Harmonic Analysis

The Frisch–Parisi formalism for fluctuations of the Schrödinger equation

Kumar, S.; Ponce Vanegas, F.

; Roncal, L.

; Vega, L.

(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

Caro, P.; Meroño, C.; Parissis, I. (2022-01-05)

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

Canto, J.

(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

Eceizabarrena, D.; Ponce Vanegas, F.

(2021-12-09)

We construct counterexamples for the fractal Schrödinger convergence problem by combining a fractal extension of Bourgain's counterexample and the intermediate space trick of Du--Kim--Wang--Zhang. We confirm that the same ...

Discrete Carleman estimates and three balls inequalities

Fernández-Bertolin, A.; Roncal, L.

; Rüland, A.; Stan, D. (2021-10-16)

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 Fujii-Wilson conditions and BMO spaces

Ombrosi, S.; Pérez, C.

; Rela, E.; Rivera-Ríos, I. (2020-07-01)

In this note we generalize the definition of the Fujii-Wilson 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 Poincare-Sobolev inequalities

Pérez, C.

; Rela, E. (2021)

Abstract. We study weighted Poincar ́e and Poincar ́e-Sobolev 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

Pérez, C.

; Picón, T.; Saari, Olli; Sousa, Mateus (2018-12-01)

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 by-product of the ...

Bilinear Spherical Maximal Functions of Product Type

Roncal, L.

; Shrivastava, S.; Shuin, K. (2021-08-12)

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

Beltran, D.; Oberlin, R.; Roncal, L.

; Stovall, B.; Seeger, A. (2021-06-22)

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

Eceizabarrena, D.; Ponce Vanegas, F.

(2021-08-24)

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 non-singular. We prove that for initial data $f\in H^s(\mathbb{R}^n)$ ...

RESTRICTED TESTING FOR POSITIVE OPERATORS

Hytönen, T.; Li, K.; Sawyer, E. (2020)

We prove that for certain positive operators T, such as the Hardy-Littlewood 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 John-Nirenberg theorem and applications

Canto, J.; Pérez, C.

(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

Lucà, R.

; Ponce Vanegas, F.

(2021-01)

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 ...