Convergence of Double Walsh-Fourier Series and Hardy Spaces

It is proved that the maximal operator of the Marczinkiewicz-Fejér meams of a double Walsh-Fourier series is bounded from the two-dimensional dyadic martingale Hardy space H _p to L _p (2/3<p<) and is of weak type (1,1). As a consequence we obtain that the Marczinkiewicz-Fejér means of a function fL ₁ converge a.e. to the function in question. Moreover, we prove that these means are uniformly bounded on H _p whenever 2/3<p<. Thus, in case fH _p , the Marczinkiewicz-Fejér means conv f in H _p norm. The same results are proved for the conjugate means, too.     

On the Uniform Convergence of Walsh-Fourier Series

We study the uniform convergence of Walsh-Fourier series of functions on the generalized Wiener class BV (p(n)↑∞) This revised version was published online in June 2006 with corrections to the Cover Date.     

Hardy Spaces of Dirichlet Series and Their Composition Operators

 In [9], Hedenmalm, Lindqvist and Seip introduce the Hilbert space of Dirichlet series with square summable coefficients , and begin its study, with modern functional and harmonic analysis tools. The space is an analogue for Dirichlet series of the space for Fourier series. We continue their study by introducing , an analogue to the spaces . Thanks to Bohr’s vision of Dirichlet series, we identify with the Hardy space of the infinite polydisk . Next, we study a variant of the Poisson semigroup for Dirichlet series. We give a result similar to the one of Weissler ([25]) about the hypercontractivity of this semigroup on the spaces . Finally, following [8], we determine the composition operators on , and we compare some properties of such an operator and of its symbol. Received October 3, 2001; in revised form January 16, 2002 Published online July 12, 2002     

On Convergence of Double Series

For the convergence of double series and iterated series, a sufficient condition is obtained. This result provides a test for the convergence of double series and iterated series.     

New Exceptional Sets and Convergence of the Square Partial Sums of Walsh–Fourier Series

For double Walsh–Fourier series and with f ∈ L~2([0, 1) × [0, 1)) we prove two almost orthogonality results relative to the linearized maximal square partial sums operator S_(N(x,y))f(x, y).Assumptions are N(x, y) non-decreasing as a function of x and of y and, roughly speaking, partial derivatives with approximately constant ratio ■≌2~(n_0) for all x and y, where n_0 is any fixed non-negative integer. Estimates, independent of N(x, y) and n_0, are then extended to L~r, 1 r 2.We give an application to the family N(x, y) = λxy on [0, 1) × [0, 1), any λ 10.     

Hardy Spaces on the Plane and Double Fourier Transforms

We provide a direct computational proof of the known inclusion where is the product Hardy space defined for example by R. Fefferman and is the classical Hardy space used, for example, by E.M. Stein. We introduce a third space of Hardy type and analyze the interrelations among these spaces. We give simple sufficient conditions for a given function of two variables to be the double Fourier transform of a function in and respectively. In particular, we obtain a broad class of multipliers on and respectively. We also present analogous sufficient conditions in the case of double trigonometric series and, as a by-product, obtain new multipliers on and respectively.     

Walsh—Fourier级数在p—级数域上的Hardy空间和Herz空间中的Cesaro可和性

讨论了二维Walsh-Fourier级数在p－级数域的整环上的两个函数空间Hardy空间和Herz空间中的Cesaro可和性。     

Parametrized Area Integrals on Hardy Spaces and Weak Hardy Spaces

In this paper, the authors prove that if Ω satisfies a class of the integral Dini condition, then the parametrized area integral μΩ,S^ρ is a bounded operator from the Hardy space H1 （R^n） to L1 （R^n） and from the weak Hardy space H^1,∞ （R^n） to L^1,∞ （R^n）, respectively. As corollaries of the above results, it is shown that μΩ,S^ρ is also an operator of weak type These conclusions are substantial improvement and （1, 1） and of type （p,p） for 1 〈 p 〈 2, respectively extension of some known results.     

调和H~p空间和混合范数空间

设Ｄ是一具有Ｃ边界的有界区域．本文就Ｄ上的调和Ｈａｒｄｙ空间和调和混合范数空间,建立了若干互相之间的连续嵌入定理．     

Discrete Hardy Spaces

We study the boundary behavior of discrete monogenic functions, i.e. null-solutions of a discrete Dirac operator, in the upper and lower half space. Calculating the Fourier symbol of the boundary operator we construct the corresponding discrete Hilbert transforms, the projection operators arising from them, and discuss the notion of discrete Hardy spaces. Hereby, we focus on the 3D-case with the generalization to the n-dimensional case being straightforward.     

Hardy空间的乘子

本文刻划了Hardy空间H^p（1≤p≤∞）到Bloch空间的乘子,推广了前人的结果。     

Convergence of logarithmic means of multiple Walsh-Fourier series

Norlund logarithmic means of multiple Walsh-Fourier series acting from space Llnd-1 L ([0, 1)d) ,d≥1 into space weak-L1([0,1)d) are studied. The maximal Orlicz space such that the Norlund logarithmic means of multiple Walsh-Fourier series for the functions from this space converge in d-dimensional measure is found.     

Hardy-Hilbert重级数定理的一个改进

利用Gram矩阵的正定性和Bernoulli不等式得到Holder不等式的一个加强的结果.由此建立了Hardy-Hilbert重级数定理的一个改进.特别,当p=2时,得到了经典的Hilbert重级数定理的一个很强的结果.     

Para-Accretive Functions and Hardy Spaces

In this paper, we study the structure of the Hardy space H^p_b{H^p_b} associated with a para-accretive function b, which was introduced in Han et al. (J. Geom. Anal. 14:291–318, 2004). The main result is to establish a new atomic decomposition of H^p_b{H^p_b} . As applications, we obtain criterions for the boundedness of operators on H^p_b{H^p_b} and a characterization of the dual space of H^p_b{H^p_b} . The classification of H^p_b{H^p_b} and Calderón–Zygmund decomposition are also discussed.     

Anisotropic Mixed-Norm Hardy Spaces

We introduce and explore Hardy spaces defined by mixed Lebesgue norms and anisotropic dilations. We prove that the definitions of these spaces in terms of smooth, non-tangential, auxiliary, grand, and Poisson maximal operators coincide. We also study the relation between anisotropic mixed-norm Hardy spaces and mixed-norm Lebesgue spaces.     

Anisotropic Local Hardy Spaces

In this paper we introduce and study the anisotropic local Hardy spaces h_A^p(\mathbbRⁿ)h_{A}^{p}(\mathbb{R}^{n}) 0<p≤1, associated with the expansive matrix A. We obtain an atomic characterization of the distributions in h_A^p(\mathbbRⁿ)h_{A}^{p}(\mathbb{R}^{n}). Also we describe the dual spaces of our local Hardy anisotropic spaces as anisotropic Campanato type spaces.     

Convergence of Walsh Series in Near- $$L^\infty $$ Spaces

We consider near- $L^\infty $ spaces that are the union of Orlicz classes and prove the convergence of Walsh series in such spaces.     

Cesàro Summability of Two-dimensional Walsh-Fourier Series

We introduce p-quasi-local operators and the two-dimensional
dyadic Hardy spaces defined by the dyadic squares. It is proved that, if a sublinear operator is p-quasi-local and bounded from to , then it is also bounded from to . As an application it is shown that the maximal operator of the Cesàro means of a martingale is bounded from to and is of weak type (1,1) provided that the supremum in the maximal operator is taken over a positive cone. So we obtain the dyadic analogue of a summability result with respect to two-dimensional trigonometric Fourier series due to Marcinkievicz and Zygmund; more exactly, the Cesàro means of a function converge a.e. to the function in question, provided again that the limit is taken over a positive cone. Finally, it is verified that if we take the supremum in a cone, but for two-powers, only, then the maximal operator of the Cesàro means is bounded from to for every .