Hardy Space Research Articles

Extended Spectra for Some Composition Operators on Weighted Hardy Spaces

Пусть $\alpha$ - комплексный скаляр, а $A$ - ограниченный линейный оператор в гильбертовом пространстве $H$. Говорят, что $\alpha$ является расширенным собственным значением оператора $A$, если существует ненулевой ограниченный линейный оператор $X$, такой, что выполняется равенство $AX=\alpha XA$. В весовых пространствах Харди, инвариантных относительно автоморфизмов, мы полностью вычисляем расширенные собственные значения операторов композиции, индуцированных дробно-линейными отображениями единичного круга $\mathbb{D}$ в себя с внутренней неподвижной точкой в $\mathbb{D}$ и еще одной неподвижной точкой вне $\overline{\mathbb{D}}$. К таким классам преобразований относятся эллиптическое и локсодромное отображения, а также гиперболическое неавтоморфное отображение.

A spectral multiplier theorem for Hardy spaces associated with Schrödinger operator on the Heisenberg group

A spectral multiplier theorem for Hardy spaces associated with Schrödinger operator on the Heisenberg group

Sparse representations of random signals

Sparse (fast) representations of deterministic signals have been well studied. Among other types, there exists one called adaptive Fourier decomposition (AFD) for functions in analytic Hardy spaces. Through the Hardy space decomposition of the L2‐space, the AFD algorithm also gives rise to sparse representations of signals of finite energy. To deal with multivariate signals, the general Hilbert space context comes into play. The multivariate counterpart of AFD in general Hilbert spaces with a dictionary has been named preorthogonal AFD (POAFD). In the present study, we generalize AFD and POAFD to random analytic signals through formulating stochastic analytic Hardy spaces and stochastic Hilbert spaces. To analyze random analytic signals, we work on two models, both being called stochastic AFD, or SAFD in brief. The two models are respectively made for (i) those expressible as the sum of a deterministic signal and an error term (SAFDI) and for (ii) those from different sources obeying certain distributive law (SAFDII). In the later part of the paper, we drop off the analyticity assumption and generalize the SAFDI and SAFDII to what we call stochastic Hilbert spaces with a dictionary. The generalized methods are named as stochastic POAFDs, SPOAFDI and SPOAFDII. Like AFDs and POAFDs for deterministic signals, the developed stochastic POAFD algorithms offer powerful tools to approximate and thus to analyze random signals.

Atomic and Wavelet Characterization of Musielak–Orlicz Hardy Spaces for Generalized Orlicz Functions

The goal of this paper is to obtain atomic and wavelet characterizations of Musielak–Orlicz Hardy spaces. Recently, in 2018, Fu and Yang obtained wavelet characterizations of Musielak–Orlicz Hardy spaces for growth functions of uniformly upper type $$p_-$$ and of uniformly lower type $$p_+$$ with $$0<p_- \le p_+\le 1$$ . What is different from the existing works is that we merely assume $$0<p_- \le p_+<\infty $$ . One of the important tools that make it possible is to refine the convexity of Orlicz functions by obtaining canonical equivalent functions. As applications of the atomic characterization, we investigate the boundedness property of singular integral operators. Especially, we obtain the boundedness property of Marcinkiewicz integral operators acting on some Musielak–Orlicz Hardy spaces which are quasi-Banach spaces.

Maximal Function Characterizations of Hardy Spaces on Rn with Pointwise Variable Anisotropy

In 2011, Dekel et al. developed highly geometric Hardy spaces Hp(Θ), for the full range 0&lt;p≤1, which were constructed by continuous multi-level ellipsoid covers Θ of Rn with high anisotropy in the sense that the ellipsoids can rapidly change shape from point to point and from level to level. In this article, when the ellipsoids in Θ rapidly change shape from level to level, the authors further obtain some real-variable characterizations of Hp(Θ) in terms of the radial, the non-tangential, and the tangential maximal functions, which generalize the known results on the anisotropic Hardy spaces of Bownik.

Universal composition operators

A Hilbert space operator U is called \textit{universal} (in the sense of Rota) if every Hilbert space operator is similar to a multiple of U restricted to one of its invariant subspaces. It follows that the \textit{invariant subspace problem} for Hilbert spaces is equivalent to the statement that all minimal invariant subspaces for U are one dimensional. In this article we characterize all linear fractional composition operators Cϕf=f∘ϕ that have universal translates on both the classical Hardy spaces H2(C+) and H2(D) of the half-plane and the unit disk, respectively. The new example here is the composition operator on H2(D) with affine symbol ϕa(z)=az+(1−a) for $0

Zeros of Gaussian power series, Hardy spaces and determinantal point processes

Given a sequence \((\xi _n)\) of standard i.i.d complex Gaussian random variables, Peres and Virág (in the paper “Zeros of the i.i.d. Gaussian power series: a conformally invariant determinantal process” Acta Math. (2005) 194, 1-35) discovered the striking fact that the zeros of the random power series \(f(z) = \sum _{n=1}^\infty \xi _n z^{n-1}\) in the complex unit disc \({\mathbb {D}}\) constitute a determinantal point process. The study of the zeros of the general random series f(z), where the restriction of independence is relaxed upon the random variables \((\xi _n)\) is an important open problem. This paper proves that if \((\xi _n)\) is an infinite sequence of complex Gaussian random variables, such that their covariance matrix is invertible and its inverse is a Toeplitz matrix, then the zero set of f(z) constitutes a determinantal point process with the same distribution as the case of i.i.d variables studied by Peres and Virág. The arguments are based on some interplays between Hardy spaces and reproducing kernels. Illustrative examples are constructed from classical Toeplitz matrices and the classical fractional Gaussian noise.

Boundedness of fractional integral operators on Musielak–Orlicz Hardy spaces

AbstractLet and let be Musielak–Orlicz functions such that , are Orlicz functions and , are Muckenhoupt weights. In this paper, we give the necessary and sufficient condition for the boundedness of the fractional integral operator from the Musielak–Orlicz Hardy space into the Musielak–Orlicz Hardy space . We also give the necessary and sufficient condition for the boundedness of from into the Musielak–Orlicz space . Our results generalize the main results in [5, 20, 21].

Variational characterizations of weighted Hardy spaces and weighted spaces

This paper obtains new characterizations of weighted Hardy spaces and certain weighted $BMO$ type spaces via the boundedness of variation operators associated with approximate identities and their commutators, respectively.

The Commutators of Strongly Singular Integral Operators on the Weighted Hardy Spaces

The Commutators of Strongly Singular Integral Operators on the Weighted Hardy Spaces

Embedding weighted Hardy spaces into Lebesgue spaces and generalized area operators

In this paper, we consider the weighted Hardy space induced by an weight ω. We characterize the positive Borel measures μ such that the identity operator is bounded when As an application, we obtain necessary and sufficient conditions for the boundedness of the generalized area operator from to

Boundedness of multilinear pseudo-differential operators with symbols in the Hörmander class S0,0

The multilinear pseudo-differential operators with symbols in the multilinear Hörmander class S0,0 are considered. A complete identification of the cases where those operators define bounded operators between local Hardy spaces is given. Some results for the boundedness between Wiener amalgam spaces are also given. These are extensions and improvements of the results known in the bilinear case.

Recovery of the Symbol of Densely Defined Toeplitz Operators over the Hardy Space

While the symbolic map for the Toeplitz collection is well studied, little work was done on a high-defined symbolic map. Operators of Toeplitz. A family of applicant symbols includes this work. To play the Toeplitz operator symbol densely defined. Sarason sub-symbols will be introduced.&#x0D; This results in a partial response. In 2008, Donald Sarason asked a question. In the limited case, an operator's Toeplitzness can be categorized according to his Sarason Sub-Symbols. This explains why the Sarason is being investigated. Sub-symbols on operators densely defined. Analytical closures are shown. Toeplitz is thickly defined.

Liouville operators over the Hardy space

The role of Liouville operators in the study of dynamical systems through the use of occupation measures has been an active area of research in control theory over the past decade. This manuscript investigates Liouville operators over the Hardy space, which encode complex ordinary differential equations in an operator over a reproducing kernel Hilbert space.

On noncommutative weak Orlicz–Hardy spaces

We introduce noncommutative weak Orlicz spaces associated with a weight and study their properties. We also define noncommutative weak Orlicz–Hardy spaces and characterize their dual spaces.

Perturbation by weakly continuous forms and semigroups on Hardy space

In the first part of the article perturbation of a closed form by a weakly continuous form is studied. This notion of weakly continuous perturbation is very handy and, as is shown in the article, leads to a new semigroup whose difference with the given semigroup consists of compact operators. We apply the results to elliptic operators on the Hardy space generalizing results from Semigroup Forum 95(2017), 281-292. A holomorphic semigroup operating on the Hardy space is obtained whose asymptotic behaviour is studied and which is compared with the semigroup generated by the elliptic operator with periodic boundary conditions on L2(0,2π).

Symbols of compact truncated Toeplitz operators

This paper characterises the dual of the model space KI1, where I is an inner function, intersected with the shifted Hardy space, zH1. With this duality result, it is then shown that every bounded truncated Toeplitz operator on the model space KI2 has a bounded symbol if and only if every compact truncated Toeplitz operator on KI2 has a symbol which is of the form I multiplied by a continuous function.

Composition of analytic paraproducts

For a fixed analytic function g on the unit disc D, we consider the analytic paraproducts induced by g, which are defined by Tgf(z)=∫0zf(ζ)g′(ζ)dζ, Sgf(z)=∫0zf′(ζ)g(ζ)dζ, and Mgf(z)=f(z)g(z). The boundedness of these operators on various spaces of analytic functions on D is well understood. The original motivation for this work is to understand the boundedness of compositions of two of these operators, for example Tg2,TgSg,MgTg, etc. Our methods yield a characterization of the boundedness of a large class of operators contained in the algebra generated by these analytic paraproducts acting on the classical weighted Bergman and Hardy spaces in terms of the symbol g. In some cases it turns out that this property is not affected by cancellation, while in others it requires stronger and more subtle restrictions on the oscillation of the symbol g than the case of a single paraproduct.

Holomorphic function spaces on the Hartogs triangle

AbstractThe definition of classical holomorphic function spaces such as the Hardy space or the Dirichlet space on the Hartogs triangle is not canonical. In this paper we introduce a natural family of holomorphic function spaces on the Hartogs triangle which includes some weighted Bergman spaces, a candidate Hardy space and a candidate Dirichlet space. For the weighted Bergman spaces and the Hardy space we study the mapping properties of Bergman and Szegő projection respectively, whereas for the Dirichlet space we prove it is isometric to the Dirichlet space on the bidisc.

OPERATORS ON BERGMAN SPACES

Bergman space theory has been at the core of complex analysis research for many years. Indeed, Hardy spaces are related to Bergman spaces. The popularity of Bergman spaces increased when functional analysis emerged. Although many researchers investigated the Bergman space theory by mimicking the Hardy space theory, it appeared that, unlike their cousins, Bergman spaces were more complex in different aspects. The issue of invariant subspace constitutes one common problem in mathematics that is yet to be resolved. For Hardy spaces, each invariant subspace for shift operators features an elegant description. However, the method for formulating particular structures for the large invariant subspace of shift operators upon Bergman spaces is still unknown. This paper aims to characterize bounded Hankel operators involving a vector-valued Bergman space compared to other different vector value Bergman spaces.