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Maximal operators on the infinite-dimensional torus
(2022-03-31)
We study maximal operators related to bases on the infinite-dimensional torus $\tom$. {For the normalized Haar measure $dx$ on $\mathbb{T}^\omega$ it is known that $M^{\mathcal{R}_0}$, the maximal operator associated with ...
Rotational smoothing
(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 ...
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 ...
Counterexamples for the fractal Schrödinger convergence problem with an Intermediate Space Trick
(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
(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 ...
Pointwise Convergence over Fractals for Dispersive Equations with Homogeneous Symbol
(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 $f\in H^s(\mathbb{R}^n)$ ...
Bilinear Spherical Maximal Functions of Product Type
(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
(2021-06-22)
We consider variation operators for the family of spherical means, with special emphasis on $L^p\to L^q$ estimates
Extensions of the John-Nirenberg 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 ...
Degenerate Poincare-Sobolev inequalities
(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 ...