Hardy Space Research Articles

Compact linear combinations of composition operators on Hardy spaces

Compact linear combinations of composition operators on Hardy spaces

On the essential norms of Toeplitz operators on abstract Hardy spaces built upon Banach function spaces

Let X be a Banach function space over the unit circle such that the Riesz projection P is bounded on X and let H[X] be the abstract Hardy space built upon X. We show that the essential norm of the Toeplitz operator T(a):H[X]→H[X]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T(a):H[X]\\rightarrow H[X]$$\\end{document} coincides with ‖a‖L∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Vert a\\Vert _{L^\\infty }$$\\end{document} for every a∈C+H∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a\\in C+H^\\infty $$\\end{document} if and only if the essential norm of the backward shift operator T(e-1):H[X]→H[X]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T(\ extbf{e}_{-1}):H[X]\\rightarrow H[X]$$\\end{document} is equal to one, where e-1(z)=z-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{e}_{-1}(z)=z^{-1}$$\\end{document}. This result extends an observation by Böttcher, Krupnik, and Silbermann for the case of classical Hardy spaces.

Mikhlin-Type Hp Multiplier Theorem on the Heisenberg Group

The classical Fourier multiplier theorem by Mikhlin in Hardy spaces is extended to the Heisenberg group. The proof relies on the theories of atom and molecule functions and the property of special Hermite functions. The main result is a Mikhlin-type Hp multiplier theorem on the Heisenberg group. If an operator-valued function M(λ) satisfies certain conditions, the right-multiplier operator TM is bounded on the Hardy space Hp(Hn), which is defined in terms of maximal functions, and elements can be decomposed into atoms or molecules. The paper also discusses the relationship with other results and open problems.

Hardy spaces and dilations on homogeneous groups

On a homogeneous group, we characterize the one-parameter groups of dilations whose associated Hardy spaces in the sense of Folland and Stein are the same.

Generalized Cesàro operators in the disc algebra and in Hardy spaces

Generalized Cesàro operators Ct\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_t$$\\end{document}, for t∈[0,1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t\\in [0,1)$$\\end{document}, are investigated when they act on the disc algebra A(D)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A({\\mathbb {D}})$$\\end{document} and on the Hardy spaces Hp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^p$$\\end{document}, for 1≤p≤∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1\\le p \\le \\infty $$\\end{document}. We study the continuity, compactness, spectrum and point spectrum of Ct\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_t$$\\end{document} as well as their linear dynamics and mean ergodicity on these spaces.

Scaled Global Operators and Fueter Variables on Non-zero Scaled Hypercomplex Numbers

In this paper we describe the rise of global operators in the scaled quaternionic case, an important extension from the quaternionic case to the family of scaled hypercomplex numbers Ht,t∈R∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {H}_t,\\, t\\in \\mathbb {R}^*$$\\end{document}, of which the H-1=H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {H}_{-1}=\\mathbb {H}$$\\end{document} is the space of quaternions and H1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {H}_{1}$$\\end{document} is the space of split quaternions. We also describe the scaled Fueter-type variables associated to these operators, developing a coherent theory in this field. We use these types of variables to build different types of function spaces on Ht\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {H}_t$$\\end{document}. Counterparts of the Hardy space and of the Arveson space are also introduced and studied in the present setting. The two different adjoints in the scaled hypercomplex numbers lead to two parallel cases in each instance. Finally we introduce and study the notion of rational function.

Complex Structure That Admits Complete Nevanlinna–Pick Spaces of Hardy Type

Abstract In this paper, we will characterize those sets, over which every irreducible complete Nevanlinna–Pick space enjoys that its multiplier and supremum norms coincide. Moreover, we will prove that, if there exists an irreducible complete Nevanlinna–Pick space of holomorphic functions on a reduced complex space $X$ whose multiplier algebra is isometrically equal to the algebra of bounded holomorphic functions (we will say that such a space is of Hardy type in this paper), then $X$ must be biholomorphic to the unit disk minus a zero analytic capacity set. This means that the Hardy space is characterized as a unique irreducible complete Nevanlinna–Pick space of Hardy type.

Invariant subspaces of analytic perturbations

Analytic perturbations are understood here as shifts of the form M z + F M_z + F , where M z M_z is the unilateral shift and F F is a finite rank operator on the Hardy space over the open unit disk. Here the term “a shift” refers to the multiplication operator M z M_z on some analytic reproducing kernel Hilbert space. In this paper, first, a natural class of finite rank operators is isolated for which the corresponding perturbations are analytic, and then a complete classification of invariant subspaces of those analytic perturbations is presented. Some instructive examples and several distinctive properties (like cyclicity, essential normality, hyponormality, etc.) of analytic perturbations are also described.

Noncommutative Nest Hardy Spaces and Martingales

Abstract Let $(\mathcal{M},\tau )$ be a finite von Neumann algebra equipped with a normalized faithful trace and let $\mathbb{A}$ be a nest of order type $\mathbb{N}$. The nest algebra is defined by $H_{\infty }^{r}(\mathbb{A})=\{x\in \mathcal{M}:ex=exe,e\in \mathbb{A}\}$, and the related noncommutative Hardy space $H_{p}^{r}(\mathbb{A})$ is the completion of $H_{\infty }^{r}(\mathbb{A})$ in $L_{p}(\mathcal{M})$. In the present paper, we make a connection between $H_{p}^{r}(\mathbb{A})$ and certain noncommutative martingale Hardy space via showing that triangular truncation is a concrete martingale transform. We include several related results for noncommutative nest Hardy spaces in the spirit of recent development of noncommutative martingale theory.

Optimal domain of generalized Volterra operators

For g in BMOA, we consider the generalized Volterra operator Tg acting on Hardy spaces Hp. This article aims to study the largest space of analytic functions, which is mapped by Tg into the Hardy space Hp. We call this space the optimal domain of Tg and we describe its structural properties. Motivation for this comes from the work of G. Curbera and W. Ricker [7] who studied the optimal domain of the classical Cesáro operator.

Relative weak compactness in infinite-dimensional Fefferman-Meyer duality

Let E be a Banach space such that E′ has the Radon-Nikodým property. The aim of this work is to connect relative weak compactness in the E-valued martingale Hardy space H1(μ,E) to a convex compactness criterion in a weaker topology, such as the topology of uniform convergence on compacts in measure. These results represent a dynamic version of the deep result of Diestel, Ruess, and Schachermayer on relative weak compactness in L1(μ,E). In the reflexive case, we obtain a Kadec-Pełczyński dichotomy for H1(μ,E)-bounded sequences, which decomposes a subsequence into a relatively weakly compact part, a pointwise weakly convexly convergent part, and a null part converging to zero uniformly on compacts in measure. As a corollary, we investigate a parameterized version of the vector-valued Komlós theorem without the assumption of H1(μ,E)-boundedness.

Sharp analytic version of Fefferman's inequality

Let T be a unit circle. Assume further that f is an element of the Hardy space H1(T) and g belongs to the analytic BMO space on T. The paper contains the identification of the optimal universal constant C in the estimate|12π∫Tf(ζ)‾g(ζ)dζ|≤C‖f‖H1(T)‖g‖BMO(T). Actually, the inequality is studied in the stronger form, involving the Littlewood-Paley function on the left and the sharp maximal function of g on the right. The proof rests on the construction of an appropriate plurisuperharmonic function on a parabolic domain and the application of probabilistic techniques.

Reverse Carleson Measures for Hardy Spaces in the Unit Ball

Reverse Carleson Measures for Hardy Spaces in the Unit Ball

Global Hessian estimate for second-order elliptic equation in Hardy spaces

Global Hessian estimate for second-order elliptic equation in Hardy spaces

A note on limiting Calderon–Zygmund theory for transformed n$n$‐Laplace systems in divergence form

AbstractWe consider rotated ‐Laplace systems on the unit ball of the form where , , and for some . We prove that with estimates. As a corollary, we obtain that solutions to , where is the Hardy space, have a higher integrability, namely, .

Linear equations with infinitely many derivatives and explicit solutions to zeta nonlocal equations

We summarize our theory on existence, uniqueness and regularity of solutions for linear equations in infinitely many derivatives of the formf(∂t)ϕ=J(t),t≥0, where f is an analytic function such as the (analytic continuation of the) Riemann zeta function. We explain how to analyse initial value problems for these equations, and we prove rigorously that the functionϕ(t)=∑n≥1μ(n)nhJ(t−ln⁡(n)), in which μ is the Möbius function and J satisfies some technical conditions to be specified in Section 4, is the solution to the zeta nonlocal equationζ(∂t+h)ϕ=J(t),t≥0, in which ζ is the Riemann zeta function and h>1. We also present explicit examples of solutions to initial value problems for this equation. Our constructions can be interpreted as highlighting the importance of the cosmological daemon functions considered by Aref'eva and Volovich (2011) [1]. Our main technical tool is the Laplace transform as a unitary operator between the Lebesgue space L2 and the Hardy space H2.

Generalized Hilbert operators arising from Hausdorff matrices

For a finite, positive Borel measure μ \mu on ( 0 , 1 ) (0,1) we consider an infinite matrix Γ μ \Gamma _\mu , related to the classical Hausdorff matrix defined by the same measure μ \mu , in the same algebraic way that the Hilbert matrix is related to the Cesáro matrix. When μ \mu is the Lebesgue measure, Γ μ \Gamma _\mu reduces to the classical Hilbert matrix. We prove that the matrices Γ μ \Gamma _\mu are not Hankel, unless μ \mu is a constant multiple of the Lebesgue measure, we give necessary and sufficient conditions for their boundedness on the scale of Hardy spaces H p , 1 ≤ p > ∞ H^p, \, 1 \leq p > \infty , and we study their compactness and complete continuity properties. In the case 2 ≤ p > ∞ 2\leq p>\infty , we are able to compute the exact value of the norm of the operator.

Local Hardy Spaces and the Tb Theorem

Local Hardy Spaces and the Tb Theorem

Random Carleson sequences for the Hardy space on the polydisc and the unit ball

We study the Kolmogorov 0−1 law for a random sequence with prescribed radii so that it generates a Carleson measure almost surely, both for the Hardy space on the polydisc and the Hardy space on the unit ball, thus providing improved versions of previous results of the first two authors and of a separate result of Massaneda. In the polydisc, the geometry of such sequences is not well understood, so we proceed by studying the random Gramians generated by random sequences, using tools from the theory of random matrices. Another result we prove, and that is of its own relevance, is the 0−1 law for a random sequence to be partitioned into M separated sequences with respect to the pseudo-hyperbolic distance, which is used also to describe the random sequences that are interpolating for the Bloch space on the unit disc almost surely.

Weighted Hardy factorizations in terms of Multilinear Calderón-Zygmund and fractional integral operators

The foundational paper of Coifman-Rochberg-Weiss provided a constructive proof of the weak factorizations of the classical Hardy space in terms of Riesz transforms. In this paper, we establish the factorization theorems for the weighted Hardy spaces. The results are also extended to the multilinear setting. Furthermore, we show that the weighted BMO space and weighted Lipschitz spaces are necessary for the weighted boundedness of commutators of the multilinear Calderón-Zygmund and fractional integral operators, respectively.