Strichartz estimates for the metaplectic representation

Abstract

Strichartz estimates are a manifestation of a dispersion phenomenon, exhibited by certain partial differential equations, which is detected by suitable Lebesgue space norms. In most cases the evolution propagator U (t) is a one parameter group of unitary operators. Motivated by the importance of decay estimates in group representation theory and ergodic theory, Strichartz-type estimates seem worth investigating when U (t) is replaced by a unitary representation of a non-compact Lie group, the group element playing the role of time. Since the Schrödinger group is a subgroup of the metaplectc group, the case of the metaplectic or oscillatory representation is of special interest in this connection. We prove uniform weak-type sharp estimates for matrix coefficients and Strichartz estimates for that representation. The crucial point is the choice of function spaces able to detect such a dispersive effect, which in general will depend on the given group action. The relevant function spaces here turn out to be the so-called modulation spaces from Time-frequency Analysis in Euclidean space, and Lebesgue spaces with respect to Haar measure on the metaplectic group. The proofs make use in an essential way of the covariance of the Wigner distribution with respect to the metaplectic representation.2010 Mathematics Subject Classification. 42B35, 22E45.

Weak-Type Estimates for the Metaplectic Representation Restricted to the Shearing and Dilation Subgroup of $$SL(2,\mathbb {R})$$

“…From this result it is easy to obtain corresponding Strichartz-type estimates, following the pattern in [1]. The results on Strichartz estimates are often applied to wellposedness and scattering of nonlinear PDEs, see [17] for further details.…”

“…In [1] we started a project aiming at studying weak-type and Strichartz estimates for unitary representations of non-compact Lie groups. There, we consider the case of the metaplectic representation, which is a faithful representation of the metaplectic group M p(n, R) in L 2 (R n ), being the double covering of the symplectic group Sp(n, R).…”

A characterization of modulation spaces by symplectic rotations

“…The integration parameters (x, ξ) of (4) live in R 2n , with dim R 2n = 2n, whereas the parameters (x, S) of (5) live in R n × U(2n, R). Recall that dim U(2n, R) = n 2 [15]; this suggests that a formula similar to (5) should hold when U(2n, R) is reduced to a suitable subgroup K ⊂ U(2n, R) of dimension n. This is indeed the case, as shown in the subsequent Theorem 1.3.…”

“…Step. We will show that (6) easily follows from (5). Indeed, the Fourier transform J = F is a metaplectic operator and we recall that the Fourier transform is a topological isomorphism F : M p (R n ) → M p (R n ), 1 ≤ p ≤ ∞, [16].…”

“…Comparing (4) and (5) we observe that in (5) the modulation operator M ξ is replaced by the metaplectic operator S and the integral on the phase space R 2n has become an integral on the cartesian product R n × U(2n, R). The integration parameters (x, ξ) of (4) live in R 2n , with dim R 2n = 2n, whereas the parameters (x, S) of (5) live in R n × U(2n, R).…”