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

Second-order Riesz Transforms and Maximal Inequalities Associated with Magnetic Schr ödinger Operators

2015 ◽  
Vol 58 (2) ◽  
pp. 432-448 ◽  
Author(s):  
Dachun Yang ◽  
Sibei Yang
Keyword(s):  
Hardy Space ◽  
Orlicz Space ◽  
Orlicz Function ◽  
Second Order ◽  
Riesz Transforms ◽  
Muckenhoupt Weights ◽  
Schrodinger Operator ◽  
Magnetic Schrödinger Operator ◽  
Maximal Inequalities ◽  
Reverse Hölder

AbstractLet be a magnetic Schrödinger operator on ℝn, wheresatisfy some reverse Hölder conditions. Let be such that ϕ(x, ·) for any given x ∊ ℝn is an Orlicz function, ϕ( ·, t) ∊ A∞(ℝn) for all t ∊ (0,∞) (the class of uniformly Muckenhoupt weights) and its uniformly critical upper type index . In this article, the authors prove that second-order Riesz transforms VA-1 and are bounded from the Musielak–Orlicz–Hardy space Hµ,A(Rn), associated with A, to theMusielak–Orlicz space Lµ(Rn). Moreover, we establish the boundedness of VA-1 on . As applications, some maximal inequalities associated with A in the scale of Hµ,A(Rn) are obtained

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 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 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 Anisotropic Hardy Spaces of Musielak-Orlicz Type with Applications to Boundedness of Sublinear Operators

2014 ◽  
Vol 2014 ◽  
pp. 1-19 ◽  
Author(s):  
Baode Li ◽  
Dachun Yang ◽  
Wen Yuan
Keyword(s):  
Hardy Space ◽  
Hardy Spaces ◽  
Orlicz Function ◽  
Real Variable ◽  
Sublinear Operator ◽  
Atomic Decompositions ◽  
Anisotropic Hardy Space ◽  
Sublinear Operators ◽  
Uniformly Bounded

Letφ:ℝn×[0,∞)→[0,∞)be a Musielak-Orlicz function andAan expansive dilation. In this paper, the authors introduce the anisotropic Hardy space of Musielak-Orlicz type,HAφ(ℝn), via the grand maximal function. The authors then obtain some real-variable characterizations ofHAφ(ℝn)in terms of the radial, the nontangential, and the tangential maximal functions, which generalize the known results on the anisotropic Hardy spaceHAp(ℝn)withp∈(0,1]and are new even for its weighted variant. Finally, the authors characterize these spaces by anisotropic atomic decompositions. The authors also obtain the finite atomic decomposition characterization ofHAφ(ℝn), and, as an application, the authors prove that, for a given admissible triplet(φ,q,s), ifTis a sublinear operator and maps all(φ,q,s)-atoms withq<∞(or all continuous(φ,q,s)-atoms withq=∞) into uniformly bounded elements of some quasi-Banach spacesℬ, thenTuniquely extends to a bounded sublinear operator fromHAφ(ℝn)toℬ. These results are new even for anisotropic Orlicz-Hardy spaces onℝn.

Carleson measure characterizations of the Campanato type space associated with Schrödinger operators on stratified Lie groups

2020 ◽  
Vol 32 (5) ◽  
pp. 1337-1373 ◽  
Author(s):  
Yixin Wang ◽  
Yu Liu ◽  
Chuanhong Sun ◽  
Pengtao Li
Keyword(s):  
Carleson Measures ◽  
Heat Semigroup ◽  
Stratified Lie Group ◽  
Poisson Semigroup ◽  
Homogeneous Dimension ◽  
Type Spaces ◽  
Stratified Lie Groups ◽  
Lusin Area Function ◽  
Reverse Hölder

AbstractLet {\mathcal{L}=-{\Delta}_{\mathbb{G}}+V} be a Schrödinger operator on the stratified Lie group {\mathbb{G}}, where {{\Delta}_{\mathbb{G}}} is the sub-Laplacian and the nonnegative potential V belongs to the reverse Hölder class {B_{q_{0}}} with {q_{0}>\mathcal{Q}/2} and {\mathcal{Q}} is the homogeneous dimension of {\mathbb{G}}. In this article, by Campanato type spaces {\Lambda^{\alpha}_{\mathcal{L}}(\mathbb{G})}, we introduce Hardy type spaces associated with {\mathcal{L}} denoted by {H^{{p}}_{\vphantom{\varepsilon}{\mathcal{L}}}(\mathbb{G})} and prove the atomic characterization of {H^{{p}}_{\vphantom{\varepsilon}{\mathcal{L}}}(\mathbb{G})}. Further, we obtain the following duality relation:\Lambda_{\mathcal{L}}^{\mathcal{Q}(1/p-1)}(\mathbb{G})=(H^{{p}}_{\vphantom{% \varepsilon}{\mathcal{L}}}(\mathbb{G}))^{\ast},\quad\mathcal{Q}/(\mathcal{Q}+% \delta)<p<1\quad\text{for}\ \delta=\min\{1,2-\mathcal{Q}/q_{0}\}.The above relation enables us to characterize {\Lambda^{\alpha}_{\mathcal{L}}(\mathbb{G})} via two families of Carleson measures generated by the heat semigroup and the Poisson semigroup, respectively. Also, we obtain two classes of perturbation formulas associated with the semigroups related to {\mathcal{L}}. As applications, we obtain the boundedness of the Littlewood–Paley function and the Lusin area function on {\Lambda^{\alpha}_{\mathcal{L}}(\mathbb{G})}.

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).

scholarly journals Higher integrability and reverse Hölder inequalities in the limit cases

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Arturo Popoli
Keyword(s):  
Hölder Inequality ◽  
Muckenhoupt Weights ◽  
Reverse Hölder Inequality ◽  
Higher Integrability ◽  
Reverse Hölder Inequalities ◽  
Reverse Hölder

Abstract We study the higher integrability of weights satisfying a reverse Hölder inequality ( ⨏ I u β ) 1 β ≤ B ⁢ ( ⨏ I u α ) 1 α {\biggl{(}\fint_{I}u^{\beta}\biggr{)}^{\frac{1}{\beta}}}\leq B{\biggl{(}\fint_{I}u^{\alpha}\biggr{)}^{\frac{1}{\alpha}}} for some B > 1 B>1 and given α < β \alpha<\beta , in the limit cases when α ∈ { - ∞ , 0 } \alpha\in\{-\infty,0\} and/or β ∈ { 0 , + ∞ } \beta\in\{0,+\infty\} . The results apply to the Gehring and Muckenhoupt weights and their corresponding limit classes.

Sign in / Sign up

Export Citation Format

Share Document