scholarly journals Variable Hardy Spaces Associated with Operators Satisfying Davies–Gaffney Estimates

2018 ◽  
Vol 61 (3) ◽  
pp. 759-810 ◽  
Author(s):  
Dachun Yang ◽  
Junqiang Zhang ◽  
Ciqiang Zhuo
Keyword(s):  
Hardy Space ◽  
Adjoint Operator ◽  
Riesz Transform ◽  
Bounded Mean Oscillation ◽  
Hölder Continuous ◽  
Measurable Coefficients ◽  
Mean Oscillation ◽  
Function Characterization ◽  
Bounded Holomorphic

AbstractLetLbe a one-to-one operator of type ω inL2(ℝn), with ω∈[0, π/2), which has a bounded holomorphic functional calculus and satisfies the Davies–Gaffney estimates. Letp(·): ℝn→(0, 1] be a variable exponent function satisfying the globally log-Hölder continuous condition. In this article, the authors introduce the variable Hardy space$H_L^{p(\cdot )} ({\open R}^n)$associated withL. By means of variable tent spaces, the authors establish the molecular characterization of$H_L^{p(\cdot )} ({\open R}^n)$. Then the authors show that the dual space of$H_L^{p(\cdot )} ({\open R}^n)$is the bounded mean oscillation (BMO)-type space${\rm BM}{\rm O}_{p(\cdot ),{\kern 1pt} L^ * }({\open R}^n)$, whereL* denotes the adjoint operator ofL. In particular, whenLis the second-order divergence form elliptic operator with complex bounded measurable coefficients, the authors obtain the non-tangential maximal function characterization of$H_L^{p(\cdot )} ({\open R}^n)$and show that the fractional integralL−αfor α∈(0, (1/2)] is bounded from$H_L^{p(\cdot )} ({\open R}^n)$to$H_L^{q(\cdot )} ({\open R}^n)$with (1/p(·))−(1/q(·))=2α/n, and the Riesz transform ∇L−1/2is bounded from$H_L^{p(\cdot )} ({\open R}^n)$to the variable Hardy spaceHp(·)(ℝn).

Variable weak Hardy spaces WH L p(·)(ℝ n ) associated with operators satisfying Davies–Gaffney estimates

2019 ◽  
Vol 31 (3) ◽  
pp. 579-605 ◽  
Author(s):  
Ciqiang Zhuo ◽  
Dachun Yang
Keyword(s):  
Hardy Space ◽  
Molecular Characterization ◽  
Atomic Decomposition ◽  
Riesz Transform ◽  
Square Function ◽  
Hölder Continuous ◽  
Weak Hardy Space ◽  
Measurable Coefficients ◽  
Bounded Holomorphic

Abstract Let {p(\,\cdot\,)\colon\mathbb{R}^{n}\to[0,1]} be a variable exponent function satisfying the globally log-Hölder continuous condition, and L a one-to-one operator of type ω in {L^{2}({\mathbb{R}}^{n})} , with {\omega\in[0,\pi/2)} , which has a bounded holomorphic functional calculus and satisfies the Davies–Gaffney estimates. In this article, we introduce the variable weak Hardy space {\mathrm{WH}^{{p(\,\cdot\,)}}_{L}(\mathbb{R}^{n})} , associated with L via the corresponding square function. Its molecular characterization is then established by means of the atomic decomposition of the variable weak tent space {\mathrm{WT}^{p(\,\cdot\,)}(\mathbb{R}_{+}^{n+1})} , which is also obtained in this article. In particular, when L is non-negative and self-adjoint, we obtain the atomic characterization of {\mathrm{WH}^{{p(\,\cdot\,)}}_{L}(\mathbb{R}^{n})} . As an application of the molecular characterization, when L is the second-order divergence form elliptic operator with complex bounded measurable coefficients, we prove that the associated Riesz transform {\nabla L^{-1/2}} is bounded from {\mathrm{WH}^{{p(\,\cdot\,)}}_{L}(\mathbb{R}^{n})} to the variable weak Hardy space {\mathrm{WH}^{p(\,\cdot\,)}(\mathbb{R}^{n})} . Moreover, when L is non-negative and self-adjoint with the kernels of {\{e^{-tL}\}_{t>0}} satisfying the Gaussian upper bound estimates, the atomic characterization of {\mathrm{WH}^{{p(\,\cdot\,)}}_{L}(\mathbb{R}^{n})} is further used to characterize this space via non-tangential 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 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).

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).

On Analytic Functions of Bergman BMO in the Ball

1999 ◽  
Vol 42 (1) ◽  
pp. 97-103 ◽  
Author(s):  
E. G. Kwon
Keyword(s):  
Hardy Space ◽  
Composition Operator ◽  
Analytic Functions ◽  
Bloch Space ◽  
Bounded Mean Oscillation ◽  
Open Unit ◽  
Bergman Metric ◽  
Open Unit Ball ◽  
Mean Oscillation ◽  
Volume Measure

AbstractLet B = Bn be the open unit ball of Cn with volume measure v, U = B1 and B be the Bloch space on , 1 ≤ α < 1, is defined as the set of holomorphic f : B → C for whichif 0 < α < 1 and , the Hardy space. Our objective of this note is to characterize, in terms of the Bergman distance, those holomorphic f : B → U for which the composition operator defined by , is bounded. Our result has a corollary that characterize the set of analytic functions of bounded mean oscillation with respect to the Bergman metric.

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 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 On the equivalence between weak BMO and the space of derivatives of the Zygmund class

2021 ◽  
Vol 54 (1) ◽  
pp. 140-150
Author(s):  
Eddy Kwessi
Keyword(s):  
Hardy Space ◽  
Dual Space ◽  
Bounded Mean Oscillation ◽  
Zygmund Class ◽  
Mean Oscillation ◽  
Derivatives Of ◽  
Class Of Functions

Abstract In this paper, we will discuss the space of functions of weak bounded mean oscillation. In particular, we will show that this space is the dual space of the special atom space, whose dual space was already known to be the space of derivative of functions (in the sense of distribution) belonging to the Zygmund class of functions. We show, in particular, that this proves that the Hardy space H 1 {H}^{1} strictly contains the special atom space.

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$w\equiv 1$.

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