Harmonic Analysis
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The Frisch–Parisi formalism for fluctuations of the Schrödinger equation
(2022)We consider the solution of the Schrödinger equation $u$ in $\mathbb{R}$ when the initial datum tends to the Dirac comb. Let $h_{\text{p}, \delta}(t)$ be the fluctuations in time of $\int\lvert x \rvert^{2\delta}\lvert ... 
Rotational smoothing
(20220105)Rotational smoothing is a phenomenon consisting in a gain of regularity by means of averaging over rotations. This phenomenon is present in operators that regularize only in certain directions, in contrast to operators ... 
Geometric Harmonic Analysis
(2021)This thesis is the compilation of the results obtained during my PhD, which started in January 2018 and is being completed in the end of 2021. The main matter is divided into ve chapters, Chapters 2 6. Each of these ... 
Counterexamples for the fractal Schrödinger convergence problem with an Intermediate Space Trick
(20211209)We construct counterexamples for the fractal Schrödinger convergence problem by combining a fractal extension of Bourgain's counterexample and the intermediate space trick of DuKimWangZhang. We confirm that the same ... 
Discrete Carleman estimates and three balls inequalities
(20211016)We prove logarithmic convexity estimates and three balls inequalities for discrete magnetic Schrödinger operators. These quantitatively connect the discrete setting in which the unique continuation property fails and the ... 
A note on generalized FujiiWilson conditions and BMO spaces
(20200701)In this note we generalize the definition of the FujiiWilson condition providing quantitative characterizations of some interesting classes of weights, such as A∞, A∞weak and Cp, in terms of BMO type spaces suited to them. ... 
Degenerate PoincareSobolev inequalities
(2021)Abstract. We study weighted Poincar ́e and Poincar ́eSobolev type in equalities with an explicit analysis on the dependence on the Ap con stants of the involved weights. We obtain inequalities of the form with different ... 
Regularity of maximal functions on Hardy–Sobolev spaces
(20181201)We prove that maximal operators of convolution type associated to smooth kernels are bounded in the homogeneous Hardy–Sobolev spaces H1,p(Rd) when p > d/(d + 1). This range of exponents is sharp. As a byproduct of the ... 
Bilinear Spherical Maximal Functions of Product Type
(20210812)In this paper we introduce and study a bilinear spherical maximal function of product type in the spirit of bilinear Calderón–Zygmund theory. This operator is different from the bilinear spherical maximal function considered ... 
Variation bounds for spherical averages
(20210622)We consider variation operators for the family of spherical means, with special emphasis on $L^p\to L^q$ estimates 
Pointwise Convergence over Fractals for Dispersive Equations with Homogeneous Symbol
(20210824)We study the problem of pointwise convergence for equations of the type $i\hbar\partial_tu + P(D)u = 0$, where the symbol $P$ is real, homogeneous and nonsingular. We prove that for initial data $f\in H^s(\mathbb{R}^n)$ ... 
RESTRICTED TESTING FOR POSITIVE OPERATORS
(2020)We prove that for certain positive operators T, such as the HardyLittlewood maximal function and fractional integrals, there is a constant D>1, depending only on the dimension n, such that the two weight norm inequality ... 
Extensions of the JohnNirenberg theorem and applications
(2021)The John–Nirenberg theorem states that functions of bounded mean oscillation are exponentially integrable. In this article we give two extensions of this theorem. The first one relates the dyadic maximal function to the ... 
Convergence over fractals for the Schrödinger equation
(202101)We consider a fractal refinement of the Carleson problem for the Schrödinger equation, that is to identify the minimal regularity needed by the solutions to converge pointwise to their initial data almost everywhere with ... 
Multilinear operatorvalued calderónzygmund theory
(2020)We develop a general theory of multilinear singular integrals with operator valued kernels, acting on tuples of UMD Banach spaces. This, in particular, involves investigating multilinear variants of the Rboundedness ... 
Endpoint estimates, extrapolation for multilinear muckenhoupt classes, and applications
(2019)In this paper we present the results announced in the recent work by the first, second, and fourth authors of the current paper concerning Rubio de Francia extrapolation for the socalled multilinear Muckenhoupt classes. ... 
Generalized PoincaréSobolev inequalities
(202012)PoincaréSobolev inequalities are very powerful tools in mathematical analysis which have been extensively used for the study of differential equations and their validity is intimately related with the geometry of the ... 
Sparse and weighted estimates for generalized Hörmander operators and commutators
(2019)In this paper a pointwise sparse domination for generalized Ho ̈rmander and also for iterated commutators with those operators is provided generalizing the sparse domination result in [24]. Relying upon that sparse domination ... 
Multilinear singular integrals on noncommutative lp spaces
(2019)We prove Lp bounds for the extensions of standard multilinear Calderón Zygmund operators to tuples of UMD spaces tied by a natural product structure. The product can, for instance, mean the pointwise product in UMD ... 
The observational limit of wave packets with noisy measurements
(2019)The authors consider the problem of recovering an observable from certain measurements containing random errors. The observable is given by a pseudodifferential operator while the random errors are generated by a Gaussian ...