Another characterization of the Hardy space with non doubling measures

2006 ◽  
Vol 279 (16) ◽  
pp. 1797-1807 ◽  
Author(s):  
Guoen Hu ◽  
Shuang Liang
Keyword(s):  
Hardy Space ◽  
Doubling Measures

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 Variational characterization of Hp

2019 ◽  
Vol 149 (5) ◽  
pp. 1123-1134 ◽  
Author(s):  
Honghai Liu
Keyword(s):  
Hardy Space ◽  
Approximate Identity ◽  
Jump Operator ◽  
Variational Characterization ◽  
Approximate Identities ◽  
Oscillation Operator

AbstractIn this paper, we obtain the variational characterization of Hardy space Hp for $p\in (((n)/({n+1})),1]$, and get estimates for the oscillation operator and the λ-jump operator associated with approximate identities acting on Hp for $p\in (((n)/({n+1})),1]$. Moreover, we give counterexamples to show that the oscillation and λ-jump associated with some approximate identity cannot be used to characterize Hp for $p\in (((n)/({n+1})),1]$.

scholarly journals A characterization of 1-rectifiable doubling measures with connected supports

2016 ◽  
Vol 9 (1) ◽  
pp. 99-109 ◽  
Author(s):  
Jonas Azzam ◽  
Mihalis Mourgoglou
Keyword(s):  
Doubling Measures

The reproducing kernel of $\mathcal H^2$ and radial eigenfunctions of the hyperbolic Laplacian

2019 ◽  
Vol 124 (1) ◽  
pp. 81-101
Author(s):  
Manfred Stoll
Keyword(s):  
Hardy Space ◽  
Unit Ball ◽  
Hypergeometric Function ◽  
Compact Subset ◽  
Spherical Harmonics ◽  
Harmonic Functions ◽  
Reproducing Kernel ◽  
Diagonal Function ◽  
Hyperbolic Harmonic Functions

In the paper we characterize the reproducing kernel $\mathcal {K}_{n,h}$ for the Hardy space $\mathcal {H}^2$ of hyperbolic harmonic functions on the unit ball $\mathbb {B}$ in $\mathbb {R}^n$. Specifically we prove that \[ \mathcal {K}_{n,h}(x,y) = \sum _{\alpha =0}^\infty S_{n,\alpha }(\lvert x\rvert )S_{n,\alpha }(\lvert y\rvert ) Z_\alpha (x,y), \] where the series converges absolutely and uniformly on $K\times \mathbb {B}$ for every compact subset $K$ of $\mathbb {B}$. In the above, $S_{n,\alpha }$ is a hypergeometric function and $Z_\alpha $ is the reproducing kernel of the space of spherical harmonics of degree α. In the paper we prove that \[ 0\le \mathcal K_{n,h}(x,y) \le \frac {C_n}{(1-2\langle x,y\rangle + \lvert x \rvert^2 \lvert y \rvert^2)^{n-1}}, \] where $C_n$ is a constant depending only on $n$. It is known that the diagonal function $\mathcal K_{n,h}(x,x)$ is a radial eigenfunction of the hyperbolic Laplacian $\varDelta_h $ on $\mathbb{B} $ with eigenvalue $\lambda _2 = 8(n-1)^2$. The result for $n=4$ provides motivation that leads to an explicit characterization of all radial eigenfunctions of $\varDelta_h $ on $\mathbb{B} $. Specifically, if $g$ is a radial eigenfunction of $\varDelta_h $ with eigenvalue $\lambda _\alpha = 4(n-1)^2\alpha (\alpha -1)$, then \[ g(r) = g(0) \frac {p_{n,\alpha }(r^2)}{(1-r^2)^{(\alpha -1)(n-1)}}, \] where $p_{n,\alpha }$ is again a hypergeometric function. If α is an integer, then $p_{n,\alpha }(r^2)$ is a polynomial of degree $2(\alpha -1)(n-1)$.

scholarly journals Wavelets Convergence and Unconditional Bases for Lp(ℝn) and H1(ℝn)

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Devendra Kumar
Keyword(s):  
Banach Space ◽  
Hardy Space ◽  
Function Space ◽  
Riesz Transform ◽  
Unconditional Bases

We prove that reasonable nice wavelets form unconditional bases in function space other than L2(ℝn, X). Moreover, characterization of convergence of wavelets series in Lp(ℝn, X) space and Hardy space H1(ℝn,X) has been obtained. Here, X is a Banach space with boundedness of Riesz transform.

scholarly journals Endpoint estimates for homogeneous Littlewood-Paleyg-functions with non-doubling measures

2009 ◽  
Vol 7 (2) ◽  
pp. 187-207 ◽  
Author(s):  
Dachun Yang ◽  
Dongyong Yang
Keyword(s):  
Hardy Space ◽  
Growth Condition ◽  
Radon Measure ◽  
Open Ball ◽  
Almost Everywhere ◽  
Doubling Measures

Letµbe a nonnegative Radon measure on ℝdwhich satisfies the growth condition that there exist constantsC0> 0 andn∈ (0, d] such that for allx∈ ℝdand r > 0,μ(B(x,r))≤C0rn, whereB(x, r) is the open ball centered atxand having radiusr. In this paper, when ℝdis not an initial cube which impliesµ(ℝd) = ∞, the authors prove that the homogeneous Littlewood-Paleyg-function of Tolsa is bounded from the Hardy spaceH1(µ) toL1(µ), and furthermore, that iff∈ RBMO (µ), then [ġ(f)]2is either infinite everywhere or finite almost everywhere, and in the latter case, [ġ(f)]2belongs to RBLO (µ) with norm no more thanC‖f‖RBMO(μ)2, whereC≻0is independent off.

The Hardy Space H1 with Non-doubling Measures and Their Applications

2013 ◽  
Author(s):  
Dachun Yang ◽  
Dongyong Yang ◽  
Guoen Hu
Keyword(s):  
Hardy Space ◽  
Doubling Measures

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.

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