Seminario de Análisis y Aplicaciones UAM-ICMAT 2020

Since their discovery three decades ago, noncommutative Khinchin inequalities have been paramount to the development of quantum analysis. Their possible generalisations have been studied in various directions and we will be interested in investigating their validity/formulation in general symmetric spaces. Those are function spaces for which the norm only depends on the distribution of functions. They generalise \(L^p\)-spaces and can be easily considered in the noncommutative framework. We will present a complete characterisation of symmetric spaces in which the noncommutative Khinchin inequalities hold, which will lead us to revisit a branch of interpolation theory developed in the 70's and the 80's.

Composition by a bi-Lipschitz measure-preserving map on the one-parameter BMO space has been applied to study the Euler equation with a BMO-type vorticity. We would like to discuss the same problem in the setting of biparameter BMO space in \(\mathbb{R}^2\). We will focus on composing by a rotation on the biparameter BMO space. This BMO space is not preserved by a rotation since it relies on the structure of axis-parallel rectangles. We will quantify this fact by interpolation inequalities. One straightforward application of the interpolation inequalities is a boundedness property of directional Hilbert transforms.

Variational estimates are refinements of maximal function estimates, in which the \(\ell^\infty\) norm is replaced by a larger \(V^r\) norm, \(1 \leq r \leq \infty\). In 2008, Jones, Seeger and Wright proved that the r-variation operator associated to the spherical averages \(\{f \ast \sigma_t \}_{t>0}\) is bounded on \(L^p(\mathbb{R}^d)\) if \(d/(d − 1) < p \leq 2d\) and \(r > 2\) or \(p > 2d\) and \(r > p/d\), and this is sharp except for the endpoint case \(r = p/d\), which remains open. In this talk I will present \(L^p(\mathbb{R}^d) \to L^q(\mathbb{R}^d) \) bounds for the local r-variation operator, that is, the associated with \(\{f \ast \sigma_t \}_{1\leq t \leq 2}\). The bounds are sharp up to endpoints (except in dimension 3), and some positive results also hold in some endpoints cases. In particular, it can be established the interesting endpoint bound \(L^{q/d}(\mathbb{R}^d) \to L^q(\mathbb{R}^d) \) for \(r = q/d\), \(q > (d^2 + 1)/(d − 1)\) if \(d \geq 3\). Our results imply associated sparse domination and weighted inequalities. This is joint work with Richard Oberlin, Luz Roncal, Andreas Seeger and Betsy Stovall.

Presentaremos perturbaciones Carleson para operadores elípticos en dominios en que existe una teoría robusta de las EDP elípticas. Tales dominios incluyen, en particular, (a) los dominios 1-sided NTA que satisfacen la capacity density condition (previamente estudiado en [Akman-Hofmann-Martell-Toro]), (b) dominios con fronteras de dimensión baja y Ahlfors-David regular, y (c) ciertos dominios con fronteras con piezas de distintas dimensiones. Nuestras perturbaciones de Carleson son generalizadas en el sentido de que, además de las clásicas perturbaciones aditivas, permitimos tomar perturbaciones multiplicativas escalares, las cuales admiten diferencias no triviales en la frontera entre la matriz perturbada y la matriz original. Finalmente, investigamos corolarios de nuestras técnicas, con implicaciones a problemas de la frontera libre y a una caracterización de \(A_\infty\) entre medidas elípticas. Es trabajo conjunto con Joseph Feneuil.

In this expository lecture we will discuss some recent results concerning fractional Poincaré and Poincaré-Sobolev inequalities with weights, the degeneracy. These results improve some well known estimates due to Fabes-Kenig-Serapioni from the 80’s in connection with the local regularity of solutions of degenerate elliptic equations and also some more recent results by Bourgain-Brezis-Minorescu. Our approach is different from the usual ones and it is based on methods that come from Harmonic Analysis and one of them is the extrapolation theorem of José Luis Rubio de Francia.

The decoupling theorem for the moment curve, due to Bourgain, Demeter, and Guth, implies the Vinogradov mean value theorem with the sharp exponent. I will present a bilinear proof of this result that avoids Brascamp-Lieb inequalities. Joint work with S. Guo, Z. Li, and P.L. Yung.

The category of nilspaces was introduced around 2010 by Antolin Camarena and Szegedy. It was originally conceived as a tool in additive combinatorics, specifically for generalizations of Szemerédi's theorem. The strength of the concept lies in its remarkable structure theory: while by definition nilspaces are compact spaces \(X\) together with closed collections of "cubes" \(C^n(X) \subset X^{2^n} , n = 1, 2, \dots\) verifying some topological axioms, under suitable assumptions a nilspace may be represented as an inverse limit of (smooth) nilmanifolds. In recent years some exciting application of nilspaces have been found in topological dynamics (maximal nilfactors) and ergodic theory (convergence of multiple ergodic averages). In the talk we will discuss some of the key results in this direction. Based on joint works with Eli Glasner, Bingbing Liang, Zhengxing Lian, Freddie Manners, Péter Varjú and XiangDong Ye.

In this talk we will discuss the regularity properties of the Hardy-Littlewood maximal operator on finite graphs, this was originally studied by Soria and Tradacete in 2014, they obtained some optimal bounds for \(0 < p \leq 1\). Recentely, in a joint work with Gonzalez-Riquelme we obtained some optimal results for \(p = 2\). Moreover we obtained some optimal results for the \(p\)-variation of the maximal operator for \(p \geq 1/2\) in the case of the star graph and for \(p > \ln 4/\ln 6\) in the case of the complete graph. We will also discuss what are the extremizers in the situations described, and some interesting open questions.

We present a simple dyadic construction that yields a new counterexample to Zygmund’s conjecture. Our result recovers F. Soria’s classical results in dimension three and four, through a different construction, and gives new ones in all other dimensions.

Desde el año 1999, la teoría de aproximación avariciosa (o greedy) se ha desarrollado en el contexto de espacios de Banach, aunque algunos autores han dado por hecho en el mundo quasi-Banach ciertos resultados que se tenían en el caso Banach. La intención de esta charla es analizar si es trivial o no la extensión del mundo de bases greedy en espacios de Banach al caso quasi-Banach. Además, daremos algunos resultados que estamos desarrollando sobre espacios de aproximación con pesos generales en combinación con las llamadas bases semi-greedy. Los resultados que mostraremos forman parte de dos trabajos conjuntos con F. Albiac, J. L. Ansorena, E. Hernández y P. Wojtaszczyk.

A general spectral theory of a linear operator does not exist, it only exists for particular subclasses of operators. One of the most celebrated theories of this type is the Nagy-Foiaş spectral theory of Hilbert space contractions, obtained in the 1960ies. It is based on the construction of a functional model, which heavily relies on certain chapters of Complex Analysis, such as the theory of Hardy spaces.
Being a contraction on a Hilbert space is characterized by a very simple operator inequality. In a landmark work of 1982, Agler showed how to pass from some other operator inequalities to a functional model of an operator by applying more general reproducing kernel Hilbert spaces instead of Hardy spaces. This work motivated an extensive research, which has become a rapidly growing branch of Operator Theory.
I will explain how to construct an explicit functional model for an operator satisfying a rather general operator inequality, and discuss the uniqueness of this model. Some spectral consequences in the spirit of the Nagy-Foiaş theory will be derived. I will also explain a new connection with the ergodic theory of linear operators.

The main goal of this talk is to present several applications in the setting of ergodic theory related with the Return Time Theorem of Bourgain via of an extension of Rubio de Francia extrapolation theorem in the setting of weighted restricted weak type inequalities.

Consider a countable discrete group acting on a separable Hilbert space via unitary represenation. If such representation is dual integrable, then the structure of the invariant subspaces and various properties of orbits can be analyzed using the corresponding bracket map. As a special case, we obtain systems of integer translates of a square integrable function. The properties of these systems have been extensively studied and among the known results is the fact that such system is \(\ell^2\) -linearly independent precisely when the periodization function is positive a.e. On the other hand, this condition is equivalent to maximality of the principal shift-invariant subspace which the system generates. Characterization of other levels of linear independence is, in most cases, still an open problem in general, however, we know that the equivalence with maximality no longer holds if we replace \(\ell^2\) with \(\ell^p\)-linear independence, for \(p\neq 2\). In this talk, after briefly recalling the main results and questions concerning this topic, we shall focus on several questions related with maximal cyclic subspaces for the previously described group setting, which are the part of a recent research in collaboration with H. Šikić.

Recall that an operator between Banach spaces is strictly singular provided it is not invertible when restricted to any (closed) infinite dimensional subspace. The class of strictly singular operators forms a closed two-sided operator ideal, containing compact operators, and was introduced by T. Kato in connection with the perturbation theory of Fredholm operators. In this talk we will focus on the interpolation properties of this class of operators acting between different \(L^p\) spaces, and the structure of strictly singular non-compact operators. In particular, by means of Riesz potential operators acting between measure spaces of different Hausdorff dimension, we will see that the set of pairs \((1/p,1/q)\) such that an operator is strictly singular but not compact from \(L^p\) to \(L^q\) can contain a line segment of any positive slope. The talk is based on joint work with F. L. Hernández and E. M. Semenov.

In this talk we discuss the structure of bounded shift-preserving operatorsacting on shift-invariant spaces of \(L^2(\mathbb{R}^d)\). For this, we work with an isometry called fiberization map and study the properties that the correspondent range function and range operator induce. We introduce a new notion of diagonalization for these operators which we call s-diagonalization and give a generalized Spectral Theorem for normal shift-preserving operators. Finally, we apply these results to a dynamical sampling problem. This work is in collaboration with A. Aguilera, C. Cabrelli and V. Paternostro.

Covariant frames and coherent states associated with unitary representations of locally compact groups are of interest in signal theory and mathematical physics. The talk introduces these concepts in harmonic analysis and their relationship with convolution Hilbert algebras and weights on the group von Neumann algebras.

The bilinear Hilbert transform is a bilinear singular integral operator (or Fourier multiplier) which is invariant not only under translations and dilations, but also under modulations. This additional symmetry turns out to make proving \(L^p\) - bounds especially difficult. I will give an overview of how time-frequency analysis is used in proving these \(L^p\)-bounds, with focus on the recently-understood setting of functions valued in UMD Banach spaces.

Maximal functions play a central role in the study of differentiation, singular integrals and almost everywhere convergence. With Sergey Tikhonov (ICREA) we recently proved some pointwise estimates for maximal functions in terms of smoothness and rearrangements. I plan to discuss the recent progress on these topics and some applications. In particular, I will discuss the Fefferman-Stein inequality for the sharp maximal function for r.i. spaces which are close to \(L^\infty\) .