arXiv subjects

Der-Chen Chang

Publications and source records attributed to Der-Chen Chang.

32 records · Page 2Linked to original sources

arXiv2016

Littlewood-Paley Characterizations of Hajłasz-Sobolev and Triebel-Lizorkin Spaces via Averages on Balls

Let $p\in(1,\infty)$ and $q\in[1,\infty)$. In this article, the authors characterize the Triebel-Lizorkin space ${F}^α_{p,q}(\mathbb{R}^n)$ with smoothness order $α\in(0,2)$ via the Lusin-area function and the $g_λ^*$-function in terms of difference between $f(x)$ and its average $B_tf(x):=\frac1{|B(x,t)|}\int_{B(x,t)}f(y)\,dy$ over a ball $B(x,t)$ centered at $x\in\mathbb{R}^n$ with radius $t\in(0,1)$. As an application, the authors obtain a series of characterizations of $F^α_{p,\infty}(\mathbb{R}^n)$ via pointwise inequalities, involving ball averages, in spirit close to Hajłasz gradients, here an interesting phenomena naturally appears that, in the end-point case when $α=2$, these pointwise inequalities characterize the Triebel-Lizorkin spaces $F^2_{p,2}(\mathbb{R}^n)$, while not $F^2_{p,\infty}(\mathbb{R}^n)$. In particular, some new pointwise characterizations of Hajłasz-Sobolev spaces via ball averages are obtained. Since these new characterizations only use ball averages, they can be used as starting points for developing a theory of Triebel-Lizorkin spaces with smoothness orders not less than $1$ on spaces of homogeneous type.

math.CA↗

arXiv2015

On the Hodge-type decomposition and cohomolgy groups of $k$-Cauchy-Fueter complexes over domains in the quaternionic space

The $k$-Cauchy-Fueter operator $ D_0^{(k) } $ on one dimensional quaternionic space $\mathbb{H}$ is the Euclidean version of helicity $\frac k 2$ massless field operator on the Minkowski space in physics. The $k$-Cauchy-Fueter equation for $k\geq 2$ is overdetermined and its compatibility condition is given by the $k$-Cauchy-Fueter complex. In quaternionic analysis, these complexes play the role of Dolbeault complex in several complex variables. We prove that a natural boundary value problem associated to this complex is regular. Then by using the theory of regular boundary value problems, we show the Hodge-type orthogonal decomposition, and the fact that the non-homogeneous $k$-Cauchy-Fueter equation $ D_0^{(k) } u=f$ on a smooth domain $Ω$ in $\mathbb{H}$ is solvable if and only if $f$ satisfies the compatibility condition and is orthogonal to the set $\mathscr H^1_{ (k) }(Ω)$ of Hodge-type elements. This set is isomorphic to the first cohomology group of the $k$-Cauchy-Fueter complex over $Ω$, which is finite dimensional, while the second cohomology group is always trivial.

math.CV↗

arXiv2014

Weak Hardy Spaces $WH_L^p({\mathbb R}^n)$ 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 weak Hardy space $WH_L^p(\mathbb{R}^n)$ associated to $L$ for $p\in (0,\,1]$ via the non-tangential square function $S_L$ and establish a weak molecular characterization of $WH_L^p(\mathbb{R}^n)$. Typical examples of such operators include the $2k$-order divergence form homogeneous elliptic operator $L_1:=(-1)^k\sum_{|α|=k=|β|}\partial^β(a_{α,β}\partial^α)$, where $\{a_{α,β}\}_{|α|=k=|β|}$ are complex bounded measurable functions, and the $2k$-order Schrödinger type operator $L_2:= (-Δ)^k+V^k$, where $Δ$ is the Laplacian operator and $0\le V\in L^k_{\mathop\mathrm{loc}}(\mathbb{R}^n)$. As applications, for $i\in\{1,\,2\}$ and $p\in(\frac{n}{n+k},\,1]$, the authors prove that the associated Riesz transform $\nabla^k (L_i^{-1/2})$ is bounded from $WH^p_{L_i}(\mathbb{R}^n)$ to the classical weak Hardy space $WH^p(\mathbb{R}^n)$ and, for all $0<p<r\le1$ and $α=n(\frac{1}{p}-\frac{1}{r})$, the fractional power $L_i^{-\fracα{2k}}$ is bounded from $WH_{L_i}^p(\mathbb{R}^n)$ to $WH_{L_i}^r(\mathbb{R}^n)$. Furthermore, the authors find the dual space of $WH_L^p(\mathbb{R}^n)$ for $p\in(0,\,1]$, which can be defined via mean oscillations based on some subtle coverings of bounded open sets and, even when $L:=-Δ$, are also previously unknown. In particular, if $L$ is a nonnegative self-adjoint operator in $L^2({\mathbb R}^n)$ satisfying the Davies-Gaffney estimates, the authors further establish the weak atomic characterization of $WH_L^p(\mathbb{R}^n)$.

math.CA↗

arXiv2014

Riesz Transform Characterizations of Musielak-Orlicz-Hardy Spaces

Let $φ$ be a Musielak-Orlicz function satisfying that, for any $(x,\,t)\in\mathbb{R}^n\times(0,\,\infty)$, $φ(\cdot,\,t)$ belongs to the Muckenhoupt weight class $A_\infty (\mathbb{R}^n)$ with the critical weight exponent $q(φ)\in[1,\,\infty)$ and $φ(x,\,\cdot)$ is an Orlicz function with $0 \frac{n-1}{n}$, and via all the Riesz transforms with the order not more than $m\in\mathbb{N}$ when $\frac{i(φ)}{q(φ)}>\frac{n-1}{n+m-1}$. Moreover, the authors also establish the Riesz transform characterizations of $H_φ(\mathbb{R}^n)$, respectively, by means of the higher order Riesz transforms defined via the homogenous harmonic polynomials or the odd order Riesz transforms. Even if when $φ(x,t):=tw(x)$ for all $x\in{\mathbb R}^n$ and $t\in [0,\infty)$, these results also widen the range of weights in the known Riesz characterization of the classical weighted Hardy space $H^1_w({\mathbb R}^n)$ obtained by R. L. Wheeden from $w\in A_1({\mathbb R}^n)$ into $w\in A_\infty({\mathbb R}^n)$ with the sharp range $q(w)\in [1,\frac n{n-1})$, where $q(w)$ denotes the critical index of the weight $w$.

math.CA↗

arXiv2012

Integrability conditions for the Grushin and Martinet distributions

We realise the first and second Grushin distributions as symmetry reductions of the 3-dimensional Heisenberg distribution and 4-dimensional Engel distribution respectively. Similarly, we realise the Martinet distribution as an alternative symmetry reduction of the Engel distribution. These reductions allow us to derive the integrability conditions for the Grushin and Martinet distributions and build certain complexes of differential operators. These complexes are well-behaved despite the distributions they resolve being non-regular.

math.DG↗

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↗

arXiv2010

Hopf fibration: geodesics and distances

Here we study geodesics connecting two given points on odd-dimensional spheres respecting the Hopf fibration. This geodesic boundary value problem is completely solved in the case of 3-dimensional sphere and some partial results are obtained in the general case. The Carnot-Carathéodory distance is calculated. We also present some motivations related to quantum mechanics.

math.DG↗

arXiv2009

Boundedness of Sublinear Operators on Product Hardy Spaces and Its Application

Let $p\in(0, 1]$. In this paper, the authors prove that a sublinear operator $T$ (which is originally defined on smooth functions with compact support) can be extended as a bounded sublinear operator from product Hardy spaces $H^p({{\mathbb R}^n\times{\mathbb R}^m})$ to some quasi-Banach space ${\mathcal B}$ if and only if $T$ maps all $(p, 2, s_1, s_2)$-atoms into uniformly bounded elements of ${\mathcal B}$. Here $s_1\ge\lfloor n(1/p-1)\rfloor$ and $s_2\ge\lfloor m(1/p-1)\rfloor$. As usual, $\lfloor n(1/p-1)\rfloor$ denotes the maximal integer no more than $n(1/p-1)$. Applying this result, the authors establish the boundedness of the commutators generated by Calderón-Zygmund operators and Lipschitz functions from the Lebesgue space $L^p({{\mathbb R}^n\times{\mathbb R}^m})$ with some $p>1$ or the Hardy space $H^p({{\mathbb R}^n\times{\mathbb R}^m})$ with some $p\le1$ but near 1 to the Lebesgue space $L^q({{\mathbb R}^n\times{\mathbb R}^m})$ with some $q>1$.

math.CA↗

arXiv2008

SubRiemannian geometry on the sphere $\mathbb{S}^3$

The present paper starts with an introduction to quaternions and then defines the 3-dimmensional sphere as the set of quaternions of length one. The quaternion group induces on $\mathbb{S}^3$ a structure of noncommutative Lie group. This group is compact and the results obtained in this case are very different than those obtained in the case of the Heisenberg group, which is a noncompact Lie group. Like in the Heisenberg group case, we introduce a nonintegrable distribution on the sphere and a metric on it using two of the noncommutative left invariant vector fields. This way $\mathbb{S}^3$ becomes a subRiemannian manifold. It is known that the group $SU(2) $ is isomorphic with the sphere $\mathbb{S}^3$ and represents an example of subRiemannian manifold where the elements are matrices. The main issue here is to study the connectivity by horizontal curves and its geodesics on this manifold. In this paper, we are using Lagrangian method to study the connectivity theorem on ${\mathbb S}^3$ by horizontal curves with minimal arc-length. We show that for any two points in ${\mathbb S}^3$, there exists such a geodesic joining these two points.

math.DG↗

arXiv2008

Modified action and differential operators on the 3-D sub-Riemannian sphere

Our main aim is to present a geometrically meaningful formula for the fundamental solutions to a second order sub-elliptic differential equation and to the heat equation associated with a sub-elliptic operator in the sub-Riemannian geometry on the unit sphere $\mathbb S^3$. Our method is based on the Hamiltonian approach, where the corresponding Hamitonian system is solved with mixed boundary conditions. A closed form of the modified action is given. It is a sub-Riemannian invariant and plays the role of a distance on $\mathbb S^3$.

math.DG↗

arXiv2008

Sub-Riemannian geodesics on the 3-D sphere

The unit sphere $\mathbb S^3$ can be identified with the unitary group SU(2). Under this identification the unit sphere can be considered as a non-commutative Lie group. The commutation relations for the vector fields of the corresponding Lie algebra define a 2-step sub-Riemannian manifold. We study sub-Riemannian geodesics on this sub-Riemannian manifold making use of the Hamiltonian formalism and solving the corresponding Hamiltonian system.

math.DG↗

arXiv2008

Sub-Lorentzian Geometry on Anti-De Sitter Space

Sub-Riemannian Geometry is proved to play an important role in many applications, e.g., Mathematical Physics and Control Theory. The simplest example of sub-Riemannian structure is provided by the 3-D Heisenberg group. Sub-Riemannian Geometry enjoys major differences from the Riemannian being a generalisation of the latter at the same time, e.g., geodesics are not unique and may be singular, the Hausdorff dimension is larger than the manifold topological dimension. There exists a large amount of literature developing sub-Riemannian Geometry. However, very few is known about its natural extension to pseudo-Riemannian analogues. It is natural to begin such a study with some low-dimensional manifolds. Based on ideas from sub-Riemannian geometry we develop sub-Lorentzian geometry over the classical 3-D anti-de Sitter space. Two different distributions of the tangent bundle of anti-de Sitter space yield two different geometries: sub-Lorentzian and sub-Riemannian. It is shown that the set of timelike and spacelike `horizontal' curves is non-empty and we study the problem of horizontal connectivity in anti-de Sitter space. We also use Lagrangian and Hamiltonian formalisms for both sub-Lorentzian and sub-Riemannian geometries to find geodesics.

math.DG↗

arXiv2006

Anisotropic quaternion Carnot groups: geometric analysis and Green's function

We construct examples of 2-step Carnot groups related to quaternions and study their fine structure and geometric properties. This involves the Hamiltonian formalism, which is used to obtain explicit equations for geodesics and the computation of the number of geodesics joining two different points on these groups. We able to find the explicit lengths of geodesics. We present the fundamental solutions of the Heat and sub-Laplace equations for these anisotropic groups and obtain some estimates for them, which may be useful.

math.DG↗

arXiv2006

Quaternion $H$-type group and differential operator $Δ_λ$

We study the relations between the quaternion $H$-type group and the boundary of the unit ball on two dimensional quaternionic space. The orthogonal projection of the space of square integrable functions defined on quaternion $H$-type group into its subspace of boundary values of $q$-holomorphic functions is consider. The precise form of Cauchy-Szego kernel and the orthogonal projection operator is obtained. The fundamental solution for the operator $Δ_λ$ is found.

math.AP↗