Local smoothing estimates for Schrödinger equations on hyperbolic space

Abstract

We establish global-in-time frequency localized local smoothing estimates for Schrödinger equations on hyperbolic space H d . In the presence of symmetric first and zeroth order potentials, which are possibly timedependent, possibly large, and have sufficiently fast polynomial decay, these estimates are proved up to a localized lower order error. Then in the timeindependent case, we show that a spectral condition (namely, absence of threshold resonances) implies the full local smoothing estimates (without any error), after projecting to the continuous spectrum. In the process, as a means to localize in frequency, we develop a general Littlewood-Paley machinery on H d based on the heat flow. Our results and techniques are motivated by applications to the problem of stability of solitary waves to nonlinear Schrödingertype equations on H d .As a testament of the robustness of approach, which is based on the positive commutator method and a heat flow based Littlewood-Paley theory, we also show that the main results are stable under small time-dependent perturbations, including polynomially decaying second order ones, and small lower order nonsymmetric perturbations.

Asymptotic Stability of Harmonic Maps on the Hyperbolic Plane under the Schrödinger Maps Evolution

“…As we discussed in Section 1.2, the main linear estimates in this paper, i.e., local smoothing estimates, or local energy decay estimates, and Strichartz estimates for the operator obtained by linearization about a harmonic map, were established in the companion paper [36]. This local smoothing estimate is the key technical ingredient for dealing with the loss of smoothing for Schrödinger equations mentioned in the previous subsection.…”

“…The harmonic map heat flow equation (1.4) To handle the presence of a derivative on the RHS, we rely on the local smoothing effect for the Schrödinger operator i@ t H . The strong linearized stability condition (Definition 1.3) allows us to apply the general theorem in the companion paper [36] to conclude that i@ t H enjoys a global-in-time local smoothing estimate with spatial weights depending on r (i.e., the distance to some fixed point); see Proposition 3.9. The global-in-time local smoothing estimate allows us to gain one derivative as needed; the price we pay is, among other things, that we need to uniformly bound r 4 A k .s/ in the region fr > 1g.…”

“…The most dangerous paradifferential term (i.e., "low-high" interaction in A k .s/r k s .s/) is treated using the global-in-time local smoothing estimate as sketched above; the sharp form of the spatial weights near r h 0 in the norms LE and LE £ (see Section 3.3) allows us to close the argument with an arbitrarily small > 0, where h 0 is optimal 4 . Handling the remaining interactions in A k .s/r k s .s/ requires additional tools, such as global-in-time Strichartz estimates for i@ t H on H 2 (Lemma 3.12, also proved in [36]), Bernstein-type inequalities adapted to the spatial weights in LE (Lemmas 3.13 and 3.14), and the decay of the expression P s .s/ off the diagonal f % sg (Lemma 6.4). We refer to Section 8 for the rigorous treatment of the magnetic interaction term.…”

“…This condition is known to hold for all equivariant harmonic maps with values in the hyperbolic plane and for a subset of those maps taking values in the sphere. One of the main technical ingredients in the paper is a global-in-time local smoothing and Strichartz estimate for the operator obtained by linearization around a harmonic map, proved in the companion paper [36].…”

Pointwise dispersive estimates for Schrödinger operators on product cones

“…See the work of Krieger-Schlag [45] and more recently Krieger-Miao-Schlag [44] for instance. See also the many works of Lawrie-Oh-Shahshahani [46,47,48,49,50,51] for treatment of geometric wave and Schrödinger equations in hyperbolic space. Pointwise decay estimates also play a role in obtaining enhanced existence times using normal form methods, see for instance recent works of Ifrim-Tataru [41] and Germain-Pusateri-Rousset [23].…”