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Journal articles on the topic 'Atomic Hardy space'

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Published: 9 March 2023

Last updated: 10 March 2023

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1

Folch-Gabayet, Magali, Martha Guzmán-Partida, and Salvador Pérez-Esteva. "Lipschitz measures and vector-valued Hardy spaces." International Journal of Mathematics and Mathematical Sciences 25, no. 5 (2001): 345–56. http://dx.doi.org/10.1155/s0161171201004549.

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We define certain spaces of Banach-valued measures called Lipschitz measures. When the Banach space is a dual spaceX*, these spaces can be identified with the duals of the atomic vector-valued Hardy spacesHXp(ℝn),0<p<1. We also prove that all these measures have Lipschitz densities. This implies that for every real Banach spaceXand0<p<1, the dualHXp(ℝn)∗can be identified with a space of Lipschitz functions with values inX*.

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2

Chai, Yan, Yaoyao Han, and Kai Zhao. "Herz-Type Hardy Spaces Associated with Operators." Journal of Function Spaces 2018 (July 17, 2018): 1–10. http://dx.doi.org/10.1155/2018/1296837.

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Suppose L is a nonnegative, self-adjoint differential operator. In this paper, we introduce the Herz-type Hardy spaces associated with operator L. Then, similar to the atomic and molecular decompositions of classical Herz-type Hardy spaces and the Hardy space associated with operators, we prove the atomic and molecular decompositions of the Herz-type Hardy spaces associated with operator L. As applications, the boundedness of some singular integral operators on Herz-type Hardy spaces associated with operators is obtained.

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3

WANG, HUA. "BOUNDEDNESS OF SEVERAL INTEGRAL OPERATORS WITH BOUNDED VARIABLE KERNELS ON HARDY AND WEAK HARDY SPACES." International Journal of Mathematics 24, no. 12 (November 2013): 1350095. http://dx.doi.org/10.1142/s0129167x1350095x.

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In this paper, by using the atomic decomposition theory of Hardy space H1(ℝn) and weak Hardy space WH1(ℝn), we give the boundedness properties of some operators with variable kernels such as singular integral operators, fractional integrals and parametric Marcinkiewicz integrals on these spaces, under certain logarithmic type Lipschitz conditions assumed on the variable kernel Ω(x, z).

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4

HYTÖNEN, TUOMAS, DACHUN YANG, and DONGYONG YANG. "The Hardy space H1 on non-homogeneous metric spaces." Mathematical Proceedings of the Cambridge Philosophical Society 153, no. 1 (December 8, 2011): 9–31. http://dx.doi.org/10.1017/s0305004111000776.

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AbstractLet (, d, μ) be a metric measure space and satisfy the so-called upper doubling condition and the geometrical doubling condition. We introduce the atomic Hardy space H1(μ) and prove that its dual space is the known space RBMO(μ) in this context. Using this duality, we establish a criterion for the boundedness of linear operators from H1(μ) to any Banach space. As an application of this criterion, we obtain the boundedness of Calderón–Zygmund operators from H1(μ) to L1(μ).

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5

Long, Long, Niyonkuru Silas, and Guangheng Xie. "Weak martingale Hardy-type spaces associated with quasi-Banach function lattice." Forum Mathematicum 34, no. 2 (January 23, 2022): 407–23. http://dx.doi.org/10.1515/forum-2021-0270.

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Abstract In this paper, the authors introduce weak martingale Hardy-type spaces associated with a quasi-Banach function lattice. The authors then establish the atomic characterizations of these weak martingale Hardy-type spaces. As applications, the authors give the sufficient conditions for the boundedness of σ-sublinear operators from weak martingale Hardy-type spaces to a quasi-Banach function lattice. Furthermore, the authors clarify the relation among different weak martingale Hardy-type spaces in the framework of a rearrangement-invariant quasi-Banach function space. Finally, the authors apply these results to the weighted Lorentz space and the generalized grand Lebesgue space.

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6

Berndt, Ryan. "Atomic Hardy space theory for unbounded singular integrals." Indiana University Mathematics Journal 55, no. 4 (2006): 1461–82. http://dx.doi.org/10.1512/iumj.2006.55.2649.

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7

Lou, Zengjian, and Shouzhi Yang. "AN ATOMIC DECOMPOSITION FOR THE HARDY-SOBOLEV SPACE." Taiwanese Journal of Mathematics 11, no. 4 (September 2007): 1167–76. http://dx.doi.org/10.11650/twjm/1500404810.

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8

KEMPPAINEN, MIKKO. "ON VECTOR-VALUED TENT SPACES AND HARDY SPACES ASSOCIATED WITH NON-NEGATIVE SELF-ADJOINT OPERATORS." Glasgow Mathematical Journal 58, no. 3 (July 21, 2015): 689–716. http://dx.doi.org/10.1017/s0017089515000415.

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AbstractIn this paper, we study Hardy spaces associated with non-negative self-adjoint operators and develop their vector-valued theory. The complex interpolation scales of vector-valued tent spaces and Hardy spaces are extended to the endpoint p=1. The holomorphic functional calculus of L is also shown to be bounded on the associated Hardy space H1L(X). These results, along with the atomic decomposition for the aforementioned space, rely on boundedness of certain integral operators on the tent space T1(X).

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9

Zhang, Yangyang, Dachun Yang, Wen Yuan, and Songbai Wang. "Real-variable characterizations of Orlicz-slice Hardy spaces." Analysis and Applications 17, no. 04 (June 10, 2019): 597–664. http://dx.doi.org/10.1142/s0219530518500318.

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In this paper, the authors first introduce a class of Orlicz-slice spaces which generalize the slice spaces recently studied by Auscher et al. Based on these Orlicz-slice spaces, the authors then introduce a new kind of Hardy-type spaces, the Orlicz-slice Hardy spaces, via the radial maximal functions. This new scale of Orlicz-slice Hardy spaces contains the variant of the Orlicz–Hardy space of Bonami and Feuto as well as the Hardy-amalgam space of de Paul Ablé and Feuto as special cases. Their characterizations via the atom, the molecule, various maximal functions, the Poisson integral and the Littlewood–Paley functions are also obtained. As an application of these characterizations, the authors establish their finite atomic characterizations, which further induce a description of their dual spaces and a criterion on the boundedness of sublinear operators from these Orlicz-slice Hardy spaces into a quasi-Banach space. Then, applying this criterion, the authors obtain the boundedness of [Formula: see text]-type Calderón–Zygmund operators on these Orlicz-slice Hardy spaces. All these results are new even for slice Hardy spaces and, moreover, for Hardy-amalgam spaces, the Littlewood–Paley function characterizations, the dual spaces and the boundedness of [Formula: see text]-type Calderón–Zygmund operators on these Hardy-type spaces are also new.

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10

Xia, Runlian, and Xiao Xiong. "Operator-valued local Hardy spaces." Journal of Operator Theory 82, no. 2 (September 15, 2019): 383–443. http://dx.doi.org/10.7900/jot.2018jun02.2191.

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This paper gives a systematic study of operator-valued local\break Hardy spaces, which are localizations of the Hardy spaces defined by Mei. We prove the h1-bmo duality and the hp-hq duality for any conjugate pair (p,q) when p∈(1,∞). We show that h1(Rd,M) and bmo(Rd,M) are also good endpoints of Lp(L∞(Rd)¯¯¯¯⊗M) for interpolation. We obtain the local version of Calder\'on--Zygmund theory, and then deduce that the Poisson kernel in our definition of the local Hardy norms can be replaced by any reasonable test function. Finally, we establish the atomic decomposition of the local Hardy space hc1(Rd,M).

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11

Vasilis, Jonatan. "Discrete Hardy Spaces Related to Powers of the Poisson Kernel." MATHEMATICA SCANDINAVICA 112, no. 2 (June 1, 2013): 240. http://dx.doi.org/10.7146/math.scand.a-15243.

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Discrete Hardy spaces $H^{1}_{\alpha}(\partial{T})$, related to powers $\alpha \ge 1/2$ of the Poisson kernels on boundaries $\partial{T}$ of regular rooted trees, are studied. The spaces for $\alpha > 1/2$ coincide with the ordinary atomic Hardy space on $\partial{T}$, which in turn is strictly smaller than $H^{1}_{1/2}(\partial{T})$. The Orlicz space $L\log\log L(\partial{T})$ characterizes the positive and increasing functions in $H^{1}_{1/2}(\partial{T})$, but there is no such characterization for general positive functions.

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12

Hao, Zhiwei. "Atomic decomposition of predictable martingale Hardy space with variable exponents." Czechoslovak Mathematical Journal 65, no. 4 (December 2015): 1033–45. http://dx.doi.org/10.1007/s10587-015-0226-x.

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13

Giga, Yoshikazu, and Xingfei Xiang. "Lorentz space estimates for vector fields with divergence and curl in Hardy spaces." Tamkang Journal of Mathematics 47, no. 2 (June 30, 2016): 249–60. http://dx.doi.org/10.5556/j.tkjm.47.2016.1932.

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In this note, we establish the estimate on the Lorentz space $L(3/2,1)$ for vector fields in bounded domains under the assumption that the normal or the tangential component of the vector fields on the boundary vanishes. We prove that the $L(3/2,1)$ norm of the vector field can be controlled by the norms of its divergence and curl in the atomic Hardy spaces and the $L^1$ norm of the vector field itself.

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14

Fu, Xing, and Dachun Yang. "Wavelet characterizations of the atomic Hardy space H 1 on spaces of homogeneous type." Applied and Computational Harmonic Analysis 44, no. 1 (January 2018): 1–37. http://dx.doi.org/10.1016/j.acha.2016.04.001.

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15

Liu, Jun, Long Huang, and Chenlong Yue. "Molecular Characterizations of Anisotropic Mixed-Norm Hardy Spaces and Their Applications." Mathematics 9, no. 18 (September 9, 2021): 2216. http://dx.doi.org/10.3390/math9182216.

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Let p→∈(0,∞)n be an exponent vector and A be a general expansive matrix on Rn. Let HAp→(Rn) be the anisotropic mixed-norm Hardy spaces associated with A defined via the non-tangential grand maximal function. In this article, using the known atomic characterization of HAp→(Rn), the authors characterize this Hardy space via molecules with the best possible known decay. As an application, the authors establish a criterion on the boundedness of linear operators from HAp→(Rn) to itself, which is used to explore the boundedness of anisotropic Calderón–Zygmund operators on HAp→(Rn). In addition, the boundedness of anisotropic Calderón–Zygmund operators from HAp→(Rn) to the mixed-norm Lebesgue space Lp→(Rn) is also presented. The obtained boundedness of these operators positively answers a question mentioned by Cleanthous et al. All of these results are new, even for isotropic mixed-norm Hardy spaces on Rn.

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16

Li, Baode, Dachun Yang, and Wen Yuan. "Anisotropic Hardy Spaces of Musielak-Orlicz Type with Applications to Boundedness of Sublinear Operators." Scientific World Journal 2014 (2014): 1–19. http://dx.doi.org/10.1155/2014/306214.

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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.

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17

Yang, Dachun, and Dongyong Yang. "Boundedness of linear operators via atoms on Hardy spaces with non-doubling measures." gmj 18, no. 2 (June 2011): 377–97. http://dx.doi.org/10.1515/gmj.2011.0018.

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Abstract Let μ be a non-negative Radon measure on which satisfies only the polynomial growth condition. Let 𝒴 be a Banach space and H 1(μ) be the Hardy space of Tolsa. In this paper, the authors prove that a linear operator T is bounded from H 1(μ) to 𝒴 if and only if T maps all (p, γ)-atomic blocks into uniformly bounded elements of 𝒴; moreover, the authors prove that for a sublinear operator T bounded from L 1(μ) to L 1, ∞(μ), if T maps all (p, γ)-atomic blocks with p ∈ (1, ∞) and γ ∈ ℕ into uniformly bounded elements of L 1(μ), then T extends to a bounded sublinear operator from H 1(μ) to L 1(μ). For the localized atomic Hardy space h 1(μ), the corresponding results are also presented. Finally, these results are applied to Calderón–Zygmund operators, Riesz potentials and multilinear commutators generated by Calderón–Zygmund operators or fractional integral operators with Lipschitz functions to simplify the existing proofs in the related papers.

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18

Mirotin, Adolf R. "Hausdorff operators on homogeneous spaces of locally compact groups." Journal of the Belarusian State University. Mathematics and Informatics, no. 2 (July 30, 2020): 28–35. http://dx.doi.org/10.33581/2520-6508-2020-2-28-35.

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Hausdorff operators on the real line and multidimensional Euclidean spaces originated from some classical summation methods. Now it is an active research area. Hausdorff operators on general groups were defined and studied by the author since 2019. The purpose of this paper is to define and study Hausdorff operators on Lebesgue and real Hardy spaces over homogeneous spaces of locally compact groups. We introduce in particular an atomic Hardy space over homogeneous spaces of locally compact groups and obtain conditions for boundedness of Hausdorff operators on such spaces. Several corollaries are considered and unsolved problems are formulated.

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19

Saibi, Khedoudj. "Intrinsic Square Function Characterizations of Variable Hardy–Lorentz Spaces." Journal of Function Spaces 2020 (February 10, 2020): 1–9. http://dx.doi.org/10.1155/2020/2681719.

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The aim of this paper is to establish the intrinsic square function characterizations in terms of the intrinsic Littlewood–Paley g-function, the intrinsic Lusin area function, and the intrinsic gλ∗-function of the variable Hardy–Lorentz space Hp⋅,qℝn, for p⋅ being a measurable function on ℝn satisfying 0<p−≔ess infx∈ℝnpx≤ess supx∈ℝnpx≕p+<∞ and the globally log-Hölder continuity condition and q∈0,∞, via its atomic and Littlewood–Paley function characterizations.

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20

Heraiz, R. "Variable Herz estimates for fractional integral operators." Ukrains’kyi Matematychnyi Zhurnal 72, no. 8 (August 18, 2020): 1034–46. http://dx.doi.org/10.37863/umzh.v72i8.6024.

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21

Zhuo, Ciqiang, and Dachun Yang. "Variable weak Hardy spaces WH L p(·)(ℝ n ) associated with operators satisfying Davies–Gaffney estimates." Forum Mathematicum 31, no. 3 (May 1, 2019): 579–605. http://dx.doi.org/10.1515/forum-2018-0125.

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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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22

Lu, Guanghui, and Shuangping Tao. "Commutators of Littlewood-Paley gκ∗ $g_{\kappa}^{*} $-functions on non-homogeneous metric measure spaces." Open Mathematics 15, no. 1 (November 13, 2017): 1283–99. http://dx.doi.org/10.1515/math-2017-0110.

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Abstract The main purpose of this paper is to prove that the boundedness of the commutator $\mathcal{M}_{\kappa,b}^{*} $ generated by the Littlewood-Paley operator $\mathcal{M}_{\kappa}^{*} $ and RBMO (μ) function on non-homogeneous metric measure spaces satisfying the upper doubling and the geometrically doubling conditions. Under the assumption that the kernel of $\mathcal{M}_{\kappa}^{*} $ satisfies a certain Hörmander-type condition, the authors prove that $\mathcal{M}_{\kappa,b}^{*} $ is bounded on Lebesgue spaces Lp(μ) for 1 < p < ∞, bounded from the space L log L(μ) to the weak Lebesgue space L1,∞(μ), and is bounded from the atomic Hardy spaces H1(μ) to the weak Lebesgue spaces L1,∞(μ).

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23

JIANG, RENJIN, and DACHUN YANG. "ORLICZ–HARDY SPACES ASSOCIATED WITH OPERATORS SATISFYING DAVIES–GAFFNEY ESTIMATES." Communications in Contemporary Mathematics 13, no. 02 (April 2011): 331–73. http://dx.doi.org/10.1142/s0219199711004221.

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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).

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24

Ruan, Jianmiao, Dashan Fan, and Chunjie Zhang. "Estimates of damped fractional wave equations." Fractional Calculus and Applied Analysis 22, no. 4 (October 27, 2019): 990–1013. http://dx.doi.org/10.1515/fca-2019-0053.

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Abstract In this paper, for the high frequency part of the solution u(x, t) to the linear fractional damped wave equation, we derive asymptotic-in-time linear estimates in Triebel-Lizorkin spaces. Thus we obtain long time decay estimates in real Hardy spaces Hp for u(x, t). The obtained results are natural extension of the known Lp estimates. Our proof is based on some basic properties of the Triebel-Lizorkin space, as well as an atomic decomposition introduced by Han, Paluszynski and Weiss.

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25

Dziubański, Jacek, and Agnieszka Hejna. "Remark on atomic decompositions for the Hardy space $H^1$ in the rational Dunkl setting." Studia Mathematica 251, no. 1 (2020): 89–110. http://dx.doi.org/10.4064/sm180618-25-11.

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26

Bui, The Anh, Jun Cao, Luong Dang Ky, Dachun Yang, and Sibei Yang. "Musielak-Orlicz-Hardy Spaces Associated with Operators Satisfying Reinforced Off-Diagonal Estimates." Analysis and Geometry in Metric Spaces 1 (February 7, 2013): 69–129. http://dx.doi.org/10.2478/agms-2012-0006.

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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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Czaja, Wojciech, and Jacek Zienkiewicz. "Atomic characterization of the Hardy space $H^1_L(\mathbb R)$ of one-dimensional Schrödinger operators with nonnegative potentials." Proceedings of the American Mathematical Society 136, no. 01 (January 1, 2008): 89–95. http://dx.doi.org/10.1090/s0002-9939-07-09096-x.

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Yang, Dachun, Dongyong Yang, and Yuan Zhou. "Localized Morrey-Campanato spaces on metric measure spaces and applications to Schrüdinger operators." Nagoya Mathematical Journal 198 (June 2010): 77–119. http://dx.doi.org/10.1215/00277630-2009-008.

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AbstractLet be a space of homogeneous type in the sense of Coifman and Weiss, and let be a collection of balls in . The authors introduce the localized atomic Hardy space the localized Morrey-Campanato space and the localized Morrey-Campanato-BLO (bounded lower oscillation) space with α ∊ ℝ and p ∊ (0, ∞) , and they establish their basic properties, including and several equivalent characterizations for In particular, the authors prove that when α > 0 and p ∊ [1, ∞), then and when p ∈(0,1], then the dual space of is Let ρ be an admissible function modeled on the known auxiliary function determined by the Schrödinger operator. Denote the spaces and , respectively, by and when is determined by ρ. The authors then obtain the boundedness from of the radial and the Poisson semigroup maximal functions and the Littlewood-Paley g-function, which are defined via kernels modeled on the semigroup generated by the Schrödinger operator. These results apply in a wide range of settings, for instance, the Schrödinger operator or the degenerate Schrödinger operator on ℝd, or the sub-Laplace Schrödinger operator on Heisenberg groups or connected and simply connected nilpotent Lie groups.

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29

Zhou, Xilin, Ziyi He, and Dachun Yang. "Real-Variable Characterizations of Hardy–Lorentz Spaces on Spaces of Homogeneous Type with Applications to Real Interpolation and Boundedness of Calderón–Zygmund Operators." Analysis and Geometry in Metric Spaces 8, no. 1 (January 1, 2020): 182–260. http://dx.doi.org/10.1515/agms-2020-0109.

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AbstractLet (𝒳, d, μ) be a space of homogeneous type, in the sense of Coifman and Weiss, with the upper dimension ω. Assume that η ∈(0, 1) is the smoothness index of the wavelets on 𝒳 constructed by Auscher and Hytönen. In this article, via grand maximal functions, the authors introduce the Hardy–Lorentz spaces H_*^{p,q}\left( \mathcal{X} \right) with the optimal range p \in \left( {{\omega \over {\omega + \eta }},\infty } \right) and q ∈ (0, ∞]. When and p \in ({\omega \over {\omega + \eta }},1]q ∈ (0, ∞], the authors establish its real-variable characterizations, respectively, in terms of radial maximal functions, non-tangential maximal functions, atoms, molecules, and various Littlewood–Paley functions. The authors also obtain its finite atomic characterization. As applications, the authors establish a real interpolation theorem on Hardy–Lorentz spaces, and also obtain the boundedness of Calderón–Zygmund operators on them including the critical cases. The novelty of this article lies in getting rid of the reverse doubling assumption of μ by fully using the geometrical properties of 𝒳 expressed via its dyadic reference points and dyadic cubes and, moreover, the results in the case q ∈ (0, 1) of this article are also new even when 𝒳 satisfies the reverse doubling condition.

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Lu, Guanghui, and Shuangping Tao. "Estimates for Parameter Littlewood-Paleygκ⁎Functions on Nonhomogeneous Metric Measure Spaces." Journal of Function Spaces 2016 (2016): 1–12. http://dx.doi.org/10.1155/2016/9091478.

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Let(X,d,μ)be a metric measure space which satisfies the geometrically doubling measure and the upper doubling measure conditions. In this paper, the authors prove that, under the assumption that the kernel ofMκ⁎satisfies a certain Hörmander-type condition,Mκ⁎,ρis bounded from Lebesgue spacesLp(μ)to Lebesgue spacesLp(μ)forp≥2and is bounded fromL1(μ)intoL1,∞(μ). As a corollary,Mκ⁎,ρis bounded onLp(μ)for1<p<2. In addition, the authors also obtain thatMκ⁎,ρis bounded from the atomic Hardy spaceH1(μ)into the Lebesgue spaceL1(μ).

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31

Yang, Dachun, Dongyong Yang, and Yuan Zhou. "Localized Morrey-Campanato spaces on metric measure spaces and applications to Schrüdinger operators." Nagoya Mathematical Journal 198 (June 2010): 77–119. http://dx.doi.org/10.1017/s0027763000009946.

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AbstractLetbe a space of homogeneous type in the sense of Coifman and Weiss, and letbe a collection of balls in. The authors introduce the localized atomic Hardy spacethe localized Morrey-Campanato spaceand the localized Morrey-Campanato-BLO (bounded lower oscillation) spacewithα∊ ℝ andp∊ (0, ∞) , and they establish their basic properties, includingand several equivalent characterizations forIn particular, the authors prove that when α > 0 andp∊ [1, ∞), thenand whenp∈(0,1], then the dual space ofisLetρbe an admissible function modeled on the known auxiliary function determined by the Schrödinger operator. Denote the spacesand, respectively, byandwhenis determined byρ. The authors then obtain the boundedness fromof the radial and the Poisson semigroup maximal functions and the Littlewood-Paleyg-function, which are defined via kernels modeled on the semigroup generated by the Schrödinger operator. These results apply in a wide range of settings, for instance, the Schrödinger operator or the degenerate Schrödinger operator on ℝd, or the sub-Laplace Schrödinger operator on Heisenberg groups or connected and simply connected nilpotent Lie groups.

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32

Weisz, F. "Atomic Hardy spaces." Analysis Mathematica 20, no. 1 (March 1994): 65–80. http://dx.doi.org/10.1007/bf01908919.

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EISNER, TANJA, and BEN KRAUSE. "(Uniform) convergence of twisted ergodic averages." Ergodic Theory and Dynamical Systems 36, no. 7 (April 13, 2015): 2172–202. http://dx.doi.org/10.1017/etds.2015.6.

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Let$T$be an ergodic measure-preserving transformation on a non-atomic probability space$(X,\unicode[STIX]{x1D6F4},\unicode[STIX]{x1D707})$. We prove uniform extensions of the Wiener–Wintner theorem in two settings: for averages involving weights coming from Hardy field functions $p$,$$\begin{eqnarray}\displaystyle \bigg\{\frac{1}{N}\mathop{\sum }_{n\leq N}e(p(n))T^{n}f(x)\bigg\}; & & \displaystyle \nonumber\end{eqnarray}$$and for ‘twisted’ polynomial ergodic averages,$$\begin{eqnarray}\displaystyle \bigg\{\frac{1}{N}\mathop{\sum }_{n\leq N}e(n\unicode[STIX]{x1D703})T^{P(n)}f(x)\bigg\} & & \displaystyle \nonumber\end{eqnarray}$$for certain classes of badly approximable$\unicode[STIX]{x1D703}\in [0,1]$. We also give an elementary proof that the above twisted polynomial averages converge pointwise$\unicode[STIX]{x1D707}$-almost everywhere for$f\in L^{p}(X),p>1,$and arbitrary$\unicode[STIX]{x1D703}\in [0,1]$.

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34

Wang, Yixin, Yu Liu, Chuanhong Sun, and Pengtao Li. "Carleson measure characterizations of the Campanato type space associated with Schrödinger operators on stratified Lie groups." Forum Mathematicum 32, no. 5 (September 1, 2020): 1337–73. http://dx.doi.org/10.1515/forum-2019-0224.

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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})}.

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35

Jiao, Yong, Lian Wu, and Lihua Peng. "Weak Orlicz–Hardy martingale spaces." International Journal of Mathematics 26, no. 08 (July 2015): 1550062. http://dx.doi.org/10.1142/s0129167x15500627.

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In this paper, several weak Orlicz–Hardy martingale spaces associated with concave functions are introduced, and some weak atomic decomposition theorems for them are established. With the help of weak atomic decompositions, a sufficient condition for a sublinear operator defined on the weak Orlicz–Hardy martingale spaces to be bounded is given. Further, we investigate the duality of weak Orlicz–Hardy martingale spaces and obtain a new John–Nirenberg type inequality when the stochastic basis is regular. These results can be regarded as weak versions of the Orlicz–Hardy martingale spaces due to Miyamoto, Nakai and Sadasue.

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36

Cho, Yong-Kum, and Joonil Kim. "Atomic decomposition on Hardy–Sobolev spaces." Studia Mathematica 177, no. 1 (2006): 25–42. http://dx.doi.org/10.4064/sm177-1-3.

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Ablé, Zobo Vincent de Paul, and Justin Feuto. "Atomic decomposition of Hardy-amalgam spaces." Journal of Mathematical Analysis and Applications 455, no. 2 (November 2017): 1899–936. http://dx.doi.org/10.1016/j.jmaa.2017.06.057.

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Pérez-Esteva, Salvador, and Hugo Ocampo-Salgado. "Atomic decomposition of vector Hardy spaces." Journal of Mathematical Analysis and Applications 403, no. 2 (July 2013): 408–22. http://dx.doi.org/10.1016/j.jmaa.2013.02.017.

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Boza, Santiago, and María J. Carro. "Hardy spaces on ZN." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 132, no. 1 (February 2002): 25–43. http://dx.doi.org/10.1017/s0308210500001517.

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The work of Coifman and Weiss concerning Hardy spaces on spaces of homogeneous type gives, as a particular case, a definition of Hp(ZN) in terms of an atomic decomposition.Other characterizations of these spaces have been studied by other authors, but it was an open question to see if they can be defined, as it happens in the classical case, in terms of a maximal function or via the discrete Riesz transforms.In this paper, we give a positive answer to this question.

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40

Ding, Wei, Meidi Qin, and Yueping Zhu. "The Boundedness on Mixed Hardy Spaces." Journal of Function Spaces 2020 (February 24, 2020): 1–12. http://dx.doi.org/10.1155/2020/5341674.

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The boundedness of operators on Hardy spaces is usually given by atomic decomposition. In this paper, we obtain the boundedness of singular integral operators in mixed Journé class on mixed Hardy spaces by a direct method.

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41

Heinig, Hans P. "Fourier operators on weighted Hardy spaces." Mathematical Proceedings of the Cambridge Philosophical Society 101, no. 1 (January 1987): 113–21. http://dx.doi.org/10.1017/s0305004100066457.

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AbstractIn this note we utilize the atomic decomposition of weighted Hardy spaces to prove weighted versions of Hardy's inequality for the Fourier transform with Muckenhoupt weight. The result extends to certain integral operators with homogeneous kernels of degree −1.

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42

Wilson, James. "On the atomic decomposition for Hardy spaces." Pacific Journal of Mathematics 116, no. 1 (January 1, 1985): 201–7. http://dx.doi.org/10.2140/pjm.1985.116.201.

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HO, Kwok-Pun. "Atomic decompositions of weighted Hardy-Morrey spaces." Hokkaido Mathematical Journal 42, no. 1 (February 2013): 131–57. http://dx.doi.org/10.14492/hokmj/1362406643.

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KAWAZOE, Takeshi. "Atomic Hardy spaces on semisimple Lie groups." Japanese journal of mathematics. New series 11, no. 2 (1985): 293–343. http://dx.doi.org/10.4099/math1924.11.293.

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45

Ho, K. P. "Atomic decompositions of martingale hardy–morrey spaces." Acta Mathematica Hungarica 149, no. 1 (February 20, 2016): 177–89. http://dx.doi.org/10.1007/s10474-016-0591-4.

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46

CONGO, Mohamed, and Marie Françoise OUEDRAOGO. "BOUNDEDNESS OF PSEUDO-DIFFERENTIAL OPERATORS ON WEIGHTED HARDY SPACES AND VARIABLE EXPONENTS HARDY LOCAL MORREY SPACES." Universal Journal of Mathematics and Mathematical Sciences 18, no. 2 (January 30, 2023): 121–43. http://dx.doi.org/10.17654/2277141723008.

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In this paper, we use the atomic decomposition to establish the boundedness of pseudo-differential operators belonging to Hörmander class on weighted Hardy spaces $H^p(\omega)$ and on variable exponents Hardy local Morrey spaces $HLM_{p(\cdot)}^{u(\cdot)}$.

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47

Song, Liang, and Chaoqiang Tan. "Hardy Spaces Associated to Schrödinger Operators on Product Spaces." Journal of Function Spaces and Applications 2012 (2012): 1–17. http://dx.doi.org/10.1155/2012/179015.

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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.

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48

Almeida, Víctor, Jorge J. Betancor, and Lourdes Rodríguez-Mesa. "Anisotropic Hardy-Lorentz Spaces with Variable Exponents." Canadian Journal of Mathematics 69, no. 6 (December 1, 2017): 1219–73. http://dx.doi.org/10.4153/cjm-2016-053-6.

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AbstractIn this paper we introduceHardy-Lorentz spaces with variable exponents associated with dilations in ℝn. We establishmaximal characterizations and atomic decompositions for our variable exponent anisotropic Hardy-Lorentz spaces.

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49

Mohsenipour, Maryam, and Ghadir Sadeghi. "Atomic decompositions of martingale Hardy-Lorentz spaces and interpolation." Filomat 31, no. 19 (2017): 5921–29. http://dx.doi.org/10.2298/fil1719921m.

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In this paper, we establish atomic decompositions for the martingale Hardy-Lorentz spaces. As an application, with the help of atomic decomposition, some interpolation theorems with a function parameter for these spaces are proved.

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Huang, Jizheng, Weiwei Li, and Yaqiong Wang. "Hardy-Sobolev Spaces Associated with Twisted Convolution." Journal of Function Spaces 2017 (2017): 1–7. http://dx.doi.org/10.1155/2017/5692746.

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We first define the Hardy-Sobolev spaces associated with twisted convolution; then we give the atomic decomposition. As an application, we consider the endpoint version of the div-curl theorem for the twisted convolution.

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