Neal Bez (he/him)
Introduction to Strichartz estimates, connection to the Fourier restriction conjecture, and preparation for the proof of Strichartz estimates (interpolation, Hardy-Littlewood-Sobolev inequality, Littlewood-Paley inequality, etc)
Dispersive estimates and proof of the non-endpoint and non-boundary Strichartz estimates
Proof of the Keel-Tao endpoint and boundary Strichartz estimates
Proof of inhomogeneous Strichartz estimates in a restricted range of exponents
Topic A-1: Proof of Strichartz estimates via analytic interpolation and extension to orthonormal Strichartz estimates [A1]
Topic A-2: Best constants and maximizers for Strichartz estimates (Schrödinger) in dimensions one and two [A2,A3]
[A1] R. Frank, J. Sabin, Restriction theorems for orthonormal functions, Strichartz inequalities, and uniform Sobolev estimates, Amer. J. Math. 139 (2017), 1649-1691
[A2] D. Foschi, Maximizers for the Strichartz inequality, J. Eur. Math. Soc. 9 (2007), 739-774
[A3] T. Ozawa, Y. Tsutsumi, Space-time estimates for null gauge forms and nonlinear Schrödinger equations, Differential Integral Equations 11 (1998), 201-222
Luca Fanelli
Introduction: Non Linear Schrödinger equation, Cauchy theory and scattering via Strichartz
Kato smoothing
Weighted L^2 resolvent estimates
Applications: time-decay and Strichartz estimates
Topic B-1: Read the two papers [B4] and [B7]. Reproduce in detail the proof of Lemma 2.1 from [B7]
Topic B-2: Using the representation formula (2.6) in [B8] study the validity of the endpoint Strichartz estimate for the Schrödinger equation with a critical (focusing) inverse-square potential in space dimension 3 and higher
[B1] Linares, F., and Ponce, G.: Introduction to Nonlinear Dispersive Equations, Springer 2014
[B2] Rodnianski, I., and Schlag, W.: Time decay for solutions of Schrödinger equations with rough and time-dependent potentials, Inventiones Math. 2004
[B3] Burq, N., Planchon, F., Stalker, J., and Tahvildar-Zadeh, S.: Strichartz estimates for the Wave and Schrodinger Equations with the Inverse-Square Potential, Journal of Functional Analysis 2003
[B4] Fanelli, L., Felli, V., Fontelos, M., and Primo, A.: Time Decay of Scaling Critical Electromagnetic Schrödinger Flows, Comm. Math. Phys. 2013
[B5] Fanelli, L., Krejcirik, D., and Vega, L.: Spectral stability of Schroedinger operators with subordinated complex potentials, Journal of Spectral Theory 2018
[B6] D'Ancona, P., and Fanelli, L.: Strichartz and Smoothing Estimates for Dispersive Equations with Magnetic Potentials, Comm. Part. Diff. Eq. 2008
[B7] Fanelli, L., Felli, V., Fontelos, M., and Primo, A.: Time decay of scaling invariant electromagnetic Schrödinger equations on the plane, Comm. Math. Phys. 2015
[B8] Fanelli, L., Felli, V., Fontelos, M., and Primo, A.: Frequency-dependent timeof Schrödinger flows, Journal of Spectral Theory, 2018
David dos Santos Ferreira
The uniform Sobolev estimates of Kenig, Ruiz and Sogge for the Euclidean Laplacian. Introduction to L^p resolvent estimates on compact manifolds
The Hadamard parametrix construction. A review of Carleson-Sjölin theory for oscillatory integral operators. Resolvent estimates
The relation with Sogge’s cluster estimates on compact Riemannian manifolds
Application to the derivation of Carleman estimates.
Topic C-1: Uniform Sobolev estimates on non-compact non-trapping manifolds [C1]
Topic C-2: Extension to higher order elliptic operators [C2]
[C1] C. Guillarmou, A. Hassell, Uniform Sobolev estimates for non-trapping metrics, J. Inst. Math. Jussieu 13 (3) (2014) 599–632
[C2] K. Krupchyk, G. Uhlmann, On Lp resolvent estimates for elliptic operators on compact manifolds, Commun. Partial Differ. Equ. 40 (3) (2015) 438–474