Parametrized Littlewood–Paley operators on Hardy and weak Hardy spaces

In this paper, we give the boundedness of the parametrized Littlewood–Paley function on the Hardy spaces and weak Hardy spaces. As the corollaries of the above results, we prove that is of weak type (1, 1) and of type (p, p) for 1 < p < 2, respectively. This results are substantial improvement and extension of some known results. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

New abstract Hardy spaces

The aim of this paper is to propose an abstract construction of spaces which keep the main properties of the (already known) Hardy spaces H¹. We construct spaces through an atomic (or molecular) decomposition. We prove some results about continuity from these spaces into L¹ and some results about interpolation between these spaces and the Lebesgue spaces. We also obtain some results on weighted norm inequalities. Finally we present partial results in order to understand a characterization of the duals of Hardy spaces.     

Hardy spaces and partial derivatives of conjugate harmonic functions

In this paper we give necessary and sufficient conditions for a harmonic vector and all its partial derivatives to belong to for all .

     

Composition operators between Bergman and Hardy spaces

We study composition operators between weighted Bergman spaces. Certain growth conditions for generalized Nevanlinna counting functions of the inducing map are shown to be necessary and sufficient for such operators to be bounded or compact. Particular choices for the weights yield results on composition operators between the classical unweighted Bergman and Hardy spaces.

     

Duality results and inequalities with respect to Hardy spaces containing function sequences

With the help of two-parameter martingales and strong martingales Hardy spaces consisting of adapted function sequences are considered. The Hardy spaces generated by the square and by the conditional square functions and their dual spaces are investigated. An inequality due to Stein and Lepingle is extended to two parameters.This research was supported by the Hungarian Scientific Research Funds No. 2085 and No. 74189 as well as by DAAD, the lattest with a stay at the Ludwig-Maximilians-Universität in München.     

Uniformly locally univalent functions and Hardy spaces

We consider the class of uniformly locally univalent functions on the unit disk with prescribed pre-Schwarzian norm. In the present paper, we show that the class is contained in the Hardy space of a specific exponent depending on the norm.     

Locally solvable vector fields and Hardy spaces

We characterize the boundary value of homegeneous solutions of planar one-sided locally solvable vector fields with analytic coefficients with the property that the Lp norm of their traces is locally uniformly bounded, 0<p?1. For p≠1/n, , the boundary value must locally belong to the local Hardy space hp(R) of Goldberg while for p=1/n, , the answer calls for a new class of atomic Hardy spaces if the vector field is of infinite type at some boundary point.     

Hörmander multipliers on two-dimensional dyadic Hardy spaces

In this paper we are interested in conditions on the coefficients of a two-dimensional Walsh multiplier operator that imply the operator is bounded on certain of the Hardy type spaces Hp, 0<p<∞. We consider the classical coefficient conditions, the Marcinkiewicz-Hörmander-Mihlin conditions. They are known to be sufficient for the trigonometric system in the one and two-dimensional cases for the spaces Lp, 1<p<∞. This can be found in the original papers of Marcinkiewicz [J. Marcinkiewicz, Sur les multiplicateurs des series de Fourier, Studia Math. 8 (1939) 78-91], Hörmander [L. Hörmander, Estimates for translation invariant operators in Lp spaces, Acta Math. 104 (1960) 93-140], and Mihlin [S.G. Mihlin, On the multipliers of Fourier integrals, Dokl. Akad. Nauk SSSR 109 (1956) 701-703; S.G. Mihlin, Multidimensional Singular Integrals and Integral Equations, Pergamon Press, 1965]. In this paper we extend these results to the two-dimensional dyadic Hardy spaces.     

Composition operators on Hardy spaces of Riemann surfaces

Our purpose is to define composition operators acting upon Hardy spaces of Riemann surfaces. In terms of counting functions related to analytic self-map on Riemann surfaces, the boundedness and compactness are characterized.     

Hardy spaces on Lie groups of polynomial growth

We give several characterizations of Hardy spaces associated with complex, second-order,subelliptic operators on Lie groups with polynomial growth.     

On zero varieties of holomorphic functions in Hardy spaces

In contrast to the famous Henkin-Skoda theorem concerning the zero varieties of holomorphic functions in the Nevanlinna class on the open unit ball B_n in , n?2, it is proved in this article that for any nonnegative, increasing, convex function ?(t) defined on , there exists satisfying such that there is no f∈H^p(B_n), 0<p<∞, with . Here N_g(ζ,1) denotes the integrated zero counting function associated with the slice function g_ζ. This means that the zero sets of holomorphic functions belonging to the Hardy spaces H^p(B_n), 0<p<∞, unlike that of the holomorphic functions in the Nevanlinna class, cannot be characterized in the above manner.     

Hadamard products with power functions and multipliers of Hardy spaces

We consider Hadamard products of power functions P(z)=(1−z)^−b with functions analytic in the open unit disk in the complex plane, and an integral representation is obtained when 0<Reb<2. Let where μ is a complex-valued measure on the closed unit disk Such sequences are shown to be multipliers of H^p for 1?p?∞. Moreover, if the support of μ is contained in a finite set of Stolz angles with vertices on the unit circle, we prove that {μ_n} is a multiplier of H^p for every p>0. When the support of μ is [0,1] we get the multiplier sequence which provides more concrete applications. We show that if the sequences {μ_n} and {ν_n} are related by an asymptotic expansion

     

Generalized Hardy spaces

Hardy spaces with generalized parameter are introduced following the maximal characterization approach. As particular cases, they include the classical Hp spaces and the Hardy-Lorentz spaces H^p,q. Real interpolation results with function parameter are obtained, Based on them, the behavior of some classical operators is studied in this generalized setting.     

Restricted summability of Fourier transforms and local Hardy spaces

A general summability method, the so-called θ-summability is considered for multi-dimensional Fourier transforms. Under some conditions on θ, it is proved that the maximal operator of the θ-means defined in a cone is bounded from the amalgam Hardy space W（hp, e∞） to W（Lp,e∞）. This implies the almost everywhere convergence of the θ-means in a cone for all f ∈ W（L1, e∞） velong to L1.     

Fourier multiplier theorem for atomic Hardy spaces on unbounded Vilenkin groups

We characterize atomic Hardy spaces on unbounded locally compact Vilenkin groups by means of a modified maximal function. The obtained Fourier multiplier theorem is more general than the corresponding results due to Kitada, Onneweer and Quek, Daly and Phillips that were proved under the boundedness assumption on the underlying group.     

Analytic Hardy spaces on the quantum torus

Analytic Hardy and BMO spaces on the quantum torus are introduced. Some basic properties of these spaces are presented. In particular, the associated H 1-BMO duality theorem is proved. Finally, we discuss some possible extensions of the obtained results.