Lectures and study topics

Topic A: Classical Strichartz estimates and restriction theorems 

Lecturer:

Neal Bez (he/him)

Content of the lectures:

1. 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)

2.  Dispersive estimates and proof of the non-endpoint and non-boundary Strichartz estimates

3.  Proof of the Keel-Tao endpoint and boundary Strichartz estimates

4.  Proof of inhomogeneous Strichartz estimates in a restricted range of exponents

Topics for student groups:

• 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]

References:

[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

Topic B:  Resolvent and Strichartz estimates, and applications to the analysis of PDEs 

Lecturer:

 Luca Fanelli

Content of the lectures:

1.  Introduction: Non Linear Schrödinger equation, Cauchy theory and scattering via Strichartz

2.  Kato smoothing

3. Weighted L^2 resolvent estimates

4. Applications: time-decay and Strichartz estimates

Topics for student groups:

• 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

References:

[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

Topic C: Resolvent estimates on compact Riemannian manifolds

Lecturer:

David dos Santos Ferreira

Content of the lectures:

1. The uniform Sobolev estimates of Kenig, Ruiz and Sogge for the Euclidean Laplacian. Introduction to L^p resolvent estimates on compact manifolds

2. The Hadamard parametrix construction. A review of Carleson-Sjölin theory for oscillatory integral operators. Resolvent estimates

3. The relation with Sogge’s cluster estimates on compact Riemannian manifolds

4. Application to the derivation of Carleman estimates.

Topics for student groups:

• Topic C-1: Uniform Sobolev estimates on non-compact non-trapping manifolds [C1]

• Topic C-2: Extension to higher order elliptic operators [C2]

References:

[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