scholarly journals The atomic Hardy space for a general Bessel operator

2021 ◽  
Author(s):  
Edyta Kania-Strojec
Keyword(s):  
Hardy Space ◽  
Hardy Spaces ◽  
Maximal Operator ◽  
Atomic Decomposition ◽  
Bessel Operator ◽  
General Context ◽  
Semigroup Of Operators ◽  
Local Type ◽  
Atomic Hardy Space ◽  
Type Parameter

AbstractWe study Hardy spaces associated with a general multidimensional Bessel operator $$\mathbb {B}_\nu $$ B ν . This operator depends on a multiparameter of type $$\nu $$ ν that is usually restricted to a product of half-lines. Here we deal with the Bessel operator in the general context, with no restrictions on the type parameter. We define the Hardy space $$H^1$$ H 1 for $$\mathbb {B}_\nu $$ B ν in terms of the maximal operator of the semigroup of operators $$\exp (-t\mathbb {B}_\nu )$$ exp ( - t B ν ) . Then we prove that, in general, $$H^1$$ H 1 admits an atomic decomposition of local type.

BOUNDEDNESS OF SEVERAL INTEGRAL OPERATORS WITH BOUNDED VARIABLE KERNELS ON HARDY AND WEAK HARDY SPACES

2013 ◽  
Vol 24 (12) ◽  
pp. 1350095 ◽  
Author(s):  
HUA WANG
Keyword(s):  
Hardy Space ◽  
Hardy Spaces ◽  
Atomic Decomposition ◽  
Integral Operators ◽  
Fractional Integrals ◽  
Variable Kernel ◽  
Marcinkiewicz Integrals ◽  
Decomposition Theory ◽  
Logarithmic Type ◽  
Lipschitz Conditions

In this paper, by using the atomic decomposition theory of Hardy space H1(ℝn) and weak Hardy space WH1(ℝn), we give the boundedness properties of some operators with variable kernels such as singular integral operators, fractional integrals and parametric Marcinkiewicz integrals on these spaces, under certain logarithmic type Lipschitz conditions assumed on the variable kernel Ω(x, z).

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 Characterizations of variable martingale Hardy spaces via maximal functions

2021 ◽  
Vol 24 (2) ◽  
pp. 393-420
Author(s):  
Ferenc Weisz
Keyword(s):  
Hardy Space ◽  
Hardy Spaces ◽  
Lorentz Space ◽  
Maximal Operator ◽  
Variable Exponent ◽  
Continuity Condition ◽  
Maximal Functions ◽  
Maximal Operators ◽  
Special Cases ◽  
New Type

Abstract We introduce a new type of dyadic maximal operators and prove that under the log-Hölder continuity condition of the variable exponent p(⋅), it is bounded on L p(⋅) if 1 < p − ≤ p + ≤ ∞. Moreover, the space generated by the L p(⋅)-norm (resp. the L p(⋅), q -norm) of the maximal operator is equivalent to the Hardy space H p(⋅) (resp. to the Hardy-Lorentz space H p(⋅), q ). As special cases, our maximal operator contains the usual dyadic maximal operator and four other maximal operators investigated in the literature.

scholarly journals Discrete Hardy Spaces Related to Powers of the Poisson Kernel

2013 ◽  
Vol 112 (2) ◽  
pp. 240
Author(s):  
Jonatan Vasilis
Keyword(s):  
Hardy Space ◽  
Orlicz Space ◽  
Hardy Spaces ◽  
Poisson Kernel ◽  
Rooted Trees ◽  
Poisson Kernels ◽  
Atomic Hardy Space ◽  
Positive Functions ◽  
Increasing Functions ◽  
Discrete Hardy Spaces

Discrete Hardy spaces $H^{1}_{\alpha}(\partial{T})$, related to powers $\alpha \ge 1/2$ of the Poisson kernels on boundaries $\partial{T}$ of regular rooted trees, are studied. The spaces for $\alpha > 1/2$ coincide with the ordinary atomic Hardy space on $\partial{T}$, which in turn is strictly smaller than $H^{1}_{1/2}(\partial{T})$. The Orlicz space $L\log\log L(\partial{T})$ characterizes the positive and increasing functions in $H^{1}_{1/2}(\partial{T})$, but there is no such characterization for general positive functions.

scholarly journals AN ATOMIC DECOMPOSITION FOR HARDY SPACES ASSOCIATED TO SCHRÖDINGER OPERATORS

2011 ◽  
Vol 91 (1) ◽  
pp. 125-144 ◽  
Author(s):  
LIANG SONG ◽  
CHAOQIANG TAN ◽  
LIXIN YAN
Keyword(s):  
Hardy Space ◽  
Maximal Function ◽  
Hardy Spaces ◽  
Atomic Decomposition ◽  
Schrödinger Operators ◽  
Nonnegative Function ◽  
Schrodinger Operators ◽  
Integrable Functions ◽  
Function Characterization ◽  
Product Domains

AbstractLetL=−Δ+Vbe a Schrödinger operator on ℝnwhereVis a nonnegative function in the spaceL1loc(ℝn) of locally integrable functions on ℝn. In this paper we provide an atomic decomposition for the Hardy spaceH1L(ℝn) associated toLin terms of the maximal function characterization. We then adapt our argument to give an atomic decomposition for the Hardy spaceH1L(ℝn×ℝn) on product domains.

scholarly journals ON VECTOR-VALUED TENT SPACES AND HARDY SPACES ASSOCIATED WITH NON-NEGATIVE SELF-ADJOINT OPERATORS

2015 ◽  
Vol 58 (3) ◽  
pp. 689-716 ◽  
Author(s):  
MIKKO KEMPPAINEN
Keyword(s):  
Hardy Space ◽  
Hardy Spaces ◽  
Atomic Decomposition ◽  
Integral Operators ◽  
Complex Interpolation ◽  
Tent Spaces ◽  
Tent Space ◽  
Holomorphic Functional Calculus ◽  
Adjoint Operators ◽  
Vector Valued

AbstractIn this paper, we study Hardy spaces associated with non-negative self-adjoint operators and develop their vector-valued theory. The complex interpolation scales of vector-valued tent spaces and Hardy spaces are extended to the endpoint p=1. The holomorphic functional calculus of L is also shown to be bounded on the associated Hardy space H1L(X). These results, along with the atomic decomposition for the aforementioned space, rely on boundedness of certain integral operators on the tent space T1(X).

scholarly journals Schrödinger Operators with Reverse Hölder Class Potentials in the Dunkl Setting and Their Hardy Spaces

2021 ◽  
Vol 27 (3) ◽  
Author(s):  
Agnieszka Hejna
Keyword(s):  
Hardy Space ◽  
Root System ◽  
Maximal Function ◽  
Hardy Spaces ◽  
Atomic Decomposition ◽  
Multiplicity Function ◽  
Reverse Hölder Class ◽  
Hölder Class ◽  
Holder Class ◽  
Reverse Hölder

AbstractFor a normalized root system R in $${\mathbb {R}}^N$$ R N and a multiplicity function $$k\ge 0$$ k ≥ 0 let $${\mathbf {N}}=N+\sum _{\alpha \in R} k(\alpha )$$ N = N + ∑ α ∈ R k ( α ) . We denote by $$dw({\mathbf {x}})=\varPi _{\alpha \in R}|\langle {\mathbf {x}},\alpha \rangle |^{k(\alpha )}\,d{\mathbf {x}}$$ d w ( x ) = Π α ∈ R | ⟨ x , α ⟩ | k ( α ) d x the associated measure in $${\mathbb {R}}^N$$ R N . Let $$L=-\varDelta +V$$ L = - Δ + V , $$V\ge 0$$ V ≥ 0 , be the Dunkl–Schrödinger operator on $${\mathbb {R}}^N$$ R N . Assume that there exists $$q >\max (1,\frac{{\mathbf {N}}}{2})$$ q > max ( 1 , N 2 ) such that V belongs to the reverse Hölder class $$\mathrm{{RH}}^{q}(dw)$$ RH q ( d w ) . We prove the Fefferman–Phong inequality for L. As an application, we conclude that the Hardy space $$H^1_{L}$$ H L 1 , which is originally defined by means of the maximal function associated with the semigroup $$e^{-tL}$$ e - t L , admits an atomic decomposition with local atoms in the sense of Goldberg, where their localizations are adapted to V.

The maximal operator of the Marcinkiewicz-Fejér means of the 2-dimensional Vilenkin-Fourier series

2008 ◽  
Vol 45 (3) ◽  
pp. 321-331
Author(s):  
István Blahota ◽  
Ushangi Goginava
Keyword(s):  
Fourier Series ◽  
Hardy Space ◽  
Maximal Operator ◽  
Fejér Means

In this paper we prove that the maximal operator of the Marcinkiewicz-Fejér means of the 2-dimensional Vilenkin-Fourier series is not bounded from the Hardy space H2/3 ( G2 ) to the space L2/3 ( G2 ).

scholarly journals Variable Anisotropic Hardy Spaces with Variable Exponents

2021 ◽  
Vol 9 (1) ◽  
pp. 65-89
Author(s):  
Zhenzhen Yang ◽  
Yajuan Yang ◽  
Jiawei Sun ◽  
Baode Li
Keyword(s):  
Hardy Spaces ◽  
Variable Exponent ◽  
Atomic Decomposition ◽  
Integral Operators ◽  
Moment Condition ◽  
Variable Exponents ◽  
Hölder Continuous ◽  
Multi Level ◽  
The Moment ◽  
Variable Anisotropy

Abstract Let p(·) : ℝ n → (0, ∞] be a variable exponent function satisfying the globally log-Hölder continuous and let Θ be a continuous multi-level ellipsoid cover of ℝ n introduced by Dekel et al. [12]. In this article, we introduce highly geometric Hardy spaces Hp (·)(Θ) via the radial grand maximal function and then obtain its atomic decomposition, which generalizes that of Hardy spaces Hp (Θ) on ℝ n with pointwise variable anisotropy of Dekel et al. [16] and variable anisotropic Hardy spaces of Liu et al. [24]. As an application, we establish the boundedness of variable anisotropic singular integral operators from Hp (·)(Θ) to Lp (·)(ℝ n ) in general and from Hp (·)(Θ) to itself under the moment condition, which generalizes the previous work of Bownik et al. [6] on Hp (Θ).

New variable martingale Hardy spaces

2021 ◽  
pp. 1-29
Author(s):  
Yong Jiao ◽  
Dan Zeng ◽  
Dejian Zhou
Keyword(s):  
Brownian Motion ◽  
Hardy Space ◽  
Hardy Spaces ◽  
Stochastic Integral ◽  
Lebesgue Spaces ◽  
Type Space ◽  
Variable Lebesgue Spaces ◽  
Martingale Inequalities

We investigate various variable martingale Hardy spaces corresponding to variable Lebesgue spaces $\mathcal {L}_{p(\cdot )}$ defined by rearrangement functions. In particular, we show that the dual of martingale variable Hardy space $\mathcal {H}_{p(\cdot )}^{s}$ with $0<p_{-}\leq p_{+}\leq 1$ can be described as a BMO-type space and establish martingale inequalities among these martingale Hardy spaces. Furthermore, we give an application of martingale inequalities in stochastic integral with Brownian motion.

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