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# Francesco Di Plinio

INdAM-COFUND Reintegration fellow between 2012-11-19 and 2014-11-18

### Project information

#### Title

TimeFrequency: Time-frequency analysis of the endpoint behavior of linear and multilinear operators with modulation invariance and related problems

#### Partner organisation

Università degli Studi di Roma “Tor Vergata”

Via della Ricerca Scientifica, 1

00133 Roma, Italy

#### Abstract

Time-frequency analysis in its modern form was brought to life in the seminal proof of boundedness of the bilinear Hilbert transform, by Lacey and Thiele. However, its roots lie in the proofs by Carleson, and then by C. Fefferman, of almost everywhere convergence of Fourier series of a square-integrable function: indeed, a more streamlined proof of this celebrated theorem, via boundedness of Carleson’s operator, was obtained by Lacey and Thiele using the same tools employed for the bilinear Hilbert transform.

The main common feature shared by the (sublinear) Carleson’s operator and the bilinear Hilbert transform is their modulation invariance. Time-frequency analysis has since then been used to prove boundedness of several fundamental operators exhibiting some sort of modulation invariance. However, the present tools are not sharp enough to completely determine the behavior of linear and multilinear operators near the endpoints of their conjectured boundedness regions.

Intrinsic difficulties (of combinatorial, analytic, and geometrical nature) arise in the application of time-frequency analysis to multiparameter analogues of Carleson’s operator. Notable examples are boundedness of maximal partial sums of two-dimensional Fourier series restricted along curves in the plane, quantitative bounds for Fourier restrictions on polygons, and boundedness of the Hilbert transform (and of the related maximal function) along smooth vector fields.

The purpose of this research project is therefore to continue the investigation of the endpoint behavior of linear and multilinear operators with modulation symmetry, and to further study the aforementioned multiparameter operators in the framework of multiparameter time-frequency analysis.

### Fellow information

#### Previous positions, awards

*2012*: Swain Fellowship, Indiana University Mathematics Department*2008-2012*: Teaching and Research Assistant, Indiana

University Mathematics Department*2008-2012*: Ph.D. in Pure Mathematics at the Indiana

University. GPA 4.00/4.00. Thesis advisors Prof. Ciprian Demeter and

Prof. Roger Temam.*2008*: College of Arts and Sciences Graduate Fellowship, Indiana University*2007-2008*: Teaching Assistant, Dipartimento di Matematica Politecnico di Milano*2007*: Research grant, Fondazione Fratelli Confalonieri

#### Publications, preprints, other works

- F. Di Plinio, V. Pata, S. Zelik,
*On the strongly damped wave equation with memory*, Indiana Univ. Math. J. 57 (2008), no. 2, 757-780. MR2414334 - F. Di Plinio, V. Pata,
*Robust exponential attractors for the strongly damped wave equation with memory I*, Russ. J. Math. Phys. 15 (2008), 301-315. MR2448344 - F. Di Plinio, V. Pata,
*Robust exponential attractors for the strongly damped wave equation with memory II*, Russ. J. Math. Phys. 16 (2009), 61-73. MR2486806 - F. Di Plinio, G. S. Duane, R. Temam,
*Time-dependent attractor for the oscillon equation*, Discrete Contin. Dyn. Syst. 29 (2011), no. 1, 141-167. MR2725285 - M. D. Chekroun, F. Di Plinio, N. E. GlattâHoltz, V. Pata,
*Asymptotics of the Coleman-Gurtin model*, Discrete Contin. Dyn. Syst. Ser. S 4 (2011), no. 2, 351-369. MR2746378 - F. Di Plinio, G. S. Duane, R. Temam,
*The 3-dimensional Oscillon Equation*, Boll. Unione Mat. Ital. Ser. IX 5 (2012), no. 1, 19-54. - F. Di Plinio, C. Demeter,
*Logarithmic L*, preprint (2012); arXiv:1203.6624, Journal of Geometric Analysis, in press^{p}bounds for maximal directional singular integrals in the plane - C. Demeter, F. Di Plinio,
*Endpoint bounds for the Quartile Operator*, preprint (2012); arXiv:1206.3798, accepted in the Journal of Fourier Analysis and Applications - C. Bardos, F. Di Plinio, R. Temam,
*The Euler equations in planar nonsmooth domains*, preprint (2012); arXiv:1212.0036, submitted - F. Di Plinio,
*Endpoint bounds for the Dirichlet problem on planar domains with corners*, in preparation - F. Di Plinio,
*Walsh-Fourier series near L*, in preparation^{1}

#### Conferences, schools, other events

*invited talk*

Talk delivered at*Oberanalysis seminar*

Hausdorff Center for Mathematics, Bonn, May 3, 2013*invited talk*

Talk delivered at*Analysis and Partial Differential Equations Seminar*

Brown University, Providence RI, USA, April 3, 2013*invited talk*

Talk delivered at*Analysis and Partial Differential Equations Seminar*

Princeton University, USA, April 1, 2013*invited talk*

Talk delivered at*Dynamical Systems and Ergodic Theory Seminar*

University of Illinois at Urbana-Champaign, USA, October 22, 2012*invited talk*

Talk delivered at*AIMS Conference on Dynamical Systems, Differential Equations and Applications, special sessions on Infinite Dimensional Dynamics and Applications (organizers J. C. Robinson, Y. You),*

The Navier-Stokes Equations and Related Problems(organizers

S. Necasova,R. Rautmann, W. Varnhorn)

Orlando, FL, USA, July 1-5, 2012*short talk*

Talk delivered at*9th International Conference on Harmonic Analysis and Partial*

Differential Equations

El Escorial, Madrid, Spain, June 11-15, 2012*invited talk*

Talk delivered at*Southeastern Analysis Meeting*

University of Alabama, Tuscaloosa, AL, USA, March 8-10, 2012*invited talk*

Talk delivered at*Dynamical Systems Seminar*

University of Toledo, OH, USA, November 15, 2011*invited talk*

Talk delivered at*AMS 2011 Spring Southeastern Section Meeting,*

special Session on Harmonic Analysis and Applications (organizers

D. Bilyk, L. De Carli, A. Stokolos, B. Wick)

Statesboro, GA, USA, March 12-13, 2011*invited talk*

Talk delivered at*Conference on Nonlinear Evolution Equations (organizers G. Grillo et al.)*

Mondello, Palermo, Italy, June 8-11, 2010*invited talk*

Talk delivered at*Conference on Mathematical Models and Analytical*

Problems for Special Materials (organizers P. Colli, C. Giorgi,

M. Grasselli et al.)

Brescia, Italy, July 9-11, 2009