LITTLEWOOD–PALEY CHARACTERIZATION OF WEIGHTED HARDY SPACES ASSOCIATED WITH OPERATORS

2016 ◽  
Vol 103 (2) ◽  
pp. 250-267 ◽  
Author(s):  
GUORONG HU
Keyword(s):  
Hardy Space ◽  
Hardy Spaces ◽  
Adjoint Operator ◽  
Weighted Hardy Space ◽  
Boundedness Condition ◽  
Weighted Hardy Spaces ◽  
Lusin Area Function ◽  
Extra Assumption ◽  
Type Decomposition

Let$(X,d,\unicode[STIX]{x1D707})$be a metric measure space endowed with a distance$d$and a nonnegative, Borel, doubling measure$\unicode[STIX]{x1D707}$. Let$L$be a nonnegative self-adjoint operator on$L^{2}(X)$. Assume that the (heat) kernel associated to the semigroup$e^{-tL}$satisfies a Gaussian upper bound. In this paper, we prove that for any$p\in (0,\infty )$and$w\in A_{\infty }$, the weighted Hardy space$H_{L,S,w}^{p}(X)$associated with$L$in terms of the Lusin (area) function and the weighted Hardy space$H_{L,G,w}^{p}(X)$associated with$L$in terms of the Littlewood–Paley function coincide and their norms are equivalent. This improves a recent result of Duonget al.[‘A Littlewood–Paley type decomposition and weighted Hardy spaces associated with operators’,J. Geom. Anal.26(2016), 1617–1646], who proved that$H_{L,S,w}^{p}(X)=H_{L,G,w}^{p}(X)$for$p\in (0,1]$and$w\in A_{\infty }$by imposing an extra assumption of a Moser-type boundedness condition on$L$. Our result is new even in the unweighted setting, that is, when$w\equiv 1$.

scholarly journals Fredholm Weighted Composition Operator on Weighted Hardy Space

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Liankuo Zhao
Keyword(s):  
Hardy Space ◽  
Composition Operator ◽  
Hardy Spaces ◽  
Weighted Composition Operator ◽  
Weighted Hardy Space ◽  
Weighted Hardy Spaces ◽  
Weighted Composition

This paper gives a unified characterization of Fredholm weighted composition operator on a class of weighted Hardy spaces.

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.

LITTLEWOOD–PALEY CHARACTERIZATIONS OF ANISOTROPIC WEAK MUSIELAK–ORLICZ HARDY SPACES

2018 ◽  
Vol 237 ◽  
pp. 39-78
Author(s):  
BO LI ◽  
RUIRUI SUN ◽  
MINFENG LIAO ◽  
BAODE LI
Keyword(s):  
Hardy Space ◽  
Maximal Function ◽  
Hardy Spaces ◽  
Growth Function ◽  
Anisotropic Growth ◽  
Area Function ◽  
Function Characterization ◽  
Lusin Area Function ◽  
Classical Hardy Space

Let $A$ be an expansive dilation on $\mathbb{R}^{n}$ and $\unicode[STIX]{x1D711}:\mathbb{R}^{n}\times [0,\infty )\rightarrow [0,\infty )$ an anisotropic growth function. In this article, the authors introduce the anisotropic weak Musielak–Orlicz Hardy space $\mathit{WH}_{A}^{\unicode[STIX]{x1D711}}(\mathbb{R}^{n})$ via the nontangential grand maximal function and then obtain its Littlewood–Paley characterizations in terms of the anisotropic Lusin-area function, $g$-function or $g_{\unicode[STIX]{x1D706}}^{\ast }$-function, respectively. All these characterizations for anisotropic weak Hardy spaces $\mathit{WH}_{A}^{p}(\mathbb{R}^{n})$ (namely, $\unicode[STIX]{x1D711}(x,t):=t^{p}$ for all $t\in [0,\infty )$ and $x\in \mathbb{R}^{n}$ with $p\in (0,1]$) are new. Moreover, the range of $\unicode[STIX]{x1D706}$ in the anisotropic $g_{\unicode[STIX]{x1D706}}^{\ast }$-function characterization of $\mathit{WH}_{A}^{\unicode[STIX]{x1D711}}(\mathbb{R}^{n})$ coincides with the best known range of the $g_{\unicode[STIX]{x1D706}}^{\ast }$-function characterization of classical Hardy space $H^{p}(\mathbb{R}^{n})$ or its weighted variants, where $p\in (0,1]$.

scholarly journals Numerical Range on Weighted Hardy Spaces as Semi Inner Product Spaces

2017 ◽  
Vol 25 (1) ◽  
pp. 87-98
Author(s):  
Mohammad Taghi Heydari
Keyword(s):  
Hardy Space ◽  
Hardy Spaces ◽  
Numerical Range ◽  
Linear Operators ◽  
Inner Product ◽  
Product Spaces ◽  
Weighted Hardy Space ◽  
Weighted Hardy Spaces ◽  
Bounded Linear ◽  
Inner Product Spaces

AbstractThe semi-inner product, in the sense of Lumer, on weighted Hardy space which generate the norm is unique. Also we will discuss some properties of the numerical range of bounded linear operators on weighted Hardy spaces.

scholarly journals Multipliers on weighted Hardy spaces over locally compact Vilenkin groups, I

1990 ◽  
Vol 48 (3) ◽  
pp. 472-496 ◽  
Author(s):  
C. W. Onneweer ◽  
T. S. Quek
Keyword(s):  
Hardy Space ◽  
Hardy Spaces ◽  
Lebesgue Space ◽  
Sufficient Conditions ◽  
Dual Group ◽  
Vilenkin Group ◽  
Weighted Lebesgue Space ◽  
Weighted Hardy Space ◽  
Weighted Hardy Spaces ◽  
Locally Compact

AbstractLet G denote a locally compact Vilenkin group with dual group Γ. We give sufficient conditions for a function ϕ ∈ L∞ (Γ) to be a multiplier from the power-weighted Hardy space to itself or the corresponding power-weighted Lebesgue space

scholarly journals Multipliers for weighted Hardy spaces on locally compact Vilenkin groups

1993 ◽  
Vol 48 (3) ◽  
pp. 441-449 ◽  
Author(s):  
Toshiyuki Kitada
Keyword(s):  
Hardy Space ◽  
Hardy Spaces ◽  
Vilenkin Group ◽  
Weighted Hardy Space ◽  
Weighted Hardy Spaces ◽  
Locally Compact ◽  
Vilenkin Groups ◽  
Hörmander Condition

Let G be a locally compact Vilenkin group. We study multipliers which satisfy a generalised Hörmander condition from power-weighted Hardy space (G) to (G) with 0 < p ≤ q < ∞, 0 < p ≤ 1, −1 < β, β′.

Non-tangential Maximal Function Characterizations of Hardy Spaces Associated with Degenerate Elliptic Operators

2015 ◽  
Vol 67 (5) ◽  
pp. 1161-1200 ◽  
Author(s):  
Junqiang Zhang ◽  
Jun Cao ◽  
Renjin Jiang ◽  
Dachun Yang
Keyword(s):  
Hardy Space ◽  
Maximal Function ◽  
Hardy Spaces ◽  
Riesz Transform ◽  
Degenerate Elliptic Operators ◽  
Degenerate Elliptic Operator ◽  
Muckenhoupt Class ◽  
Degenerate Elliptic ◽  
Function Characterization

AbstractLet w be either in the Muckenhoupt class of A2(ℝn) weights or in the class of QC(ℝn) weights, and let be the degenerate elliptic operator on the Euclidean space ℝn, n ≥ 2. In this article, the authors establish the non-tangential maximal function characterization of the Hardy space associated with , and when with , the authors prove that the associated Riesz transform is bounded from to the weighted classical Hardy space .

scholarly journals Musielak-Orlicz-Hardy Spaces Associated with Operators Satisfying Reinforced Off-Diagonal Estimates

2013 ◽  
Vol 1 ◽  
pp. 69-129 ◽  
Author(s):  
The Anh Bui ◽  
Jun Cao ◽  
Luong Dang Ky ◽  
Dachun Yang ◽  
Sibei Yang
Keyword(s):  
Hardy Space ◽  
Orlicz Function ◽  
Riesz Transform ◽  
Hölder Inequality ◽  
Muckenhoupt Weights ◽  
Order Elliptic Operator ◽  
Lusin Area Function ◽  
Reverse Hölder ◽  
Conjugate Exponent

Abstract Let X be a metric space with doubling measure and L a one-to-one operator of type ω having a bounded H∞ -functional calculus in L2(X) satisfying the reinforced (pL; qL) off-diagonal estimates on balls, where pL ∊ [1; 2) and qL ∊ (2;∞]. Let φ : X × [0;∞) → [0;∞) be a function such that φ (x;·) is an Orlicz function, φ(·;t) ∊ A∞(X) (the class of uniformly Muckenhoupt weights), its uniformly critical upper type index l(φ) ∊ (0;1] and φ(·; t) satisfies the uniformly reverse Hölder inequality of order (qL/l(φ))′, where (qL/l(φ))′ denotes the conjugate exponent of qL/l(φ). In this paper, the authors introduce a Musielak-Orlicz-Hardy space Hφ;L(X), via the Lusin-area function associated with L, and establish its molecular characterization. In particular, when L is nonnegative self-adjoint and satisfies the Davies-Gaffney estimates, the atomic characterization of Hφ,L(X) is also obtained. Furthermore, a sufficient condition for the equivalence between Hφ,L(ℝn) and the classical Musielak-Orlicz-Hardy space Hv(ℝn) is given. Moreover, for the Musielak-Orlicz-Hardy space Hφ,L(ℝn) associated with the second order elliptic operator in divergence form on ℝn or the Schrödinger operator L := −Δ + V with 0 ≤ V ∊ L1loc(ℝn), the authors further obtain its several equivalent characterizations in terms of various non-tangential and radial maximal functions; finally, the authors show that the Riesz transform ∇L−1/2 is bounded from Hφ,L(ℝn) to the Musielak-Orlicz space Lφ(ℝn) when i(φ) ∊ (0; 1], from Hφ,L(ℝn) to Hφ(ℝn) when i(φ) ∊ (; 1], and from Hφ,L(ℝn) to the weak Musielak-Orlicz-Hardy space WHφ(ℝn) when i(φ)=is attainable and φ(·; t) ∊ A1(X), where i(φ) denotes the uniformly critical lower type index of φ

scholarly journals ORLICZ–HARDY SPACES ASSOCIATED WITH OPERATORS SATISFYING DAVIES–GAFFNEY ESTIMATES

2011 ◽  
Vol 13 (02) ◽  
pp. 331-373 ◽  
Author(s):  
RENJIN JIANG ◽  
DACHUN YANG
Keyword(s):  
Hardy Space ◽  
Adjoint Operator ◽  
Riesz Transform ◽  
Concave Function ◽  
Maximal Functions ◽  
Heat Semigroup ◽  
Area Function ◽  
Poisson Semigroup ◽  
Function Characterization ◽  
Lusin Area Function

Let [Formula: see text] be a metric space with doubling measure, L a nonnegative self-adjoint operator in [Formula: see text] satisfying the Davies–Gaffney estimate, ω a concave function on (0, ∞) of strictly lower type pω∈(0, 1] and ρ(t) = t-1/ω-1(t-1) for all t∈(0, ∞). The authors introduce the Orlicz–Hardy space [Formula: see text] via the Lusin area function associated to the heat semigroup, and the BMO-type space [Formula: see text]. The authors then establish the duality between [Formula: see text] and [Formula: see text]; as a corollary, the authors obtain the ρ-Carleson measure characterization of the space [Formula: see text]. Characterizations of [Formula: see text], including the atomic and molecular characterizations and the Lusin area function characterization associated to the Poisson semigroup, are also presented. Let [Formula: see text] and L = -Δ+V be a Schrödinger operator, where [Formula: see text] is a nonnegative potential. As applications, the authors show that the Riesz transform ∇L-1/2 is bounded from Hω, L(ℝn) to L(ω). Moreover, if there exist q1, q2∈(0, ∞) such that q1<1<q2 and [ω(tq2)]q1 is a convex function on (0, ∞), then several characterizations of the Orlicz–Hardy space Hω, L(ℝn), in terms of the Lusin-area functions, the non-tangential maximal functions, the radial maximal functions, the atoms and the molecules, are obtained. All these results are new even when ω(t) = tp for all t ∈ (0, ∞) and p ∈ (0, 1).

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