Dispersive and Strichartz estimates for the three-dimensional wave equation with a scaling-critical class of potentials

Characteristic Strichartz Estimates for the Wave Equation

Dispersive estimates for the three-dimensional Schrödinger equation with rough potentials

The three-dimensional Schrödinger propogator e , H = −△+V , is a bounded map from L to L∞ with norm controlled by |t|−3/2 provided the potential satisfies two conditions: An integrability condition limiting the singularities and decay of V , and a zero-energy spectral condition on H . This is shown by expressing the spectral measure of H in terms of its resolvents and proving a family of L mapp...

Dispersive bounds for the three-dimensional Schrödinger equation with almost critical potentials

We prove a dispersive estimate for the time-independent Schrödinger operator H = −∆ + V in three dimensions. The potential V (x) is assumed to lie in the intersection L(R) ∩ L(R), p < 3 2 < q, and also to satisfy a generic zero-energy spectral condition. This class, which includes potentials that have pointwise decay |V (x)| ≤ C(1+ |x|)−2−ε, is nearly critical with respect to the natural scalin...

Strichartz Estimates for the Magnetic Schrödinger Equation with Potentials V of Critical Decay

We study the Strichartz estimates for the magnetic Schrödinger equation in dimension n ≥ 3. More specifically, for all Schöldinger admissible pairs (r, q), we establish the estimate ‖eitHf‖Lqt (R;Lx(R)) ≤ Cn,r,q,H‖f‖L2(Rn) when the operator H = −∆A + V satisfies suitable conditions. In the purely electric case A ≡ 0, we extend the class of potentials V to the FeffermanPhong class. In doing so, ...

Strichartz Estimates for the Wave Equation on Manifolds with Boundary

Strichartz estimates are well established on flat Euclidean space, where M = R and gij = δij . In that case, one can obtain a global estimate with T = ∞; see for example Strichartz [27], Ginibre and Velo [9], Lindblad and Sogge [16], Keel and Tao [14], and references therein. However, for general manifolds phenomena such as trapped geodesics and finiteness of volume can preclude the development...