Research
interests |

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- Partial Differential Equations: fractional nonlocal
equations, regularity, Harnack inequalities, heat kernel,
free boundary and transmission problems.
- Harmonic Analysis: orthogonal expansions, singular
integrals, Hardy, Campanato and BMO spaces, discrete
harmonic analysis, one-sided weighted estimates and
fractional differential equations.
- Functional Analysis: geometry of Banach spaces,
semigroups of operators, fractional power operators.
- Geometric Analysis: nonlocal Monge-Ampère equations,
operators on Riemannian manifolds, surfaces of minimum
mean curvature variation.
- 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.
Book
- Regularity
Techniques for Elliptic PDEs and the Fractional Laplacian.
With a foreword by Luis A. Caffarelli and José L. Torrea.
Chapman & Hall, CRC Press, 334 pp, 2024. A book preview
can be found here.
Refereed papers
- Extension
problem and Harnack's inequality for some fractional
operators, Comm. Partial Differential Equations
35 (2010), 2092--2122 (with J. L. Torrea).
- Regularity
theory for the fractional harmonic oscillator, J.
Funct. Anal. 260 (2011), 3097--3131 (with J. L.
Torrea).
- Square
functions associated to Schrödinger operators, Studia
Math. 203 (2011), 171--194 (with I. Abu-Falahah
and J. L. Torrea).
- Regularity
properties of Schrödinger operators, J. Math. Anal.
Appl. 388 (2012), 817--837 (with T. Ma, J. L.
Torrea and C. Zhang).
- Fractional
integrals on compact Riemannian symmetric spaces of rank one,
Adv. Math. 235 (2013), 627--647 (with Ó.
Ciaurri and L. Roncal).
- Harnack's
inequalities for fractional nonlocal equations, Discrete
Contin. Dyn. Syst. 33 (2013), 3153--3170 (with
C. Zhang).
- 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).
- 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).
- Extension
problem for fractional operators: semigroups and wave
equations, J. Evol. Equ. 13 (2013),
343--368 (with J. E. Galé and P. J. Miana).
- 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).
- 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).
- The
fractional Bessel equation in Hölder spaces,
J. Approx. Theory 184
(2014), 55--99 (with J. J. Betancor
and A. J. Castro).
- 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).
- Fractional
semilinear Neumann problems arising from a fractional
Keller--Segel model, Calc. Var. Partial Differential
Equations 54 (2015), 1009--1042 (with B.
Volzone).
- Riesz
transforms on compact Riemannian symmetric spaces of rank
one, Milan J. Math. 83 (2015), 345--370
(with Ó. Ciaurri and L. Roncal).
- 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).
- Fractional
Laplacian on the torus, Commun. Contemp. Math. 18
(2016), 1550033, 26pp. (with L. Roncal).
- Hardy
spaces for Fourier--Bessel expansions, J. Anal.
Math. 128 (2016), 261--287 (with J. Dziubański,
M. Preisner and L. Roncal).
- Fractional
elliptic equations, Caccioppoli estimates and regularity,
Ann. Inst. H. Poincaré Anal. Non Linéaire 33
(2016), 767--807 (with L. A. Caffarelli).
- 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).
- Harnack
inequality for the fractional nonlocal linearized
Monge--Ampère equation, Calc. Var. Partial
Differential Equations 56 (2017), 56:103
(with D. Maldonado).
- 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).
- 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).
- The
fractional nonlocal Ornstein--Uhlenbeck equation, Gaussian
symmetrization and regularity, Discrete Contin. Dyn.
Syst. 38 (2018), 3269--3298 (with F. Feo and B.
Volzone).
- Fractional
Laplacians on the sphere, the Minakshisundaram zeta function
and semigroups, New Developments in the Analysis of
Nonlocal Operators, 167--189, Contemp. Math. 723,
Amer. Math. Soc., Providence, RI, 2019 (with P. L. De Nápoli).
- User's guide to the fractional Laplacian and
the method of semigroups, Handbook of Fractional
Calculus with Applications, Anatoly Kochubei, Yuri
Luchko (Eds.), Fractional Differential Equations,
235--266, Berlin, Boston, De Gruyter, 2019.
- One-sided fractional derivatives, fractional
Laplacians and weighted Sobolev spaces, Nonlinear
Anal. 193 (2020), 111505 (with M. Vaughan).
- The obstacle problem for a fractional
Monge--Ampère equation, Comm. Partial Differential
Equations 45 (2020), 457--482 (with Y.
Jhaveri).
- Hölder, Sobolev, weak-type and BMO estimates
in mixed-norm with weights for parabolic equations, Sci.
China Math. 64 (2021), 129--154 (with J.
L. Torrea).
- Regularity for C1,α
interface transmission problems, Arch. Ration. Mech.
Anal. 240 (2021), 265--294 (with L. A.
Caffarelli and M. Soria-Carro).
- Regularity estimates for nonlocal space-time
master equations in bounded domains, J. Evol. Equ. 21
(2021), 503--565 (with A. Biswas).
- Harnack inequalities and Hölder estimates
for master equations, SIAM J. Math. Anal. 53
(2021), 2319--2348 (with A. Biswas and M. De
León-Contreras).
- Fractional elliptic
equations in nondivergece form: definition, applications and
Harnack inequality, J. Math. Pures Appl. 156
(2021), 245--306 (with M. Vaughan).
- Fractional
derivatives: Fourier, elephants, memory effects,
viscoelastic materials, and anomalous diffusions, Notices
Amer. Math. Soc. 70 (2023), no. 4, 576--587.
- Regularity of
viscosity solutions to fully nonlinear elliptic transmission
problems, Adv. Math. 435 (2023),
109353, 52 pp (with M. Soria-Carro).
- Sharp extension
problem characterizations for higher fractional power
operators in Banach spaces, J. Funct. Anal. 287
(2024), 110474, 27 pp (with A. Biswas)
- A PDE approach to the
existence and regularity of surfaces of minimum mean
curvature variation, Arch. Ration. Mech. Anal. 248
(2024), Paper No. 70, 17 pp (with L. A. Caffarelli and H.
Vivas).
- Interior Schauder
estimates for fractional elliptic equations in nondivergence
form, SIAM J. Math. Anal. (2025), to appear
(with M. Vaughan).
Submitted papers
- Fractional powers of
first order differential operators and new families of
polynomials associated to inverse measures, (2022)
submitted (with M. Mazzitelli and J. L. Torrea).
Video lectures
- The Signorini problem, fractional
Laplacians and the language of semigroups, Newton
Institute, Cambridge UK (2014).
- On nonlocal Monge--Ampère equations,
ICERM - Brown University (2018).
- Ecuaciones de Monge--Ampère no locales,
Café Matemático - Universidad Nacional de Mar del Plata
(2020).
- Nonlocal
fractional Monge--Ampère equations, AIMS-Cameroon
Research Centre Colloquium (2021).
- Regularity for C1,α
interface transmission problems, Two
Nonlinear Days in Urbino (2021).
- Harnack inequality for fractional
nondivergence form elliptic equations, Virtual Analysis
and PDE Seminar (2022).
- Fractional
derivatives: Fourier, elephants, memory effects,
viscoelastic materials, and anomalous diffusions, Center
for Mathematics and Artificial Intelligence Colloquium (2023).
- Abel, Caffarelli and
fractional nonlocal equations, presentation given at the
Evening of the Abel
Prize 2023 -- Highlighting the work of Luis Caffarelli
at the Royal Netherlands Academy of Arts and Sciences (KNAW),
Amsterdam (2024).
Preprints
- Fractional
discrete Laplacian versus discretized fractional Laplacian,
(2015) preprint (with Ó. Ciaurri, L. Roncal, J. L. Torrea and
J. L. Varona).
- Boolos' Hardest Logic Puzzle Ever in its
purest form, (2018) preprint (with J. J.
Colomina-Almiñana)
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