Harmonic Analysis
http://hdl.handle.net/20.500.11824/3
Sat, 20 Aug 2022 02:07:22 GMT2022-08-20T02:07:22ZA∞ condition for general bases revisited: complete classification of definitions
http://hdl.handle.net/20.500.11824/1504
A∞ condition for general bases revisited: complete classification of definitions
Kosz, D.
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 are
considered. For cubes in Rd every two conditions are known to be equivalent,
while for general bases we have a trichotomy: equivalence, one-way implication,
or no dependency may occur. In most cases the relations between different
conditions have already been established. Here all the unsolved cases are
treated and, as a result, a full diagram of the said relations is presented.
Fri, 27 May 2022 00:00:00 GMThttp://hdl.handle.net/20.500.11824/15042022-05-27T00:00:00ZCorrigendum to: An extension problem and trace Hardy inequality for the sublaplacian on H-type groups
http://hdl.handle.net/20.500.11824/1503
Corrigendum to: An extension problem and trace Hardy inequality for the sublaplacian on H-type groups
Roncal, L.; Thangavelu, S.
Recently we have found a couple of errors in our paper entitled An extension problem
and trace Hardy inequality for the sub-Laplacian on $H$-type groups, Int. Math. Res.
Not. IMRN (2020), no. 14, 4238--4294. They concern Propositions 3.12--3.13, and Theorem
1.5, Corollary 1.6 and Remark 4.10. The purpose of this corrigendum is to point out the
errors and supply necessary modifications where it is applicable.
Wed, 10 Mar 2021 00:00:00 GMThttp://hdl.handle.net/20.500.11824/15032021-03-10T00:00:00ZMaximal 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.
Thu, 31 Mar 2022 00:00:00 GMThttp://hdl.handle.net/20.500.11824/14842022-03-31T00:00:00ZThe 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 $R$.
Sat, 01 Jan 2022 00:00:00 GMThttp://hdl.handle.net/20.500.11824/14292022-01-01T00:00:00Z