

CACCIAFESTA FEDERICO
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Curriculum Vitae
Research areas
My research activity is mainly devoted to dispersive partial differential equations. In particular, I am (or I have been) focused on the following themes.
 The Dirac equation: study of dispersive (local smoothing, Strichartz, time decay...) estimates for perturbations of the Dirac equation; in particular, scaling critical potential perturbations (Coulomb or AharonovBohm potentials) and the Dirac equation on curved space (how does the underlying geometry affect dispersive dynamics?). Applications to the dynamics of relativistic models.  Variable coefficients equation: the application of the Morawetz multiplier method seems to give interesting result in the contest of variable coefficients dispersive PDEs. In particular, we have adapted it to obtain quantitative result in the contest of the Helmholtz equation on exterior domains (AgmonHormander type estimates, Limiting Absorption Principle, Sommerfeld radiation condition...) and of the Dirac equation on nonflat manifolds (asymptotically flat, wrapped products...)  Invariant measures: existence and properties of invariant measures for dispersive flows. In particular, the construction of invariant measures for the cubic NLS (and more general nonlinear equations) on the real line, via "direct" methods and by relying on probabilistic tools.  Weighted functional inequalities: study of relevant inequalities with direct applications to dispersive PDEs, as weighted L^p inequalities for powers of electromagnetic Schroedinger operators, or weighted estimates for singular integrals. Publications
 Dispersive estimates for the Dirac equation in an AharonovBohm field, with L. Fanelli, submitted.
 Hankel transforms and weak dispersion, with L. Fanelli, submitted.  A limiting absorption principle for the Helmholtz equation with variable coefficients, with P. D'Ancona and R. Lucà, submitted.  Invariance of Gibbs measures under the flows of Hamiltonain equations on the real line, with A.S. De Suzzoni, submitted.  Weak dispersive estimates for fractional AharonovBohmSchroedinger groups, with L. Fanelli, submitted.  Weak dispersion for the Dirac equation on curved spacetime, with A.S. De Suzzoni, submitted. Singular integrals with angular integrability, with R. Lucà, Proc. Amer. Math. Soc. 144 (2016), 34133418.  Stability of invariant measures and continuity of the KdV flow, Bull. Braz. Math. Soc. New series 47 (1) 110.  Local smoothing estimates for the massless Dirac equation in 2 and 3 dimensions, with E. Seré, J. Funct. Anal 271 (2016) no.8, 23392358.  On Gibbs measure and weak flow for the cubic NLS with nonlocalised initial data, with A.S. De Suzzoni.  Helmholtz and dispersive equations with variable coefficients on exterior domains, with P. D'Ancona and R. Lucà, SIAM J. Math. Anal. 48 (2016), no.3 17981832.  Invariant measure for the Schrödinger equation on the real line, with A.S. De Suzzoni, J. Funct. Anal. 269 (2015) no. 1, 271–324.  Continuity of the flow of KdV with regard to the Wasserstein metrics and application to an invariant measure, with A.S. de Suzzoni, J. Differential equations 259 (2015), no. 3, 1024–1067.  Smoothing estimates for variable coefficients Schroedinger equation with electromagnetic potential, J. Math. Anal. Appl. 402 (2013), pp. 286296.  Endpoint estimates and global existence for the nonlinear Dirac equation with a potential, with P. D'Ancona, J. Differential equations 254 (2013), pp. 22332260.  The cubic nonlinear Dirac equation, Actes de Journées EDP 2012, Biarritz.  Weighted L^p estimates for powers of selfadjoint operators, with P. D'Ancona, Advances in Mathematics 229 (2012), pp. 501530.  Virial identity and dispersive estimates for the ndimensional Dirac equation, J. Math. Sci. Univ. Tokyo 18 (2011), pp. 123.  Global small solutions to the critical Dirac equation with potential, Nonlinear Analysis 74 (2011), pp. 60606073. List of taught course units in A.Y. 2019/20


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