相关论文: Optimal dispersive estimates for the wave equation…

We prove optimal (that is, without loss of derivatives) dispersive estimates for the Schrodinger group $e^{it(-\Delta+V)}$ for a class of real-valued potentials $V\in C^k(R^n)$ with $k>(n-3)/2$, where $n=4,5$.

We prove dispersive estimates for solutions to the wave equation with a real-valued potential $V\in L^\infty({\bf R}^n)$, $n\ge 4$, satisfying $V(x)=O(|x|^{-(n+1)/2-\epsilon})$, $|x|>1$, $\epsilon>0$.

We prove optimal dispersive estimates at high frequency for the Schrodinger group with real-valued potentials $V(x)=O(|x|^{-\delta})$, $\delta>n-1$, and $V\in C^k({\bf R}^n$, $k>k_n$, where $n\ge 4$ and $(n-3)/2\le k_n<n/2$. We also give a…

We prove dispersive estimates for solutions to the Schrodinger equation with a real-valued potential $V\in L^\infty({\bf R}^n)$, $n\ge 4$, satisfying $V(x)=O(|x|^{-(n+2)/2-\epsilon})$, $|x|>1$, $\epsilon>0$.

We prove dispersive estimates at low frequency in dimensions n greater or equal to 4 for the wave equation for a very large class of real-valued potentials, provided the zero is neither an eigenvalue nor a resonance.

We study the wave equation with potential $u_{tt}-\Delta u+Vu=0$ in two spatial dimensions, with $V$ a real-valued, decaying potential. With $H=-\Delta+V$, we study a variety of mapping estimates of the solution operators, $\cos(t\sqrt{H})$…

We prove the dispersive and Strichartz estimates for solutions to the wave equation with a class of many-electric potentials in spatial dimension three. To obtain the desired dispersive estimate, based on the spectral properties of the…

We prove an optimal dispersive $L^{\infty}$ decay estimate for a three dimensional wave equation perturbed with a large non smooth potential belonging to a particular Kato class. The proof is based on a spectral representation of the…

In this paper we use a unified way studying the decay estimate for a class of dispersive semigroup given by $e^{it\phi(\sqrt{-\Delta})}$, where $\phi: \mathbb{R}^+\to \mathbb{R}$ is smooth away from the origin. Especially, the decay…

We prove dispersive bounds for fractional Schr\"odinger operators on $\mathbb R^n$ of the form $H=(-\Delta)^{\alpha}+V$ with $V$ a real-valued, decaying potential and $\alpha \notin\mathbb N$. We derive pointwise bounds on the resolvent…

We prove a family of dispersive estimates for the higher order Schr\"odinger equation $iu_t=(-\Delta)^mu +Vu$ for $m\in \mathbb N$ with $m>1$ and $2m<n<4m$. Here $V$ is a real-valued potential belonging to the closure of $C_0$ functions…

We study the insulated conductivity problem with inclusions embedded in a bounded domain in $\mathbb{R}^n$. The gradient of solutions may blow up as $\varepsilon$, the distance between inclusions, approaches to $0$. It was known that the…

We prove optimal high-frequency resolvent estimates for perturbations of the Laplacian by large long-range magnetic and electric potentials in all dimensions $n\ge 3$. As an application, we prove dispersive estimates for the corresponding…

We prove dispersive estimates for the linear Schr\"odinger evolution associated to an operator -\Delta + V, where the potential is a signed measure of fractal dimension at least 3/2.

We provide optimal bounds for $\alpha (\beta \circ \gamma \circ \dots )$ in $4$-distributive varieties, as well as some further partial generalizations.

We obtain a decay estimate for solutions to the linear dispersive equation $iu_t-(-\Delta)^{1/4}u=0$ for $(t,x)\in\mathbb{R}\times\mathbb{R}$. This corresponds to a factorization of the linearized water wave equation…

Let $\Delta_\kappa$ be the Dunkl Laplacian on $\mathbb{R}^n$ and $\phi: \mathbb{R}^+ \to \mathbb{R}$ is a smooth function. The aim of this manuscript is twofold. First, we study the decay estimate for a class of dispersive semigroup of the…

We extend several fundamental estimates regarding spectral multipliers for the free Laplacian on $\mathbb R^3$ to the case of perturbed Hamiltonians of the form $-\Delta+V$, where $V$ is a scalar real-valued potential. In this paper, we…

Let $H=-\Delta+V$, where $V$ is a real valued potential on $\R^2$ satisfying $|V(x)|\les \la x\ra^{-3-}$. We prove that if zero is a regular point of the spectrum of $H=-\Delta+V$, then $$ \|w^{-1} e^{itH}P_{ac}f\|_{L^\infty(\R^2)}\les…

For the wave equation $\partial_t^2-\Delta+V$ on $\mathbb{R}^d$ with compactly supported, real valued potential $V$, we establish a sharp relation between Sobolev regularity of $V$ and the existence of finite order expansions as…