Harmonic Analysis
http://hdl.handle.net/20.500.11824/3
2022-07-12T03:28:53ZMaximal operators on the infinite-dimensional torus
http://hdl.handle.net/20.500.11824/1484
Maximal operators on the infinite-dimensional torus
Roncal, Luz; Kosz, D.; Martínez-Perales, J.; Paternostro, V.; Rela, E.; Roncal, L.
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 the dyadic basis $\mathcal{R}_0$, is of weak type $(1,1)$, but $M^{\mathcal{R}}$, the operator associated with the natural general basis $\mathcal{R}$, is not. We extend the latter result to all $q \in [1,\infty)$. Then we find a wide class of intermediate bases $\mathcal{R}_0 \subset \mathcal{R}' \subset \mathcal{R}$, for which maximal functions have controlled, but sometimes very peculiar behavior.} Precisely, for given $q_0 \in [1, \infty)$ we construct $\mathcal{R}'$ such that $M^{\mathcal{R}'}$ is of restricted weak type $(q,q)$ if and only if $q$ belongs to a predetermined range of the form $(q_0, \infty]$ or $[q_0, \infty]$. Finally, we study the weighted setting, considering the Muckenhoupt $A_p^\mathcal{R}(\mathbb{T}^\omega)$ and reverse H\"older $\mathrm{RH}_r^\mathcal{R}(\mathbb{T}^\omega)$ classes of weights associated with $\mathcal{R}$. For each $p \in (1, \infty)$ and each $w \in A_p^\mathcal{R}(\mathbb{T}^\omega)$ we obtain that $M^{\mathcal{R}}$ is not bounded on $L^q(w)$ in the whole range $q \in [1,\infty)$. Since we are able to show that
\[
\bigcup_{p \in (1, \infty)}A_p^\mathcal{R}(\mathbb{T}^\omega) = \bigcup_{r \in (1, \infty)} \mathrm{RH}_r^\mathcal{R}(\mathbb{T}^\omega),
\]
the unboundedness result applies also to all reverse H\"older weights.
2022-03-31T00:00:00ZThe Frisch–Parisi formalism for fluctuations of the Schrödinger equation
http://hdl.handle.net/20.500.11824/1429
The Frisch–Parisi formalism for fluctuations of the Schrödinger equation
Kumar, S.; Ponce Vanegas, F.; Roncal, L.; Vega, L.
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 $\int\lvert x \rvert^{2\delta}\lvert u(x,t) \rvert^2\,dx$, for $0 < \delta < 1$, after removing a smooth background. We prove that the Frisch--Parisi formalism holds for $H_\delta(t) = \int_{[0,t]}h_{\text{p}, \delta}(2s)\,ds$, which is morally a simplification of the Riemann's non-differentiable curve $R$. Our motivation is to understand the evolution of the vortex filament equation of polygonal filaments, which are related to $R$.
2022-01-01T00:00:00ZRotational smoothing
http://hdl.handle.net/20.500.11824/1413
Rotational smoothing
Caro, P.; Meroño, C.; Parissis, I.
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 regularizing in all directions. The gain of regularity is the result of rotating the directions where the corresponding operator performs the smoothing effect. In this paper we carry out a systematic study of the rotational smoothing for a class of operators that includes $k$-vector-space Riesz potentials in $\mathbb{R}^n$ with $k<n$, and the convolution with fundamental solutions of elliptic constant-coefficient differential operators acting on $k$-dimensional linear subspaces. Examples of the latter type of operators are the planar Cauchy transform in $\mathbb{R}^n$, or a solution operator for the transport equation in $\mathbb{R}^n$. The analysis of rotational smoothing is motivated by the resolution of some inverse problems under low-regularity assumptions.
2022-01-05T00:00:00ZGeometric Harmonic Analysis
http://hdl.handle.net/20.500.11824/1402
Geometric Harmonic Analysis
Canto, J.
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 chapters has its own introductory
part, some longer some shorter. This chapter is intended to be an introduction to the
whole thesis. Without going into technical details, in this Chapter we will not only
motivate the results and the content of the dissertation, but we also explain how and
why these results came to be studied. We also introduce the main notation and some
preliminary concepts that will be used throughout the dissertation.
2021-01-01T00:00:00Z