Related papers: $L^p$ estimates for fractional schrodinger operato…

We improve an $L^2\times L^2\to L^2$ estimate for a certain bilinear operator in the finite field of size $p$, where $p$ is a prime sufficiently large. Our method carefully picks the variables to apply the Cauchy-Schwarz inequality. As a…

In the paper \cite{KLMR} the $L^p$-realization $L_p$ of the matrix Schr\"odinger operator $\mathcal{L}u=div(Q\nabla u)+Vu$ was studied. The generation of a semigroup in $L^p(\R^d,\C^m)$ and characterization of the domain $D(L_p)$ has been…

Given a semi-convex potential V on a convex and bounded domain $\Omega$, we consider the Jordan-Kinderlehrer-Otto scheme for the Fokker-Planck equation with potential V, which defines, for fixed time step $\tau$ > 0, a sequence of densities…

An initial-boundary value problem with a Caputo time derivative of fractional order $\alpha\in(0,1)$ is considered, solutions of which typically exhibit a singular behaviour at an initial time. An L2-type discrete fractional-derivative…

In this work, we prove an optimal global-in-time L p --L q estimate for solutions to the Kramers-Fokker-Planck equation with short range potential in dimension three. Our result shows that the decay rate as t $\rightarrow$ +$\infty$ is the…

We use the mirror coupling of Brownian motion to show that under a $\beta\in (0,1)$-dependent Kato type assumption (which is satisfied under a suitable $L^q$-assumption on the electro-magnetic potential, where $q$ depends on $\beta$ and the…

For $2\leq p<\infty$, $\alpha'>2/p$, and $\delta>0$, we construct Cantor-type measures on $\mathbb{R}$ supported on sets of Hausdorff dimension $\alpha<\alpha'$ for which the associated maximal operator is bounded from $L^p_\delta…

This paper investigates $L^p$-estimates for solutions to the wave equation perturbed by a scaling-critical partial inverse-square potential. We study a model in which the singularity of the potential appears only in a subset of the…

We prove a dispersive estimate for the time-independent Schrodinger operator H = -\Delta + V in three dimensions. The potential V(x) is assumed to lie in the intersection L^p(R^3) \cap L^q(R^3), p < 3/2 < q, and also to satisfy a generic…

We consider a self-adjoint non-negative operator $H$ in a Hilbert space $\mathsf{L}^2(X,{\rm d}\mu)$. We assume that the semigroup $(\mathrm{e}^{-t H})_{t>0}$ is defined by an integral kernel, $p$, which allows an estimate of the form…

Let $\{u_\lambda\}$ be a sequence of $L^2$-normalized Laplacian eigenfunctions on a compact two-dimensional smooth Riemanniann manifold $(M,g)$. We seek to get an $L^p$ restriction bounds of the Neumann data $ \lambda^{-1} \partial_\nu…

Let $H:=-\Delta+V$ be a nonnegative Schr\"odinger operator on $L^2({\bf R}^N)$, where $N\ge 2$ and $V$ is an inverse square potential. In this paper we obtain sharp decay estimates of the operator norms of $e^{-tH}$ and $\nabla e^{-tH}$ in…

$L^p$ to $L^p_{\beta}$ boundedness theorems are proven for translation invariant averaging operators over hypersurfaces in Euclidean space. The operators can either be Radon transforms or averaging operators with multiparameter fractional…

Let $L$ be a non-negative self-adjoint operator acting on $L^2(X)$ where $X$ is a space of homogeneous type with a dimension $n$. Suppose that the heat kernel of $L$ satisfies the Davies-Gaffney estimates of order $m\geq 2$. Let $H^1_L(X)$…

We consider fractional Schr\"odinger operators with possibly singular potentials and derive certain spatially averaged estimates for its complex-time heat kernel. The main tool is a Phragm\'en-Lindel\"of theorem for polynomially bounded…

We consider the higher order Schr\"odinger operator $H=(-\Delta)^m+V(x)$ in $n$ dimensions with real-valued potential $V$ when $n>2m$, $m\in \mathbb N$. We adapt our recent results for $m>1$ to show that the wave operators are bounded on…

Semilinear stochastic partial differential equations on bounded domains $\mathscr{D}$ are considered. The semilinear term may have arbitrary polynomial growth as long as it is continuous and monotone except perhaps near the origin. Typical…

Let $L$ be a non-negative self-adjoint operator acting on $L^2(X)$, where $X$ is a space of homogeneous type with a dimension $n$. Suppose that the heat operator $e^{-tL}$ satisfies the generalized Gaussian $(p_0, p'_0)$-estimates of order…

In 1998, V. Liskevich and Y. Semenov showed sharp Gaussian upper bounds for Schr\"odinger semigroups on $\mathbb R^3$ with potentials satisfying a global Kato class condition. Using similar basic ideas we show sharp Gaussian upper bounds…

In this article we study some Kramers-Fokker-Planck operators with a polynomial potential $V(q)$ of degree greater than two having quadratic limiting behavior. This work provides an accurate global subelliptic estimate for KFP operators…