402 research outputs found

    Real-Variable Characterizations Of Hardy Spaces Associated With Bessel Operators

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    Let Ξ»>0\lambda>0, p\in((2\lz+1)/(2\lz+2), 1], and β–³Ξ»β‰‘βˆ’d2dx2βˆ’2Ξ»xddx\triangle_\lambda\equiv-\frac{d^2}{dx^2}-\frac{2\lambda}{x} \frac d{dx} be the Bessel operator. In this paper, the authors establish the characterizations of atomic Hardy spaces Hp((0,∞),dmΞ»)H^p((0, \infty), dm_\lambda) associated with β–³Ξ»\triangle_\lambda in terms of the radial maximal function, the nontangential maximal function, the grand maximal function, the Littlewood-Paley gg-function and the Lusin-area function, where dmΞ»(x)≑x2λ dxdm_\lambda(x)\equiv x^{2\lambda}\,dx. As an application, the authors further obtain the Riesz transform characterization of these Hardy spaces.Comment: Anal. Appl. (Singap.) (to appear

    Real-variable Characterizations of Orlicz-Hardy Spaces on Strongly Lipschitz Domains of Rn\mathbb{R}^n

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    Let Ξ©\Omega be a strongly Lipschitz domain of Rn\mathbb{R}^n, whose complement in Rn\mathbb{R}^n is unbounded. Let LL be a second order divergence form elliptic operator on L2(Ξ©)L^2 (\Omega) with the Dirichlet boundary condition, and the heat semigroup generated by LL have the Gaussian property (Gdiam(Ξ©))(G_{\mathrm{diam}(\Omega)}) with the regularity of their kernels measured by μ∈(0,1]\mu\in(0,1], where diam(Ξ©)\mathrm{diam}(\Omega) denotes the diameter of Ξ©\Omega. Let Ξ¦\Phi be a continuous, strictly increasing, subadditive and positive function on (0,∞)(0,\infty) of upper type 1 and of strictly critical lower type pΦ∈(n/(n+ΞΌ),1]p_{\Phi}\in(n/(n+\mu),1]. In this paper, the authors introduce the Orlicz-Hardy space HΞ¦, r(Ξ©)H_{\Phi,\,r}(\Omega) by restricting arbitrary elements of the Orlicz-Hardy space HΞ¦(Rn)H_{\Phi}(\mathbb{R}^n) to \boz and establish its atomic decomposition by means of the Lusin area function associated with {eβˆ’tL}tβ‰₯0\{e^{-tL}\}_{t\ge0}. Applying this, the authors obtain two equivalent characterizations of H_{\Phi,\,r}(\boz) in terms of the nontangential maximal function and the Lusin area function associated with the heat semigroup generated by LL.Comment: 65 pages, Rev. Mat. Iberoam. (to appear

    Musielak-Orlicz Hardy Spaces Associated with Operators and Their Applications

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    Let X\mathcal{X} be a metric space with doubling measure and LL a nonnegative self-adjoint operator in L2(X)L^2(\mathcal{X}) satisfying the Davies-Gaffney estimates. Let Ο†: XΓ—[0,∞)β†’[0,∞)\varphi:\,\mathcal{X}\times[0,\infty)\to[0,\infty) be a function such that Ο†(x,β‹…)\varphi(x,\cdot) is an Orlicz function, Ο†(β‹…,t)∈A∞(X)\varphi(\cdot,t)\in A_{\infty}(\mathcal{X}) (the class of Muckenhoupt weights) and its uniformly critical lower type index i(Ο†)∈(0,1]i(\varphi)\in(0,1]. In this paper, the authors introduce a Musielak-Orlicz Hardy space HΟ†, L(X)H_{\varphi,\,L}(\mathcal{X}) by the Lusin area function associated with the heat semigroup generated by LL, and a Musielak-Orlicz BMO\mathop\mathrm{BMO}-type space BMOΟ†, L(X)\mathop\mathrm{BMO}_{\varphi,\,L}(\mathcal{X}) which is further proved to be the dual space of HΟ†, L(X)H_{\varphi,\,L}(\mathcal{X}); as a corollary, the authors obtain the Ο†\varphi-Carleson measure characterization of BMOΟ†, L(X)\mathop\mathrm{BMO}_{\varphi,\,L}(\mathcal{X}). Characterizations of HΟ†, L(X)H_{\varphi,\,L}(\mathcal{X}), including the atom, the molecule and the Lusin area function associated with the Poisson semigroup of LL, are presented. Using the atomic characterization, the authors characterize HΟ†, L(X)H_{\varphi,\,L}(\mathcal{X}) in terms of gΞ», Lβˆ—g^\ast_{\lambda,\,L}. As further applications, the authors obtain several equivalent characterizations of the Musielak-Orlicz Hardy space HΟ†, L(Rn)H_{\varphi,\,L}(\mathbb{R}^n) associated with the Schr\"odinger operator L=βˆ’Ξ”+VL=-\Delta+V, where 0≀V∈Lloc1(Rn)0\le V\in L^1_{\mathrm{loc}}(\mathbb{R}^n) is a nonnegative potential, in terms of the Lusin-area function, the non-tangential maximal function, the radial maximal function, the atom and the molecule.Comment: J. Geom. Anal. (to appear

    Boundedness of Linear Operators via Atoms on Hardy Spaces with Non-doubling Measures

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    Let μ\mu be a non-negative Radon measure on Rd{\mathbb R}^d which only satisfies the polynomial growth condition. Let Y{\mathcal Y} be a Banach space and H1(μ)H^1(\mu) the Hardy space of Tolsa. In this paper, the authors prove that a linear operator TT is bounded from H1(μ)H^1(\mu) to Y{\mathcal Y} if and only if TT maps all (p,γ)(p, \gamma)-atomic blocks into uniformly bounded elements of Y{\mathcal Y}; moreover, the authors prove that for a sublinear operator TT bounded from L1(μ)L^1(\mu) to L1,∞(μ)L^{1, \infty}(\mu), if TT maps all (p,γ)(p, \gamma)-atomic blocks with p∈(1,∞)p\in(1, \infty) and γ∈N\gamma\in{\mathbb N} into uniformly bounded elements of L1(μ)L^1(\mu), then TT extends to a bounded sublinear operator from H1(μ)H^1(\mu) to L1(μ)L^1(\mu). For the localized atomic Hardy space h1(μ)h^1(\mu), corresponding results are also presented. Finally, these results are applied to Calder\'on-Zygmund operators, Riesz potentials and multilinear commutators generated by Calder\'on-Zygmund operators or fractional integral operators with Lipschitz functions, to simplify the existing proofs in the corresponding papers.Comment: Georgian Math. J. (to appear

    Maximal Function Characterizations of Variable Hardy Spaces Associated with Non-negative Self-adjoint Operators Satisfying Gaussian Estimates

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    Let p(β‹…):Β Rnβ†’(0,1]p(\cdot):\ \mathbb R^n\to(0,1] be a variable exponent function satisfying the globally log⁑\log-H\"older continuous condition and LL a non-negative self-adjoint operator on L2(Rn)L^2(\mathbb R^n) whose heat kernels satisfying the Gaussian upper bound estimates. Let HLp(β‹…)(Rn)H_L^{p(\cdot)}(\mathbb R^n) be the variable exponent Hardy space defined via the Lusin area function associated with the heat kernels {eβˆ’t2L}t∈(0,∞)\{e^{-t^2L}\}_{t\in (0,\infty)}. In this article, the authors first establish the atomic characterization of HLp(β‹…)(Rn)H_L^{p(\cdot)}(\mathbb R^n); using this, the authors then obtain its non-tangential maximal function characterization which, when p(β‹…)p(\cdot) is a constant in (0,1](0,1], coincides with a recent result by Song and Yan [Adv. Math. 287 (2016), 463-484] and further induces the radial maximal function characterization of HLp(β‹…)(Rn)H_L^{p(\cdot)}(\mathbb R^n) under an additional assumption that the heat kernels of LL have the H\"older regularity.Comment: 32 pages, submitted. arXiv admin note: text overlap with arXiv:1512.0595

    Musielak-Orlicz BMO-Type Spaces Associated with Generalized Approximations to the Identity

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    Let X\mathcal{X} be a space of homogenous type and $\varphi:\ \mathcal{X}\times[0,\infty) \to[0,\infty)agrowthfunctionsuchthat a growth function such that \varphi(\cdot,t)isaMuckenhouptweightuniformlyin is a Muckenhoupt weight uniformly in tand and \varphi(x,\cdot)anOrliczfunctionofuniformlyuppertype1andlowertype an Orlicz function of uniformly upper type 1 and lower type p\in(0,1].Inthisarticle,theauthorsintroduceanewMusielakβˆ’OrliczBMOβˆ’typespace. In this article, the authors introduce a new Musielak-Orlicz BMO-type space \mathrm{BMO}^{\varphi}_A(\mathcal{X})associatedwiththegeneralizedapproximationtotheidentity,giveoutitsbasicpropertiesandestablishitstwoequivalentcharacterizations,respectively,intermsofthespaces associated with the generalized approximation to the identity, give out its basic properties and establish its two equivalent characterizations, respectively, in terms of the spaces \mathrm{BMO}^{\varphi}_{A,\,\mathrm{max}}(\mathcal{X})and and \widetilde{\mathrm{BMO}}^{\varphi}_A(\mathcal{X}).Moreover,twovariantsoftheJohnβˆ’Nirenberginequalityon. Moreover, two variants of the John-Nirenberg inequality on \mathrm{BMO}^{\varphi}_A(\mathcal{X})areobtained.Asanapplication,theauthorsfurtherprovethatthespace are obtained. As an application, the authors further prove that the space \mathrm{BMO}^{\varphi}_{\sqrt{\Delta}}(\mathbb{R}^n),associatedwiththePoissonsemigroupoftheLaplaceoperator, associated with the Poisson semigroup of the Laplace operator \Deltaon on \mathbb{R}^n,coincideswiththespace, coincides with the space \mathrm{BMO}^{\varphi}(\mathbb{R}^n)$ introduced by L. D. Ky.Comment: Acta Math. Sin. (Engl. Ser.) (to appear

    Boundedness and compactness characterizations of Cauchy integral commutators on Morrey spaces

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    Let CΞ“C_\Gamma be the Cauchy integral operator on a Lipschitz curve Ξ“\Gamma. In this article, the authors show that the commutator [b,CΞ“][b,C_\Gamma] is bounded (resp., compact) on the Morrey space Lp, λ(R)L^{p,\,\lambda}(\mathbb R) for any (or some) p∈(1,∞)p\in(1, \infty) and λ∈(0,1)\lambda\in(0, 1) if and only if b∈BMO(R)b\in {\rm BMO}(\mathbb R) (resp., CMO(R){\rm CMO}(\mathbb R)). As an application, a factorization of the classical Hardy space H1(R)H^1(\mathbb R) in terms of CΞ“C_\Gamma and its adjoint operator is obtained.Comment: 26 page

    Lusin Area Function and Molecular Characterizations of Musielak-Orlicz Hardy Spaces and Their Applications

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    Lusin Area Function and Molecular Characterizations of Musielak-Orlicz Hardy Spaces and Their ApplicationsLet Ο†:RnΓ—[0,∞)β†’[0,∞)\varphi: \mathbb R^n\times [0,\infty)\to[0,\infty) be a growth function such that Ο†(x,β‹…)\varphi(x,\cdot) is nondecreasing, Ο†(x,0)=0\varphi(x,0)=0, Ο†(x,t)>0\varphi(x,t)>0 when t>0t>0, lim⁑tβ†’βˆžΟ†(x,t)=∞\lim_{t\to\infty}\varphi(x,t)=\infty, and Ο†(β‹…,t)\varphi(\cdot,t) is a Muckenhoupt A∞(Rn)A_\infty(\mathbb{R}^n) weight uniformly in tt. In this paper, the authors establish the Lusin area function and the molecular characterizations of the Musielak-Orlicz Hardy space HΟ†(Rn)H_\varphi(\mathbb{R}^n) introduced by Luong Dang Ky via the grand maximal function. As an application, the authors obtain the Ο†\varphi-Carleson measure characterization of the Musielak-Orlicz BMO{\mathop\mathrm{BMO}}-type space BMOΟ†(Rn)\mathop\mathrm{BMO}_{\varphi}(\mathbb{R}^n), which was proved to be the dual space of HΟ†(Rn)H_\varphi(\mathbb{R}^n) by Luong Dang Ky

    Characterizations of BMO Associated with Gauss Measures via Commutators of Local Fractional Integrals

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    Let dΞ³(x)β‰‘Ο€βˆ’n/2eβˆ’βˆ£x∣2dxd\gamma(x)\equiv\pi^{-n/2}e^{-|x|^2}dx for all x∈Rnx\in{\mathbb R}^n be the Gauss measure on Rn{\mathbb R}^n. In this paper, the authors establish the characterizations of the space BMO(Ξ³)(\gamma) of Mauceri and Meda via commutators of either local fractional integral operators or local fractional maximal operators. To this end, the authors first prove that such a local fractional integral operator of order Ξ²\beta is bounded from Lp(Ξ³)L^p(\gamma) to Lp/(1βˆ’pΞ²)(Ξ³)L^{p/(1-p\beta)}(\gamma), or from the Hardy space H1(Ξ³)H^1(\gamma) of Mauceri and Meda to L1/(1βˆ’Ξ²)(Ξ³)L^{1/(1-\beta)}(\gamma) or from L1/Ξ²(Ξ³)L^{1/\beta}(\gamma) to BMO(Ξ³)(\gamma), where β∈(0,1)\beta\in(0, 1) and p∈(1,1/Ξ²)p\in(1, 1/\beta).Comment: 25 pages; Israel J. Math. (to appear

    Weighted LpL^p Estimates of Kato Square Roots Associated to Degenerate Elliptic Operators

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    Let ww be a Muckenhoupt A2(Rn)A_2(\mathbb{R}^n) weight and Lw:=βˆ’wβˆ’1div(Aβˆ‡)L_w:=-w^{-1}\mathop\mathrm{div}(A\nabla) the degenerate elliptic operator on the Euclidean space Rn\mathbb{R}^n, nβ‰₯2n\geq 2. In this article, the authors establish some weighted LpL^p estimates of Kato square roots associated to the degenerate elliptic operators LwL_w. More precisely, the authors prove that, for w∈Ap(Rn)w\in A_{p}(\mathbb{R}^n), p∈(2nn+1, 2]p\in(\frac{2n}{n+1},\,2] and any f∈Cc∞(Rn)f\in C^\infty_c(\mathbb{R}^n), βˆ₯Lw1/2(f)βˆ₯Lp(w, Rn)∼βˆ₯βˆ‡fβˆ₯Lp(w, Rn)\|L_w^{1/2}(f)\|_{L^p(w,\,\mathbb{R}^n)}\sim \|\nabla f\|_{L^p(w,\,\mathbb{R}^n)}, where Cc∞(Rn)C_c^\infty(\mathbb{R}^n) denotes the set of all infinitely differential functions with compact supports.Comment: 40 pages, Submitte
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