This paper studies the operator-valued Hardy spaces introduced and studied by Tao Mei. Our principal result shows that the Poisson kernel in Mei's definition of these spaces can be replaced by any reasonable test function. As an application, we get a general characterization of Hardy spaces on quantum tori. The latter characterization plays a key role in our recent study of Triebel-Lizorkin spaces on quantum tori.
Introduction and main resultsThis paper is devoted to the study of operator-valued Hardy spaces introduced by Mei [14]. Motivated by the development of noncommutative martingale inequalities (see, for instance, [4,11,12,13,17,19,21,22]) and the Littlewood-Paley-Stein theory of quantum Markov semigroups (cf. [6, 7, 8]), Mei developed a remarkable theory of operator-valued Hardy spaces on R d . These spaces are shown to be very useful for many aspects of noncommutative harmonic analysis (cf. e.g. [9,10]). They are defined by the Littlewood-Paley g-function or Lusin area integral function associated to the Poisson kernel. However, it is a classical result in the scalar case that the Poisson kernel does not play any special role and can be replaced by any (reasonable) test function with mild conditions. This extension is not only interesting of its own right but also crucial for applications; for instance, it plays an important role in the part of harmonic analysis related to the Littlewood-Paley decomposition as well as in the applications of harmonic analysis to PDE.Recently, we were led to extending this classical result to the noncommutative setting in our study of Triebel-Lizorkin spaces on quantum tori in [29] (see also the announcement [28]). This noncommutative extension is a key ingredient for the part of [29] on Triebel-Lizorkin spaces. To our best knowledge, all existing proofs of this result use maximal functions in a crucial way. Because of the lack of the noncommutative analogue of the pointwise maximal function, they do not extend to the operator-valued setting. We will investigate the problem via duality as in Mei's work [14], combined with the operator-valued Calderón-Zygmund theory. We show that the main arguments of [14] can be adapted to general test functions in place of the Poisson kernel. This adaptation sometimes is quite straightforward, sometimes requires significantly extra efforts. One of the major differences is the lack of harmonicity of the convolution function by a general test kernel. This harmonicity is useful for some arguments in [14]; for example, it permits one to easily see the majoration of the Littlewood-Paley (radial) square function by the Lusin (conic) square function. In the general case, we have a variant of this result whose proof is, however, more elaborated. It should be also pointed out that both radial and conic square functions are important for the theory: the former is simpler and readily extends to the setting of semigroups; because of the non-tangential nature of the cone used, the latter controls other related functions and is required for the H 1 -BM...
