Jie Xiao

Memorial University of Newfoundland · Department of Mathematics and Statistics

Starting with the fractional Schrödinger equation and Fourier analysis, this paper presents a new Lp-uncertainty principle for the positively-ordered Laplace pair {(−Δ)α2,(−Δ)β2}.

Given n ≥ 2 and \(\alpha > \tfrac{1}{2}\), we obtained an improved upbound of Hausdorff’s dimension of the fractional Schrödinger operator; that is, $$\mathop {\sup }\limits_{f \in {H^s}({\mathbb{R}^n})} {\dim _H}\left\{ {x \in {{\mathbb{R}^n}}:\;\mathop {\lim }\limits_{t \to 0} {e^{{\rm{i}}t{{( - \Delta )}^\alpha }}}f(x) \ne f(x)} \right\} \le n +...

This paper shows $$ \sup_{f\in H^s(\mathbb{R}^n)}\dim _H\left\{x\in\mathbb{R}^n:\ \lim_{t\rightarrow0}e^{it(-\Delta)^\alpha}f(x)\neq f(x)\right\}\leq n+1-\frac{2(n+1)s}{n}\ \ \text{under}\ \ \begin{cases} n\geq2;\\ \alpha>\frac12; \frac{n}{2(n+1)}<s\leq\frac{n}{2} . \end{cases}

This paper addresses the so-called conformal capacities in $\mathbb R^n$, $n\ge 3$, through comparing three existing definitions (due to Betsakos, Colesanti-Cuoghi, Anderson-Vamananmurthy-Fuglede respectively) and studying their associated iso-capacitary inequalities with connection to half-diameter, mean-width, mean-curvature and ADM-mass, Hadamar...

The title compound, C22H18N2O, was synthesized from naphthalen-2-ol, benzaldehyde and pyridin-2-amine. In the crystal, mol­ecules are linked into centrosymmetric R 2 2(16) dimers by pairs of O—H⋯N hydrogen bonds. The mol­ecular conformation is stabilized by an N—H⋯O hydrogen bond. The dihedral angle between the naphthylene ring system and the pheny...

In the title compound, C(10)H(10)N(4)O(2)·H(2)O, the dihedral angle between the tetra-zole and benzene rings is 63.24 (11)°. The crystal structure is stabilized by intra-molecular O-H⋯N and O-H⋯O hydrogen bonds.

The in situ hydrothermal reactions of 4-nitrobenzonitrile with Cd(ClO4)2 and CdCl2 afforded two novel CdII–tetrazole coordination polymers, 2D network {[Cd(H2O)2(4-nptz)2](H2O)2}n (1) and 1D double chains [Cd(H2O)2(4-nptz)2]n (2). Their synthesis, solid-state structure, and XRPD patterns are reported.

In the title complex, [Zn(C(4)H(3)N(6))(2)(H(2)O)(2)], the metal centre lies on an inversion centre and displays a distorted octa-hedral ZnN(4)O(2) coordination geometry. The organic ligand is not planar; the dihedral angle between the imidazole and tetra-zole rings is 8.39 (9)°. An extended network of inter-molecular N-H⋯N and O-H⋯N hydrogen bonds...

In the mol-ecule of the title compound, C(15)H(13)N(3)O(4), the dihedral angle between the pyrazole and benzene ring planes is 67.7 (1)°. The crystal structure is stabilized by an intra-molecular C-H⋯O hydrogen bond and two weak inter-molecular C-H⋯O inter-actions.

In the mol-ecule of the title compound, C(15)H(13)N(3)O(4), the dihedral angle between the pyrazole and benzene rings is 79.89 (6)°. An intra-molecular C-H⋯O hydrogen bond is present. The crystal structure is stabilized by π-π stacking inter-actions between centrosymmetrically related pyrazole rings with a centroid-centroid distance of 3.500 (3) Å.

The title compound, C(15)H(13)N(3)O(4), was synthesized from dimethyl 1H-pyrazole-3,5-dicarboxyl-ate and 4-(bromo-meth-yl)benzonitrile. The inter-planar angle between the pyrazole and cyano-benzyl ring planes is 71.74 (17)° and an intramolecular C-H⋯O interaction occurs. The crystal structure is stabilized by π-π stacking inter-actions between the...

The title compound, C(11)H(10)N(4)O(4)center dot H(2)O, was synthesized from 1-azidomethyl-3-nitrobenzene and ethyl acetylacetate. Single-crystal X-ray analysis reveals that the dihedral angle between the triazole and benzene ring planes is 84.80 (2)degrees. The packing of the molecules is stabilized by strong O-H center dot center dot center dot O...

In this paper we establish the local and global well-posedness of the real valued fifth order Kadomtsev–Petviashvili I equation in the anisotropic Sobolev spaces with nonnegative indices. In particular, our local well-posedness improves Saut–Tzvetkov's one and our global well-posedness gives an affirmative answer to Saut–Tzvetkov's L2-data conjectu...

The title compound, C(16)H(12)N(2)O, was accidentally synthesized by the reaction of 4-(bromo-meth-yl)benzonitrile and penta-erythritol. The dihedral angle between the benzene rings is 57.39 (9)°. In the crystal structure, mol-ecules are linked by inter-molecular C-H⋯N hydrogen-bonding inter-actions to form chains running parallel to the b axis.

In the title compound, C(11)H(9)N(5)S, the dihedral angle between the mean planes of the thione-substituted triazole ring and benzonitrile ring is 4.28 (3)°. Inter-molecular N-H⋯S hydrogen bonds link the mol-ecules together into characteristic dimers.

In the title mol­ecule, C12H11NO4, the two acetyl groups are inclined by 71.3 (1) and 46.2 (1)° to the benzene ring. In the crystal structure, mol­ecules are linked into a chain along the c axis by C—H⋯O hydrogen bonds.

The title compound, C(37)H(26)N(2)O(2), was synthesized from 1,1'-methyl-enedinaphthalen-2-ol and 3-(bromo-methyl)-benzo-nitrile. The two naphthyl systems are almost perpendicular to each other [dihedral angle 83.3 (9)°] and the two cyano-benz-yloxy rings approximately parallel to each other [dihedral angle 15.5 (2)°].

In the title mol-ecule, C(12)H(11)NO(4), the two acetyl groups are inclined by 71.3 (1) and 46.2 (1)° to the benzene ring. In the crystal structure, mol-ecules are linked into a chain along the c axis by C-H⋯O hydrogen bonds.

Let μ be a nonnegative Borel measure on the open unit disk D⊂C. This note shows how to decide that the Möbius invariant space Qp, covering BMOA and B, is boundedly (resp., compactly) embedded in the quadratic tent-type space Tp∞(μ). Interestingly, the embedding result can be used to determine the boundedness (resp., compactness) of the Volterra-typ...

The aim of this article is: (a) To establish the existence of the best isoperimetric constants for the $(H^1,BMO)$-normal conformal metrics $e^{2u}|dx|^2$ on $\mathbb R^n$, $n\ge 3$, i.e., the conformal metrics with the Q-curvature orientated conditions $$ (-\Delta)^{n/2}u\in H^1(\mathbb R^n) & \ u(x)=\hbox{const.}+\frac{\int_{\mathbb R^n}(\log\fra...

Let $\mu$ be a nonnegative Borel measure on the open unit disk $\mathbb{D}\subset\mathbb{C}$. This note shows how to decide that the M\"obius invariant space $\mathcal{Q}_p$, covering $\mathcal{BMOA}$ and $\mathcal{B}$, is boundedly (resp., compactly) embedded in the quadratic tent-type space $T^\infty_p(\mu)$. Interestingly, the embedding result c...

This paper shows that each of the sharp (endpoint) Sobolev inequality and the isoperimetric inequality can be split into two sharp and stronger inequalities through either the 1-variational capacity or the 1-integral affine surface area. Furthermore, some related sharp analytic and geometric inequalities are also explored.

Given p ∈ [1,∞) and λ ∈ (0, n), we study Morrey space \(L^{p,\lambda}({\Bbb R}^n)\) of all locally integrable complex-valued functions f on \({\Bbb R}^n\) such that for every open Euclidean ball B ⊂ \({\Bbb R}^n\) with radius rB there are numbers C = C(f ) (depending on f ) and c = c(f,B) (relying upon f and B) satisfying $$r^{-\lambda}_B\sum_B \ve...

Two embeddings of a homogeneous endpoint Besov space are established via the Hausdorff capacity and the heat equation. Meanwhile, a co-capacity formula and a trace inequality are derived from the Besov space.

This note completely describes the bounded or compact Riemann-Stieltjes integral operators $T_g$ acting between the weighted Bergman space pairs $(A^p_\alpha,A^q_\beta)$ in terms of particular regularities of the holomorphic symbols $g$ on the open unit ball of $\Bbb C^n$.

It is proved that for $\alpha\in (0,1)$, $Q_\alpha(\rn)$, not only as an intermediate space of $W^{1,n}(\rn)$ and $BMO(\rn)$ but also as an affine variant of Sobolev space $\dot{L}^{2}_\alpha(\rn)$ which is sharply imbedded in $L^{\frac{2n}{n-2\alpha}}(\rn)$, is isomorphic to a quadratic Morrey space under fractional differentiation. At the same ti...

Let u(t,x) be the solution of the heat equation (∂t-Δx)u(t,x)=0 on R+1+n=(0,∞)×Rn subject to u(0,x)=f(x) on Rn. The main goal of this paper is to characterize such a nonnegative measure μ on R+1+n that f(x)↦u(t2,x) induces a bounded embedding from the Sobolev space W˙1,p(Rn), p∈[1,n) into the Lebesgue space Lq(R+1+n,μ), q∈(0,∞).

We establish a sharp Sobolev trace inequality for the fractional-order derivatives. As a close connection with this best estimate, we show a fractional-order logarithmic Sobolev trace inequality with the asymptotically optimal constant, but also sharpen the Poincaré embedding for the conformal invariant energy and BMO spaces.

This paper characterizes the so-called Möbius invariant QK spaces in terms of Carleson-type measures, boundary values, inner factors and absolute values of analytic functions on the unit disk.

The purpose of this paper is to continue the study of the so-called holomorphic Q classes via two conformal deformations of the fractional image area of a holomorphic function on the open unit disk. Results include fractional order Carleson measures, embeddings of Choquet spaces with respect to Hausdorff capacities, preduals, strong isoperimetric i...

This paper contains several results relating $Q$ spaces in several real variables with their dyadic counterparts, which are analogues of theorems for BMO and for $Q$ spaces on the circle. In addition, it gives an atomic (or quasi-orthogonal) decomposition for these $Q$ spaces in terms of the same type of atoms used to decompose BMO.

We introduce the so-called Bloch-Sobolev function spaces and show that these spaces have nice closure properties. We also characterize the boundedness and compactness of a composition operator C φ (with an-alytic symbol φ between two subdomains Ω, Ω R 2) acting between two Bloch-Sobolev spaces. As a by-product we obtain a characterization of those...

Several duality questions for fractional Carleson measures and the spaces are resolved using a new type of tent spaces. These tent spaces are defined in terms of Choquet integrals with respect to Hausdorff capacity. A predual for is then defined as a space of distributions containing the Hardy space H1, and an atomic decomposition is proved.

For, let Qp be a proper subspace of BMOA defined by means of a modified Garcia norm. We give a boundary value criterion for a function to be in Qp and a necessary and sufficient condition in terms of P-Carleson measure for an inner function to be in Qp. There are applications to Lipschitz spaces, gap series and partial sum operators. We discuss als...

For p∈(0,+∞) let D p be the Dirichlet type space of functions f analytic in the unit disk U={z:|z|<1} for which |f| D p 2 :=∫∫ U f ' (z) 2 1 - |z| 2 p dxdy<∞· Furthermore let Q p be the Möbius invariant subspace of D p consisting of those f∈D p with sup w∈U |f∘φ w | D p <∞, where φ w (z)=(w-z)/(1-w ¯z). In particular, let Q p,0 ={f∈Q p :lim |w|→1 |...