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Kato–Ponce estimates for fractional sublaplacians in the Heisenberg group
(2022-11-04)
We give a proof of commutator estimates for fractional powers of the sublaplacian
on the Heisenberg group. Our approach is based on pointwise and $L^p$ estimates involving square
fractional integrals and Littlewood--Paley ...
A∞ condition for general bases revisited: complete classification of definitions
(2022-05-27)
We refer to the discussion on different characterizations of the
A∞ class of weights, initiated by Duoandikoetxea, Martín-Reyes, and Ombrosi
[Math. Z. 282 (2016), pp. 955–972]. Twelve definitions of the A∞ condition ...
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 ...
Polynomial averages and pointwise ergodic theorems on nilpotent groups
(2022)
We establish pointwise almost everywhere convergence for ergodic averages along polynomial sequences in nilpotent groups of step two of measure-preserving transformations on $\sigma$-finite measure spaces. We also establish ...
Notes on $H^{\log}$: structural properties, dyadic variants, and bilinear $H^1$-$BMO$ mappings
(2022)
This article is devoted to a study of the Hardy space $H^{\log} (\mathbb{R}^d)$ introduced by Bonami, Grellier, and Ky. We present an alternative approach to their result relating the product of a function in the real Hardy ...
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 $f\in H^s(\mathbb{R}^n)$ ...