

Published

Vol. 66, no. 1 (2023)

Front matter  
Preface.  i–ii 
Remembering Eleonor Harboure / Recordando a Eleonor Harboure.  iii–x 
Variation
and oscillation operators on weighted Morrey–Campanato spaces in the
Schrödinger setting.
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 $\{t^k\partial
_t^ke^{t\mathcal L}\}_{t > 0}$, $k\in \mathbb{N}$.

1–34 
Boundedness of fractional operators
associated with Schrödinger operators on weighted variable Lebesgue spaces
via extrapolation.
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.

35–67 
Twoweighted estimates of
the multilinear fractional integral operator between weighted Lebesgue
and Lipschitz spaces with optimal parameters.
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.

69–90 
Avatars of
Stein's theorem in the complex setting.
In this paper, we establish some variants of Stein's theorem, which states that
a nonnegative 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 $A^1(\mathbb{C}_+)$.

91–115 
Vertical
Littlewood–Paley functions related to a Schrödinger operator.
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 reverseHölder
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 reverseHölder
property. Using a subordination formula we extend these results to the
corresponding quadratic function associated to the semigroup related to
$\mathcal{L}^\alpha$, $0 < \alpha < 1$.

117–139 
Learning the model from the data.
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 finitedimensional vector spaces, functional spaces such as
$L^2(\mathbb{R}^d)$, and other general situations. We highlight the invariant
properties of these subspacebased models that make them suitable for
diverse applications, particularly in the field of image processing.

141–152 
Characterizations
of local $A_{\infty}$ weights and applications
to local singular integrals.
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 oneweight
boundedness of local singular integrals from $L^{\infty}$ to BMO.

153–175 
Extrapolation of
compactness for certain pseudodifferential operators.
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.

177–186 
Pointwise
convergence of fractional powers of Hermite type operators.
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 welldefined 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
$R > 0$.

187–205 
Remarks on
a boundary value problem for a matrix valued $\overline{\partial}$ equation.
In this short note, we discuss a boundary value problem for a matrix
valued $\overline{\partial}$ equation.

207–211 
Upper
endpoint estimates and extrapolation for commutators.
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.

213–228 
Mixed weak type
inequalities for the fractional maximal operator.
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}fu \right)^q\,
\end{equation*}
for the fractional maximal operator
\[
M_\alpha f(x)=\sup_{h > 0}B(x,h)^{\alpha/n1}\int_{B(x,h)} f
\]
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.

229–242 
Hermite Besov and
Triebel–Lizorkin spaces and applications.
We present an overview of Besov and Triebel–Lizorkin spaces in the Hermite
setting and applications on boundedness properties of Hermite
pseudomultipliers and fractional Leibniz rules in such spaces. We also
give a new weighted estimate for Hermite multipliers for weights related to
Hermite operators.

243–263 
A tribute to
Pola Harboure: Isoperimetric inequalities and the HMS extrapolation
theorem.
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
HurriSyrjänen, MartínezPerales, 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.

265–280 
On fractional
operators with more than one singularity.
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 \[K(x,y)=k_1(xA_1y)k_2(xA_2y)\dots
k_m(xA_my),\] 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.

281–295 
Segal–Bargmann
transforms and holomorphic Sobolev spaces of fractional order.
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.

297–310 
$L^p(\mathbb{R}^n)$dimension free
estimates of the Riesz transforms.
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 $1 < p < \infty$.

311–323 
Published by the Unión Matemática Argentina
ISSN 16699637 (online), ISSN 00416932 (print)
Registro Nacional de la Propiedad Intelectual nro. 180.863
Contact: revuma(at)criba.edu.ar