Research interests

My main research interests lie in the area of Euclidean harmonic analysis and its interactions with dispersive PDE, geometric measure theory and analytic number theory. Particular examples are questions related to the Fourier restriction conjecture, decoupling inequalities, local smoothing estimates, Bochner--Riesz means, radial Fourier multipliers, averages along manifolds, maximal and variation norm Radon transforms, the Kakeya conjecture and extremisers for Strichartz estimates. I am also interested in the recent developments on sparse operators that have led to optimal results in classical weighted harmonic analysis, in questions related to the regularity of classical maximal functions and in the geometric aspects of oscillatory Fourier multipliers, pseudodifferential operators and Fourier integral operators.

Publications and preprints

• On a planar Pierce--Yung operator, (with S. Guo and J. Hickman), submitted, arXiv

• Localised variants of multilinear restriction, (with J. Duncan and J. Hickman), submitted, arXiv

• Bochner--Riesz means at the critical index: weighted and sparse bounds, (with J. Roos and A. Seeger), Math. Annalen, arXiv

• Off-diagonal estimates for the helical maximal function, (with J. Duncan and J. Hickman), Proc. Lond. Math. Soc., arXiv

• On sharp isoperimetric inequalities on the hypercube, (with P. Ivanisvili and J. Madrid), submitted, arXiv

• Endpoint sparse domination for classes of multiplier transformations, (with J. Roos and A. Seeger), Math. Z. , arXiv

• L^p-L^q local smoothing estimates for the wave equation via k-broad Fourier restriction, (with O. Saari), J. Fourier Anal. Appl., arXiv

• Continuity of the gradient of the fractional maximal operator in W^{1,1}(R^d), (with C. González-Riquelme, J. Madrid and J. Weigt), Math. Res. Lett., arXiv.

• Sobolev improving for averages over curves in R^4, (with S. Guo, J. Hickman and A. Seeger), Adv. Math., arXiv 

• Sharp L^p bounds for the helical maximal function, (with S. Guo, J. Hickman and A. Seeger), to appear in Am. J. Math. , arXiv

• Variation bounds for spherical averages, (with R. Oberlin, L. Roncal, A. Seeger and B. Stovall), Math. Annalen, arXiv

• Multi-scale sparse domination, (with J. Roos and A. Seeger), Mem. Amer. Math. Soc., arXiv.

• The circular maximal operator on Heisenberg radial functions, (with S. Guo, J. Hickman and A. Seeger), Ann. Sc. Norm. Super. Pisa Cl. Sci., arXiv.

• Regularity of the centered fractional maximal function on radial functions, (with J. Madrid), J. Funct. Anal., arXiv

• Bilinear identities involving the k-plane transform and Fourier extension operators, (with L. Vega), Proc. A Royal Soc. Edinburgh, arXiv

• Endpoint Sobolev continuity of the fractional maximal function in higher dimensions, (with J. Madrid), Int. Math. Res. Not. IMRN, arXiv

• Regularity of fractional maximal functions through Fourier multipliers, (with J.P. Ramos and O. Saari), J. Funct. Anal., arXiv

• Variable coefficient Wolff-type inequalities and sharp local smoothing estimates for wave equations on manifolds, (with J. Hickman and C. D. Sogge), Anal. PDE, arXiv

• Sparse bounds for pseudodifferential operators, (with L. Cladek), J. Anal. Math., arXiv

• Control of pseudodifferential operators by maximal functions via weighted inequalities, Trans. Amer. Math. Soc., arXiv

• Subdyadic square functions and applications to weighted harmonic analysis, (with J. Bennett), Adv. Math., arXiv

• A Fefferman-Stein inequality for the Carleson operator, Rev. Mat. Iberoam., arXiv

Expository papers

• A note on endpoint Bochner--Riesz estimates, (with J. Roos and A. Seeger), Oberwolfach Preprint, also available here

• Sharp local smoothing estimates for Fourier integral operators, (with J. Hickman and C. D. Sogge), conference proceedings of 'Geometric Aspects of Harmonic Analysis', Springer INdAM series 45, Chapter 2, on the occasion of Fulvio Ricci's 70th birthday, arXiv

Teaching

A la Universitat de València

• Spring 2024 (2n quadrimestre):

  • Anàlisi Harmònica, Grau en Matemàtiques; Dilluns-Dimecres 17.00-19.00, aula 0.5  (Facultat de Matemàtiques, Bloc G). Fitxa de l'assignatura Página de la asignatura: Aula Virtual o aquí

  • Valentia Matematica; summer school, 17-20 Juny, Facultat de Matemàtiques.  Mini-curs en Geometria d'incidències

• Fall 2023 (1r quadrimestre): 

  • Fundamentos de matemática avanzada (Teoria de la medida),  Máster Universitario en Investigación Matemática; Martes-Jueves 15.30-17.45, aula 1.5  (Facultat de Matemàtiques, Bloc G). Fitxa de l'assignatura

• Spring 2022 (2n quadrimestre):

  • Matemàtiques II,  Grau en Enginyeria Electrònica de Telecomunicacions; aula AE 2.1.7 E.T.S.E. Calendari

• Fall 2022 (1r quadrimestre):

  • Matemàtiques I, Grup C, Grau en Química; Teoria. 1pm-2pm a l'aula F1.1 (Facultat de Química, Bloc F) els dies que apareixen al calendari. Per a seminaris i pràctiques d'informatica, consulteu ací en funció del vostre grup.  Enllaços a Aula Virtual, Guía Docent i Dates Exàmens

At UW-Madison

• Spring 2022:

  • MATH 629 - Introduction to Measure and Integration, MWF: 12.05pm-12.55pm, VV B135. Course webpage: Canvas or here

• Fall 2021:

  • MATH 521, Section 001 - Analysis I; MWF: 9.55am-10.45am, VV B119. Course webpage: Canvas or here

  • MATH 521, Section 002 - Analysis I; MWF: 11.00am-11.50am, VV B119. Course webpage: Canvas or here

• Spring 2021:

  • MATH 521, Section 001 - Analysis I; MWF: 9.55am-10.45am, (online). Course webpage: Canvas

  • MATH 629 - Introduction to Measure and Integration, MWF: 12.05pm-12.55pm, (online). Course webpage: Canvas

• Fall 2020:

• • MATH 421, Section 005 - The Theory of Single Variable Calculus; MWF 5.40pm-6.30pm, (online). Course webpage: Canvas or here

• Spring 2020: 

• • MATH 521, Section 001 - Analysis I; MWF: 9.55am-10.45am, SOC SCI 6102. Course webpage: Canvas  

  • MATH 521, Section 002 - Analysis I; MWF: 12.05pm-12.55pm, VV B119. Course webpage: Canvas 

• Fall 2019: 

• • MATH 627 - Introduction to Fourier Analysis; MWF 1.20pm-2.10pm, VV B231. Course webpage: Canvas or here