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 Hardy Spaces Associated to Schrödinger Operators on Product Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Liang Song ◽  
Chaoqiang Tan
Keyword(s):  
Hardy Spaces ◽  
Atomic Decomposition ◽  
Schrödinger Operators ◽  
Nonnegative Function ◽  
Maximal Functions ◽  
Schrodinger Operators ◽  
Schrodinger Operator ◽  
Product Spaces ◽  
Area Integral ◽  
Lusin Area Integral

LetL=−Δ+Vbe a Schrödinger operator onℝn, whereV∈Lloc1(ℝn)is a nonnegative function onℝn. In this article, we show that the Hardy spacesLon product spaces can be characterized in terms of the Lusin area integral, atomic decomposition, and maximal functions.

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 Weighted Weak Local Hardy Spaces Associated with Schrödinger Operators

2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
Hua Zhu
Keyword(s):  
Hardy Spaces ◽  
Atomic Decomposition ◽  
Critical Radius ◽  
Schrödinger Operators ◽  
Riesz Transforms ◽  
Schrodinger Operators ◽  
Local Hardy Spaces ◽  
Radius Function

We characterize the weighted weak local Hardy spacesWhρp(ω)related to the critical radius functionρand weightsω∈A∞ρ,∞(Rn)which locally behave as Muckenhoupt’s weights and actually include them, by the atomic decomposition. As an application, we show that localized Riesz transforms are bounded on the weighted weak local Hardy spaces.

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 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.

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