Concerning LpL^p resolvent estimates for simply connected manifolds of constant curvature

We prove families of uniform $(L^r,L^s)$ resolvent estimates for simply connected manifolds of constant curvature (negative or positive) that imply the earlier ones for Euclidean space of Kenig, Ruiz and the second author \cite{KRS}. In the case of the sphere we take advantage of the fact that the half-wave group of the natural shifted Laplacian is periodic. In the case of hyperbolic space, the key ingredient is a natural variant of the Stein-Tomas restriction theorem.Comment: 25 pages, 2 figure

Corrosion Resistance and Electrocatalytic Properties of Metallic Glasses

Metallic glasses exhibit excellent corrosion resistance and electrocatalytic properties, and present extensive potential applications as anticorrosion, antiwearing, and catalysis materials in many industries. The effects of minor alloying element, microstructure, and service environment on the corrosion resistance, pitting corrosion, and electrocatalytic efficiency of metallic glasses are reviewed. Some scarcities in corrosion behaviors, pitting mechanism, and eletrocatalytic reactive activity for hydrogen are discussed. It is hoped that the overview is beneficial for some researcher paying attention to metallic glasses

Heisenberg Uniqueness Pairs and the wave equation

Given a curve $\Gamma$ and a set $\Lambda$ in the plane, the concept of the Heisenberg uniqueness pair $(\Gamma, \Lambda)$ was first introduced by Hedenmalm and Motes-Rodr\'{\i}gez (Ann. of Math. 173(2),1507-1527, 2011, \cite{HM}) as a variant of the uncertainty principle for the Fourier transform. The main results of Hedenmalm and Motes-Rodr\'{\i}gez concern the hyperbola $\Gamma_{\epsilon}=\{(x_1, x_2)\in \mathbb{R}^2,\, x_1x_2=\epsilon\}$ ($0\ne\epsilon\in \mathbb{R}$) and lattice-crosses $\Lambda_{\alpha\beta}=(\alpha\mathbb{Z}\times \{0\})\cup(\{0\}\times \beta\mathbb{Z})$ ($\alpha, \beta>0$), where it's proved that $(\Gamma_{\epsilon}, \Lambda_{\alpha\beta})$ is a Heisenberg uniqueness pair if and only if $\alpha\beta\leq 1/|\epsilon|$. In this paper, we aim to study the endpoint case (i.e., $\epsilon=0$ in $\Gamma_{\epsilon}$) and investigate the following problem: what's the minimal amount of information required on $\Lambda$ (the zero set) to form a Heisenberg uniqueness pair? When $\Lambda$ is contained in the union of two curves in the plane, we give characterizations in terms of some dynamical system conditions. The situation is quite different in higher dimensions and we obtain characterizations in the case that $\Lambda$ is the union of two hyperplanes.Comment: 24 page

Dispersive estimates for the Schr\"{o}dinger equation with finite rank perturbations

In this paper, we investigate dispersive estimates for the time evolution of Hamiltonians $$H=-\Delta+\sum_{j=1}^N\langle\cdot\,, \varphi_j\rangle \varphi_j\quad\,\,\,\text{in}\,\,\,\mathbb{R}^d,\,\, d\ge 1,$$ where each $\varphi_j$ satisfies certain smoothness and decay conditions. We show that, under a spectral assumption, there exists a constant $C=C(N, d, \varphi_1,\ldots, \varphi_N)>0$ such that $$\|e^{-itH}\|_{L^1-L^{\infty}}\leq C t^{-\frac{d}{2}}, \,\,\,\text{for}\,\,\, t>0.$$ As far as we are aware, this seems to provide the first study of $L^1-L^{\infty}$ estimates for finite rank perturbations of the Laplacian in any dimension. We first deal with rank one perturbations ($N=1$). Then we turn to the general case. The new idea in our approach is to establish the Aronszajn-Krein type formula for finite rank perturbations. This allows us to reduce the analysis to the rank one case and solve the problem in a unified manner. Moreover, we show that in some specific situations, the constant $C(N, d, \varphi_1,\ldots, \varphi_N)$ grows polynomially in $N$. Finally, as an application, we are able to extend the results to $N=\infty$ and deal with some trace class perturbations.Comment: 78 page