On smoothing estimates in modulation spaces and the nonlinear Schrödinger equation with slowly decaying initial data

Wellposedness of NLS in Modulation Spaces

“…To this end we first extend the almost conserved energies constructed in [26] to the range p = 2, 1 ≤ q < 2 and then use the principle of persistence of regularity to see that for a restricted range of 1 ≤ p, q ≤ 2, the newly constructed local solutions are also global. Finally, we prove as in [29,16] that in the defocusing case when we take s ≥ 1, we obtain global solutions in M 1 p,1 for any 2 < p < ∞. In fact, the same technique shows global well-posedness in M s p,q for any 2 < p < ∞ if s > 2 − 1/q is large enough.…”

“…In the case u 0 ∈ M p,1 with 2 < p < ∞, we want to use techniques inspired by [16]. Similar results were obtained for p = 4 and p = 6 in [29]. Note though that the spaces M s 4,2 and M s 6,2 with s > 3/2 embed into M 1 4,2 and M 1 6,2 in which we will prove global wellposedness.…”

“…This is done by a Banach fixed point argument using multilinear interpolation on the estimates obtained in [19] and the trivial estimates for q = 1. From this we obtain local wellposedness in a range of (p, q), comprising all of the aforementioned range for s = 0 for which local well-posedness results were shown, except for the point (p, q) = (4, 2) from [29]. The regularity s = 0 is sharp if we aim for analytic well-posedness by the considerations we provide in Section 6.…”

“…Taking into account the well-posedness in M 4,2 from [29], these results can be slightly strengthened to include the line 1/q = 1 − 2/p, 4 ≤ p ≤ ∞. Indeed, in [29] the estimate…”

“…Global solutions for initial data in M p,p ′ with p sufficiently close to 2 were constructed in [11], though we note that these solutions were allowed take value in a different space M p,q for t > 0. Using decoupling techniques, Schippa [29] recently proved L p smoothing estimates and extended the range of local wellposedness results for p ∈ {4, 6} and also, inspired by the work [16], gave global results for q = 2, 2 ≤ p < ∞, s > 3/2. Finally we want to mention the preprint [30] in which Schippa very recently considered the energy-critical NLS with initial data in modulation spaces.…”