### Session S13 - Harmonic Analysis, Fractal Geometry, and Applications

## Talks

Thursday, July 15, 16:00 ~ 16:30 UTC-3

## Local dimensions of self-similar measures with overlap

### Kathryn Hare

#### University of Waterloo, Canada - This email address is being protected from spambots. You need JavaScript enabled to view it.

The local dimension of a measure is a way to quantify its local behaviour. For self-similar measures that satisfy a suitable separation condition, it is well known that the set of attainable local dimensions is a closed interval, but for measures which fail to satisfy this condition the situation is more complicated and less well-understood. We will show that for a large class of self-similar measures on R with ‘controlled’ overlap the set of local dimensions is a finite union of (possibly singleton) compact intervals, the number of which is bounded by geometric properties of the underlying IFS.

Thursday, July 15, 16:35 ~ 17:05 UTC-3

## Completeness, exponentially completeness, and approximate orthogonality on the ball

### Azita Mayeli

#### City University of New York , The Graduate Center and Queensborough, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.

Let $\Omega\subset \Bbb R^d$ be a bounded domain with positive Lebesgue measure, and assume that $\mathcal F\subset L^2(\Omega)$ is non-empty. We say $\mathcal F$ is {\it exponentially complete} if for all $\xi\in\Bbb R^d$, there is a function $f\in \mathcal F$ such that $$\langle f, e^{2\pi i x\cdot \xi} \rangle \neq 0 .$$ Let $B_d\subset \Bbb R^d$, $d>1$, denote the unit ball. It is known that $L^2(B_d)$ dose not admit any Parseval frame of exponentials. Motivated by this fact, we will investigate the exponentially completeness of exponential functions $\mathcal F=\mathcal E(A)\subset L^2(B_d)$, where $$\mathcal E(A):=\{f_a(x):=e^{2\pi i x\cdot a}: ~ a\in A\subset \Bbb R^d\},$$ and $A$ is countable. In particular, we show that for any set $A$ of size $\sharp A=2$, the set $\mathcal E(A)$ is exponentially incomplete in $L^2(B_d)$, $d\geq 2$, and also there are exponentially complete sets $\mathcal E(A)$ for any large size $A$ in $\Bbb R^d, ~d\geq 3$.\\

In the second half of the talk, we weaken the orthogonality condition and show that there is no set $A$ with positive and finite upper density such that the exponentials $\mathcal E(A)$ are mutually $\phi$-approximately orthogonal on the ball. More precisely, given a bounded domain $\Omega$, and a bounded measurable function $\phi:[0,\infty) \to [0, \infty)$ with $\phi(t)\to 0$ as $t\to \infty$, we say that $e^{2\pi i x\cdot a}$ and $e^{2\pi i x\cdot a'}$, $a\neq a'$, are {\it $\phi$-approximately orthogonal} if $$|\widehat{\chi_\Omega}(a-a')|\leq \phi(|a-a|).$$ We prove that if $\phi$ decays faster than $(1+t)^{-\frac{d+1}{2}}$ as $t\to \infty$, then the unit ball can not admit any $\phi$-approximate orthogonal basis of exponentials. \\

Joint work with Alex Iosevich (University of Rochester, USA).

Thursday, July 15, 17:10 ~ 17:40 UTC-3

## Conformal removability is hard

### Christopher Bishop

#### Stony Brook University, United States of America - This email address is being protected from spambots. You need JavaScript enabled to view it.

Suppose E is a compact set in the complex plane and U is its complement. The set E is called removable for a property P, if any holomorphic function on U with this property extends to be holomorphic on the whole plane. This is an important concept with applications in complex analysis, dynamics and probability. Tolsa famously characterized removable sets for bounded holomorphic functions, but such a characterization remains unknown for conformal maps on U that extend homeomorphically to the boundary. We offer an explanation for why the latter problem is actually harder: the collection of removable sets for bounded holomorphic maps is a G-delta set in the space of compact planar sets with the Hausdorff metric, but the collection of conformally removable sets is not even a Borel subset of this space. These results follow from known facts, but they suggest a number of new questions about fractals, removable curves and conformal welding.

Thursday, July 15, 17:45 ~ 18:15 UTC-3

## Intermediate Assouad-like dimensions

### Ignacio García

#### CEMIM - IFIMAR, Universidad Nacional de Mar del Plata and CONICET, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.

Many notions of dimension have been introduced to help understand the geometry of (often `small') subsets of metric spaces, such as subsets of $\mathbb{R}^{n}$ of Lebesgue measure zero. Hausdorff and box dimensions are well known examples of such notions, and possibly less known are the upper and lower Assouad dimensions of a set $E$, which quantify the `thickest' or `thinnest' part of the space.

In the talk I will review some results related to a general class of `intermediate dimensions', referred to as the upper and lower $\Phi $-dimensions, which can roughly be thought of as local refinements of the box-counting dimensions where one takes the most extreme local behaviour, at scales `ruled' by a function $\Phi$. These $\Phi$-dimensions provide a range of bi-Lipschitz invariant dimensions between the box and Assouad dimensions. As the box and Assouad dimensions for a given set can all be different, the intermediate $\Phi$-dimensions provide more refined information about the local geometry of the set, such as detailed information about the scales at which one can observe extreme local behaviour.

Joint work with Kathryn Hare (University of Waterloo, Canada) and Kranklin Mendivil (Acadia University, Canada).

Thursday, July 15, 18:30 ~ 19:00 UTC-3

## Sets with rational linear patterns

### Malabika Pramanik

#### University of British Columbia , Canada - This email address is being protected from spambots. You need JavaScript enabled to view it.

How large does a set have to be in order to include a nontrivial solution of a translation-invariant linear equation with integer coefficients? Is it possible for a large set to avoid nontrivial solutions of many such equations? We will discuss answers to these questions under varying definitions of size, and report on ongoing work in this direction.

Joint work with Yiyu Liang (Beijing Jiaotong University).

Thursday, July 15, 19:05 ~ 19:35 UTC-3

## Pseudo-multipliers and smooth molecules on Hermite Besov and Hermite Triebel-Lizorkin spaces

### Virginia Naibo

#### Kansas State University, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.

We will present boundedness properties of pseudo-multipliers with symbols of Hörmander-type in function spaces associated to the Hermite operator. The main tools in the proofs involve new molecular decompositions and molecular synthesis estimates for Hermite Besov and Hermite Triebel-Lizorkin spaces, which allow to obtain boundedness results on spaces for which the smoothness allowed includes non-positive values. In particular, we obtain continuity results for pseudo-multipliers on Lebesgue and Hermite local Hardy spaces.

Joint work with Fu Ken Ly (The University of Sydney).

Thursday, July 15, 19:40 ~ 20:10 UTC-3

## Fefferman-Stein inequalities for the Hardy-Littlewood maximal function on the infinite rooted $k$-ary tree

### Sheldy Javier Ombrosi

#### Universidad Nacional del Sur, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.

In this talk we present weighted endpoint estimates for the Hardy-Littlewood maximal function on the infinite rooted $k$-ary tree. Namely, the following Fefferman-Stein estimate \[ w\left(\left\{ x\in T\,:\,Mf(x)>\lambda\right\} \right)\leq c_{s}\frac{1}{\lambda}\int_{T}|f(x)|M(w^{s})(x)^{\frac{1}{s}}dx\qquad s>1 \] is settled and moreover it is shown it is sharp, in the sense that it does not hold in general if $s=1$. This result is a generalization of the unweighted case ($w\equiv 1$) independently obtained by Naor-Tao and Cowling-Meda-Setti. Some examples of non trivial weights such that the weighted weak type $(1,1)$ estimate holds are provided. A strong Fefferman-Stein type estimate and as a consequence some vector valued extensions are obtained.

Joint work with Israel Rivera-Ríos (Universidad Nacional del Sur) and Martín Dario Safe (Universidad Nacional del Sur).

Thursday, July 15, 20:15 ~ 20:45 UTC-3

## NECESSARY CONDITION ON THE WEIGHT FOR MAXIMAL AND INTEGRAL OPERATORS WITH ROUGH KERNELS

### María Silvina Riveros

#### Universidad Nacional de Córdoba - CIEM-CONICET, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.

Let $0\leq \alpha

Bibliography:

[1]. Iba\~nez-Firnkorn, Gonzalo H. and Riveros, Mar{\'\i}a Silvina. \textit{Certain fractional type operators with H\"ormander conditions}, Ann. Acad. Sci. Fenn. Math. Vol (43) 913--929 (2018)

Joint work with Gonzalo Iba\~nez-Firnkorn ( Universidad Nacional de Córdoba - CIEM-CONICET) and Raúl Vidal ( Universidad Nacional de Córdoba - CIEM-CONICET).

Friday, July 16, 11:00 ~ 11:30 UTC-3

## Zeros of Gaussian Weyl-Heisenberg functions

### José Luis Romero

#### University of Vienna, Austria - This email address is being protected from spambots. You need JavaScript enabled to view it.

GWHF are Gaussian random functions on the plane whose stochastics are invariant under the Weyl-Heisenberg group (twisted stationarity). Main examples include the short-time Fourier transform (STFT) of complex white noise, and higher order derivatives of Gaussian entire functions. I will present basic statistics for zero sets of GWHF and applications to the STFT.

Joint work with Antti Haimi (Austrian Academy of Sciences) and Guenther Koliander (Austrian Academy of Sciences).

Friday, July 16, 11:35 ~ 12:05 UTC-3

## Fourier frames for singular measures and pure type phenomena

### Nir Lev

#### Bar-Ilan University, Israel - This email address is being protected from spambots. You need JavaScript enabled to view it.

Let $\mu$ be a positive, finite measure on $\mathbb{R}^d$. When is it possible to construct a system of exponential functions that constitutes a frame in the Hilbert space $L^2(\mu)$ ? In the talk, I will explain the notion of a frame, discuss what is known about the problem, and present some recent results.

Friday, July 16, 12:10 ~ 12:40 UTC-3

## Learning group representations in brain's visual cortex

### Davide Barbieri

#### Universidad Autónoma de Madrid, Spain - This email address is being protected from spambots. You need JavaScript enabled to view it.

Human vision has inspired several advances in harmonic analysis, especially wavelet analysis, and it has been the main source of heuristics for the development of neural network architectures devoted to image processing. One of the most studied neural structures in brain's visual cortex is area V1, where neurons perform a wavelet-like analysis that is generally considered to be associated with the group structure of rotations and translations. It is indeed possible to model part of the perceptual behavior of the network of neural cells in V1 as a projection of the image onto one, or more, orbits of that group, and consequently associate to each neuron in V1 a parameter of the group. However, due to the physical constraint of having a neural displacement onto a two dimensional layer, the group is not fully, nor uniformly, represented in V1. The represented subset of the group has however a characteristic geometric structure, that has been modeled over the physiological measurements of what are called orientation preference maps. A natural question posed by this empirical observation is whether the missing part of the group, and of the corresponding wavelet coefficients, has perceptual consequences, and if, on the other hand, it is possible to recover or estimate in some stable way the missing information. The ability to perform such a task would allow one to effectively learn a full group representation from a partial set of well positioned detectors. We will propose such a mechanism, and briefly discuss its possible physical implementation.

Friday, July 16, 12:45 ~ 13:15 UTC-3

## Necessary conditions for interpolation by multivariate polynomials

### Jorge Antezana

#### UNLP & IAM-CONICET , Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.

Let $\Omega$ be a smooth, bounded, convex domain in $\mathbb R^n$, and let $\mathcal{P}_k$ be the vector space of of multivariate real polynomials of degree at most $k$. In these spaces we will consider the Hilbert structure given by the $L^2$ norm associated to the Lebesgue measure restricted to $\Omega$.

Consider a sequence $\{\Lambda_k\}_{k\geq 0}$ consisting of finite subsets of $\Omega$. In this talk, we will discuss some necessary geometric conditions for the set $\Lambda_k$ to be interpolating for $\mathcal{P}_k$ and with uniform bounds. Taking as prototype the results about interpolation in spaces of holomorphic functions, the necessary conditions are expressed in terms of an appropriate separation condition, a Carleson condition, and a density condition. On the other hand, in the particular case of the unit ball, we will show that there is not an orthogonal basis of reproducing kernels in the space $\mathcal{P}_k$, when $k$ is big enough.

Joint work with Jordi Marzo (Universidad de Barcelona, España) and Joaquim Ortega Cerdà (Universidad de Barcelona, España).

Friday, July 16, 13:45 ~ 14:15 UTC-3

## Frames generated by the action of a discrete group

### Victoria Paternostro

#### IMAS (CONICET) - University of Buenos Aires, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.

In this talk we shall discuss the structure of subspaces of a Hilbert space that are invariant under unitary representations of discrete groups. We will study in depth the reproducing properties (this is, being a Riesz basis or a frame) of countable families of orbits. In particular we shall see that every separable Hilbert space $\mathcal{H}$ for which there exists a dual integrable representation $\Pi$ of a discrete group $\Gamma$ on $\mathcal{H}$, admits a Parseval frame of the form \[\{\Pi(\gamma)\phi:\,\gamma\in\Gamma, \phi\in\Phi\} \] where $\Phi\subseteq \mathcal{H}$ is an at most countable set. Our results extend those that already exist in the euclidean case to this more general context.

Joint work with Davide Barbieri (Universidad Autónoma de Madrid, Spain) and Eugenio Hernández (Universidad Autónoma de Madrid, Spain).

Friday, July 16, 14:20 ~ 14:50 UTC-3

## Boundary value problems for $p$-elliptic systems

### Jill Pipher

#### Brown University, US - This email address is being protected from spambots. You need JavaScript enabled to view it.

We discuss joint work with M. Dindos and J. Li on boundary value problems for elliptic systems, with applications to the Lam\'e systems and homogenization. This paper explores, in the context of elliptic systems, the structural condition of $p$-ellipticity that was independently introduced for complex coefficient divergence form elliptic equations by Dindos-Pipher and Carbonaro-Dragicevic.

Joint work with Martin Dindos (U. of Edinburgh) and Jungang Li (Brown University).

Friday, July 16, 14:55 ~ 15:25 UTC-3

## Adjacent Dyadic Systems

### Theresa Anderson

#### Purdue University, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.

A very useful tool in analysis and applications, called by many names, is the "1/3" trick, which says that any ball in Euclidean space is contained in a dyadic cube of roughly the same size, where the dyadic cube comes from one of a finite number of dyadic grids. For $\mathbb{R}^d$, Conde showed that the optimal number of grids to perform this trick is $d+1$. In recent joint work, we completely classify all grids that allow this property, termed "adjacent dyadic systems", and discuss an interesting connection to number theory that arises.

Joint work with Bingyang Hu (Purdue University, USA).

Friday, July 16, 15:30 ~ 16:00 UTC-3

## Improving Estimates in Discrete Settings

### Michael Lacey

#### Georgia Institute of Technology, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.

Averages improve smoothness and integrability of functions. This well known principle is very familiar in continuous settings. There is a wealth of results and approaches to questions of this type in a variety of continuous settings. The investigation of related questions in the discrete setting is much more recent. We will survey recent results in this area, outlining the types of results that are possible, some that have been established, and the range of techniques used. We will touch on averages over the integers, as well as finite field settings.

Joint work with Rui Han, Louisiana State University, Fan Yang, Australian National University, Hamed Mousavi, Georgia Tech, Christina Gianitsi, Georgia Tech and Jacob Rahami, Georgia Tech.

Tuesday, July 20, 16:00 ~ 16:30 UTC-3

## Tiling the integers with translates of one tile: the Coven-Meyerowitz tiling conditions for three prime factors

### Izabella Laba

#### University of British Columbia, Canada - This email address is being protected from spambots. You need JavaScript enabled to view it.

It is well known that if a finite set of integers $A$ tiles the integers by translations, then the translation set must be periodic, so that the tiling is equivalent to a factorization $A\oplus B=\mathbb{Z}_M$ of a finite cyclic group. Coven and Meyerowitz (1998) proved that when the tiling period $M$ has at most two distinct prime factors, each of the sets $A$ and $B$ can be replaced by a highly ordered "standard" tiling complement. It is not known whether this behaviour persists for all tilings with no restrictions on the number of prime factors of $M$.

In joint work with Itay Londner, we proved that this is true when $M=(pqr)^2$ is odd. (We are currently finalizing the even case.) In my talk I will discuss this problem and introduce the main ingredients in the proof.

Joint work with Itay Londner (University of British Columbia, Canada).

Tuesday, July 20, 16:35 ~ 17:05 UTC-3

## Some recent progress on Fourier frames and Riesz bases for singular measures

### Chun-Kit Lai

#### San Francisco State University, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.

In this talk, we give some recent results concerning the construction and existence of exponential frames and Riesz bases on different singular measures on ${\mathbb R}^d$ such as Cantor-type measures and measures supported on union of subspaces.

Tuesday, July 20, 17:10 ~ 17:40 UTC-3

## Stein-Weiss inequality in $L^{1}$ norm for vector fields

### Tiago Picon

#### University of São Paulo, Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.

In this talk, we investigate the limit case $p=1$ of the Stein-Weiss inequality for the Riesz potential. Our main result is a characterization of this inequality for a special class of vector fields associated to cocanceling operators. As application, we recovered some classical div-curl inequalities in literature. In addition, we also discussed a two-weight inequality with general weights extending the previous result due to Sawyer for the scalar case.

Joint work with Pablo De Nápoli (Universidad de Buenos Aires , Argentina).

Tuesday, July 20, 17:45 ~ 18:15 UTC-3

## Some topics in harmonic analysis over locally compact abelian groups

### Emily King

#### Colorado State University, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.

Many results in harmonic analysis over Euclidean space rely on the existence of full rank lattices in $\mathbb{R}^d$ to generate translation groups. Thus, it is not always possible to simply port over proofs from harmonic analysis over $\mathbb{R}^d$ to harmonic analysis over locally compact abelian groups without lattices, like vector spaces over the $p$-adic numbers $\mathbb{Q}_p^d$. In this talk, some results about wavelet and Gabor systems over such groups with connections to fractal geometry will be presented.

Joint work with Victoria Paternostro (Universidad de Buenos Aires, Argentina) and Maria Skopina (Saint Petersburg State University, Russia).

Tuesday, July 20, 18:30 ~ 19:00 UTC-3

## Sampling and Interpolation of Cumulative Distribution Functions of Cantor Sets in $[0,1]$

### Eric Weber

#### Iowa State University, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.

We consider the class of Cantor sets that are constructed from affine iterated function systems on the real line. These Cantor sets possess a cumulative distribution function (CDF), constructed from the invariant Borel probability measure associated with their iterated function systems. Under appropriate assumptions, we identify sampling schemes of such CDFs, meaning that the underlying Cantor set can be reconstructed from sufficiently many samples of its CDF. To this end, we prove that two Cantor sets have almost-nowhere intersection with respect to their corresponding invariant measures.

Joint work with Allison Byars (University of Wisconsin), Evan Camrud (Iowa State University), Steven Nathan Harding (Milwaukee School of Engineering), Sarah McCarty (Iowa State University) and Keith Sullivan (Concordia College).

Tuesday, July 20, 19:05 ~ 19:35 UTC-3

## Nonnegativity and spanning structure

### Alex Powell

#### Vanderbilt University, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.

We investigate nonnegativity as an obstruction to various types of spanning structure in Lp spaces. For example, if a system of functions is pointwise nonnegative, then it cannot be a frame or Riesz basis for L2, or an unconditional quasibasis for Lp. On the other hand, there exist pointwise nonnegative Markushevich bases and conditional quasibases in Lp, and in recent work we show that L2 admits a nonnegative Schauder basis.

Joint work with Dan Freeman (Saint Louis University, USA), Mitchell Taylor (University of California, Berkeley, USA) and Anneliese Spaeth (Huntingdon College, USA).

Tuesday, July 20, 19:40 ~ 20:10 UTC-3

## The Tur\'an problem for a ball centered at the origin in $\mathbb{R}^d$ and its dual.

### Jean-Pierre Gabardo

#### McMaster University, Canada - This email address is being protected from spambots. You need JavaScript enabled to view it.

The Tur\'an problem for $B_r$, the open ball of radius $r$ centered at the origin in $\mathbb{R}^d$, consists in computing the supremum of the integrals of positive definite functions supported on that ball and taking the value $1$ at the origin. Siegel proved, in the thirties, that this supremum has the value $c_r=2^{-d}\,|B_r|$, where $|\cdot|$ denotes the Lebesgue measure and is reached by the function $f_r=c_r^{-1}\,\chi_{B_{r/2}}*\chi_{B_{r/2}}$. Several proofs of this result are know and, in this talk, we will outline a new proof of it based on the explicit construction of a maximizer for the dual Tur\'an problem, which is a positive-definite distribution $T_r$ equal to the Dirac delta function $\delta_0$ on $B_r$ and which satisfies $f_r*T_r=1$ on $\mathbb{R}^d$. As an intermediary step needed for this result, we construct new families of Parseval frames, consisting of Bessel functions, on the interval $[0,r]$.

Tuesday, July 20, 20:15 ~ 20:45 UTC-3

## Dynamical sampling: A source term inverse problem

### Akram Aldroubi

#### Vanderbilt University, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.

Dynamical sampling is a term describing an emerging set of problems related to recovering signals and evolution operators from space-time samples. For example, one problem is to recover a function $f$ from space-time samples $\{(A_{t_i}f)(x_i)\} $ of its time evolution $f_t=(A_{t}f)(x)$, where $\{A_t\}_{t \in \mathcal T }$ is a known evolution operator and $\{(x_i,t_i)\}\subset \mathbb R^d\times \mathbb R^+.$

Applications of dynamical sampling include inverse problems in sensor networks, and source term recovery from physically driven evolutionary systems.

Dynamical sampling problems are tightly connected to frame theory as well as more classical areas of mathematics such as approximation theory, and functional analysis. In this talk, I will present a situation in which a function $u(t,x)$ is evolving under the action of a sum of two unknown terms: a bursts-like term and a background source term. The problem is to recover the driving bursts-like source term from noisy space-time samples of of $u$.

Joint work with Longxiu Huang (UCLA, USA), Keri Kornelson (University of Oklahoma, USA) and Ilya Krishtal (Northern Illinois University, USA).

## Posters

## A model for frames of iterations of two operators

### Alejandra Patricia Aguilera Aguilera

#### Universidad de Buenos Aires, IMAS-CONICET, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.

We consider frames in a separable Hilbert space $\mathcal{H}$ of the form $\{T^{k}L^{j}v: k\in\mathbb{Z}, j=0,1,2,...\}$ with a bounded operator $L$, a unitary operator $T$ that commutes with $L$, and a vector $v\in\mathcal{H}$.

We observe that $\mathcal{H}$ is mapped isomorphically to a subspace $\mathfrak{N}$ of the Hilbert space of measurable vector-valued functions $L^{2}(\mathbb{T},H^{2})$, where $H^{2}:=H^{2}(\mathbb{D})$ is the Hardy space on the unit disc. Then we show that the system $\{T^{k}L^{j}v: k\in\mathbb{Z}, j\in j=0,1,2,...\}$ corresponds to the set $\{\mathcal{U}^{k} \hat{A}^{j}\psi: k\in\mathbb{Z}, j\in j=0,1,2,...\}$ via the underlying isomorphism between $\mathcal{H}$ and $\mathfrak{N}$. Here, $\psi$ is some function in $\mathfrak{N}$, $\mathcal{U}$ is the bilateral shift with multiplicity in $L^{2}(\mathbb{T},H^{2})$, and $\hat{A}$ commutes with $\mathcal{U}$ and acts pointwise as the compression of the unilateral shift on model subspaces of the Hardy space $H^2$.

We also give a characterization of all vectors $v$ such that the system $\{T^{k}L^{j}v: k\in\mathbb{Z}, j=0,1,2,...\}$ is a frame for $\mathcal{H}$ assuming that there is some $v_{0}\in\mathcal{H}$ with that property.

Joint work with Carlos Cabrelli (Universidad de Buenos Aires, IMAS-CONICET), Diana Carbajal (Universidad de Buenos Aires, IMAS-CONICET) and Victoria Paternostro (Universidad de Buenos Aires, IMAS-CONICET).

## One-bit Quantization for Phase Retrieval

### Dylan Domel-White

#### Vanderbilt University, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.

Abstractly, phase retrieval is the problem of recovering a vector in a real or complex Hilbert space (up to a global phase factor) from magnitudes of linear functionals applied to the vector. In other words, rather than having access to linear functional measurements of the form $x \mapsto \left\langle x,v \right\rangle$ as in many sensing applications, we have nonlinear measurements $x \mapsto \left|\left\langle x,v \right\rangle\right|$. If $v$ is a unit vector, then $\left|\left\langle x,v \right\rangle\right|$ is the norm of the projection of $x$ onto the span of $v$, and so norms of projections onto higher dimensional subspaces are a natural generalization of typical phase retrieval measurements as described above. We present a measurement and recovery algorithm along with theoretical performance guarantees for \textit{one-bit} phase retrieval, where only one bit of information is recorded from each projection norm. For our algorithm, we obtain one-bit information by thresholding the norms of projections onto random subspaces of dimension equal to half that of the Hilbert space, and recover by finding the principal component of an auxiliary matrix constructed using the one-bit measurements.

Joint work with Bernhard Bodmann (University of Houston, TX, United States).

## Preconditioned Gradient Descent Algorithm for Inverse Filtering on Spatially Distributed Networks

### Nazar Emirov

#### Boston College, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.

Graph filters and their inverses have been widely used in denoising, smoothing, sampling, interpolating and learning. Implementation of an inverse filtering procedure on spatially distributed networks (SDNs) is a remarkable challenge, as each agent on an SDN is equipped with a data processing subsystem with limited capacity and a communication subsystem with confined range due to engineering limitations. In this work, we introduce a preconditioned gradient descent algorithm to implement the inverse filtering procedure associated with a graph filter having small geodesic-width. The proposed algorithm converges exponentially, and it can be implemented at vertex level and applied to time-varying inverse filtering on SDNs.

Joint work with Cheng Cheng (Sun Yat-sen University, China) and Qiyu Sun (University of Central Florida, USA).

## Differential Transform of Cesàro averages in product spaces.

### Cecilia Ferrari Freire

#### Universidad Nacional del Comahue - IITCI (CONICET), Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.

In this poster we present convergence results for the series of differences of Cesàro averages along lacunary sequences in product spaces. These results give some information about how the Cesàro averages converge. From a result in [D] y [DRF] and following the ideas in [BMR] for the lacunar maximal operator in the lateral context we obtain new results in the product spaces context.

References

[BMR] Bernardis, A.L.; Mart\'\i n-Reyes, F.J.; \textit{Differential Transforms of Cesàro averages in weighted Spaces.} Publ. Math. 52 (2008), 101-127.

[D] Duoandikoetxea, J.; Multiple singular integrals and Maximal functions along hypersurfaces. Ann. Inst.Fourier, Grenoble. 36, 4(1986), 185-206.

[DRF] Duoandikoetxea, J. and Rubio de Francia, J.L.; Maximal and singular integral operators via Fourier transform estimates. Invent. Math. 84, 541-561 (1986).

Joint work with Raquel Crescimbeni (Universidad Nacional del Comahue - IITCI (CONICET), Argentina).

## Properties of balanced frames

### Sigrid Heineken

#### IMAS, UBA-CONICET, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.

In this work we consider balanced frames, i.e. those frames which sum is zero, and in particular balanced unit norm tight frames, in finite dimensional Hilbert spaces.

So far there has not been paid attention to frames that are balanced. Here we present several advantages of balanced unit norm tight frames in signal processing. They provide an exact reconstruction in the presence of systematic errors in the transmitted coefficients, and are optimal when these coefficients are corrupted with noises that can have non-zero mean. Moreover, using balanced frames we can know that the transmitted coefficients were perturbed, and we also have an indication of the source of the error.

We analyze various properties of these types of frames. We define an equivalence relation in the set of the dual frames of a balanced frame. This allows to show that we can obtain all the duals from the balanced ones. We investigate the problem of finding the nearest balanced frame to a given frame, characterizing completely its existence and giving its expression. We introduce and study a new concept of complement for balanced frames. Finally, we present examples and methods for constructing balanced unit norm tight frames.

Joint work with Patricia Morillas (IMASL, UNSL-CONICET) and Pablo Tarazaga (IMASL, UNSL-CONICET)}.

## Random sampling and reconstruction of concentrated signals in a reproducing kernel space

### YAXU LI

#### Hangzhou Normal University, China - This email address is being protected from spambots. You need JavaScript enabled to view it.

We consider (random) sampling of signals concentrated on a bounded Corkscrew domain $\Omega$ of a metric measure space, and reconstructing concentrated signals approximately from their (un)corrupted sampling data taken on a sampling set contained in $\Omega$. We establish a weighted stability of bi-Lipschitz type for a (random) sampling scheme on the set of concentrated signals in a reproducing kernel space. The weighted stability of bi-Lipschitz type provides a weak robustness to the sampling scheme, however due to the nonconvexity of the set of concentrated signals, it does not imply the unique signal reconstruction. From (un)corrupted samples taken on a finite sampling set contained in $\Omega$, we propose an algorithm to find approximations to signals concentrated on a bounded Corkscrew domain $\Omega$. Random sampling is a sampling scheme where sampling positions are randomly taken according to a probability distribution. Next we show that, with high probability, signals concentrated on a bounded Corkscrew domain $\Omega$ can be reconstructed approximately from their uncorrupted (or randomly corrupted) samples taken at i.i.d. random positions drawn on $\Omega$, provided that the sampling size is at least of the order $\mu(\Omega)\ln(\mu(\Omega))$, where $\mu(\Omega)$ is the measure of the concentrated domain $\Omega$. Finally, we demonstrate the performance of proposed approximations to the original concentrated signal when the sampling procedure is taken either with small density or randomly with large size.

Joint work with Qiyu Sun (University of Central Florida) and Jun Xian (Sun Yat-sen University).

## Characterization of Simple Solids using the G-Particle-Hole Equation and the Fourier Transform in Finite Atomic Systems

### Gustavo Ernesto Massaccesi

#### Universidad de Buenos Aires, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.

The natural space to analyze the electron distribution in a linear crystal is generated by the functions corresponding to the lowest energy orbitals associated with the atoms of the system. The periodic repetition of the atoms makes this a Shift Invariant Space. A similar structure is associated with one-dimensional cyclic crystals in which the translational symmetry is analogous to rotations.

Previous works have described treatments which combine the G-particle-hole Hypervirial equation and the method of Equations of Motion. In this work we formulate a symmetry-adapted version of the combined algorithm for Abelian groups, in particular the $C_N$ one. The introduction of the point group symmetry within this hybrid framework provides a remarkable computational improvement. The results obtained in selected sets of small-to-medium-sized cyclic one-dimensional chains, used as prototype to describe solid models, reveal a significant computational saving in both floating-point operations and memory storage.

Joint work with Gustavo E. Massaccesi (Departamento de Ciencias Exactas, Ciclo Básico Común, Universidad de Buenos Aires; Argentina), Juan J. Torres‑Vega (Centro de Investigaciones Tecnológicas, Biomédicas y Medioambientales; Peru), Elías Ríos (Instituto de Investigaciones Fisicoquímicas Teóricas y Aplicadas, Universidad Nacional de La Plata, Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina), Alberto Camjayi (Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires; Argentina / Instituto de Física de Buenos Aires, Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina), Alicia Torre (Departamento de Química Física, Facultad de Ciencia y Tecnología, Universidad del País Vasco; Spain), Luis Lain (Departamento de Química Física, Facultad de Ciencia y Tecnología, Universidad del País Vasco; Spain), Ofelia B. Oña (Instituto de Investigaciones Fisicoquímicas Teóricas y Aplicadas, Universidad Nacional de La Plata, Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina), William Tiznado (Departamento de Química, Facultad de Ciencias Exactas, Universidad Andres Bello; Chile) and Diego R. Alcoba (Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires; Argentina / Instituto de Física de Buenos Aires, Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina).

## Self-improving Poincaré-Sobolev type functionals in product spaces

### Carolina Alejandra Mosquera

#### Universidad de Buenos Aires e IMAS CONICET, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.

In this poster we give a geometric condition which ensures that $(q,p)$-Poincaré-Sobolev inequalities are implied from generalized $(1,1)$-Poincaré inequalities related to $L^1$ norms in the context of product spaces. The concept of eccentricity plays a central role in the paper. We provide several $(1,1)$-Poincaré type inequalities adapted to different geometries and then show that our selfimproving method can be applied to obtain special interesting Poincar\'e-Sobolev estimates. Among other results, we prove that for each rectangle $R$ of the form $R=I_1\times I_2 \subset \mathbb{R}^{n}$ where $I_1\subset \mathbb{R}^{n_1}$ and $I_2\subset \mathbb{R}^{n_2}$ are cubes with sides parallel to the coordinate axes, we have that

\begin{equation*} \left( \frac{1}{w(R)}\int_{ R } |f -f_{R}|^{p_{\delta,w}^*} \,wdx\right)^{\frac{1}{p_{\delta,w}^*}} \leq c\,\delta^{\frac1p}(1-\delta)^{\frac1p}\,[w]_{A_{1,\mathcal{R}}}^{\frac1p}\, \Big(a_1(R)+a_2(R)\Big), \end{equation*}

where $\delta \in (0,1)$, $w \in A_{1,\mathcal{R}}$, $\frac{1}{p} -\frac{1}{ p_{\delta,w}^* }= \frac{\delta}{n} \, \frac{1}{1+\log [w]_{A_{1,\mathcal{R}}}}$ and $a_i(R)$ are bilinear analog of the fractional Sobolev seminorms $[u]_{W^{\delta,p}(Q)}$. This is a biparameter weighted version of the celebrated fractional Poincaré-Sobolev estimates with the gain $\delta^{\frac1p}(1-\delta)^{\frac1p}$.

Joint work with Eugenia Cejas (Universidad de La Plata, Argentina), Carlos Pérez (Universidad del País Vasco y BCAM, España) and Ezequiel Rela (Universidad de Buenos Aires e IMAS-CONICET, Argentina).

## PROPIEDADES ANALÍTICAS Y GEOMÉTRICAS DE ESPACIOS DE HERZ EN ESPACIOS DE TIPO HOMOGÉNEO

### Alejandra Perini

#### Universidad Nacional del Comahue, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.

In the Euclidean context of $R^n$ the Herz spaces, initially introduced by Herz [H] in the context of the study of Berstein-type theorems and Lipschitz spaces, was characterized by an equivalent norm by a later work by Johnson [J]. Today, this characterization is used as the definition of Herz spaces may be apt to extend the definition of these spaces to the context of other metrics in $R^n$. Indeed, Ragusa [R] defines the Herz spaces associated with parabolic metrics in $R^n$ and studies their applications in the context of obtaining regularity results of weak solutions for parabolic differential equations in divergence form. In this work we extend the definition of Herz spaces to the abstract context of homogeneous type spaces making use of ChristŽs dyadic families [Ch]. We also address some analytical and geometric problems of these spaces related to the characterization and unconditionality of wavelet bases.

More precisely. If $ D $ is a dyadic family of type Christ, defined on $ (X, d, \ mu) $ space of homogeneous type, we give the following definition:

Let $(X, d, \mu )$ be a homogeneous type space, $ 1

As a first result we establish that the spaces $ K_{p, q} $ are Banach spaces and that the norm does not depend on the choice of cubes $ Q_{n}^{i} $ in the sense that for two of such sequences the norms are equivalents. On the other hand, some of the results that we obtain can be summarized in the following theorem:

Theorem: Let $ (X, d \mu) $ space of homogeneous type $ D $ be a dyadic family and $ H $ an associated Haar system. Then the Haar system truncated at the level of zero resolution turns out to be an unconditional basis for Herz spaces and a characterization of the norms $ K_{p, q} $ of a function $ f $ in terms of its wavelet coefficients is valid.

We also explore the geometric conditions on the space $ (X, d, \mu ) $ for obtain that the democracy in the Herz space $ K_{p, q} $ of the truncated Haar system implies that Herz space it is a Lebesgue space.

[Ch] Christ M., A T(b) theorem with remarks on analytic capacity and the Cauchy integral}, Colloq. Math. 60/61 (2), 601--628 (1990).

[H] Herz, C.S., Lipschitz Spaces and BernsteinŽs Theorem and absolutely convergent Fourier Transform, Journal of Math. and Mechanics, Vol 18, No 4 (1968).

[J] Johnson, R., Lipschitz Spaces - Littlewood Paley Spaces and convoluteurs. Procceding London Math., Soc (3) 29 127-141 (1974).

[R] Ragusa M. A., Parabolic Herz Spaces and their applications. Applied Math. Letters 25 1270-1273 (2012).

Joint work with Dr. Luis Nowak (Universidad Nacional del Comahue) and Prof Daniela Fernandez (Universidad Nacional del Comahue).

## Boundedness and compactness for commutators of singular integrals related to a critical radius function

### Pablo Quijano

#### IMAL (UNL - CONICET), Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.

We work in the general framework of a family of singular integrals with kernels controlled in terms of a critical radius function $\rho$. This family models the harmonic analysis derived from the Schrödinger operator $L= -\Delta+V$, where the non-negative potential $V$ satisfies an appropriate reverse H\"older condition. For their commutators, we find sufficient conditions on the symbols for boundedness and/or compactness when acting on weighted $L^p$ spaces. In all cases, the classes of symbols and weights are larger than their classical counterparts, $\textup{BMO}$, $\textup{CMO}$ and $A_p$. When these general results are applied to the Schrödinger context, we obtain boundedness and compactness for commutators of operators like $\nabla L^{-1/2}$, $\nabla^2 L^{-1}$, $V^{1/2} L^{-1/2}$, $V^{1/2} \nabla L^{-1}$, $VL^{-1}$ and $L^{i\alpha}$. As in Uchiyama's classical paper, we give a full description of the class for compactness, $\textup{CMO}^\infty_\rho$, assuming $\rho$ to be bounded. Finally, we provide examples showing that $\textup{CMO}$ is strictly contained in $\textup{CMO}^\infty_\rho$ for any $\rho$, bounded or not.

Joint work with Bruno Bongioanni IMAL (UNL - CONICET) and and Eleonor Harboure IMAL (UNL - CONICET).

## Quantitative matrix weighted estimates

### Israel Pablo Rivera Ríos

#### Universidad Nacional del Sur - INMABB, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.

In this poster we will show the recent progress in quantitative weigthed estimates in the matrix setting.