scholarly journals Bilinear forms on vector Hardy spaces

1997 ◽  
Vol 39 (3) ◽  
pp. 371-378
Author(s):  
Gordon Blower
Keyword(s):  
Hilbert Space ◽  
Hardy Space ◽  
Bilinear Form ◽  
Hardy Spaces ◽  
Carleson Measure ◽  
Reproducing Kernel ◽  
Bilinear Forms ◽  
Sufficient Condition

AbstractLet φ: ℋ → be a bilinear form on vector Hardy space. Introduce the symbol φ of Φ by (φ (Z1, Z2), a ⊗ b) = Φ (K21 ⊗ a, K22 ⊗ b ), where Kw is the reproducing kernel for w ∈ D. We show that Φ extends to a bounded bilinear form on provided that the gradient defines a Carleson measure in the bidisc D2. We obtain a sufficient condition for Φ to extend to a Hilbert space. For vectorial bilinear Hankel forms we obtain an analogue of Nehari's Theorem.

Modulus of continuity and convergence of subsequences of Vilenkin–Fejér means in martingale Hardy spaces

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Giorgi Tutberidze
Keyword(s):  
Hardy Space ◽  
Modulus Of Continuity ◽  
Hardy Spaces ◽  
Lebesgue Space ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Fejér Means ◽  
Necessary And Sufficient

Abstract In this paper, we find a necessary and sufficient condition for the modulus of continuity for which subsequences of Fejér means with respect to Vilenkin systems are bounded from the Hardy space H p {H_{p}} to the Lebesgue space L p {L_{p}} for all 0 < p < 1 2 {0<p<\frac{1}{2}} .

Non-degenerate Invariant Bilinear Forms on Lie Color Algebras

2010 ◽  
Vol 17 (03) ◽  
pp. 365-374 ◽  
Author(s):  
Song Wang ◽  
Linsheng Zhu
Keyword(s):  
Lie Algebras ◽  
Bilinear Form ◽  
Bilinear Forms ◽  
Sufficient Condition ◽  
Invariant Bilinear Form ◽  
Lie Color Algebras ◽  
Double Extension

In this paper, we study Lie color algebras 𝔤 with a non-degenerate color-symmetric, 𝔤-invariant bilinear form B, such a (𝔤,B) is called a quadratic Lie color algebra. Our first result generalizes the notion of double extensions to quadratic Lie color algebras. This notion was introduced by Medina and Revoy to study quadratic Lie algebras. In the second theorem, we give a sufficient condition for a quadratic Lie color algebra to be a quadratic Lie color algebra by double extension. At last, we generalize the notion of T*-extensions to Lie color algebras.

On Dirichlet Spaces With a Class of Superharmonic Weights

2018 ◽  
Vol 70 (4) ◽  
pp. 721-741 ◽  
Author(s):  
Guanlong Bao ◽  
Nihat Gokhan Göğüş ◽  
Stamatis Pouliasis
Keyword(s):  
Hardy Spaces ◽  
Carleson Measure ◽  
Reproducing Kernel ◽  
Boundary Behavior ◽  
Composition Operators ◽  
Open Unit ◽  
Dirichlet Spaces ◽  
Weighted Hardy Spaces ◽  
Borel Measures ◽  
Infinite Sum

AbstractIn this paper, we investigate Dirichlet spaces with superharmonic weights induced by positive Borel measures μ on the open unit disk. We establish the Alexander-Taylor-Ullman inequality for spaces and we characterize the cases where equality occurs. We define a class of weighted Hardy spaces via the balayage of the measure μ. We show that is equal to if and only if μ is a Carleson measure for . As an application, we obtain the reproducing kernel of when μ is an infinite sum of point-mass measures. We consider the boundary behavior and innerouter factorization of functions in . We also characterize the boundedness and compactness of composition operators on .

scholarly journals Musielak–Orlicz Hardy Spaces Associated with Divergence Form Elliptic Operators Without Weight Assumptions

2014 ◽  
Vol 216 ◽  
pp. 71-110 ◽  
Author(s):  
Tri Dung Tran
Keyword(s):  
Hardy Space ◽  
Hardy Spaces ◽  
Carleson Measure ◽  
Dual Space ◽  
Orlicz Function ◽  
Adjoint Operator ◽  
Riesz Transform ◽  
Divergence Form ◽  
Measurable Coefficients ◽  
Nirenberg Inequality

AbstractLet L be a divergence form elliptic operator with complex bounded measurable coefficients, let ω be a positive Musielak-Orlicz function on (0, ∞) of uniformly strictly critical lower-type pω ∈ (0, 1], and let ρ(x,t) = t−1/ω−1 (x,t−1) for x ∈ ℝn, t ∊ (0, ∞). In this paper, we study the Musielak-Orlicz Hardy space Hω,L(ℝn) and its dual space BMOρ,L* (ℝ n), where L* denotes the adjoint operator of L in L2 (ℝ n). The ρ-Carleson measure characterization and the John-Nirenberg inequality for the space BMOρ,L (ℝn) are also established. Finally, as applications, we show that the Riesz transform ∇L−1/2 and the Littlewood–Paley g-function gL map Hω,L(ℝn) continuously into L(ω).

scholarly journals Musielak–Orlicz Hardy Spaces Associated with Divergence Form Elliptic Operators Without Weight Assumptions

2014 ◽  
Vol 216 ◽  
pp. 71-110
Author(s):  
Tri Dung Tran
Keyword(s):  
Hardy Space ◽  
Hardy Spaces ◽  
Carleson Measure ◽  
Dual Space ◽  
Orlicz Function ◽  
Adjoint Operator ◽  
Riesz Transform ◽  
Divergence Form ◽  
Measurable Coefficients ◽  
Nirenberg Inequality

AbstractLetLbe a divergence form elliptic operator with complex bounded measurable coefficients, letωbe a positive Musielak-Orlicz function on (0, ∞) of uniformly strictly critical lower-typepω∈ (0, 1], and letρ(x,t) = t−1/ω−1(x,t−1) forx∈ ℝn, t∊ (0, ∞). In this paper, we study the Musielak-Orlicz Hardy spaceHω,L(ℝn) and its dual space BMOρ,L* (ℝn), whereL*denotes the adjoint operator ofLinL2(ℝn). Theρ-Carleson measure characterization and the John-Nirenberg inequality for the space BMOρ,L(ℝn) are also established. Finally, as applications, we show that the Riesz transform ∇L−1/2and the Littlewood–Paleyg-functiongLmapHω,L(ℝn) continuously intoL(ω).

Walsh-Marcinkiewicz means and Hardy spaces

2014 ◽  
Vol 12 (8) ◽  
Author(s):  
Károly Nagy ◽  
George Tephnadze
Keyword(s):  
Hardy Space ◽  
Strong Convergence ◽  
Hardy Spaces ◽  
Convergence Theorem ◽  
Maximal Operator ◽  
Strong Convergence Theorem ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Marcinkiewicz Means ◽  
Necessary And Sufficient

AbstractThe main aim of this paper is to investigate the Walsh-Marcinkiewicz means on the Hardy space H p, when 0 < p < 2/3. We define a weighted maximal operator of Walsh-Marcinkiewicz means and establish some of its properties. With its aid we provide a necessary and sufficient condition for convergence of the Walsh-Marcinkiewicz means in terms of modulus of continuity on the Hardy space H p, and prove a strong convergence theorem for the Walsh-Marcinkiewicz means.

scholarly journals LUSIN AREA FUNCTION AND MOLECULAR CHARACTERIZATIONS OF MUSIELAK–ORLICZ HARDY SPACES AND THEIR APPLICATIONS

2013 ◽  
Vol 15 (06) ◽  
pp. 1350029 ◽  
Author(s):  
SHAOXIONG HOU ◽  
DACHUN YANG ◽  
SIBEI YANG
Keyword(s):  
Hardy Space ◽  
Maximal Function ◽  
Hardy Spaces ◽  
Carleson Measure ◽  
Dual Space ◽  
Growth Function ◽  
Area Function ◽  
Type Space ◽  
Lusin Area Function

Let φ : ℝn× [0,∞) → [0,∞) be a growth function such that φ(x, ⋅) is nondecreasing, φ(x, 0) = 0, φ(x, t) > 0 when t > 0, limt→∞φ(x, t) = ∞, and φ(⋅, t) is a Muckenhoupt A∞(ℝn) weight uniformly in t. In this paper, the authors establish the Lusin area function and the molecular characterizations of the Musielak–Orlicz Hardy space Hφ(ℝn) introduced by Luong Dang Ky via the grand maximal function. As an application, the authors obtain the φ-Carleson measure characterization of the Musielak–Orlicz BMO-type space BMOφ(ℝn), which was proved to be the dual space of Hφ(ℝn) by Luong Dang Ky.

scholarly journals Power series of several variables with condition of logarithmical convexity

2021 ◽  
Vol 8 (1) ◽  
pp. 49-62
Author(s):  
Alexandr V. Zheleznyak ◽  
Keyword(s):  
Hilbert Space ◽  
Hardy Space ◽  
Power Series ◽  
Unit Disc ◽  
Reproducing Kernel ◽  
Space Of Analytic Functions ◽  
Several Variables ◽  
Positive Coefficients ◽  
Classical Hardy Space ◽  
Reproducing Kernel Function

We obtain a new version of Hardy theorem about power series of several variables reciprocal to the power series with positive coefficients. We prove that if the sequence {as} = as1,s2,...,sn, ||s|| ≥ K satisfies condition of logarithmically convexity and the first coefficient a0 is sufficiently large then reciprocal power series has only negative coefficients {bs} = bs1,s2,...,sn, except b0,0,...,0 for any K. The classical Hardy theorem corresponds to the case K = 0, n = 1. Such results are useful in Nevanlinna - Pick theory. For example, if function k(x, y) can be represented as power series Σn≥0 an(x-y)n, an > 0, and reciprocal function 1/k(x,y) can be represented as power series Σn≥0 bn(x-y)n such that bn < 0, n > 0, then k(x, y) is a reproducing kernel function for some Hilbert space of analytic functions in the unit disc D with Nevanlinna-Pick property. The reproducing kernel 1/1-x-y of the classical Hardy space H2(D) is a prime example for our theorems.

Non-degenerate Invariant Bilinear Forms on Leibniz Algebras

2015 ◽  
Vol 22 (04) ◽  
pp. 711-720
Author(s):  
Song Wang ◽  
Linsheng Zhu
Keyword(s):  
Lie Algebras ◽  
Bilinear Form ◽  
Leibniz Algebra ◽  
Bilinear Forms ◽  
Sufficient Condition ◽  
Invariant Bilinear Form ◽  
Leibniz Algebras ◽  
Double Extension

In this paper, we study Leibniz algebras [Formula: see text] with a non-degenerate Leibniz-symmetric [Formula: see text]-invariant bilinear form B, such a pair [Formula: see text] is called a quadratic Leibniz algebra. Our first result generalizes the notion of double extensions to quadratic Leibniz algebras. This notion was introduced by Medina and Revoy to study quadratic Lie algebras. In the second theorem, we give a sufficient condition for a quadratic Leibniz algebra to be a quadratic Leibniz algebra by double extension.

scholarly journals Toeplitz Operators on the Symmetrized Bidisc

2020 ◽  
Author(s):  
Tirthankar Bhattacharyya ◽  
B Krishna Das ◽  
Haripada Sau
Keyword(s):  
Hilbert Space ◽  
Hardy Space ◽  
Unit Disc ◽  
Toeplitz Operators ◽  
Reproducing Kernel ◽  
Multiplication Operators ◽  
Algebraic Characterization ◽  
Hilbert Modules ◽  
Symmetrized Bidisc

Abstract The symmetrized bidisc has been a rich field of holomorphic function theory and operator theory. A certain well-known reproducing kernel Hilbert space of holomorphic functions on the symmetrized bidisc resembles the Hardy space of the unit disc in several aspects. This space is known as the Hardy space of the symmetrized bidisc. We introduce the study of those operators on the Hardy space of the symmetrized bidisc that are analogous to Toeplitz operators on the Hardy space of the unit disc. More explicitly, we first study multiplication operators on a bigger space (an $L^2$-space) and then study compressions of these multiplication operators to the Hardy space of the symmetrized bidisc and prove the following major results. (1) Theorem I analyzes the Hardy space of the symmetrized bidisc, not just as a Hilbert space, but as a Hilbert module over the polynomial ring and finds three isomorphic copies of it as $\mathbb D^2$-contractive Hilbert modules. (2) Theorem II provides an algebraic, Brown and Halmos-type characterization of Toeplitz operators. (3) Theorem III gives several characterizations of an analytic Toeplitz operator. (4) Theorem IV characterizes asymptotic Toeplitz operators. (5) Theorem V is a commutant lifting theorem. (6) Theorem VI yields an algebraic characterization of dual Toeplitz operators. Every section from Section 2 to Section 7 contains a theorem each, the main result of that section.

Sign in / Sign up

Export Citation Format

Share Document