28 research outputs found
Variable coefficient Wolff-type inequalities and sharp local smoothing estimates for wave equations on manifolds
The sharp Wolff-type decoupling estimates of Bourgain--Demeter are extended to the variable coefficient setting. These results are applied to obtain new sharp local smoothing estimates for wave equations on compact Riemannian manifolds, away from the endpoint regularity exponent. More generally, local smoothing estimates are established for a natural class of Fourier integral operators; at this level of generality the results are sharp in odd dimensions, both in terms of the regularity exponent and the Lebesgue exponent
Sharp local smoothing estimates for Fourier integral operators
The theory of Fourier integral operators is surveyed, with an emphasis on local smoothing estimates and their applications. After reviewing the classical background, we describe some recent work of the authors which established sharp local smoothing estimates for a natural class of Fourier integral operators. We also show how local smoothing estimates imply oscillatory
integral estimates and obtain a maximal variant of an oscillatory integral estimate of Stein. Together with an oscillatory integral counterexample of Bourgain, this shows that our local smoothing estimates are sharp in odd spatial dimensions. Motivated by related counterexamples, we formulate local smoothing conjectures which take into account natural geometric assumptions
arising from the structure of the Fourier integrals
Global pointwise decay estimates for defocusing radial nonlinear wave equations
We prove global pointwise decay estimates for a class of defocusing
semilinear wave equations in dimensions restricted to spherical symmetry.
The technique is based on a conformal transformation and a suitable choice of
the mapping adjusted to the nonlinearity. As a result we obtain a pointwise
bound on the solutions for arbitrarily large Cauchy data, provided the
solutions exist globally. The decay rates are identical with those for small
data and hence seem to be optimal. A generalization beyond the spherical
symmetry is suggested.Comment: 9 pages, 1 figur
On Existence and Scattering with Minimal Regularity for Semilinear Wave Equations
AbstractWe prove existence and scattering results for semilinear wave equations with low regularity data. We also determine the minimal regularity that is needed to ensure local existence and well-posedness, and we give counterexamples to well-posedness. More specifically, we show that equations of the type □ u= |u| p, with initial data (u, ut) in Ḣγ(Rn) × Ḣγ − 1(Rn), have a local solution if γ ≥ γ(p, n), and we construct counterexamples if γ < γ(p, n). The existence results rely on mixed-norm space-time estimates of Strichartz-type
L^{2}-restriction bounds for eigenfunctions along curves in the quantum completely integrable case
We show that for a quantum completely integrable system in two dimensions,the
-normalized joint eigenfunctions of the commuting semiclassical
pseudodifferential operators satisfy restriction bounds ofthe form for generic
curves on the surface. We also prove that the maximal restriction
bounds of Burq-Gerard-Tzvetkov are always attained for certain exceptional
subsequences of eigenfunctions.Comment: Correct some typos and added some more detail in section
Carleman estimates and absence of embedded eigenvalues
Let L be a Schroedinger operator with potential W in L^{(n+1)/2}. We prove
that there is no embedded eigenvalue. The main tool is an Lp Carleman type
estimate, which builds on delicate dispersive estimates established in a
previous paper. The arguments extend to variable coefficient operators with
long range potentials and with gradient potentials.Comment: 26 page
Lower bounds for nodal sets of Dirichlet and Neumann eigenfunctions
Let \phi\ be a Dirichlet or Neumann eigenfunction of the Laplace-Beltrami
operator on a compact Riemannian manifold with boundary. We prove lower bounds
for the size of the nodal set {\phi=0}.Comment: 7 page
On Nonlinear Functionals of Random Spherical Eigenfunctions
We prove Central Limit Theorems and Stein-like bounds for the asymptotic
behaviour of nonlinear functionals of spherical Gaussian eigenfunctions. Our
investigation combine asymptotic analysis of higher order moments for Legendre
polynomials and, in addition, recent results on Malliavin calculus and Total
Variation bounds for Gaussian subordinated fields. We discuss application to
geometric functionals like the Defect and invariant statistics, e.g.
polyspectra of isotropic spherical random fields. Both of these have relevance
for applications, especially in an astrophysical environment.Comment: 24 page
The global wave front set of tempered oscillatory integrals with inhomogeneous phase functions
We study certain families of oscillatory integrals ,
parametrised by phase functions and amplitude functions globally
defined on , which give rise to tempered distributions, avoiding
the standard homogeneity requirement on the phase function. The singularities
of are described both from the point of view of the lack of
smoothness as well as with respect to the decay at infinity. In particular, the
latter will depend on a version of the set of stationary points of ,
including elements lying at the boundary of the radial compactification of
. As applications, we consider some properties of the two-point
function of a free, massive, scalar relativistic field and of classes of global
Fourier integral operators on , with the latter defined in terms
of kernels of the form .Comment: 30 pages, 2 figures, mistakes and typos correctio
Near Sharp Strichartz estimates with loss in the presence of degenerate hyperbolic trapping
We consider an -dimensional spherically symmetric, asymptotically
Euclidean manifold with two ends and a codimension 1 trapped set which is
degenerately hyperbolic. By separating variables and constructing a
semiclassical parametrix for a time scale polynomially beyond Ehrenfest time,
we show that solutions to the linear Schr\"odiner equation with initial
conditions localized on a spherical harmonic satisfy Strichartz estimates with
a loss depending only on the dimension and independent of the degeneracy.
The Strichartz estimates are sharp up to an arbitrary loss. This is
in contrast to \cite{ChWu-lsm}, where it is shown that solutions satisfy a
sharp local smoothing estimate with loss depending only on the degeneracy of
the trapped set, independent of the dimension