Related papers: On local smoothing estimates for wave equations

We use functions of a bicomplex variable to unify the existing constructions of harmonic morphisms from a 3-dimensional Euclidean or pseudo-Euclidean space to a Riemannian or Lorentzian surface. This is done by using the notion of…

We study optimization over Riemannian embedded submanifolds, where the objective function is relatively smooth in the ambient Euclidean space. Such problems have broad applications but are still largely unexplored. We introduce two…

The purpose of this article is twofold. First we give a very robust method for proving sharp time decay estimates for the most classical three models of dispersive Partial Differential Equations, the wave, Klein-Gordon and Schr{\"o}dinger…

This note is devoted to optimal spectral estimates for Schr\"odinger operators on compact connected Riemannian manifolds without boundary. These estimates are based on the use of appropriate interpolation inequalities and on some recent…

We derive estimates relating the values of a solution at any two points to the distance between the points, for quasilinear isotropic elliptic equations on compact Riemannian manifolds, depending only on dimension and a lower bound for the…

We establish new Strichartz estimates for orthonormal systems on compact Riemannian manifolds in the non-sharp admissible region of exponents, covering wave, Klein-Gordon, and fractional Schr\"odinger equations. Our approach combines the…

Multilinear embedding estimates for the fractional Laplacian are obtained in terms of functionals defined over a hyperbolic surface. Convolution estimates used in the proof enlarge the classical framework of the convolution algebra for…

In this note, we obtain the sharp estimates for the first eigenvalue of Paneitz operator for $4$-dimensional compact submanifolds in Euclidean space. Since unit spheres and projective spaces can be canonically imbedded into Euclidean space,…

This paper shows how abstract resolvent estimates imply local smoothing for solutions to the Schr\"odinger equation. If the resolvent estimate has a loss when compared to the optimal, non-trapping estimate, there is a corresponding loss in…

We obtain sharp estimates for heat kernels and Green's functions on complete noncompact Riemannian manifolds with Euclidean volume growth and nonnegative Ricci curvature. We will then apply these estimates to obtain sharp Moser-Trudinger…

For the Hamiltonian operator H = -{\Delta}+V(x) of the Schr\"odinger Equation with a repulsive potential, the problem of local decay is considered. It is analyzed by a direct method, based on a new, L^2 bounded, propagation observable. The…

We prove localized energy estimates for the wave equation in domains with a strictly concave boundary when homogeneous Dirichlet or Neumann conditions are imposed. By restricting the solution to small, frequency dependent, space time…

In this paper, we investigate the $L^p$ bilinear quasimode estimates on compact Riemannian manifolds. We obtain results in the full range $p\ge2$ on all $n$-dimensional manifolds with $n\ge2$. This in particular implies the $L^p$ bilinear…

We consider wave equations on Lorentzian manifolds in case of low regularity. We first extend the classical solution theory to prove global unique solvability of the Cauchy problem for distributional data and right hand side on smooth…

We consider Morawetz estimates for weighted energy decay of solutions to the wave equation on scattering manifolds (i.e., those with large conic ends). We show that a Morawetz estimate persists for solutions that are localized at low…

We prove a family of sharp bilinear space-time estimates for the half-wave propagator. As a consequence, for radially symmetric initial data, we establish sharp estimates of this kind for a range of exponents beyond the classical range.

We consider the solution operator for the wave equation on the flat Euclidean cone over the circle of radius $\rho > 0$, the manifold $\mathbb{R}_+ \times \mathbb{R} / 2 \pi \rho \mathbb{Z}$ equipped with the metric $\g(r,\theta) = dr^2 +…

We consider solutions to the linear wave equation on non-compact Riemannian manifolds without boundary when the geodesic flow admits a filamentary hyperbolic trapped set. We obtain a polynomial rate of local energy decay with exponent…

We derive various eigenvalue estimates for the Hodge Laplacian acting on differential forms on weighted Riemannian manifolds. Our estimates unify and extend various results from the literature and we provide a number of geometric…

We present a local Fourier slice equation that enables local and sparse projection of a signal. Our result exploits that a slice in frequency space is an iso-parameter set in spherical coordinates. Therefore, the projection of suitable…