Seminario de Análisis y Aplicaciones UAM-ICMAT 2021

23 de Abril de 2021, 11:30 : ONLINE - URL https://zoom.us/j/96993911826 Jorge Abel Antezana, Universidad Nacional de La Plata and Instituto Argentino de Matemática
Necessary conditions for interpolationby multivariate polynomialsPDF,

Let \(\Omega\) be a smooth, bounded, convex domain in \(\mathbb{R}^n\) and let \(\{\Lambda_k\}_{k\geq 0}\) be a sequence of finite subsets of \(\Omega\).
Denote by \(\mathcal{P}_k\) 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.
In this talk we will discuss some necessary geometric conditions that assure that \(\Lambda_k\) is interpolating for \(\mathcal{P}_k\).
At each level \(k\), the interpolating condition is simply the linear independence of the corresponding reproducing kernels.
So, we are interested in asymptotic results in \(k\).
In particular, we will present density conditions that match precisely the necessary geometric conditions that sampling sets are known to satisfy.
These density conditions are expressed in terms of the equilibrium potential of the convex set.
If time permits, we will show that in the particular case of the unit ball, there is not an orthogonal basis of reproducing kernels in the space \(\mathcal{P}_k\),
when \(k\) is big enough.
This talk is based in a joint work with Jordi Marzo and Joaquim Ortega Cerdá.

16 de Abril de 2021, 11:30 : ONLINE - URL https://zoom.us/j/94404294982 Marina Iliopoulou, University of Kent
Sharp \(L^p\) estimates for oscillatory integral operators of arbitrary signaturePDF,

The restriction problem in harmonic analysis asks for \(L^p\) bounds on the Fourier transform of functions defined on curved surfaces.
In this talk, we will present improved restriction estimates for hyperbolic paraboloids, that depend
on the signature of the paraboloids. These estimates still hold, and are sharp,
in the variable coefficient regime. This is joint work with Jonathan Hickman.

9 de Abril de 2021, 11:30 : ONLINE - URL https://zoom.us/j/96974398786 Luz Roncal, Basque Center for Applied Mathematics
Directional square functionsPDF,

Charles Fefferman’s counterexample for the ball multiplier is intimately linked to
square function estimates for directional singular integrals along all possible directions.
Quantification of such a failure of the boundedness of the ball
multiplier is measured, for instance, through \(L^p\)-bounds for the \(N\)-gon multiplier
which provide information in terms of \(N\).
We present a general approach, developed in collaboration with N. Accomazzo, F. Di Plinio,
P. Hagelstein, and I. Parissis, based on a directional embedding theorem for Carleson sequences,
to study time-frequency model square functions associated to conical or directional Fourier multipliers.
The estimates obtained for these square functions are applied to obtain sharp or quantified
bounds for directional Rubio de Francia type square functions. In particular, a
precise logarithmic bound for the polygon multiplier is shown, improving previous results.

26 de Marzo de 2021, 11:30 : ONLINE - URL https://zoom.us/j/96082403168 Zoe Nieraeth, Basque Center for Applied Mathematics
Vector-valued extensions of operators through sparse domination and a multilinear UMD conditionPDF,

Vector-valued extensions of important operators in harmonic analysis have
been actively studied in the past decades. A centerpoint of the theory is the
result of Burkholder and Bourgain that the Hilbert transform extends to a bounded
operator on \(L^p(\mathbb{R}; X)\) if and only if the Banach space \(X\) has the so-called
UMD property. In the specific case where X is a Banach function space, it is a
deep result of Bourgain and Rubio de Francia that the UMD property is equivalent to the
Hardy-Littlewood maximal operator having a bounded extension
to both \(X\) and \(X'\). In turn, this leads to power vector-valued extrapolation methods.
In this talk I will place these ideas in the context of the more modern
technique of domination by sparse forms. These forms are intimately related
to Muckenhoupt weight classes and the multilsubinear Hardy-Littlewood maximal operator.
Moreover, I will discuss some of the current progress in extending
the UMD property to a multilinear setting.
This talk is based on joint work with Emiel Lorist.

12 de Marzo de 2021, 11:30 : ONLINE - URL https://zoom.us/j/97951668479 Martí Prats, Universitat de Barcelona
The two-phase problem for harmonic measure in VMO via jump formulas for the Riesz transformPDF,

Let \(\Omega^+ \subset \mathbb{R}^{n+1}\) be an NTA domain and let \(\Omega^- = \mathbb{R}^{n+1} \setminus \overline{\Omega^+}\) be an NTA domain
as well. Denote by \(\omega^+\) and \(\omega^-\) their respective harmonic measures. Assume
that \(\Omega^+\) is a \(\delta\)-Reifenberg flat domain for some \(\delta > 0\) small enough.
In a joint work with X. Tolsa we show that \(\log \frac{d\omega^-}{d\omega^+} \in \textrm{VMO}(\omega^+)\) if and only if \(\Omega^+\) is
vanishing Reifenberg flat, \(\Omega^+\) and \(\Omega^+\) have joint big pieces of chord-arc subdomains, and the inner unit normal of \(\Omega^+\)
has vanishing oscillation with respect to the approximate normal.
This result can be considered as a two-phase counterpart of a more well known related one-phase problem for harmonic measure solved by Kenig and Toro.

5 de Marzo de 2021, 11:30 : ONLINE - URL https://zoom.us/j/93006435602 Karlheinz Gröchenig, Universität Wien
New Function Spaces Associated to Representations of Nilpotent Lie GroupsPDF,

To every irreducible, unitary, square-integrable representation of a locally
compact group one can associate a family of function spaces. Many families
of function spaces, notably the Besov-Triebel-Lizorkin spaces, Bergman spaces
on the disk, and modulation spaces arise in this way. In this talk we focus on
the function spaces that are related to square-integrable, irreducible, unitary
representations of several low-dimensional nilpotent Lie groups. These are new
examples of coorbit theory and yield new families of function spaces on \(\mathbb{R}^d\).
The mathematical challenge is to decide when such spaces are different.

26 de Febrero de 2021, 11:30 : ONLINE - URL https://zoom.us/j/97431429382 Carmelo Puliatti, Euskal Herriko Unibersitatea
Blow-ups of caloric measure and applications to two-phase problemsPDF,

Let \(\Omega^+\) and \(\Omega^-\) be disjoint time-varying domains in \(\mathbb{R}^n_x \times \mathbb{R}^n_t, n \geq 2\),
and let \(\omega^\pm\) denote their associated caloric measures. Under appropriate mild non-degeneracy and regularity
hypotheses on \(\Omega^\pm\), mutual absolute continuity of \(\omega^+\) and \(\omega^-\) on
\(E \subset \partial\Omega^+ \cap \partial\Omega^- \cap \textrm{supp}\, \omega^+\) implies that the parabolic Hausdorff
dimension of \(\omega^+|_E\) is \(n + 1\) and the parabolic blow-ups of \(\omega^+\) at \(\omega^+ - \textrm{a.e.}\) point of
\(E\) are equal to a constant multiple of the parabolic \((n + 1)\)-Hausdorff measure
restricted to hyperplanes containing a line parallel to the time-axis.
This is a parabolic analogue of a result of Kenig, Preiss and Toro, and its
proof involves a set of techniques based on parabolic tangent measures. These methods, which I am going to discuss
in my talk, also have other geometric applications, amongst which a caloric version of a theorem of Tsirelson about
triple-points. This is a joint work with Mihalis Mourgoglou.

12 de Febrero de 2021, 11:30 : ONLINE - URL https://zoom.us/j/99697372954 David Pérez García, Universidad Complutense de Madrid
Sobolev-type inequalities in position based cryptographyPDF,
,
Youtube

The goal of this talk is to present a new setup where quantum information, high energy physics and functional inequalities meet:
position based cryptography. In the field of position based cryptography one aims to develop cryptographic tasks using the
geographical position of an agent as its only credential. Once the agent proved to the verifier that he/she is in fact at the
claimed position, they interact considering the identity of the agent as guaranteed. This proposal is appealing for practical
applications and it is also of fundamental interest since it presents a way to prevent man-in-the-middle attacks
without the need of a secure private channel. Furthermore, since the study of position based cryptography entered into the
quantum domain approximately a decade ago, beautiful and striking connections were established with
topics ranging from classical complexity theory to the AdS/CFT holographic
correspondence. I this talk, I will present a new connection with geometric
functional analysis that allows us to use a Sobolev-type inequality due to Pisier
for vector-valued functions on the boolean hypercube. Using it as a key tool,
we will provide new lower bounds on the entanglement consumption needed
to break position based cryptography.
(Joint work with Marius Junge, Aleksander M. Kubicki and Carlos Palazuelos.)

29 de Enero de 2021, 11:00 : ONLINE - URL https://conectaha.csic.es/b/jos-vqj-olj-lpt Maria Vallarino, Politecnico di Torino
Calderón-Zygmund theory and Hardy spaces on trees with nondoubling flow measuresPDF,

The classical Calderón-Zygmund theory was developed in the Euclidean
space and, more generally, on spaces of homogeneous type, which are measure
metric spaces with the doubling property.
In this talk we consider trees endowed with flow measures, which are non-doubling
measures of at least exponential growth. In this setting, we develop
a Calderón-Zygmund theory and we define BMO and Hardy spaces, proving
a number of desired results extending the corresponding theory as known in
the classical setting.
This is a joint work with Matteo Levi, Federico Santagati and Anita Tabacco.

22 de Enero de 2021, 11:30 : ONLINE - URL https://conectaha.csic.es/b/jos-vqj-olj-lpt Spyridon Kakaroumpas, Julius-Maximilians-Universität Würzburg
Dyadic product BMO in the Bloom settingPDF,

Ó. Blasco and S. Pott showed that the supremum of operator norms over
\(L\) of all bicommutators (with the same symbol) of one-parameter Haar multipliers
dominates the biparameter dyadic product BMO norm of the symbol itself.
In this talk we present recent work extending this result to the two-weight
Bloom setting, and to any exponent \(1 < p < \infty\). The proof relies on new two-weight
John–Nirenberg inequalities for Bloom dyadic product BMO, analogous
to those for usual one-parameter BMO due to I. Holmes, M. Lacey and B. Wick,
and those for little BMO due to I. Holmes, S. Petermichl and B. Wick.
This is joint work with Odí Soler i Gibert (Julius-Maximilians-Universität Würzburg).