A noncommutative weak type (1,1) estimate for a square function from ergodic theory☆
Introduction
Inspired by quantum mechanics and probability, noncommutative harmonic analysis has become an independent field of mathematical research. By using new functional analytic methods from operator space theory and quantum probability, various problems in noncommutative harmonic analysis have been investigated (see, for instance, [10], [24], [25], [26], [27], [43], [45], [38], [35]). Especially, Parcet et al. developed a remarkable operator-valued Calderón-Zygmund theory. More precisely, Parcet [38] formulated a noncommutative version of Calderón-Zygmund decomposition using the theory of noncommutative martingales and developed a pseudo-localization principle for singular integrals which is new even in classical theory (see [15] for more results on this principle). As a result, Parcet obtained the weak type estimates of Calderón-Zygmund operators acting on operator-valued functions. This result played an important role in the perturbation theory [6], where the weak type estimates were exploited to solve the Nazarov-Peller conjecture.
Later on, Mei and Parcet [35] proved a weak type estimate for a large class of noncommutative square functions, see [13] for more related results. However, it seems that Mei and Parcet's weak type estimate could not be used to get estimate (for ) by interpolation, since the decomposition does not linearly depend on the original functions. This drawback could be revised through operator-valued Calderón-Zygmund theory—Proposition 4.3 in [2] where the author proposed a simplified version of Parcet's arguments [38], together with noncommutative Khintchine's inequality as considered in [41], [42]. Moreover, using Khintchine's inequality for weak space considered in [3], one gets another kind of weak type inequality for the Calderón-Zygmund operators with Hilbert valued kernels acting on operator valued functions.
Note that the arguments in [38], [35], [2] depend heavily on the Lipschitz's regularity of the underlying kernel. In this paper, motivated by the study of noncommutative maximal inequality, we will establish a weak type estimate for a square function from ergodic theory. And this square function is different from the class of Calderón-Zygmund operators considered in the previous papers [38], [35], [2] since the associated kernel does not enjoy any regularity.
To illustrate our motivation and present the main results, we need to set up some notions and notations, and refer the reader to Section 2 for more detailed information. Let be a von Neumann algebra equipped with a normal semi-finite faithful (n.s.f.) trace τ and be the tensor von Neumann algebra with the tensor trace . Let be locally integrable, where is the subset of with τ-finite support. For , denote to be the open ball centered at the origin 0 with radius equal to . Then we define the averaging operator on as Given , denote the k-th conditional expectation associated to the sigma algebra generated by the standard dyadic cubes with side-length equal to . The sequence of operators that we are going to investigate in the present paper is defined as follows: In the scalar-valued case, that is, replacing by the set of complex numbers , the square function plays an important role in deducing variational inequalities for ergodic averages or averaging operators from the ones for martingales.
The variational inequalities are much stronger than the maximal inequalities and imply pointwise convergence immediately without knowing a priori pointwise convergence on a dense subclass of functions, which are absent in some models of dynamical systems. Let us recall briefly the history of the development of the variational inequalities. This line of research started with Lépingle's work [29] on martingales which improved the classical Doob maximal inequality. The first variational inequality for the ergodic averages of a dynamical system proved by Bourgain [1] has opened up a new research direction in ergodic theory and harmonic analysis. Bourgain's work has been extended to many other kinds of operators in ergodic theory and harmonic analysis. For instance, Campbell et al. [4], [5] first proved the variational inequalities associated with singular integrals. The reader is referred to [18], [19], [28], [20], [8], [31], [32], [33], [21], [37], [14] and references therein for more information on the development of ergodic theory and harmonic analysis in this direction of research.
The square function (1.2) appeared in most of the above references on variational inequalities, and plays an important role. In the present paper, similarly, using the noncommutative square function estimates, we provide another proof of the noncommutative Hardy-Littlewood maximal inequalities (or ergodic maximal inequalities) combined with the noncommutative Doob's maximal inequalities, see Corollary 1.4.
The statement of our result requires the so-called column and row function spaces [44]. We refer the reader to Section 2 for definitions of noncommutative spaces and weak space—. Let , and be a finite sequence in . Define Then define (resp. ) to be the completion of all finite sequences in with respect to (resp. ). The space is defined as follows.
- •
If , equipped with the intersection norm:
- •
If , equipped with the sum norm: where the infimum runs over all decompositions with and in .
We also recall the definitions of BMO spaces associated to the von Neumann algebra tensor product with the tensor trace where tr is the canonical trace on . Let stand for the ⁎-algebra of ψ-measurable operators affiliated with . According to [35], we define the dyadic BMO space as a subspace of with where the row and column dyadic norms are given by where is the set of all standard dyadic cubes in . We refer the reader to [34] for more precise definitions and relative properties of .
Let () be defined as in (1.1). The following is our main result. Theorem 1.1 Let . Then the following assertions are true with a positive constant depending only on p and the dimension d: for , for , for ,
Remark 1.2 The three estimates in Theorem 1.1 for infinite 's or summations over should be understood as the consequences of the corresponding uniform estimates for all finite subsequences of operators 's and some standard approximation arguments (see for instance Section 6.A of [24]). For this reason, as in [35] we will not explain the convergence of infinite sums appearing in the whole paper when there is no ambiguity.
The assertion (iii) could be regarded as a result in vector-valued harmonic analysis with the underlying Banach spaces being noncommutative spaces, but seems new even in that setting since the kernel of does not enjoy regularity; while some regularity assumption is required in the theory of vector-valued Calderón-Zygmund singular integrals, see e.g. the book [16] and the references therein.
If we set and , then together with the noncommutative Burkholder-Gundy inequality [40], [39], Theorem 1.1 finds its first application: Corollary 1.3 For , we have where the positive constant depends only on the dimension d. Corollary 1.4 For , there exists a projection with where the positive constant depends only on the dimension d.
Remark 1.5 (i) The -versions of the two corollaries () also hold true if we appeal to the noncommutative Burkholder-Gundy inequalities [43] and Doob maximal inequalities [22]. Moreover, as Theorem 1.1 (iii), the -versions of Corollary 1.3 seem new even in the framework of vector-valued harmonic analysis. (ii) Replacing the domain by in Theorem 1.1, Corollary 1.3, Corollary 1.4, similar results hold also true (see e.g. [14], [19] and the references therein). Then by the noncommutative Calderón transference principle [11], we provide another proof of ergodic maximal inequalities associated with actions of groups and (see [12] for more results). (iii) Note that in [34], the author established the result in Corollary 1.4 by appealing to noncommutative martingales, while our method involves only one martingale which certainly have further application.
However, with a moment's thought, there are many difficulties to adapt the arguments in [38], [35], [2] to our setting. Indeed, it is obvious that the kernel associated with T (or ) does not enjoy Lipschitz's regularity while the methods in [38], [35], [2] depend heavily on this smoothness condition. This prompted us to look for some new methods. It turns out that the main ingredient in showing the strong type estimate—an almost orthogonality principle plays an important role in overcoming these difficulties. But numerous modifications are necessary in establishing the noncommutative endpoint estimates.
We end our introduction with a brief description of the organization of the paper. In Section 2, we present some preliminaries on noncommutative -spaces and introduce some notations. A large portion of Section 3 is devoted to the proof of conclusion (i) of Theorem 1.1 while Corollary 1.3 and Corollary 1.4 will be proved at the end of this section. The estimate is proved in Section 4. In Section 5, we give the proof of conclusion (iii) of Theorem 1.1.
Section snippets
Preliminaries
This section collects all the necessary preliminaries for the whole paper. The reader is referred to [44], [9] for more information on noncommutative -spaces and noncommutative martingales.
Weak type estimates
In this section, we first prove conclusion (i) of Theorem 1.1 and Corollary 1.3 as well as Corollary 1.4 will be shown in the last subsection. By decomposing with positive and for , we assume that f is positive in order to avoid unnecessary computations. Let us work on the following dense subset of Here means the support of f as an operator-valued function on . That is to say,
estimate
In this section, we examine the estimate. The dyadic BMO spaces were defined in the Introduction.
In order to prove conclusion (ii), it suffices to show Indeed, (4.1) is equivalent to and Using the fact and taking the adjoint of both sides in (4.2), we have Similarly, we use (4.3) to get
Strong type estimates
In this section, we show the strong type estimate of for . Proposition 5.1 Let . Then is bounded from to .
Proof The result for is just Lemma 3.13. For the case of , using the weak type estimate of T obtained in Section 3 and Lemma 3.13, we conclude that T is bounded from to by real interpolation. Thus is bounded from to thanks to noncommutative Khintchine's inequalities [41], [42]. We now turn to the case of . If we
Acknowledgements
We are very grateful to the referees for a careful reading of the manuscript and many useful suggestions.
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