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

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

Canadian Journal of Mathematics ◽

10.4153/cjm-2014-038-1 ◽

2015 ◽

Vol 67 (5) ◽

pp. 1161-1200 ◽

Cited By ~ 7

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 .

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

Analysis and Geometry in Metric Spaces ◽

10.2478/agms-2012-0006 ◽

2013 ◽

Vol 1 ◽

pp. 69-129 ◽

Cited By ~ 34

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

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Variable Hardy Spaces Associated with Operators Satisfying Davies–Gaffney Estimates

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091517000414 ◽

2018 ◽

Vol 61 (3) ◽

pp. 759-810 ◽

Cited By ~ 7

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).

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

Forum Mathematicum ◽

10.1515/forum-2018-0125 ◽

2019 ◽

Vol 31 (3) ◽

pp. 579-605 ◽

Cited By ~ 2

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.

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Two Results About H∞ Functional Calculus on Analytic umd Banach Spaces

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700003360 ◽

2003 ◽

Vol 74 (3) ◽

pp. 351-378 ◽

Cited By ~ 3

Author(s):

Christian Le Merdy

Keyword(s):

Banach Space ◽

Hardy Space ◽

Banach Spaces ◽

Functional Calculus ◽

Sectorial Operators ◽

Derivation Operator ◽

Umd Banach Spaces ◽

Vector Valued

AbstractLet X be a Banach space with the analytic UMD property, and let A and B be two commuting sectorial operators on X which admit bounded H∞ functional calculi with respect to angles θ1 and θ2 satisfying θ1 + θ2 > π. It was proved by Kalton and Weis that in this case, A + B is closed. The first result of this paper is that under the same conditions, A + B actually admits a bounded H∞ functional calculus. Our second result is that given a Banach space X and a number 1 ≦ p < ∞, the derivation operator on the vector valued Hardy space Hp (R; X) admits a bounded H∞ functional calculus if and only if X has the analytic UMD property. This is an ‘analytic’ version of the well-known characterization of UMD by the boundedness of the H∞ functional calculus of the derivation operator on vector valued Lp-spaces Lp (R; X) for 1 < p < ∞ (Dore-Venni, Hieber-Prüss, Prüss).

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Molecular Characterization of Hardy Spaces Associated with Twisted Convolution

Journal of Function Spaces ◽

10.1155/2014/326940 ◽

2014 ◽

Vol 2014 ◽

pp. 1-6 ◽

Cited By ~ 3

Author(s):

Jizheng Huang ◽

Yu Liu

Keyword(s):

Hardy Space ◽

Molecular Characterization ◽

Hardy Spaces ◽

Riesz Transform ◽

Twisted Convolution

We give a molecular characterization of the Hardy space associated with twisted convolution. As an application, we prove the boundedness of the local Riesz transform on the Hardy space.

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Entropy numbers of embedding maps between Besov spaces with an application to eigenvalue problems

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500015341 ◽

1981 ◽

Vol 90 (1-2) ◽

pp. 63-70 ◽

Cited By ~ 5

Author(s):

Bernd Carl

Keyword(s):

Banach Space ◽

Asymptotic Behaviour ◽

Besov Spaces ◽

Eigenvalue Problems ◽

Function Spaces ◽

Sequence Spaces ◽

Entropy Numbers ◽

Diagonal Operators

SynopsisIn this paper we determine the asymptotic behaviour of entropy numbers of embedding maps between Besov sequence spaces and Besov function spaces. The results extend those of M. Š. Birman, M. Z. Solomjak and H. Triebel originally formulated in the language of ε-entropy. It turns out that the characterization of embedding maps between Besov spaces by entropy numbers can be reduced to the characterization of certain diagonal operators by their entropy numbers.Finally, the entropy numbers are applied to the study of eigenvalues of operators acting on a Banach space which admit a factorization through embedding maps between Besov spaces.The statements of this paper are obtained by results recently proved elsewhere by the author.

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