arXiv subjects
Publications and source records attributed to Xianghong Chen.
In this note, we present a characterization of semistable unitary operators on $L^2(\mathbb{R})$, under the assumption that the operator is (i) translation-invariant, (ii) symmetric, and (iii) locally uniformly continuous (LUC) under dilation. As a consequence, we characterize one-parameter groups formed by such operators, which are of the form $e^{iβt|{d}/{dx}|^α}$, with $α,β\in\mathbb R$.
For $d\ge 1$, we construct a compact subset $K\subseteq \mathbb {R}^{d+1}$ containing a $d$-sphere of every radius between $1$ and $2$, such that for every $δ\in (0,1)$, the $δ$-neighbourhood of $K$ has Lebesgue measure $\lesssim |\log δ|^{-2/d}$. This is the smallest possible order when $d=2$, and improves a result of Kolasa-Wolff (Pacific J. Math., 190(1):111-154, 1999). Our construction also generalises to Holder-continuous families of $C^{2,α}$ hypersurfaces with nonzero Gaussian curvature.
Keich (1999) showed that the sharp gauge function for the generalized Hausdorff dimension of Besicovitch sets in $\mathbb R^2$ is between $r^2\log 1/r$ and $r^2(\log 1/r) (\log\log 1/r)^{2+\varepsilon}$ by refining an argument of Bourgain (1991). It is not known whether the iterated logarithms in Keich's bound are necessary. In this paper we construct a family of Besicovitch line sets whose sharp gauge function is smaller than $r^2(\log 1/r) (\log\log 1/r)^{\varepsilon}$. Moreover, these Besicovitch sets are minimal in the sense that there is essentially only one line in the set pointing in each direction.
Cyclocarbon molecules are critical in understanding the carbon structure formation and the nature of the interaction between carbon atoms. In cyclocarbons, light elements such as H, O and N may interplay with rings to form doped cyclocarbon molecules. Such molecules show unique optical properties that have never been reported before. In this study, density functional theory with a tuned PBE functional (39% HF exchange) is employed to study the ground and excited states of a cyclo[18] carbon molecule (C18) and its doped variants C18M (M = H, Be, B, N, and O). The doping is shown to either make the UV-Vis spectra of C18 blue- or red-shifted depending on the spin brought by the dopant. Furthermore, introducing extra-atoms is found to cause the emergence of a new set of infrared modes in the structure under investigation. Finally, applying the molecular dynamic simulations enables one to observe the terahertz characteristics of C18M with frequencies up to 1.5 THz due to the propagation of a particular pattern along the carbon ring.
Let $Δ_{\mathbb{T}^m\times \mathbb{R}^n}$ denote the Laplace-Beltrami operator on the product spaces $\mathbb{T}^m\times \mathbb{R}^n$. In this article we show that $$ \left\|e^{itΔ_{\mathbb{T}^m\times \mathbb{R}^n}}f\right\|_{L^p(\mathbb{T}^m\times \mathbb{R}^n\times [0,1])} \leq C \|f\|_{W^{α,p}(\mathbb{T}^m\times\mathbb{R}^n)} $$ holds if $p\geq 2(m+n+2)/(m+n)$ and $α> (m+2n)(1/2-1/p)-2/p$. Furthermore, we apply the $\ell^2$-decoupling inequalities to establish local $L^p$-smoothing estimates for the Schrödinger operator $e^{itΔ_{\mathbb{T}^m\times\mathbb{R}^n}}$ in modulation spaces $M_{p,q}^α(\mathbb{T}^m\times\mathbb{R}^n)$: $$ \|e^{itΔ_{\mathbb{T}^m\times\mathbb{R}^n}}f\|_{L^p(\mathbb{T}^m\times\mathbb{R}^n\times [0,1])}\leq C \|f\|_{M_{p,q}^α(\mathbb{T}^m\times\mathbb{R}^n)} $$ for some range of $α$ and $p, q$. The smoothing estimates in $L^p$-Sobolev and modulation spaces are sharp up to the endpoint regularity, in a certain range of $p$ and $q$.
Let $Δ_{\mathbb S^n}$ denote the Laplace-Beltrami operator on the $n$-dimensional unit sphere $\mathbb S^n$. In this paper we show that $$ \| e^{it Δ_{\mathbb S^n}}f \|_{L^4([0, 2π) \times \mathbb S^n)} \leq C \| f\|_{W^{α, 4} (\mathbb S^n)} $$ holds provided that $n\geq 2$, $α> {(n-2)/4}.$ The range of $α$ is sharp up to the endpoint. As a consequence, we obtain space-time estimates for the Schrödinger propagator $e^{it Δ_{\mathbb S^n}}$ on the $L^p$ spaces for $2\leq p\leq \infty.$ We also prove that for zonal functions on ${\mathbb S}^n$, the Schrödinger maximal operator $\sup_{0\leq t<2π} |e^{itΔ_{\mathbb S^n}} f|$ is bounded from $W^{α, 2}(\mathbb S^n) $ to $L^{\frac{6n}{3n-2}}(\mathbb S^n)$ whenever $α>{1/3}$.
We show that there exists an integrable function on the $n$-sphere $(n\ge 2)$, whose Cesàro (C,$\frac{n-1}{2}$) means with respect to the spherical harmonic expansion diverge unboundedly almost everywhere. By studying equivalence theorems, we also obtain the corresponding results for Riesz and Bochner-Riesz means. This extends results of Stein (1961) for flat tori and complements the work of Taibleson (1985) for spheres.
We study the signed Bernoulli convolution $$ν_β^{(n)}=*_{j=1}^n \left (\frac12δ_{β^{-j}}-\frac12δ_{-β^{-j}}\right ),\ n\ge 1$$ where $β>1$ satisfies $$β^m=β^{m-1}+\cdots+β+1$$ for some integer $m\ge 2$. When $m$ is odd, we show that the variation $|ν_β^{(n)}|$ coincides the unsigned Bernoulli convolution $$μ_β^{(n)}=*_{j=1}^n \left (\frac12δ_{β^{-j}}+\frac12δ_{-β^{-j}}\right ).$$ When $m$ is even, we obtain the exact asymptotic of the total variation $\|ν_β^{(n)}\|$ as $n\rightarrow\infty$.
Let $ϕ$ be a smooth function on a compact interval $I$. Let $$γ(t)=\left (t,t^2,\cdots,t^{n-1},ϕ(t)\right).$$ In this paper, we show that $$\left(\int_I \big|\hat f(γ(t))\big|^q \big|ϕ^{(n)}(t)\big|^{\frac{2}{n(n+1)}} dt\right)^{1/q}\le C\|f\|_{L^p(\mathbb R^n)}$$ holds in the range $$1\le p<\frac{n^2+n+2}{n^2+n},\quad 1\le q<\frac{2}{n^2+n}p'.$$ This generalizes an affine restriction theorem of Sjölin (1974) for $n=2$. Our proof relies on ideas of Sjölin (1974) and Drury (1985), and more recently Bak-Oberlin-Seeger (2008) and Stovall (2016), as well as a variation bound for smooth functions.
Let $G$ be a connected, simply connected, compact semisimple Lie group of dimension $n$. It has been shown by Clerc \cite{Clerc1974} that, for any $f\in L^1(G)$, the Bochner-Riesz mean $S_R^δ(f)$ converges almost everywhere to $f$, provided $δ>(n-1)/2$. In this paper, we show that, at the critical index $δ=(n-1)/2$, there exists an $f\in L^1(G)$ such that $$\limsup_{R\rightarrow\infty} \big|S_{R}^{(n-1)/2}(f)(x)\big|=\infty, \ \text{a.e.}\ x\in G.$$ This is an analogue of a well-known result of Kolmogorov \cite{Kolmogoroff1923} for Fourier series on the circle, and a result of Stein \cite{Stein1961} for Bochner-Riesz means on the tori $\mathbb T^{n}, n\geq 2$. We also study localization properties of the Bochner-Riesz mean $S_{R}^{(n-1)/2}(f)$ for $f\in L^1(G)$.
We study spectral properties of the transfer operators $L$ defined on the circle $\mathbb T=\mathbb R/\mathbb Z$ by $$(Lu)(t)=\frac{1}{d}\sum_{i=0}^{d-1} f\left(\frac{t+i}{d}\right)u\left(\frac{t+i}{d}\right),\ t\in\mathbb T$$ where $u$ is a function on $\mathbb T$. We focus in particular on the cases $f(t)=|\cos(πt)|^q$ and $f(t)=|\sin(πt)|^q$, which are closely related to some classical Fourier-analytic questions. We also obtain some explicit computations, particularly in the case $d=2$. Our study extends work of Strichartz \cite{Strichartz1990} and Fan and Lau \cite{FanLau1998}.
We study the regularity of convolution powers for measures supported on Salem sets, and prove related results on Fourier restriction and Fourier multipliers. In particular we show that for $α$ of the form ${d}/{n}, n=2,3,\cdots$ there exist $α$-Salem measures for which the $L^2$ Fourier restriction theorem holds in the range $p\le \frac{2d}{2d-α}$. The results rely on ideas of Körner. We extend some of his constructions to obtain upper regular $α$-Salem measures, with sharp regularity results for $n$-fold convolutions for all $n\in \mathbb N$.
We prove a Fourier restriction estimate under the assumption that certain convolution power of the measure admits an $r$-integrable density.
We construct Salem sets on the real line with endpoint Fourier decay and near-endpoint regularity properties. This complements a result of Łaba and Pramanik, who obtained near-endpoint Fourier decay and endpoint regularity properties. We then modify the construction to extend a theorem of Hambrook and Łaba to show sharpness of the $L^2$-Fourier restriction estimate by Mockenhaupt and Bak-Seeger, including the case where the Hausdorff and Fourier dimension do not coincide.