Research

My mathematical interests include analysis, number theory, geometry and mathematical physics. In particular, I have worked on the critical surface quasigeostrophic equation, pointwise inequalities for fractional laplacians, the isometric embedding problem with low regularity and certain questions about eigenfunctions on manifolds. This did not prevent me from eventual unfruitful excursions to other areas.

Here you can find a list of my contributions:

(with A. Córdoba) A pointwise inequality for fractional laplacians, Advances in Mathematics Volume 280, 6 August 2015, Pages 79-85, arXiv:1502.01635
Eigenfunction maxima and spherical means. (Chapter 5 of my PhD dissertation, also arXiv:2205.14489 )
(with D. Alonso-Orán and A. Córdoba) Integral representation for fractional Laplace-Beltrami operators, Advances in Mathematics Volume 328, 13 April 2018, Pages 436-445, arXiv:1704.06126.
(with D. Alonso-Orán and A. Córdoba) Continuity of weak solutions of the critical quasigeostrophic equation on the sphere, Advances in Mathematics Volume 328, 13 April 2018, Pages 264-299, arXiv:1704.06129.
(with D. Alonso-Orán and A. Córdoba) Global well-posedness of critical surface quasigeostrophic equation on the sphere, Advances in Mathematics Volume 328, 13 April 2018, Pages 248-263, arXiv:1704.06132.
(with D. Alonso-Orán, F. Chamizo and A. Mas) Pointwise monotonicity of heat kernels, arXiv:1807.11072, to appear in Revista Matemática Complutense (2021).
(with P-Y. Bienvenu and D. González-Sánchez) A note on the Bilinear Bogolyubov Theorem: Transverse and bilinear sets, Proc. Amer. Math. Soc. 148 (2020), pp. 23-31 1, arXiv:1811.09853.
Córdoba's differentiation theorem: revisited, Annales de l'Institut Fourier, Volume 73 (2023) no. 2, pp. 559-565., arXiv:2205.15309 .
$C^{1,1/3-\epsilon}$ surface isometric embeddings in $\mathbb{R}^6$, preprint not intended for publication.
$C^{1,1-\epsilon}$ isometric embeddings, accepted in Transactions of the American Mathematical Society. Contains as an Appendix: An abstract generalization of Källen-Kröner's iteration, an originally independent paper.
(with D. Spector) An improvement to the John-Nirenberg inequality for functions in Critical Lorentz spaces, Advances in Nonlinear Analysis 10 (2021), pp. 877-894, arXiv:2007.04576.
(with F. Torres de Lizaur) Distribution symmetry of toral eigenfunctions, Revista Matemática Iberoamericana 38 num. 4 (2022) pp. 1371-1382, arXiv:2205.14491 .
(with F. Torres de Lizaur) Sign equidistribution of Legendre polynomials, arXiv:2205.14493, accepted for publication in Analysis and Mathematical Physics.
(with D. Alonso-Orán) On the blow up of a non-local transport equation in compact manifolds, to appear in the Proc. Amer. Math. Soc, arXiv:2205.15310.
On Quasisymmetry and blow ups of eigenfunctions, submitted.
(with A. Burchard) On monotonicity of heat kernels: a new example and counterexamples, submitted.
On divisors in many congruence classes, submitted.

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