,j)} − ŜF′_{(i,j−1)} + T̂ŜF′_{(i−1,j−1)}]dm = lim_{v} ∫ [\frac{1}{v^{2}} ∑_{i,j=0}^{v−1} F′_{(i,j)} − F′_{(i−1,j)} − F′_{(i,j−1)} + F′_{(i−1,j−1)}]dm = lim_{v} \frac{1}{v^{2}} ∫ F′_{(v−1,v−1)}dm = γ_{F′}..

This equality, combined with the inequality above, implies that

0 ≤ ∫ lim sup_{n} \frac{1}{r_{n}^{2}} T̂^{a_{n}^{1}} Ŝ^{a_{n}^{2}}(F′_{r_{n}} − H′_{r_{n}})dm < γ_{F′} − (γ_{F′} − ϵ) = ϵ.

Hence lim_{n} \frac{1}{r_{n}^{2}} T̂^{a_{n}^{1}} Ŝ^{a_{n}^{2}}F′_{r_{n}} exists m-a.e., which in turn, shows that lim_{n} \frac{1}{r_{n}^{2}} T^{a_{n}^{1}}S^{a_{n}^{2}}F_{r_{n}} exists μ-a.e. □

Remark 6. The assertion of Theorem 3.2 is also valid for positive power bounded Lamperti operators (See [JO_{1}] for details).

4. Concluding Discussions

In two parameter case, for a MAS w = {(a_{n}, r_{n})} and commuting MPTs T and S if we define

ν_{n}f(x) = \frac{1}{|r_{n}|} ∑_{i=0}^{r_{n}^{1}−1} ∑_{j=0}^{r_{n}^{2}−1} T^{a_{n}^{1}+i}S^{a_{n}^{2}+j}f(x),

then lim_{n} ν̂_{n}(γ, δ) = 0 for all (γ, δ) ≠ (1, 1), |γ| = 1, |δ| = 1. Hence, this is the same conclusion of Corollary 2.2 when T and S are MPTs. One way of extending this result to L_{p}-contractions, 1 < p < ∞, or to Dunford-Schwartz operators on L_{1}, would be via multidimensional version of Theorem A. However, multidimensional Theorem A is not available yet, due to the fact that Theorem A uses Akcoglu-Sucheston Dilation Theorem [AS_{2}], which is known in one parameter case only.

In a related matter, it is not known whether the condition (∗) is the only assumption that guarantees the existence of a dominant for a strongly superadditive process in L_{p}-spaces, 1 < p < ∞. (See also Remark 10 below.) It seems that, in order to obtain a dominant for multiparameter processes in an alternative way, either one has to derive a method of “reduction of order” for B-sequences, as in [Ç_{1}, EH], or one has to employ a dilation argument to reduce the problem to the case of processes with respect to commuting invertible isometries of L_{p}-spaces.

If a two-parameter extension of Akcoglu-Sucheston Dilation Theorem were available, it would yield some nice results mentioned above as well as a dominated estimate for moving averages of two-parameter superadditive processes over B-cubes, as in [Ç_{2}]. (Such a dominated estimate would yield to mean convergence, provided that one has the right (exact) dominant.) The only multiparameter dilation result known in the literature is due to Ando, and is valid only for the two-parameter case in the Hilbert space setting:

Theorem B. [SF, Theorem I.6.4] Let T and S be commuting contractions of a Hilbert space ℍ. Then there exists a larger Hilbert space ℍ′ and commuting unitary operators Q and R on ℍ′ such that, for all n ≥ 1, T^{n} = P Q^{n}I and S^{n} = P R^{n}I, where I is the natural imbedding of ℍ into ℍ′ and P : ℍ′ → ℍ′ is the orthogonal projection onto the subspace ℍ.

Using this theorem, one easily obtains a dominated estimate for moving averages of two-parameter superadditive processes over B-cubes via the dominated estimate of Jones and Olsen [JO_{1}] for the moving averages of multiparameter additive processes relative to positive L_{p}-contractions. Since the proof is very similar to the analogous statement in [Ç_{2}], it will be omitted.

Convergence of Moving Averages 147

Theorem 4.1. Let T and S be positive L_{2}-contractions, F ⊂ L_{2} be a positive (T, S)-superadditive process with a dominant ϕ, and let {(a_{n}, r_{n})} be a B-sequence. Then

∥\frac{1}{|r_{n}|}T^{a_{n}^{1}}S^{a_{n}^{2}}F_{r_{n}}∥_{2} ≤ C∥ϕ∥_{2},

where C is a constant independent of the sequence and the process.

Remark 7. It has been shown in [SF] that Theorem B cannot be extended to n-parameters, when n > 2.

Remark 8. Obtaining a “reduction of order” procedure for the moving averages of multiparameter (super)additive processes is an open problem.

Remark 9. The proofs of Theorems 2.5 and 3.2 depend upon the assumption of strong superadditivity, which is defined in two parameter case only. Strong superadditivity is needed for the existence of an exact dominant.

Remark 10. Akcoglu and Krengel gave an example of two-parameter bounded superadditive process (in L_{1}), which is not strongly superadditive [AK]. (Since, for instance F_{(11,10)} = 18, while F_{(6,5)} + F_{(5,10)} + F_{(11,5)} − F_{(5,5)} = 19.) On the other hand, that process is dominated by the additive process \{G_{I}\}, where G_{I} = \frac{1}{6}|I|. Actually, this additive process is generated by g = \frac{1}{6}I_{0}, with I_{0} =[0,1]×[0,1]. Since γ_{F} = \frac{1}{6}, g is an exact dominant for F.

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Department of Mathematics, North Dakota State University, Fargo, ND 58105-5075, USA

comez@plains.nodak.edu http://hypatia.math.ndsu.NoDak.edu/faculty/comez/

This paper is available via http://nyjm.albany.edu:8000/j/1998/3A-11.html.