Revista de la
Unión Matemática Argentina

Vol. 66, no. 1 (2023)
Special volume in honour of Eleonor Harboure (1948–2022)

September 2023

Photo courtesy of Marcela Porta
Guest editors: Bruno Bongioanni, Carlos Cabrelli, Ursula Molter, and Beatriz Viviani.
Front matter
Preface. Bruno Bongioanni, Carlos Cabrelli, Ursula Molter, and Beatriz Viviani i–ii
Remembering Eleonor Harboure / Recordando a Eleonor Harboure. Bruno Bongioanni, Oscar Salinas, and Beatriz Viviani iii–x
Variation and oscillation operators on weighted Morrey–Campanato spaces in the Schrödinger setting. Víctor Almeida, Jorge J. Betancor, Juan C. Fariña, and Lourdes Rodríguez-Mesa
We denote by $\mathcal{L}$ the Schrödinger operator with potential $V$, that is, $\mathcal{L}=-\Delta+V$, where it is assumed that $V$ satisfies a reverse Hölder inequality. We consider weighted Morrey–Campanato spaces ${\mathrm{BMO}}_{\mathcal{L},w}^\alpha (\mathbb{R}^d)$ and ${\mathrm{BLO}}_{\mathcal{L},w}^\alpha (\mathbb{R}^d)$ in the Schrödinger setting. We prove that the variation operator $V_\sigma (\{T_t\}_{t > 0})$, $\sigma > 2$, and the oscillation operator $O(\{T_t\}_{t > 0},\{t_j\}_{j\in\mathbb{Z}})$, where $t_j < t_{j+1}$, $j\in \mathbb{Z}$, $\displaystyle \lim_{j\rightarrow +\infty}t_j=+\infty$ and $\displaystyle\lim_{j\rightarrow -\infty }t_j=0$, being $T_t=t^k\partial _t^ke^{-t\mathcal{L}}$, $t > 0$, with $k\in \mathbb{N}$, are bounded operators from ${\rm BMO}_{\mathcal{L},w}^\alpha (\mathbb{R}^d)$ into ${\rm BLO}_{\mathcal{L},w}^\alpha (\mathbb{R}^d)$. We also establish the same property for the maximal operators defined by $\{t^k\partial _t^ke^{-t\mathcal L}\}_{t > 0}$, $k\in \mathbb{N}$.
Boundedness of fractional operators associated with Schrödinger operators on weighted variable Lebesgue spaces via extrapolation. Rocío Ayala and Adrian Cabral
In this work we obtain boundedness results for fractional operators associated with Schrödinger operators $\mathcal{L}=-\Delta+V$ on weighted variable Lebesgue spaces. These operators include fractional integrals and their respective commutators. In particular, we obtain weighted inequalities of the type $L^{p(\cdot)}$-$L^{q(\cdot)}$ and estimates of the type $L^{p(\cdot)}$-Lipschitz variable integral spaces. For this purpose, we developed extrapolation results that allow us to obtain boundedness results of the type described above in the variable setting by starting from analogous inequalities in the classical context. Such extrapolation results generalize what was done by Harboure, Macías, and Segovia [Amer. J. Math. 110 no. 3 (1988), 383–397], and by Bongioanni, Cabral, and Harboure [Potential Anal. 38 no. 4 (2013), 1207–1232], for the classic case, that is, $V\equiv0$ and $p(\cdot)$ constant, respectively.
Two-weighted estimates of the multilinear fractional integral operator between weighted Lebesgue and Lipschitz spaces with optimal parameters. Fabio Berra, Gladis Pradolini, and Wilfredo Ramos
Given an $m$-tuple of weights $\vec{v}=(v_1,\dots,v_m)$, we characterize the classes of pairs $(w,\vec{v})$ involved in the boundedness properties of the multilinear fractional integral operator from $\prod_{i=1}^mL^{p_i}\left(v_i^{p_i}\right)$ into suitable Lipschitz spaces associated to a parameter $\delta$, $\mathcal{L}_w(\delta)$. Our results generalize some previous estimates not only for the linear case but also for the unweighted problem in the multilinear context. We emphasize the study related to the range of the parameters involved in the problem described above, which is optimal in the sense that they become trivial outside of the region obtained. We also exhibit nontrivial examples of pairs of weights in this region.
Avatars of Stein's theorem in the complex setting. Aline Bonami, Sandrine Grellier, and Benoît Sehba
In this paper, we establish some variants of Stein's theorem, which states that a non-negative function belongs to the Hardy space $H^1(\mathbb{T})$ if and only if it belongs to $L\log L(\mathbb{T})$. We consider Bergman spaces of holomorphic functions in the upper half plane and develop avatars of Stein's theorem and relative results in this context. We are led to consider weighted Bergman spaces and Bergman spaces of Musielak–Orlicz type. Eventually, we characterize bounded Hankel operators on $A^1(\mathbb{C}_+)$.
Vertical Littlewood–Paley functions related to a Schrödinger operator. Bruno Bongioanni, Eleonor Harboure, and Pablo Quijano
In this work we consider the Littlewood–Paley quadratic function associated to the Schrödinger operator $\mathcal{L}= -\Delta+V$ involving spatial derivatives of the semigroup's kernel. Under an appropriate reverse-Hölder condition on the potential we show boundedness on weighted $L^p$ spaces for $1 < p < p_0$, where $p_0$ depends on the order of the reverse-Hölder property. Using a subordination formula we extend these results to the corresponding quadratic function associated to the semigroup related to $\mathcal{L}^\alpha$, $0 < \alpha < 1$.
Learning the model from the data. Carlos Cabrelli and Ursula Molter
The task of approximating data with a concise model comprising only a few parameters is a key concern in many applications, particularly in signal processing. These models, typically subspaces belonging to a specific class, are carefully chosen based on the data at hand. In this survey, we review the latest research on data approximation using models with few parameters, with a specific emphasis on scenarios where the data is situated in finite-dimensional vector spaces, functional spaces such as $L^2(\mathbb{R}^d)$, and other general situations. We highlight the invariant properties of these subspace-based models that make them suitable for diverse applications, particularly in the field of image processing.
Characterizations of local $A_{\infty}$ weights and applications to local singular integrals. Federico Campos, Oscar Salinas, and Beatriz Viviani
In a general geometric setting, we prove different characterizations of a local version of Muckenhoupt $A_{\infty}$ weights. As an application, we obtain conclusions about the relationship between this class and the one-weight boundedness of local singular integrals from $L^{\infty}$ to BMO.
Extrapolation of compactness for certain pseudodifferential operators. María J. Carro, Javier Soria, and Rodolfo H. Torres
A recently developed extrapolation of compactness on weighted Lebesgue spaces is revisited and a new application to a class of compact pseudodifferential operators is presented.
Pointwise convergence of fractional powers of Hermite type operators. Guillermo Flores, Gustavo Garrigós, Teresa Signes, and Beatriz Viviani
When $L$ is the Hermite or the Ornstein–Uhlenbeck operator, we find minimal integrability and smoothness conditions on a function $f$ so that the fractional power $L^\sigma f(x_0)$ is well-defined at a given point $x_0$. We illustrate the optimality of the conditions with various examples. Finally, we obtain similar results for the fractional operators $(-\Delta+R)^\sigma$, with $R > 0$.
Remarks on a boundary value problem for a matrix valued $\overline{\partial}$ equation. Carlos E. Kenig
In this short note, we discuss a boundary value problem for a matrix valued $\overline{\partial}$ equation.
Upper endpoint estimates and extrapolation for commutators. Kangwei Li, Sheldy Ombrosi, and Israel P. Rivera-Ríos
In this note we revisit the upper endpoint estimates for commutators following the line by Harboure, Segovia, and Torrea [Illinois J. Math. 41 no. 4 (1997), 676–700]. Relying upon the suitable BMO subspace suited for the commutator that was introduced in Accomazzo's PhD thesis (2020), we obtain a counterpart for commutators of the upper endpoint extrapolation result by Harboure, Macías and Segovia [Amer. J. Math. 110 no. 3 (1988), 383–397]. Multilinear counterparts are provided as well.
Mixed weak type inequalities for the fractional maximal operator. María Lorente and Francisco J. Martín-Reyes
Let $0\leq \alpha < n$, $q=\frac{n}{n-\alpha}$. We establish the mixed weak type inequality \begin{equation*} \sup_{\lambda > 0}\lambda^q \int_{\{x\in\mathbb{R}^n:M_{\alpha}f(x) > \lambda v(x) \}}u^qv^q\leq C\left(\int_{\mathbb{R}^n}|f|u \right)^q\, \end{equation*} for the fractional maximal operator \[ M_\alpha f(x)=\sup_{h > 0}|B(x,h)|^{\alpha/n-1}\int_{B(x,h)} |f| \] under the following assumptions: (a) $u^q$ belongs to the Muckenhoupt $A_1$ class, (b) $v$ is essentially constant over dyadic annuli, and (c) $(\lambda v)^q$ satisfies a certain condition $C_p(\gamma)$ for all $\lambda > 0$. The last condition is fulfilled by any Muckenhoupt weight but it is also satisfied by some non Muckenhoupt weights. Our approach is based on the study of the same kind of inequalities for the local fractional maximal operator, a Hardy type operator and its adjoint.
Hermite Besov and Triebel–Lizorkin spaces and applications. Fu Ken Ly and Virginia Naibo
We present an overview of Besov and Triebel–Lizorkin spaces in the Hermite setting and applications on boundedness properties of Hermite pseudo-multipliers and fractional Leibniz rules in such spaces. We also give a new weighted estimate for Hermite multipliers for weights related to Hermite operators.
A tribute to Pola Harboure: Isoperimetric inequalities and the HMS extrapolation theorem. Carlos Pérez and Ezequiel Rela
We give a simpler proof of the Gagliardo estimate with a measure obtained by Franchi, Pérez, and Wheeden [Proc. London Math. Soc. (3) 80 no. 3 (2000), 665–689], and improved by Pérez and Rela [Trans. Amer. Math. Soc. 372 no. 9 (2019), 6087–6133]. This result will be further improved using fractional Poincaré type inequalities with the extra bonus of Bourgain–Brezis–Mironescu as done by Hurri-Syrjänen, Martínez-Perales, Pérez, and Vähäkangas [Internat. Math. Res. Notices (2022), rnac246] with a new argument. This will be used with the HMS extrapolation theorem to get $L^p$ type result.
On fractional operators with more than one singularity. María Silvina Riveros and Raúl E. Vidal
Let $0\leq \alpha < n$, $m\in \mathbb{N}$ and let $T_{\alpha,m}$ be an integral operator given by a kernel of the form \[K(x,y)=k_1(x-A_1y)k_2(x-A_2y)\dots k_m(x-A_my),\] where $A_i$ are invertible matrices and each $k_i$ satisfies a fractional size and a generalized fractional Hörmander condition that depends on $\alpha$. In this survey, written in honour to Eleonor Harboure, we collect several results about boundedness in different spaces of the operator $T_{\alpha,m}$, obtained along the last 35 years by several members of the Analysis Group of FAMAF, UNC.
Segal–Bargmann transforms and holomorphic Sobolev spaces of fractional order. Sundaram Thangavelu
In this note we investigate the image of Sobolev spaces of fractional order on a compact Lie group $K$ under the Segal–Bargmann transform. We show that the image can be characterised in terms of certain weighted Bergman spaces of holomorphic functions on the complexification extending a theorem of Hall and Lewkeeratiyutkul. We also treat the heat kernel transform associated to the Hermite operator.
$L^p(\mathbb{R}^n)$-dimension free estimates of the Riesz transforms. José L. Torrea
In this note we describe some known results about dimension free boundedness in $L^p(\mathbb{R}^n)$ of the Riesz transforms, for $p$ in the range $1 < p < \infty$.