arXiv subjects

Sibei Yang

Publications and source records attributed to Sibei Yang.

99 records · Page 6Linked to original sources

arXiv2014

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

Let $\mathcal{X}$ be a space of homogenous type and $φ:\ \mathcal{X}\times[0,\infty) \to[0,\infty)$ a growth function such that $φ(\cdot,t)$ is a Muckenhoupt weight uniformly in $t$ and $φ(x,\cdot)$ 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 $\mathrm{BMO}^φ_A(\mathcal{X})$ 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 $\mathrm{BMO}^φ_{A,\,\mathrm{max}}(\mathcal{X})$ and $\widetilde{\mathrm{BMO}}^φ_A(\mathcal{X})$. Moreover, two variants of the John-Nirenberg inequality on $\mathrm{BMO}^φ_A(\mathcal{X})$ are obtained. As an application, the authors further prove that the space $\mathrm{BMO}^φ_{\sqrtΔ}(\mathbb{R}^n)$, associated with the Poisson semigroup of the Laplace operator $Δ$ on $\mathbb{R}^n$, coincides with the space $\mathrm{BMO}^φ(\mathbb{R}^n)$ introduced by L. D. Ky.

math.CA↗

arXiv2013

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

Let $\mathcal{X}$ be a metric space with doubling measure and $L$ a one-to-one operator of type $ω$ having a bounded $H_\infty$-functional calculus in $L^2(\mathcal{X})$ satisfying the reinforced $(p_L, q_L)$ off-diagonal estimates on balls, where $p_L\in[1,2)$ and $q_L\in(2,\infty]$. Let $φ:\,\mathcal{X}\times[0,\infty)\to[0,\infty)$ be a function such that $φ(x,\cdot)$ is an Orlicz function, $φ(\cdot,t)\in {\mathbb A}_{\infty}(\mathcal{X})$ (the class of uniformly Muckenhoupt weights), its uniformly critical upper type index $I(φ)\in(0,1]$ and $φ(\cdot,t)$ satisfies the uniformly reverse Hölder inequality of order $(q_L/I(φ))'$. In this paper, the authors introduce a Musielak-Orlicz-Hardy space $H_{φ,\,L}(\mathcal{X})$, 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 $H_{φ,\,L}(\mathcal{X})$ is also obtained. Furthermore, a sufficient condition for the equivalence between $H_{φ,\,L}(\mathbb{R}^n)$ and the classical Musielak-Orlicz-Hardy space $H_φ(\mathbb{R}^n)$ is given. Moreover, for the Musielak-Orlicz-Hardy space $H_{φ,\,L}(\mathbb{R}^n)$ associated with the second order elliptic operator in divergence form on $\rn$ or the Schrödinger operator $L:=-Δ+V$ with $0\le V\in L^1_{\mathrm{loc}}(\mathbb{R}^n)$, the authors further obtain its several equivalent characterizations in terms of various non-tangential and radial maximal functions; finally, the authors discuss the boundedness of the Riesz transform $\nabla L^{-1/2}$.

math.CA↗

arXiv2012

Orlicz-Hardy Spaces Associated with Divergence Operators on Unbounded Strongly Lipschitz Domains of $\mathbb{R}^n$

Let $Ω$ be either $\mathbb{R}^n$ or an unbounded strongly Lipschitz domain of $\mathbb{R}^n$, and $Φ$ be a continuous, strictly increasing, subadditive and positive function on $(0,\infty)$ of upper type 1 and of strictly critical lower type $p_Φ\in(n/(n+1),1]$. Let $L$ be a divergence form elliptic operator on $L^2 (Ω)$ with the Neumann boundary condition and the heat semigroup generated by $L$ have the Gaussian property $(G_{\infty})$. In this paper, the authors introduce the Orlicz-Hardy space $H_{Φ,\,L}(Ω)$ via the nontangential maximal function associated with $\{e^{-t\sqrt{L}}\}_{t\ge0}$, and establish its equivalent characterization in terms of the Lusin area function associated with $\{e^{-t\sqrt{L}}\}_{t\ge0}$. The authors also introduce the "geometrical" Orlicz-Hardy space $H_{Φ,\,z}(Ω)$ via the classical Orlicz-Hardy space $H_Φ(\mathbb{R}^n)$, and prove that the spaces $H_{Φ,\,L}(Ω)$ and $H_{Φ,\,z}(Ω)$ coincide with equivalent norms, from which, characterizations of $H_{Φ,\,L}(Ω)$, including the vertical and the nontangential maximal function characterizations associated with $\{e^{-tL}\}_{t\ge0}$, and the Lusin area function characterization associated with $\{e^{-tL}\}_{t\ge0}$, are deduced. All the above results generalize the well-known results of P. Auscher and E. Russ by taking $Φ(t)\equiv t$ for all $t\in(0,\infty)$.

math.CA↗

arXiv2012

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 $φ:\,\mathcal{X}\times[0,\infty)\to[0,\infty)$ be a function such that $φ(x,\cdot)$ is an Orlicz function, $φ(\cdot,t)\in A_{\infty}(\mathcal{X})$ (the class of Muckenhoupt weights) and its uniformly critical lower type index $i(φ)\in(0,1]$. In this paper, the authors introduce a Musielak-Orlicz Hardy space $H_{φ,\,L}(\mathcal{X})$ by the Lusin area function associated with the heat semigroup generated by $L$, and a Musielak-Orlicz $\mathop\mathrm{BMO}$-type space $\mathop\mathrm{BMO}_{φ,\,L}(\mathcal{X})$ which is further proved to be the dual space of $H_{φ,\,L}(\mathcal{X})$; as a corollary, the authors obtain the $φ$-Carleson measure characterization of $\mathop\mathrm{BMO}_{φ,\,L}(\mathcal{X})$. Characterizations of $H_{φ,\,L}(\mathcal{X})$, 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 $H_{φ,\,L}(\mathcal{X})$ in terms of $g^\ast_{λ,\,L}$. As further applications, the authors obtain several equivalent characterizations of the Musielak-Orlicz Hardy space $H_{φ,\,L}(\mathbb{R}^n)$ associated with the Schrödinger operator $L=-Δ+V$, 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.

math.CA↗

arXiv2012

Weighted Local Orlicz-Hardy Spaces on Domains and Their Applications in Inhomogeneous Dirichlet and Neumann Problems

Let $Ω$ be either $\mathbb{R}^n$ or a strongly Lipschitz domain of $\mathbb{R}^n$, and $ω\in A_{\infty}(\mathbb{R}^n)$ (the class of Muckenhoupt weights). Let $L$ be a second order divergence form elliptic operator on $L^2 (Ω)$ with the Dirichlet or Neumann boundary condition, and assume that the heat semigroup generated by $L$ has the Gaussian property $(G_1)$ with the regularity of their kernels measured by $μ\in(0,1]$. Let $Φ$ be a continuous, strictly increasing, subadditive, positive and concave function on $(0,\infty)$ of critical lower type index $p_Φ^-\in(0,1]$. In this paper, the authors introduce the "geometrical" weighted local Orlicz-Hardy spaces $h^Φ_{ω,\,r}(Ω)$ and $h^Φ_{ω,\,z}(Ω)$ via the weighted local Orlicz-Hardy spaces $h^Φ_ω(\mathbb{R}^n)$, and obtain their two equivalent characterizations in terms of the nontangential maximal function and the Lusin area function associated with the heat semigroup generated by $L$ when $p_Φ^-\in(n/(n+μ),1]$. As applications, the authors prove that the operators $\nabla^2{\mathbb G}_D$ are bounded from $h^Φ_{ω,\,r}(Ω)$ to the weighted Orlicz space $L^Φ_ω(Ω)$, and from $h^Φ_{ω,\,r}(Ω)$ to itself when $Ω$ is a bounded semiconvex domain in $\mathbb{R}^n$ and $p_Φ^-\in(\frac{n}{n+1},1]$, and the operators $\nabla^2{\mathbb G}_N$ are bounded from $h^Φ_{ω,\,z}(Ω)$ to $L^Φ_ω(Ω)$, and from $h^Φ_{ω,\,z}(Ω)$ to $h^Φ_{ω,\,r}(Ω)$ when $Ω$ is a bounded convex domain in $\mathbb{R}^n$ and $p_Φ^-\in(\frac{n}{n+1},1]$, where ${\mathbb G}_D$ and ${\mathbb G}_N$ denote, respectively, the Dirichlet Green operator and the Neumann Green operator.

math.CA↗

arXiv2012

Endpoint Boundedness of Riesz Transforms on Hardy Spaces Associated with Operators

Let $L_1$ be a nonnegative self-adjoint operator in $L^2({\mathbb R}^n)$ satisfying the Davies-Gaffney estimates and $L_2$ a second order divergence form elliptic operator with complex bounded measurable coefficients. A typical example of $L_1$ is the Schrödinger operator $-Δ+V$, where $Δ$ is the Laplace operator on ${\mathbb R}^n$ and $0\le V\in L^1_{\mathop\mathrm{loc}} ({\mathbb R}^n)$. Let $H^p_{L_i}(\mathbb{R}^n)$ be the Hardy space associated to $L_i$ for $i\in\{1,\,2\}$. In this paper, the authors prove that the Riesz transform $D (L_i^{-1/2})$ is bounded from $H^p_{L_i}(\mathbb{R}^n)$ to the classical weak Hardy space $WH^p(\mathbb{R}^n)$ in the critical case that $p=n/(n+1)$. Recall that it is known that $D (L_i^{-1/2})$ is bounded from $H^p_{L_i}(\mathbb{R}^n)$ to the classical Hardy space $H^p(\mathbb{R}^n)$ when $p\in(n/(n+1),\,1]$.

math.CA↗

arXiv2012

Local Hardy Spaces of Musielak-Orlicz Type and Their Applications

Let $ϕ: \mathbb{R}^n\times[0,\fz)\rightarrow[0,\fz)$ be a function such that $ϕ(x,\cdot)$ is an Orlicz function and $ϕ(\cdot,t)\in A^{\mathop\mathrm{loc}}_{\infty}(\mathbb{R}^n)$ (the class of local weights introduced by V. S. Rychkov). In this paper, the authors introduce a local Hardy space $h_ϕ(\mathbb{R}^n)$ of Musielak-Orlicz type by the local grand maximal function, and a local $\mathop\mathrm{BMO}$-type space $\mathop\mathrm{bmo}_ϕ(\mathbb{R}^n)$ which is further proved to be the dual space of $h_ϕ(\mathbb{R}^n)$. As an application, the authors prove that the class of pointwise multipliers for the local $\mathop\mathrm{BMO}$-type space $\mathop\mathrm{bmo}^ϕ(\mathbb{R}^n)$, characterized by E. Nakai and K. Yabuta, is just the dual of $L^1(\rn)+h_{Φ_0}(\mathbb{R}^n)$, where $ϕ$ is an increasing function on $(0,\infty)$ satisfying some additional growth conditions and $Φ_0$ a Musielak-Orlicz function induced by $ϕ$. Characterizations of $h_ϕ(\mathbb{R}^n)$, including the atoms, the local vertical and the local nontangential maximal functions, are presented. Using the atomic characterization, the authors prove the existence of finite atomic decompositions achieving the norm in some dense subspaces of $h_ϕ(\mathbb{R}^n)$, from which, the authors further deduce some criterions for the boundedness on $h_ϕ(\mathbb{R}^n)$ of some sublinear operators. Finally, the authors show that the local Riesz transforms and some pseudo-differential operators are bounded on $h_ϕ(\mathbb{R}^n)$.

math.CA↗

arXiv2011

Weighted Local Orlicz-Hardy Spaces with Applications to Pseudo-differential Operators

Let $Φ$ be a concave function on $(0,\infty)$ of strictly lower type $p_Φ\in(0,1]$ and $ω\in A^{\mathop\mathrm{loc}}_{\infty}(\mathbb{R}^n)$. We introduce the weighted local Orlicz-Hardy space $h^Φ_ω(\mathbb{R}^n)$ via the local grand maximal function. Let $ρ(t)\equiv t^{-1}/Φ^{-1}(t^{-1})$ for all $t\in(0,\infty)$. We also introduce the $\mathop\mathrm{BMO}$-type space $\mathop\mathrm{bmo}_{ρ,\,ω}(\mathbb{R}^n)$ and establish the duality between $h^Φ_ω(\mathbb{R}^n)$ and $\mathop\mathrm{bmo}_{ρ,\,ω}(\mathbb{R}^n)$. Several real-varaiable characterizations of $h^Φ_ω(\mathbb{R}^n)$ are presented. Using the atomic characterization, we prove the existence of finite atomic decompositions achieving the norm in some dense subspaces of $h^Φ_ω(\mathbb{R}^n)$. As applications, we show that the local Riesz transforms are bounded on $h^Φ_ω(\mathbb{R}^n)$, the local fractional integrals are bounded from {\normalsize$h^p_{ω^p}(\mathbb{R}^n)$} to {\normalsize$L^q_{ω^q}(\mathbb{R}^n)$} when $q>1$ and from {\normalsize$h^p_{ω^p}(\mathbb{R}^n)$} to {\normalsize$h^q_{ω^q}(\mathbb{R}^n)$} when $q\le 1$, and some pseudo-differential operators are also bounded on both $h^Φ_ω(\mathbb{R}^n)$. All results for any general $Φ$ even when $ω\equiv 1$ are new.

math.CA↗

arXiv2011

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

Let $Ω$ 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 $L^2 (Ω)$ with the Dirichlet boundary condition, and the heat semigroup generated by $L$ have the Gaussian property $(G_{\mathrm{diam}(Ω)})$ with the regularity of their kernels measured by $μ\in(0,1]$, where $\mathrm{diam}(Ω)$ denotes the diameter of $Ω$. Let $Φ$ be a continuous, strictly increasing, subadditive and positive function on $(0,\infty)$ of upper type 1 and of strictly critical lower type $p_Φ\in(n/(n+μ),1]$. In this paper, the authors introduce the Orlicz-Hardy space $H_{Φ,\,r}(Ω)$ by restricting arbitrary elements of the Orlicz-Hardy space $H_Φ(\mathbb{R}^n)$ 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 $H_{Φ,\,r}(\boz)$ in terms of the nontangential maximal function and the Lusin area function associated with the heat semigroup generated by $L$.

math.CA↗