arXiv subjects

Dachun Yang

Publications and source records attributed to Dachun Yang.

At least 199 records · Page 11Linked to original sources

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

The Hardy Space $H^1$ on Non-homogeneous Metric Spaces

Let $({\mathcal X}, d, μ)$ be a metric measure space and satisfy the so-called upper doubling condition and the geometrical doubling condition. In this paper, we introduce the atomic Hardy space $H^1(μ)$ and prove that its dual space is the known space ${\rm RBMO}(μ)$ in this context. Using this duality, we establish a criterion for the boundedness of linear operators from $H^1(μ)$ to any Banach space. As an application of this criterion, we obtain the boundedness of Calderón--Zygmund operators from $H^1(μ)$ to $L^1(μ)$.

math.CA↗

arXiv2012

New Characterizations of Besov-Triebel-Lizorkin-Hausdorff Spaces Including Coorbits and Wavelets

In this paper, the authors establish new characterizations of the recently introduced Besov-type spaces $\dot{B}^{s,τ}_{p,q}({\mathbb R}^n)$ and Triebel-Lizorkin-type spaces $\dot{F}^{s,τ}_{p,q}({\mathbb R}^n)$ with $p\in (0,\infty]$, $s\in{\mathbb R}$, $τ\in [0,\infty)$, and $q\in (0,\infty]$, as well as their preduals, the Besov-Hausdorff spaces $B\dot{H}^{s,τ}_{p,q}(\R^n)$ and Triebel-Lizorkin-Hausdorff spaces $F\dot{H}^{s,τ}_{p,q}(\R^n)$, in terms of the local means, the Peetre maximal function of local means, and the tent space (the Lusin area function) in both discrete and continuous types. As applications, the authors then obtain interpretations as coorbits in the sense of H. Rauhut in [Studia Math. 180 (2007), 237-253] and discretizations via the biorthogonal wavelet bases for the full range of parameters of these function spaces. Even for some special cases of this setting such as $\dot F^s_{\infty,q}({\mathbb R}^n)$ for $s\in{\mathbb R}$, $q\in (0,\infty]$ (including $\mathop\mathrm{BMO} ({\mathbb R}^n)$ when $s=0$, $q=2$), the $Q$ space $Q_α({\mathbb R}^n)$, the Hardy-Hausdorff space $HH_{-α}({\mathbb R}^n)$ for $α\in (0,\min\{\frac n2,1\})$, the Morrey space ${\mathcal M}^u_p({\mathbb R}^n)$ for $1<p\le u<\infty$, and the Triebel-Lizorkin-Morrey space $\dot{\mathcal{E}}^s_{upq}({\mathbb R}^n)$ for $0<p\le u<\infty$, $s\in{\mathbb R}$ and $q\in(0,\infty]$, some of these results are new.

math.FA↗

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

Relations among Besov-Type Spaces, Triebel-Lizorkin-Type Spaces and Generalized Carleson Measure Spaces

In this paper, the authors construct some counterexamples to show that the generalized Carleson measure space and the Triebel-Lizorkin-type space are not equivalent for certain parameters, which was claimed to be true in [Taiwanese J. Math. 15 (2011), 919-926]. Moreover, the authors show that for some special parameters, the generalized Carleson measure space, the Triebel-Lizorkin-type space and the Besov-type space coincide with certain Triebel-Lizorkin space, which answers a question posed in Remark 6.11(i) of [Lecture Notes in Mathematics 2005, Springer-Verlag, Berlin, 2010, xi+281 pp.].

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↗

arXiv2012

Hardy Spaces $H_L^p({\mathbb R}^n)$ Associated to Operators Satisfying $k$-Davies-Gaffney Estimates

Let $L$ be a one to one operator of type $ω$ having a bounded $H_\infty$ functional calculus and satisfying the $k$-Davies-Gaffney estimates with $k\in{\mathbb N}$. In this paper, the authors introduce the Hardy space $H_L^p(\mathbb{R}^n)$ with $p\in (0,\,1]$ associated to $L$ in terms of square functions defined via $\{e^{-t^{2k}L}\}_{t>0}$ and establish their molecular and generalized square function characterizations. Typical examples of such operators include the $2k$-order divergence form homogeneous elliptic operator $L_1$ with complex bounded measurable coefficients and the $2k$-order Schrödinger type operator $L_2\equiv (-Δ)^k+V^k$, where $Δ$ is the Laplacian and $0\le V\in L^k_{\mathop\mathrm{loc}}(\mathbb{R}^n)$. Moreover, as applications, for $i\in\{1,\,2\}$, the authors prove that the associated Riesz transform $\nabla^k(L_i^{-1/2})$ is bounded from $H_{L_i}^p(\mathbb{R}^n)$ to $H^p(\mathbb{R}^n)$ for $p\in(n/(n+k),\,1]$ and establish the Riesz transform characterizations of $H_{L_1}^p(\mathbb{R}^n)$ for $ p\in(rn/(n+kr),\,1]$ if $\{e^{-tL_1}\}_{t>0}$ satisfies the $L^r-L^2$ $k$-off-diagonal estimates with $r\in (1,2]$. These results when $k\equiv1$ and $L\equiv L_1$ are known.

math.CA↗

arXiv2012

New Real-Variable Characterizations of Musielak-Orlicz Hardy Spaces

Let $φ: {\mathbb R^n}\times [0,\infty)\to[0,\infty)$ be such that $φ(x,\cdot)$ is an Orlicz function and $φ(\cdot,t)$ is a Muckenhoupt $A_\infty({\mathbb R^n})$ weight. The Musielak-Orlicz Hardy space $H^φ(\mathbb R^n)$ is defined to be the space of all $f\in{\mathcal S}'({\mathbb R^n})$ such that the grand maximal function $f^*$ belongs to the Musielak-Orlicz space $L^φ(\mathbb R^n)$. Luong Dang Ky established its atomic characterization. In this paper, the authors establish some new real-variable characterizations of $H^φ(\mathbb R^n)$ in terms of the vertical or the non-tangential maximal functions, or the Littlewood-Paley $g$-function or $g_λ^\ast$-function, via first establishing a Musielak-Orlicz Fefferman-Stein vector-valued inequality. Moreover, the range of $λ$ in the $g_λ^\ast$-function characterization of $H^φ(\mathbb R^n)$ coincides with the known best results, when $H^φ(\mathbb R^n)$ is the classical Hardy space $H^p(\mathbb R^n)$, with $p\in (0,1]$, or its weighted variant.

math.CA↗

arXiv2011

Equivalent Characterizations for Boundedness of Maximal Singular Integrals on $ax+b$\,--Groups

Let $(S, d, ρ)$ be the affine group $\mathrm{R}^n \ltimes \mathrm{R}^+$ endowed with the left-invariant Riemannian metric $d$ and the right Haar measure $ρ$, which is of exponential growth at infinity. In this paper, for any linear operator $T$ on $(S, d, ρ)$ associated with a kernel $K$ satisfying certain integral size condition and Hörmander's condition, the authors prove that the following four statements regarding the corresponding maximal singular integral $T^\ast$ are equivalent: $T^\ast$ is bounded from $L_c^\infty$ to $\mathrm{BMO}$, $T^\ast$ is bounded on $L^p$ for all $p\in(1, \infty)$, $T^\ast$ is bounded on $L^p$ for certain $p\in(1, \infty)$ and $T^\ast$ is bounded from $L^1$ to $L^{1,\,\infty}$. As applications of these results, for spectral multipliers of a distinguished Laplacian on $(S, d, ρ)$ satisfying certain Mihlin-Hörmander type condition, the authors obtain that their maximal singular integrals are bounded from $L_c^\infty$ to $\mathrm{BMO}$, from $L^1$ to $L^{1,\,\infty}$, and on $L^p$ for all $p\in(1, \infty)$.

math.CA↗

arXiv2011

Vanishing Mean Oscillation Spaces Associated with Operators Satisfying Davies-Gaffney Estimates

Let $(\mathcal{X}, d, μ)$ be a metric measure space, $L$ a linear operator which has a bounded $H_\infty$ functional calculus and satisfies the Davies-Gaffney estimate, $Φ$ a concave function on $(0,\infty)$ of critical lower type $p_Φ^-\in(0,1]$ and $ρ(t)\equiv t^{-1}/Φ^{-1}(t^{-1})$ for all $t\in(0,\infty)$. In this paper, the authors introduce the generalized VMO space ${\mathrm {VMO}}_{ρ,L}({\mathcal X})$ associated with $L$, and establish its characterization via the tent space. As applications, the authors show that $({\mathrm {VMO}}_{ρ,L}({\mathcal X}))^*=B_{Φ,L^*}({\mathcal X})$, where $L^*$ denotes the adjoint operator of $L$ in $L^2({\mathcal X})$ and $B_{Φ,L^*}({\mathcal X})$ the Banach completion of the Orlicz-Hardy space $H_{Φ,L^*}({\mathcal X})$.

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↗

arXiv2011

Real-Variable Characterizations Of Hardy Spaces Associated With Bessel Operators

Let $λ>0$, $p\in((2\lz+1)/(2\lz+2), 1]$, and $\triangle_λ\equiv-\frac{d^2}{dx^2}-\frac{2λ}{x} \frac d{dx}$ be the Bessel operator. In this paper, the authors establish the characterizations of atomic Hardy spaces $H^p((0, \infty), dm_λ)$ associated with $\triangle_λ$ 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 $dm_λ(x)\equiv x^{2λ}\,dx$. As an application, the authors further obtain the Riesz transform characterization of these Hardy spaces.

math.CA↗

arXiv2010

Spaces of Type BLO on Non-homogeneous Metric Measure Spaces

Let $({\mathcal X}, d, μ)$ be a metric measure space and satisfy the so-called upper doubling condition and the geometrically doubling condition. In this paper, the authors introduce the space ${\mathop\mathrm{RBLO}}(μ)$ and prove that it is a subset of the known space ${\mathop\mathrm{RBMO}}(μ)$ in this context. Moreover, the authors establish several useful characterizations for the space ${\mathop\mathrm{RBLO}}(μ)$. As an application, the authors obtain the boundedness of the maximal Calderón-Zygmund operators from $L^\infty(μ)$ to ${\mathop\mathrm{RBLO}}(μ)$.

math.CA↗

arXiv2010

Boundedness of Calderón-Zygmund Operators on Non-homogeneous Metric Measure Spaces

Let $({\mathcal X}, d, μ)$ be a separable metric measure space satisfying the known upper doubling condition, the geometrical doubling condition and the non-atomic condition that $μ(\{x\})=0$ for all $x\in{\mathcal X}$. In this paper, we show that the boundedness of a Calderón-Zygmund operator $T$ on $L^2(μ)$ is equivalent to that of $T$ on $L^p(μ)$ for some $p\in (1, \infty)$, and that of $T$ from $L^1(μ)$ to $L^{1,\,\infty}(μ).$ As an application, we prove that if $T$ is a Calderón-Zygmund operator bounded on $L^2(μ)$, then its maximal operator is bounded on $L^p(μ)$ for all $p\in (1, \infty)$ and from the space of all complex-valued Borel measures on ${\mathcal X}$ to $L^{1,\,\infty}(μ)$. All these results generalize the corresponding results of Nazarov et al. on metric spaces with measures satisfying the so-called polynomial growth condition.

math.CA↗

arXiv2010

Dyadic Sets, Maximal Functions and Applications on $ax+b$ --Groups

Let $S$ be the Lie group $\mathrm{R}^n\ltimes \mathrm{R}^+$ endowed with the left-invariant Riemannian symmetric space structure and the right Haar measure $ρ$, which is a Lie group of exponential growth. Hebisch and Steger in [Math. Z. 245(2003), 37--61] proved that any integrable function on $(S,ρ)$ admits a Calderón--Zygmund decomposition which involves a particular family of sets, called Calderón--Zygmund sets. In this paper, we first show the existence of a dyadic grid in the group $S$, which has {nice} properties similar to the classical Euclidean dyadic cubes. Using the properties of the dyadic grid we shall prove a Fefferman--Stein type inequality, involving the dyadic maximal Hardy--Littlewood function and the dyadic sharp dyadic function. As a consequence, we obtain a complex interpolation theorem involving the Hardy space $H^1$ and the $BMO$ space introduced in [Collect. Math. 60(2009), 277--295].

math.CA↗