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: WongZakai 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
 Extension
problem and Harnack's inequality for some fractional
operators, Comm. Partial Differential Equations
35 (2010), 20922122 (with J. L. Torrea).
 Regularity
theory for the fractional harmonic oscillator, J.
Funct. Anal. 260 (2011), 30973131 (with J. L.
Torrea).
 Square
functions associated to Schrödinger operators, Studia
Math. 203 (2011), 171194 (with I. AbuFalahah
and J. L. Torrea).
 Regularity
properties of Schrödinger operators, J. Math. Anal.
Appl. 388 (2012), 817837 (with T. Ma, J. L.
Torrea and C. Zhang).
 Fractional
integrals on compact Riemannian symmetric spaces of rank one,
Adv. Math. 235 (2013), 627647 (with Ó.
Ciaurri and L. Roncal).
 Harnack's
inequalities for fractional nonlocal equations, Discrete
Contin. Dyn. Syst. 33 (2013), 31533170 (with
C. Zhang).
 A
T1 criterion for HermiteCalderónZygmund operators
on the BMO_H(R^n) space and applications, Ann.
Sc. Norm. Super. Pisa Cl. Sci. (5) XII (2013),
157187 (with J. J. Betancor, R. Crescimbeni, J. C.
Fariña and J. L. Torrea).
 Rate
of convergence of WongZakai 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), 95130, Birkhäuser Basel, 2013 (with I. Gyöngy).
 Extension
problem for fractional operators: semigroups and wave
equations, J. Evol. Equ. 13 (2013),
343368 (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), 195202 (with I. AbuFalahah and J. L. Torrea).
 Regularity
estimates in Hölder spaces for Schrödinger operators via a T1
theorem, Ann. Mat. Pura Appl. 193 (2014),
561589 (with T. Ma, J. L. Torrea and C. Zhang).
 The
fractional Bessel equation in Hölder spaces,
J. Approx. Theory 184
(2014), 5599 (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),
203212 (with L. Roncal).
 Fractional
semilinear Neumann problems arising from a fractional
KellerSegel model, Calc. Var. Partial Differential
Equations 54 (2015), 10091042 (with B.
Volzone).
 Riesz
transforms on compact Riemannian symmetric spaces of rank
one, Milan J. Math. 83 (2015), 345370
(with Ó. Ciaurri and L. Roncal).
 Maximum
principles, extension problem and inversion for nonlocal
onesided equations, J. Differential Equations 260
(2016), 63336362 (with A. Bernardis, F. J. MartínReyes
and J. L. Torrea).
 Fractional
Laplacian on the torus, Commun. Contemp. Math. 18
(2016), 1550033, 26pp. (with L. Roncal).
 Hardy
spaces for FourierBessel expansions, J. Anal.
Math. 128 (2016), 261287 (with J. Dziubański,
M. Preisner and L. Roncal).
 Fractional
elliptic equations, Caccioppoli estimates and regularity,
Ann. Inst. H. Poincaré Anal. Non Linéaire 33
(2016), 767807 (with L. A. Caffarelli).
 On
weighted mixednorm Sobolev estimates for some basic
parabolic equations, Commun. Pure Appl. Anal. 16
(2017), 855882 (with L. Ping and J. L. Torrea).
 Harnack
inequality for the fractional nonlocal linearized
MongeAmpère 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), 38933924 (with J. L.
Torrea).
 Nonlocal
discrete diffusion equations and the fractional discrete
Laplacian, regularity and applications, Adv. Math.
330 (2018), 688738 (with Ó. Ciaurri, L. Roncal, J.
L. Torrea and J. L. Varona).
 The
fractional nonlocal OrnsteinUhlenbeck equation, Gaussian
symmetrization and regularity, Discrete Contin. Dyn.
Syst. 38 (2018), 32693298 (with F. Feo and B.
Volzone).
 Fractional
Laplacians on the sphere, the Minakshisundaram zeta function
and semigroups, to appear in Contemp. Math.
(with P. L. De Nápoli).
 User's guide to the fractional Laplacian and
the method of semigroups, to appear in Handbook of
Fractional Calculus with Applications.
Submitted papers
 Fractional
discrete Laplacian versus discretized fractional Laplacian,
(2015) preprint (with Ó. Ciaurri, L. Roncal, J. L. Torrea and
J. L. Varona).
 The obstacle problem for a fractional
MongeAmpère equation, (2017) submitted (with Y.
Jhaveri).
 Boolos' Hardest Logic Puzzle Ever in its
purest form, (2018) submitted (with J. J.
ColominaAlmiñana).
 Harnack inequalities and Hölder estimates
for master equations, (2018) submitted (with A. Biswas
and M. De LeónContreras).
 Hölder, Sobolev, weaktype and BMO estimates
in mixednorm with weights for parabolic equations,
(2018) submitted (with J. L. Torrea).
Video lectures
 The
Signorini problem, fractional Laplacians and the language of
semigroups, Newton Institute, Cambridge UK (2014).
 On nonlocal MongeAmpère equations,
ICERM  Brown University (2018).
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