Research interests

- Partial Differential Equations: fractional Laplacians, regularity theory, Harnack inequalities, heat kernel, free boundary problems.
- Harmonic Analysis: orthogonal expansions, singular integrals, Hardy, Campanato and BMO spaces, discrete harmonic analysis.
- Functional Analysis: geometry of Banach spaces, semigroups of operators.
- Geometric Analysis: operators on compact Riemannian manifolds.
- Stochastic Analysis: Wong-Zakai approximations.
- Logic.

You can find all my papers and preprints on arXiv here.
You can find my Google Scholar profile here.
Here is my PhD Thesis.

List of publications

  1. Extension problem and Harnack's inequality for some fractional operators, Comm. Partial Differential Equations 35 (2010), 2092--2122 (with J. L. Torrea).
  2. Regularity theory for the fractional harmonic oscillator, J. Funct. Anal. 260 (2011), 3097--3131 (with J. L. Torrea).
  3. Square functions associated to Schrödinger operators, Studia Math. 203 (2011), 171--194 (with I. Abu-Falahah and J. L. Torrea).
  4. Regularity properties of Schrödinger operators, J. Math. Anal. Appl. 388 (2012), 817--837 (with T. Ma, J. L. Torrea and C. Zhang).
  5. Fractional integrals on compact Riemannian symmetric spaces of rank one, Adv. Math. 235 (2013), 627--647 (with Ó. Ciaurri and L. Roncal).
  6. Harnack's inequalities for fractional nonlocal equations, Discrete Contin. Dyn. Syst. 33 (2013), 3153--3170 (with C. Zhang).
  7. A T1 criterion for Hermite-Calderón-Zygmund operators on the BMO_H(R^n) space and applications, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) XII (2013), 157--187 (with  J. J. Betancor, R. Crescimbeni, J. C. Fariña and J. L. Torrea).
  8. Rate of convergence of Wong--Zakai approximations for stochastic partial differential equations, Proceedings of the Seventh Seminar on Stochastic Analysis, Random Fields and Applications VII, Ascona 2011, in: Progress in Probability 67, R. Dalang, M. Dozzi and F. Russo (Eds), 95--130, Birkhäuser Basel, 2013 (with I. Gyöngy).
  9. Extension problem for fractional operators: semigroups and wave equations, J. Evol. Equ. 13 (2013), 343--368 (with J. E. Galé and P. J. Miana).
  10. A note on the almost everywhere convergence to initial data for some evolution equations, Potential Anal. 40 (2014), 195--202 (with I. Abu-Falahah and J. L. Torrea).
  11. Regularity estimates in Hölder spaces for Schrödinger operators via a T1 theorem, Ann. Mat. Pura Appl. 193 (2014), 561--589 (with T. Ma, J. L. Torrea and C. Zhang).
  12. The fractional Bessel equation in Hölder spaces, J. Approx. Theory 184 (2014), 55--99 (with J. J. Betancor and A. J. Castro).
  13. Transference of fractional Laplacian regularity, Special Functions, Partial Differential Equations and Harmonic Analysis. In honor of Calixto P. Calderón, in: Springer Proceedings in Mathematics and Statistics 108, A. M. Stokolos, C. Georgakis and W. Urbina (Eds.) (2014), 203--212 (with L. Roncal).
  14. Fractional semilinear Neumann problems arising from a fractional Keller--Segel model, Calc. Var. Partial Differential Equations 54 (2015), 1009--1042 (with B. Volzone).
  15. Riesz transforms on compact Riemannian symmetric spaces of rank one, Milan J. Math. 83 (2015), 345--370 (with Ó. Ciaurri and L. Roncal).
  16. Maximum principles, extension problem and inversion for nonlocal one-sided equations, J. Differential Equations 260 (2016), 6333--6362 (with A. Bernardis, F. J. Martín-Reyes and J. L. Torrea).
  17. Fractional Laplacian on the torus, Commun. Contemp. Math. 18 (2016), 1550033, 26pp. (with L. Roncal).
  18. Hardy spaces for Fourier--Bessel expansions, J. Anal. Math. 128 (2016), 261--287 (with J. Dziubański, M. Preisner and L. Roncal).
  19. Fractional elliptic equations, Caccioppoli estimates and regularity, Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016), 767--807 (with L. A. Caffarelli).
  20. On weighted mixed-norm Sobolev estimates for some basic parabolic equations, Commun. Pure Appl. Anal. 16 (2017), 855--882 (with L. Ping and J. L. Torrea).
  21. Harnack inequality for the fractional nonlocal linearized Monge--Ampère equation, Calc. Var. Partial Differential Equations 56 (2017), 56:103 (with D. Maldonado).
  22. Regularity theory and extension problem for fractional nonlocal parabolic equations and the master equation, SIAM J. Math. Anal. 49 (2017), 3893--3924 (with J. L. Torrea).
  23. Nonlocal discrete diffusion equations and the fractional discrete Laplacian, regularity and applications, Adv. Math. 330 (2018), 688--738 (with Ó. Ciaurri, L. Roncal, J. L. Torrea and J. L. Varona).
  24. The fractional nonlocal Ornstein--Uhlenbeck equation, Gaussian symmetrization and regularity, Discrete Contin. Dyn. Syst. 38 (2018), 3269--3298 (with F. Feo and B. Volzone).
  25. Fractional Laplacians on the sphere, the Minakshisundaram zeta function and semigroups, to appear in Contemp. Math. (with P. L. De Nápoli).
  26. User's guide to the fractional Laplacian and the method of semigroups, to appear in Handbook of Fractional Calculus with Applications.
Submitted papers
  1. Fractional discrete Laplacian versus discretized fractional Laplacian, (2015) preprint (with Ó. Ciaurri, L. Roncal, J. L. Torrea and J. L. Varona).
  2. The obstacle problem for a fractional Monge--Ampère equation, (2017) submitted (with Y. Jhaveri).
  3. Boolos' Hardest Logic Puzzle Ever in its purest form, (2018) submitted (with J. J. Colomina-Almiñana).
  4. Harnack inequalities and Hölder estimates for master equations, (2018) submitted (with A. Biswas and M. De León-Contreras).
  5. Hölder, Sobolev, weak-type and BMO estimates in mixed-norm with weights for parabolic equations, (2018) submitted (with J. L. Torrea).

Video lectures

  1. The Signorini problem, fractional Laplacians and the language of semigroups, Newton Institute, Cambridge UK (2014).
  2. On nonlocal Monge--Ampère equations, ICERM - Brown University (2018).

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