Real-Variable Characterizations Of Hardy Spaces Associated With Bessel Operators

Let $\lambda>0$, p\in((2\lz+1)/(2\lz+2), 1], and $\triangle_\lambda\equiv-\frac{d^2}{dx^2}-\frac{2\lambda}{x} \frac d{dx}$ be the Bessel operator. In this paper, the authors establish the characterizations of atomic Hardy spaces $H^p((0, \infty), dm_\lambda)$ associated with $\triangle_\lambda$ in terms of the radial maximal function, the nontangential maximal function, the grand maximal function, the Littlewood-Paley $g$-function and the Lusin-area function, where $dm_\lambda(x)\equiv x^{2\lambda}\,dx$. As an application, the authors further obtain the Riesz transform characterization of these Hardy spaces.Comment: Anal. Appl. (Singap.) (to appear

Musielak-Orlicz Hardy Spaces Associated with Operators and Their Applications

Let $\mathcal{X}$ be a metric space with doubling measure and $L$ a nonnegative self-adjoint operator in $L^2(\mathcal{X})$ satisfying the Davies-Gaffney estimates. Let $\varphi:\,\mathcal{X}\times[0,\infty)\to[0,\infty)$ be a function such that $\varphi(x,\cdot)$ is an Orlicz function, $\varphi(\cdot,t)\in A_{\infty}(\mathcal{X})$ (the class of Muckenhoupt weights) and its uniformly critical lower type index $i(\varphi)\in(0,1]$. In this paper, the authors introduce a Musielak-Orlicz Hardy space $H_{\varphi,\,L}(\mathcal{X})$ by the Lusin area function associated with the heat semigroup generated by $L$, and a Musielak-Orlicz $\mathop\mathrm{BMO}$-type space $\mathop\mathrm{BMO}_{\varphi,\,L}(\mathcal{X})$ which is further proved to be the dual space of $H_{\varphi,\,L}(\mathcal{X})$; as a corollary, the authors obtain the $\varphi$-Carleson measure characterization of $\mathop\mathrm{BMO}_{\varphi,\,L}(\mathcal{X})$. Characterizations of $H_{\varphi,\,L}(\mathcal{X})$, including the atom, the molecule and the Lusin area function associated with the Poisson semigroup of $L$, are presented. Using the atomic characterization, the authors characterize $H_{\varphi,\,L}(\mathcal{X})$ in terms of $g^\ast_{\lambda,\,L}$. As further applications, the authors obtain several equivalent characterizations of the Musielak-Orlicz Hardy space $H_{\varphi,\,L}(\mathbb{R}^n)$ associated with the Schr\"odinger operator $L=-\Delta+V$, where $0\le V\in L^1_{\mathrm{loc}}(\mathbb{R}^n)$ 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

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

Let $\Omega$ be a strongly Lipschitz domain of $\mathbb{R}^n$, whose complement in $\mathbb{R}^n$ is unbounded. Let $L$ be a second order divergence form elliptic operator on $L^2 (\Omega)$ with the Dirichlet boundary condition, and the heat semigroup generated by $L$ have the Gaussian property $(G_{\mathrm{diam}(\Omega)})$ with the regularity of their kernels measured by $\mu\in(0,1]$, where $\mathrm{diam}(\Omega)$ denotes the diameter of $\Omega$. Let $\Phi$ be a continuous, strictly increasing, subadditive and positive function on $(0,\infty)$ of upper type 1 and of strictly critical lower type $p_{\Phi}\in(n/(n+\mu),1]$. In this paper, the authors introduce the Orlicz-Hardy space $H_{\Phi,\,r}(\Omega)$ by restricting arbitrary elements of the Orlicz-Hardy space $H_{\Phi}(\mathbb{R}^n)$ to \boz and establish its atomic decomposition by means of the Lusin area function associated with $\{e^{-tL}\}_{t\ge0}$. 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 $L$.Comment: 65 pages, Rev. Mat. Iberoam. (to appear

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

Let $\mu$ be a non-negative Radon measure on ${\mathbb R}^d$ which only satisfies the polynomial growth condition. Let ${\mathcal Y}$ be a Banach space and $H^1(\mu)$ the Hardy space of Tolsa. In this paper, the authors prove that a linear operator $T$ is bounded from $H^1(\mu)$ to ${\mathcal Y}$ if and only if $T$ maps all $(p, \gamma)$-atomic blocks into uniformly bounded elements of ${\mathcal Y}$; moreover, the authors prove that for a sublinear operator $T$ bounded from $L^1(\mu)$ to $L^{1, \infty}(\mu)$, if $T$ maps all $(p, \gamma)$-atomic blocks with $p\in(1, \infty)$ and $\gamma\in{\mathbb N}$ into uniformly bounded elements of $L^1(\mu)$, then $T$ extends to a bounded sublinear operator from $H^1(\mu)$ to $L^1(\mu)$. For the localized atomic Hardy space $h^1(\mu)$, 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

Let $p(\cdot):\ \mathbb R^n\to(0,1]$ be a variable exponent function satisfying the globally $\log$-H\"older continuous condition and $L$ a non-negative self-adjoint operator on $L^2(\mathbb R^n)$ whose heat kernels satisfying the Gaussian upper bound estimates. Let $H_L^{p(\cdot)}(\mathbb R^n)$ be the variable exponent Hardy space defined via the Lusin area function associated with the heat kernels $\{e^{-t^2L}\}_{t\in (0,\infty)}$. In this article, the authors first establish the atomic characterization of $H_L^{p(\cdot)}(\mathbb R^n)$; using this, the authors then obtain its non-tangential maximal function characterization which, when $p(\cdot)$ is a constant in $(0,1]$, coincides with a recent result by Song and Yan [Adv. Math. 287 (2016), 463-484] and further induces the radial maximal function characterization of $H_L^{p(\cdot)}(\mathbb R^n)$ under an additional assumption that the heat kernels of $L$ 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

Let $\mathcal{X}$ be a space of homogenous type and $\varphi:\ \mathcal{X}\times[0,\infty) \to[0,\infty)$a growth function such that$\varphi(\cdot,t)$is a Muckenhoupt weight uniformly in$t$and$\varphi(x,\cdot)$an Orlicz function of uniformly upper type 1 and lower type$p\in(0,1]$. In this article, the authors introduce a new Musielak-Orlicz BMO-type space$\mathrm{BMO}^{\varphi}_A(\mathcal{X})$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$\widetilde{\mathrm{BMO}}^{\varphi}_A(\mathcal{X})$. Moreover, two variants of the John-Nirenberg inequality on$\mathrm{BMO}^{\varphi}_A(\mathcal{X})$are obtained. As an application, the authors further prove that the space$\mathrm{BMO}^{\varphi}_{\sqrt{\Delta}}(\mathbb{R}^n)$, associated with the Poisson semigroup of the Laplace operator$\Delta$on$\mathbb{R}^n$, 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

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

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

Lusin Area Function and Molecular Characterizations of Musielak-Orlicz Hardy Spaces and Their ApplicationsLet $\varphi: \mathbb R^n\times [0,\infty)\to[0,\infty)$ be a growth function such that $\varphi(x,\cdot)$ is nondecreasing, $\varphi(x,0)=0$, $\varphi(x,t)>0$ when $t>0$, $\lim_{t\to\infty}\varphi(x,t)=\infty$, and $\varphi(\cdot,t)$ is a Muckenhoupt $A_\infty(\mathbb{R}^n)$ weight uniformly in $t$. In this paper, the authors establish the Lusin area function and the molecular characterizations of the Musielak-Orlicz Hardy space $H_\varphi(\mathbb{R}^n)$ introduced by Luong Dang Ky via the grand maximal function. As an application, the authors obtain the $\varphi$-Carleson measure characterization of the Musielak-Orlicz ${\mathop\mathrm{BMO}}$-type space $\mathop\mathrm{BMO}_{\varphi}(\mathbb{R}^n)$, which was proved to be the dual space of $H_\varphi(\mathbb{R}^n)$ by Luong Dang Ky

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

Let $w$ be a Muckenhoupt $A_2(\mathbb{R}^n)$ weight and $L_w:=-w^{-1}\mathop\mathrm{div}(A\nabla)$ the degenerate elliptic operator on the Euclidean space $\mathbb{R}^n$, $n\geq 2$. In this article, the authors establish some weighted $L^p$ estimates of Kato square roots associated to the degenerate elliptic operators $L_w$. More precisely, the authors prove that, for $w\in A_{p}(\mathbb{R}^n)$, $p\in(\frac{2n}{n+1},\,2]$ and any $f\in C^\infty_c(\mathbb{R}^n)$, $\|L_w^{1/2}(f)\|_{L^p(w,\,\mathbb{R}^n)}\sim \|\nabla f\|_{L^p(w,\,\mathbb{R}^n)}$, where $C_c^\infty(\mathbb{R}^n)$ denotes the set of all infinitely differential functions with compact supports.Comment: 40 pages, Submitte

Atomic Characterizations of Weak Martingale Musielak--Orlicz Hardy Spaces and Their Applications

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space and $\varphi:\ \Omega\times[0,\infty)\to [0,\infty)$be a Musielak--Orlicz function. In this article, the authors establish the atomic characterizations of weak martingale Musielak--Orlicz Hardy spaces$WH_{\varphi}^s(\Omega)$,$WH_{\varphi}^M(\Omega)$,$WH_{\varphi}^S(\Omega)$,$WP_{\varphi}(\Omega)$and$WQ_{\varphi}(\Omega)$. Using these atomic characterizations, the authors then obtain the boundedness of sublinear operators from weak martingale Musielak--Orlicz Hardy spaces to weak Musielak--Orlicz spaces, and some martingale inequalities which further clarify the relationships among$WH_{\varphi}^s(\Omega)$,$WH_{\varphi}^M(\Omega)$,$WH_{\varphi}^S(\Omega)$,$WP_{\varphi}(\Omega)$and$WQ_{\varphi}(\Omega)$. All these results improve and generalize the corresponding results on weak martingale Orlicz--Hardy spaces. Moreover, the authors also improve all the known results on weak martingale Musielak--Orlicz Hardy spaces. In particular, both the boundedness of sublinear operators and the martingale inequalities, for the weak weighted martingale Hardy spaces as well as for the weak weighted martingale Orlicz--Hardy spaces, are new.Comment: 28 pages; Submitted. arXiv admin note: text overlap with arXiv:1810.0500