Harmonic Analysishttp://hdl.handle.net/20.500.11824/32022-11-28T12:38:35Z2022-11-28T12:38:35ZA∞ condition for general bases revisited: complete classification of definitionsKosz, D.http://hdl.handle.net/20.500.11824/15042022-08-18T22:20:44Z2022-05-27T00:00:00ZA∞ 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.
2022-05-27T00:00:00ZCorrigendum to: An extension problem and trace Hardy inequality for the sublaplacian on H-type groupsRoncal, L.Thangavelu, S.http://hdl.handle.net/20.500.11824/15032022-08-18T22:20:43Z2021-03-10T00:00:00ZCorrigendum 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.
2021-03-10T00:00:00ZMaximal operators on the infinite-dimensional torusRoncal, LuzKosz, D.Martínez-Perales, J.Paternostro, V.Rela, E.Roncal, L.http://hdl.handle.net/20.500.11824/14842022-07-06T22:20:02Z2022-03-31T00:00:00ZMaximal 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 equationKumar, S.Ponce Vanegas, F.Roncal, L.Vega, L.http://hdl.handle.net/20.500.11824/14292022-02-23T00:19:37Z2022-01-01T00:00:00ZThe 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 $R$.
2022-01-01T00:00:00Z