Theory Of Hardy Spaces Research Articles

Hardy Spaces and Canonical Kernels on Quadric CR Manifolds

CR functions on an embedded quadric M always extend holomorphically to M+iΓM\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M+i\\Gamma _M$$\\end{document} where ΓM\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma _M$$\\end{document} is the closure of the convex hull of the image of the Levi form. When ΓM\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma _M$$\\end{document} is a closed polygonal cone, we show that the Bergman kernel on the interior of M+iΓM\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M+i\\Gamma _M$$\\end{document} is a derivative of the Szegö kernel. Moreover, we develop the Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document} Hardy space theory which turns out to be particularly robust. We provide examples that show that it is unclear how to formulate a corresponding relationship between the Bergman and Szegö kernels on a wider class of quadrics.

Fredholm theory of the Toeplitz algebra on the space of all entire functions

AbstractWe prove that an aggregate Toeplitz operator on the Fréchet space of all entire functions is a Fredholm operator if and only if its symbol does not vanish. The result is motivated by and closely resembles the classical result of Gohberg and Douglas from the Hardy space theory of Toeplitz operators. There are however some subtle differences which we also discuss.

Orlicz-Lorentz-Karamata Hardy martingale spaces: inequalities and fractional integral operators

Let 0<q≤∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0<q\\le \\infty $$\\end{document}, b be a slowly varying function and Φ:[0,∞)⟶[0,∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\Phi : [0,\\infty ) \\longrightarrow [0,\\infty ) $$\\end{document} be an increasing function with Φ(0)=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Phi (0)=0$$\\end{document} and limr→∞Φ(r)=∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lim \\limits _{r \\rightarrow \\infty }\\Phi (r)=\\infty $$\\end{document}. In this paper, we introduce a new class of function spaces LΦ,q,b\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_{\\Phi ,q,b}$$\\end{document} which unify and generalize the Lorentz-Karamata spaces with Φ(t)=tp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Phi (t)=t^p$$\\end{document} and the Orlicz-Lorentz spaces with b≡1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$b\\equiv 1$$\\end{document}. Based on the new spaces, we introduce five new Hardy spaces containing martingales, the so-called Orlicz-Lorentz-Karamata Hardy martingale spaces and then develop a theory of these martingale Hardy spaces. To be precise, we first investigate several properties of Orlicz-Lorentz-Karamata spaces and then present Doob’s maximal inequalities by using Hardy’s inequalities. The characterization of these Hardy martingale spaces are constructed via the atomic decompositions. As applications of the atomic decompositions, martingale inequalities and the relation of the different martingale Hardy spaces are presented. The dual theorems and a new John-Nirenberg type inequality for the new framework are also established. Moreover, we study the boundedness of fractional integral operators on Orlicz-Lorentz-Karamata Hardy martingale spaces. The results obtained here generalize the previous results for Lorentz-Karamata Hardy martingale spaces as well as for Orlicz-Lorentz Hardy martingales spaces. Especially, we remove the condition that b is non-decreasing as in [38, 39] and the condition qΦ-1<1/q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q_{\\Phi ^{-1}}<1/q$$\\end{document} in [24], respectively.

Octonionic monogenic and slice monogenic Hardy and Bergman spaces

Abstract In this paper we discuss some basic properties of octonionic Bergman and Hardy spaces. In the first part we review some fundamental concepts of the general theory of octonionic Hardy and Bergman spaces together with related reproducing kernel functions in the monogenic setting. We explain how some of the fundamental problems in well-defining a reproducing kernel can be overcome in the non-associative setting by looking at the real part of an appropriately defined para-linear octonion-valued inner product. The presence of a weight factor of norm 1 in the definition of the inner product is an intrinsic new ingredient in the octonionic setting. Then we look at the slice monogenic octonionic setting using the classical complex book structure. We present explicit formulas for the slice monogenic reproducing kernels for the unit ball, the right octonionic half-space and strip domains bounded in the real direction. In the setting of the unit ball we present an explicit sequential characterization which can be obtained by applying the special Taylor series representation of the slice monogenic setting together with particular octonionic calculation rules that reflect the property of octonionic para-linearity.

Generalized grand Lorentz martingale spaces

In this article, we define a new class of function spaces L_{{p,q}),\theta} which unify and generalize the Iwaniec–Sbordone spaces with q=p , grand Lorentz spaces \theta=1 and classical Lorentz spaces \theta=0 . Based on the new space, we introduce a kind of Hardy-type space, H^s_{{p,q}),\theta} , via the martingale operators and then develop a theory of these martingale Hardy spaces. More specifically, the atomic decompositions of H^s_{{p,q}),\theta} (0&lt;p\leq1 , 1&lt;q&lt;\infty) are established. As applications, the dual theorems for the new framework are provided. We also obtain a new John–Nirenberg-type inequality by the duality. The results extend the very recent work of Jiao et al. (2017) to the case of generalized grand Lorentz spaces. Our results are new, even for the grand Lorentz spaces.

Heat kernels and theory of Hardy spaces associated to Schrödinger operators on stratified groups

Let G be a stratified group and let ΔG be a sub-Laplacian on G. In this paper, we consider the Schrödinger operator L=ΔG+V, where the potential V is a nonnegative polynomial. We first prove the upper bound of the higher order derivatives of the heat kernel of L. We then establish a theory of Hardy spaces HLp(G) associated to L for the full range p∈(0,1]. Particularly, we prove that for any p∈(0,1], the Hardy space HLp(G) introduced in terms of nontangential or radial maximal functions associated to the semigroup e−tL admits a new atomic decomposition. Moreover, we provide the description of the dual spaces of these new Hardy spaces in terms of certain local Campanato spaces related to the potential V. As a byproduct, we obtain the maximal function characterizations for the local Hardy spaces associated to an arbitrary critical function.

Variable Anisotropic Singular Integral Operators

We introduce the class of variable anisotropic singular integral operators associated to a continuous multi-level ellipsoid cover $$\Theta $$ of $$\mathbb {R}^n$$ introduced by Dahmen et al. (Constr Approx 31:149–194, 2010). This is an extension of the classical isotropic singular integral operators on $${\mathbb R}^n$$ of arbitrary smoothness and their anisotropic analogues for general expansive matrices introduced by the first author Bownik (Mem Am Math Soc 164(781):1–122, 2003). We establish the boundedness of variable anisotropic singular integral operators T on the Hardy spaces with pointwise variable anisotropy $$H^p(\Theta )$$ , which were developed by Dekel et al. (J Fourier Anal Appl 17:1066–1107, 2011). In contrast with the general theory of Hardy spaces on spaces of homogenous type, our results work in the full range $$0<p\le 1$$ .

New fractional maximal operators in the theory of martingale Hardy and Lebesgue spaces with variable exponents

We generalize the usual Doob maximal operator as well as the fractional maximal operator and introduce M_{gamma ,s,alpha }, a new fractional maximal operator for martingales. We prove that under the log-Hölder continuity condition of the variable exponents p(cdot ) and q(cdot ), the maximal operator M_{gamma ,s,alpha } is bounded from the variable Lebesgue space L_{q(cdot )} to L_{p(cdot )} and from the variable Hardy space H_{q(cdot )} to L_{p(cdot )}, whenever 0 le alpha <1, 0<q_-le q_+ le 1/alpha , 0<gamma ,s<infty , 1/p(cdot )= 1/q(cdot )- alpha and 1/p_- - 1/p_+ < gamma +s. Moreover, for alpha =0, the operator M_{gamma ,s,0} generates equivalent quasi-norms on the Hardy spaces H_{p(cdot )}.

Mixed-norm Herz spaces and their applications in related Hardy spaces

In this paper, the authors introduce a class of mixed-norm Herz spaces, [Formula: see text], which is a natural generalization of mixed-norm Lebesgue spaces and some special cases of which naturally appear in the study of the summability of Fourier transforms on mixed-norm Lebesgue spaces. The authors also give their dual spaces and obtain the Riesz–Thorin interpolation theorem on [Formula: see text]. Applying these Riesz–Thorin interpolation theorem and using some ideas from the extrapolation theorem, the authors establish both the boundedness of the Hardy–Littlewood maximal operator and the Fefferman–Stein vector-valued maximal inequality on [Formula: see text]. As applications, the authors develop various real-variable theory of Hardy spaces associated with [Formula: see text] by using the existing results of Hardy spaces associated with ball quasi-Banach function spaces. These results strongly depend on the duality of [Formula: see text] and the non-trivial constructions of auxiliary functions in the Riesz–Thorin interpolation theorem.

Generic Versions of a Result in the Theory of Hardy Spaces

We show generic existence of functions f in the Hardy space \(H^p(0<p<1)\) on the open unit disc whose primitive F(f) satisfies the following. For every \(a>\dfrac{p}{1-p}\) and every \(A,B\in {\mathbb {R}}\), \(A<B\) it holds $$\begin{aligned} \sup _{0<r<1}\int ^B_A|F(f)(re^{i\theta })|^ad\theta =+\infty . \end{aligned}$$Results of similar nature are valid when the space \(H^p\) is replaced by localized versions of it, \(0<p<1\), or intersections of such spaces.

Noncommutative rational Clark measures

Abstract We characterize the noncommutative Aleksandrov–Clark measures and the minimal realization formulas of contractive and, in particular, isometric noncommutative rational multipliers of the Fock space. Here, the full Fock space over $\mathbb {C} ^d$ is defined as the Hilbert space of square-summable power series in several noncommuting (NC) formal variables, and we interpret this space as the noncommutative and multivariable analogue of the Hardy space of square-summable Taylor series in the complex unit disk. We further obtain analogues of several classical results in Aleksandrov–Clark measure theory for noncommutative and contractive rational multipliers. Noncommutative measures are defined as positive linear functionals on a certain self-adjoint subspace of the Cuntz–Toeplitz algebra, the unital $C^*$ -algebra generated by the left creation operators on the full Fock space. Our results demonstrate that there is a fundamental relationship between NC Hardy space theory, representation theory of the Cuntz–Toeplitz and Cuntz algebras, and the emerging field of noncommutative rational functions.

Littlewood–Paley–Rubio De Francia Inequality for the Two-Parameter Walsh System

A version of Littlewood–Paley–Rubio de Francia inequality for the two-parameter Walsh system is proved: for any family of disjoint rectangles $$ {I}_k={I}_k^1\times {I}_k^2 $$ in ℤ+ × ℤ+ and a family of functions fk with Walsh spectrum inside Ik the following is true: $$ {\left\Vert \sum_k{f}_k\right\Vert}_{L^p}\le {C}_p{\left\Vert {\left(\sum_k{\left|{f}_k\right|}^2\right)}^{1/2}\right\Vert}_{L^p},\kern2em 1<p\le 2, $$ where Cp does not depend on the choice of rectangles {Ik} or functions {fk}. The arguments are based on the atomic theory of two-parameter martingale Hardy spaces. In the course of the proof, a two-parameter version of Gundy’s theorem on the boundedness of operators taking martingales to measurable functions is formulated, which might be of independent interest.

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.

The Regularity Problem for Uniformly Elliptic Operators in Weighted Spaces

This paper studies the regularity problem for block uniformly elliptic operators in divergence form with complex bounded measurable coefficients. We consider the case where the boundary data belongs to Lebesgue spaces with weights in the Muckenhoupt classes. Our results generalize those of S. Mayboroda (and those of P. Auscher and S. Stahlhut employing the first order method) who considered the unweighted case. To obtain our main results we use the weighted Hardy space theory associated with elliptic operators recently developed by the last two named authors. One of the novel contributions of this paper is the use of an “inhomogeneous” vertical square function which is shown to be controlled by the gradient of the function to which is applied in weighted Lebesgue spaces.

A mean counting function for Dirichlet series and compact composition operators

We introduce a mean counting function for Dirichlet series, which plays the same role in the function theory of Hardy spaces of Dirichlet series as the Nevanlinna counting function does in the classical theory. The existence of the mean counting function is related to Jessen and Tornehave's resolution of the Lagrange mean motion problem. We use the mean counting function to describe all compact composition operators with Dirichlet series symbols on the Hardy–Hilbert space of Dirichlet series, thus resolving a problem which has been open since the bounded composition operators were described by Gordon and Hedenmalm. The main result is that such a composition operator is compact if and only if the mean counting function of its symbol satisfies a decay condition at the boundary of a half-plane.

Hardy Spaces for Boundary Value Problems of Elliptic Systems with Block Structure

We present recent results on elliptic boundary value problems where the theory of Hardy spaces associated with operators plays a key role.

Surface waves in the magnetopause: Regularity and time evolution

The equation describing surface waves in a one-dimensional magnetohydrodynamic discontinuity involves the Hilbert transform and is therefore nonlocal. It is found that by taking the analytic function whose real part is the first order perturbation one obtains a local equation. This equation yields several integral equalities which may be studied with the help of certain classical results of the theory of Hardy spaces. In particular this method provides information on the relation between the smoothness of the solution and its possible blow up in time.

Lebesgue Decomposition of Non-Commutative Measures

AbstractWe extend the Lebesgue decomposition of positive measures with respect to Lebesgue measure on the complex unit circle to the non-commutative (NC) multi-variable setting of (positive) NC measures. These are positive linear functionals on a certain self-adjoint subspace of the Cuntz–Toeplitz $C^{\ast }-$algebra, the $C^{\ast }-$algebra of the left creation operators on the full Fock space. This theory is fundamentally connected to the representation theory of the Cuntz and Cuntz–Toeplitz $C^{\ast }-$algebras; any *−representation of the Cuntz–Toeplitz $C^{\ast }-$algebra is obtained (up to unitary equivalence), by applying a Gelfand–Naimark–Segal construction to a positive NC measure. Our approach combines the theory of Lebesgue decomposition of sesquilinear forms in Hilbert space, Lebesgue decomposition of row isometries, free semigroup algebra theory, NC reproducing kernel Hilbert space theory, and NC Hardy space theory.

BMO Spaces on Weighted Homogeneous Trees

We consider an infinite homogeneous tree $${\mathcal {V}}$$ endowed with the usual metric d defined on graphs and a weighted measure $$\mu $$ . The metric measure space $$({\mathcal {V}},d,\mu )$$ is nondoubling and of exponential growth, hence the classical theory of Hardy and BMO spaces does not apply in this setting. We introduce a space $$BMO(\mu )$$ on $$({\mathcal {V}},d,\mu )$$ and investigate some of its properties. We prove in particular that $$BMO(\mu )$$ can be identified with the dual of a Hardy space $$H^1(\mu )$$ introduced in a previous work and we investigate the sharp maximal function related with $$BMO(\mu )$$ .

Calculating the spectral factorization and outer functions by sampling-based approximations—Fundamental limitations

This paper considers the problem of approximating the spectral factor of continuous spectral densities with finite Dirichlet energy based on finitely many samples of these spectral densities. Although there exists a closed form expression for the spectral factor, this formula shows a very complicated behavior because of the non-linear dependency of the spectral factor from spectral density and because of a singular integral in this expression. Therefore approximation methods are usually applied to calculate the spectral factor.It is shown that there exists no sampling-based method which depends continuously on the samples and which is able to approximate the spectral factor for all densities in this set. Instead, to any sampling-based approximation method there exists a large set of spectral densities so that the approximation method does not converge to the spectral factor for every spectral density in this set as the number of available sampling points is increased. The paper will also show that the same results hold for sampling-based algorithms for the calculation of the outer function in the theory of Hardy spaces.