Integral representations in Hardy classes and best approximations in certain functional spaces

For the case of a simply connected domain in the plane one proves necessary and sufficient conditions for the representation of functions of the Hardy class H¹ by an integral with respect to the harmonic measure of its boundary values. A theorem is given, characterizing the rate of decrease of the best polynomial approximations of an entire function in Hardy classes by the order and the type of this function.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 3, pp. 342–347, March, 1991.     

Estimates for the Norms of the Carleman--Goluzin--Krylov Operators in the Disk-Algebra and the Hardy Space H1

The Carleman--Goluzin--Krylov formula expressing an analytic function of the Hardy class in the unit disk is considered. Analogs of the Patil theorem asserting the convergence in this formula are discussed in the cases of the disk-algebra and the Hardy space H¹. Bibliography: 14 titles. Illustrations: 2 figures.     

Hardy space and Bergman space on the Octonions

In this paper, square-integrable Octonion-valued function spaces on R³ and R⁷ are decomposed into the direct sum of Octonion Hardy and conjugate Hardy spaces, and square-integrable Octonion function spaces on the upper half spaces R ₊ ⁴ and R ₊ ⁸ are decomposed into infinity direct sum of subspaces in which the first components are just the Octonion Bergman spaces. The Octonion Hardy Spaces on R³ and R⁷ are characterized.     

Hardy spaces and the<Emphasis Type="Italic">Tb</Emphasis> theorem

It is well-known that Calderón-Zygmund operators T are bounded on H^p for\(\frac{n}{{n + 1}}< p \leqslant 1\) provided T*(1) = 0. In this article, it is shown that if T*(b) = 0, where b is a para-accretive function, T is bounded from the classical Hardy space H^p to a new Hardy space H _b ^p . To develop an H _b ^p theory, a discrete Calderón-type reproducing formula and Plancherel-Pôlya-type inequalities associated to a para-accretive function are established. Moreover, David, Journé, and Semmes’ result [9] about the L^P, 1 < p < ∞, boundedness of the Littlewood-Paley g function associated to a para-accretive function is generalized to the case of p ≤ 1. A new characterization of the classical Hardy spaces by using more general cancellation adapted to para-accretive functions is also given. These results complement the celebrated Calderón-Zygmund operator theory.     

Hausdorff Operators on Function Spaces

The authors mainly study the Hausdorff operators on Euclidean space Rn.They establish boundedness of the Hausdorff operators in various function spaces,such as Lebesgue spaces,Hardy spaces,local Hardy ...     

Fourier transform and quasi-analytic classes of functions of bounded type on tubular domains

A condition for a function of bounded type to belong to the Hardy class H ¹ in terms of the Fourier transform of the boundary values of this function on R ⁿ is found. Applications of the obtained result to the theories of Hardy classes and of quasi-analytic classes of functions are given.     

A Hardy space for Fourier integral operators

We introduce a new function space, denoted by H _FIO ¹ (ℝⁿ), which is preserved by the algebra of Fourier integral operators of order 0 associated to canonical transformations. A subspace of L¹ (ℝ_n), this space in many aspects resembles the real Hardy space of Fefferman-Stein. In particular, we obtain an atomic characterization of H _FIO ¹ (ℝ_n). In contrast to the standard Hardy space, these atoms are localized in frequency space as well as in real space.     

On Fourier transforms of functions of bounded type in tubular domains

We obtain a necessary and sufficient condition in terms of the Fourier transform under which an analytic function of bounded type in a tubular domain belongs to the Hardy class H ¹(?₊ ⁿ ).     

Hardy and BMO spaces associated to divergence form elliptic operators

Consider a second order divergence form elliptic operator L with complex bounded measurable coefficients. In general, operators based on L, such as the Riesz transform or square function, may lie beyond the scope of the Calderón–Zygmund theory. They need not be bounded in the classical Hardy, BMO and even some L ^p spaces. In this work we develop a theory of Hardy and BMO spaces associated to L, which includes, in particular, a molecular decomposition, maximal and square function characterizations, duality of Hardy and BMO spaces, and a John–Nirenberg inequality. S. Hofmann was supported by the National Science Foundation.     

Commuting relations for Hausdorff operators and Hilbert transforms on real Hardy spaces

We consider Hausdorff operators generated by a function ϕ integrable in Lebesgue"s sense on either R or R ², and acting on the real Hardy space H ¹(R), or the product Hardy space H ¹¹(R×R), or one of the hybrid Hardy spaces H ¹⁰(R ²) and H ⁰¹(R ²), respectively. We give a necessary and sufficient condition in terms of ϕ that the Hausdorff operator generated by it commutes with the corresponding Hilbert transform. This revised version was published online in June 2006 with corrections to the Cover Date.     

Erratum to: “The Mellin Transform of Hardy’s Function is Entire”

We prove that an appropriately modified Mellin transform of the Hardy function Z(x) is an entire function. The proof is based on the fact that the function (2^1?s ? 1)ζ(s) is entire.     

EXTENSIONS OF HILBERT MODULES AND HANKEL OPERATORS

1.IntroductionInthepresentpaper,wewillcontinuethestudyofHilbertmodules[1-5].LetHbeaHilbertspaceandAafunctionalgebra.WesaythatHisaHilbertmoduleoverAifthereisamultiplication(a,f)-affromAxHtoH,makingHintoanXmoduleandif,inaddition,theactionisjointlycontinuousinthesup-normonAandtheHilbertspacenormonH.ThecategoryHofallHilbertNmodulesisanaturalsettingfornumerousquestionsinoperatortheory.In[9],DouglasandPaulseninitiatedasystematicstudyofHilbertmodules'theywereabletotranslatemanyconceptsandpro…     

INTERPOLATION SPACES BETWEEN H^1 AND L^∞ ON SPACES OF HOMOGENEOUS TYPE

Using the maximal function characterization of Hardy spaces,we study the interpolation spaces between H^1 and L^∞on spaces of homogeneous type.     

Radial maximal function characterizations of Hardy spaces on RD-spaces and their applications

Let ${{\mathcal X}}Let X{{\mathcal X}} be an RD-space with m(X)=￥{\mu({\mathcal X})=\infty}, which means that X{{\mathcal X}} is a space of homogeneous type in the sense of Coifman and Weiss and its measure has the reverse doubling property. In this paper, we characterize the atomic Hardy spaces H^p_at(X){H^p_{\rm at}({\mathcal X})} of Coifman and Weiss for p ? (n/(n+1),1]{p\in(n/(n+1),1]} via the radial maximal function, where n is the “dimension” of X{{\mathcal X}}, and the range of index p is the best possible. This completely answers the question proposed by Ronald R. Coifman and Guido Weiss in 1977 in this setting, and improves on a deep result of Uchiyama in 1980 on an Ahlfors 1-regular space and a recent result of Loukas Grafakos et al in this setting. Moreover, we obtain a maximal function theory of localized Hardy spaces in the sense of Goldberg on RD-spaces by generalizing the above result to localized Hardy spaces and establishing the links between Hardy spaces and localized Hardy spaces. These results have a wide range of applications. In particular, we characterize the Hardy spaces H^p_at(M){H^p_{\rm at}(M)} via the radial maximal function generated by the heat kernel of the Laplace-Beltrami operator Δ on complete noncompact connected manifolds M having a doubling property and supporting a scaled Poincaré inequality for all p ? (n/(n+a),1]{p\in(n/(n+\alpha),1]}, where α represents the regularity of the heat kernel. This extends some recent results of Russ and Auscher-McIntosh-Russ.     

Interpolation and harmonic majorants in big Hardy-Orlicz spaces

Free interpolation in Hardy spaces is characterized by the well-known Carleson condition. The result extends to Hardy-Orlicz spaces contained in the scale of classical Hardy spaces H ^p, p > 0. For the Smirnov and the Nevanlinna classes, interpolating sequences have been characterized in a recent paper in terms of the existence of harmonic majorants (quasi-bounded in the case of the Smirnov class). Since the Smirnov class can be regarded as the union over all Hardy-Orlicz spaces associated with a so-called strongly convex function, it is natural to ask how the condition changes from the Carleson condition in classical Hardy spaces to harmonic majorants in the Smirnov class. The aim of this paper is to narrow down this gap from the Smirnov class to “big” Hardy-Orlicz spaces. More precisely, we characterize interpolating sequences for a class of Hardy-Orlicz spaces that carry an algebraic structure and are strictly bigger than ⋃ _p>0 H ^p . It turns out that the interpolating sequences are again characterized by the existence of quasi-bounded majorants, but now the functions defining these quasi-bounded majorants have to be in suitable Orlicz spaces. The existence of harmonic majorants defined by functions in such Orlicz spaces is also discussed in the general situation. We finish the paper with a class of examples of separated Blaschke sequences which are interpolating for certain Hardy-Orlicz spaces without being interpolating for slightly smaller ones.     

The mellin transform of Hardy’s function is entire

We prove that an appropriately modified Mellin transform of the Hardy function Z(x) Is en entire function. The proof is based on the fact that the function (2^1?s ? 1)ζ(s) is integer.     

Iterated maximal functions in variable exponent Lebesgue spaces

In this paper we study the iterated Hardy?CLittlewood maximal operator in variable exponent Lebesgue spaces with exponent allowed to reach the value 1. We use modulars where the L ^p(·)-modular is perturbed by a logarithmic-type function, and the results hold also in the more general context of such Musielak?COrlicz spaces.     

The molecular characterization of weighted Hardy spaces

The molecular characterization of the weighted atomic Hardy space H_w ^p,q,s is given. As an application, the boundedness of Hilbert transform on the weighted Hardy space is proved.     

非齐次粗糙核参数型Marcinkiewicz算子的H~p有界性

设Ω是球面上函数,b是径向函数,ρ是实部正的复数;设Ψ为C~2([0,∞))的递增凸函数,Ψ(0)=0.本文研究非齐次粗糙核参数型Marcinkiewicz算子μ_(Ω,b)~ρ,以及旋转曲面上的非齐次粗糙核参数型Marcinkiewicz算子μ_(Ω,Ψ,b)~ρ,给出非齐次粗糙核Ω和b的最小光滑性条件,建立算子μ_(Ω,b)~ρ和μ_(Ω,Ψ,b)~ρ在Hardy空间和弱Hardy空间上的有界性.本文结果推进了先前b≡1情形的已有工作.     

Hardy–Carleson Measures and Their Dual Poisson–Szegö Transforms

Recently Choe et al. have introduced the notion of dual Berezin transforms and used it to obtain new characterizations of the Carleson measures for the weighted Bergman spaces over the unit ball in C ⁿ . Continuing our investigation on the Hardy spaces, we obtain new characterizations of the Carleson measures for the Hardy spaces by means of the dual Poisson–Szegö transforms introduced by Koosis. Compared with the results for the weighted Bergman spaces, our results for the Hardy spaces not only show an similarity, but also reveal a new characterization.