On Blaschke products associated with n-widths

Journal of Functional Analysis, 1988

The spaces of boundary values of vector-valued functions in Hardy spaces defined by either holomorphic functions on the disk or harmonic functions with maximal function in LJ' are characterized in terms of vector-valued measures of bounded p-variation. We extend to the case p = 1 a characterization of the Radon Nikodym property based on the existence of limits at the boundary for harmonic functions with maximal function in L'. In the case 0 <p< 1 we find the UMD property as the necessary and sufficient condition to make the spaces defined by maximal function and hy conjugate Poisson kernel coincide. '('1 1988 Academc Press, Inc. In this paper we are concerned with Hardy spaces of vector-valued functions on the disk. Our main objectives are: To extend several definitions of these spaces to the vector-valued setting, to study their relationships depending on the geometry of the Banach space, and to find representations for the boundary values of functions in these different Hardy spaces when we do not require any condition on the Banach space. The paper is divided into three sections. In the first one we show that the Hardy space of B-valued holomorphic functions H$(D) is isometric (via Poisson integral) to certain space of B-valued measures, the so-called measures of bounded p-variation. With this result we can regard the analytic RadonNikodym property in an equivalent way, which allows us to give another formulation of this property. The second section is devoted to solving the same question but for B-valued harmonic functions whose maximal function belongs to Lp (1 < p < co). We find now the Radon Nikodym property as the right condition to make the classical result remain * Supported by the C.

Transactions of the American Mathematical Society, 2011

In this work we characterize the pair of weights (w, v) such that the one-sided Hardy-Littlewood maximal function in dimension two is of weaktype (p, p), 1 ≤ p < ∞, with respect to the pair (w, v). As an application of this result we obtain a generalization of the classic Dunford-Schwartz Ergodic Maximal Theorem for bi-parameter flows of null-preserving transformations.

Contemporary Mathematics, 2009

Let (X, B, µ, T ) be an ergodic dynamical system on a non-atomic finite measure space. We assume without loss of generality that µ(X) = 1. Consider the We obtain the following maximal inequality. For each 1 < p ≤ ∞ there exists a finite constant C p such that for each λ > 0, and nonnegative functions f ∈ L p and g ∈ L We also show that for each α > 2 the maximal function R * (f, g) is a.e. finite for pairs of functions (f, g) ∈ (L(log L) 2α , L 1 ).

Bulletin des Sciences Mathématiques, 2005

We consider the Hardy-Littlewood maximal operator M on Musielak-Orlicz Spaces L ϕ (R d). We give a necessary condition for the continuity of M on L ϕ (R d) which generalizes the concept of Muckenhoupt classes. In the special case of generalized Lebesgue spaces L p(•) (R d) we show that this condition is also sufficient. Moreover, we show that the condition is "left-open" in the sense that not only M but also M q is continuous for some q > 1, where M q f = (M(|f | q)) 1/q .

2015

Abstract: Motivated by G. H. Hardy’s 1939 results [4] on functions orthogonal with respect to their real zeros λn, n = 1, 2,..., we will consider, within the same general conditions imposed by Hardy, functions satisfying an orthogonality with respect to their zeros with Jacobi weights on the interval (0, 1), that is, the functions f(z) = zνF (z), ν ∈ R, where F is entire and∫ 1 0 f(λnt)f(λmt)t α(1 − t)βdt = 0, α&gt; −1 − 2ν, β&gt; −1, when n 6 = m. Considering all possible functions on this class we are lead to the discovery of a new family of generalized Bessel functions including Bessel and Hy-perbessel functions as special cases.