The aim of this session is to bring together young researchers who develop their work in Mathematical Analysis, mainly Harmonic Analysis and Operator Theory. We want to place special emphasis on the relationship between the aforementioned areas and stimulate interaction and debate of problems from different points of view.
In particular, recent problems of Harmonic Analysis and Operator Theory will be shown, related to singular integrals, weighted inequalities, geometric measure theory, spectral properties, invariant subspaces, among others. In addition, the relationship of these topics with other areas, such as Complex Analysis and Partial Differential Equations, will be shown.
El objetivo de esta sesión es reunir a jóvenes investigadores que desarrollan su trabajo en Análisis Matemático, siguiendo como líneas principales el Análisis Armónico y la Teoría de Operadores. Se quiere hacer especial hincapié en la relación entre las áreas mencionadas y estimular la interacción y el debate de los problemas desde diferentes puntos de vista.
En particular, se mostrarán problemas recientes del Análisis Armónico y Teoría de Operadores, relacionados con integrales singulares, desigualdades con peso, teoría de la medida geométrica, propiedades espectrales, subespacios invariantes, entre otros. Además se mostrará la relación de estos temas con otras áreas, como el Análisis Complejo y Ecuaciones en Derivadas Parciales.
30-XX; 42-XX; 46-XX
(secondary)
1.A (0.7)
1.B (0.7)
1.C (0.7)
It is well known that the celebrated Gagliardo estimate can be viewed as an extension of the classical isoperimetric inequality although the best constant was not obtained by Gagliardo. In this talk, we will explore some generalizations of this result involving weights, and we will discuss how it can be extended beyond smooth domains.
Ongoing joint wotk with Carlos Pérez and Ezequiel Rela.
The celebrated work of Carleson characterizes Carleson measures on the unit disc in terms of the so called one-box condition. In the polydisc, a geometric characterization of Carleson measures is much more elusive. In this talk, we will consider random atomic measures in the polydisc, and we will discuss their probability of being Carleson measures. We will discuss a well known re-formulation of the problem in terms of Gram matrices, and then use tools from the theory of random matrices.
Joint work with Nikolaos Chalmoukis and Giuseppe Lamberti.
Extrapolation techniques allow us to determine bounds for an operator in different spaces based on known bounds on other spaces. The aim of this talk is to use various multilinear extrapolation techniques that extend existing linear results. The techniques are based on the extrapolation theorems by Yano and Rubio de Francia, which use weights from the Muckenhoupt classes.
Joint work with M.J. Carro and T. Luque.
The Rubio de Francia square function is the square function formed by frequency projections over a collection of disjoint intervals of the real line. In this talk, we will show a new pointwise sparse bound for the Rubio de Francia square functions. These sparse bounds lead to quantified weighted inequalities. In the first part of the talk, we will give the background of the problem. In the second part, we will explain the new results mentioned above.
Joint work with Francesco Di Plinio, Ioannis Parissis and Luz Roncal.
We study the boundedness and spectral properties of a generalized Cesàro-Hardy operator in L^p[0,1]. Using semigroup theory, we express these operators and their duals via subordination with composition semigroups. Through functional calculus, we transfer the spectral properties of the semigroups' generators to the generalized Cesàro-Hardy operators. Additionally, we prove the universality of certain translations of the semigroup, providing new insights into the Invariant Subspace Problem.
Joint work with Luciano Abadías.
A semigroup of bounded operators $(T(t))_{t\geq0}$ on a Hilbert space $H$ is said to be similar to a contraction semigroup if there exists an equivalent Hilbertian norm $\|\cdot\|_e$ on $H$ for which $\|T(t)\|_e\leq1$, $t\geq0$. In this talk we present a new characterization of semigroups similar to contraction semigroups in terms of quasi-contraction semigroups. We also provide new counterexamples for this theory, namely a (quasi-)nilpotent semigroup which is not similar to a contraction one.
Joint work with Yuri Tomilov.
We exhibit the existence of continuous (and even invertible) linear operators acting on Banach (and even Hilbert) spaces whose restriction to their respective closed linear subspaces of chain recurrent vectors are not chain recurrent operators. This example completely solves in the negative a problem recently posed by N. C. Bernardes Jr. and A. Peris on chain recurrence in Linear Dynamics.
Joint work with Dimitris Papathanasiou.
We deal with the existence and localization of solutions for operator systems by means of the fixed point index. We emphasize that our results are coexistence fixed point theorems for operator systems, that means that every component of the fixed points obtained is non-trivial. Finally, these coexistence fixed point theorems are applied to obtain results concerning the existence of positive solutions for systems of Hammerstein integral equations.
The Volterra operator $T_g$, although well understood in the unit disc setting, its study on Banach spaces of Dirichlet series is quite recent. In this talk, we focus on the boundedness of the operator $T_g$ acting on some Bergman-type spaces of Dirichlet series: the $\mathcal{A}_p^{\mu}$-spaces. To do so, we shall introduce a new family of Bloch spaces of Dirichlet series, the Blochμ-spaces. Using these spaces we shall provide a sufficient and necessary condition for the operator $T_g$ to act boundedly on the spaces $\mathcal{A}_p^{\mu}$.
Joint work with Professor Pascal Lefèvre and Professor Hervé Queffélec.
We study the constant $\mathscr{C}_p$ defined as the smallest constant $C$ such that $|f(0)|^p \leq C\|f\|_p^p$ holds for every function $f$ in the Paley--Wiener space $PW^p$. Brevig, Chirre, Ortega-Cerdà, and Seip have recently shown that $\mathscr{C}_p
2$. We improve this bound for $2
The Marcinkiewicz interpolation theorem states that an operator of weak type (p0,p0) and (p1,p1) is of strong type (p,p) for all the p's in between p0 and p1. It's non-commutative version is true, though the proof much more involved. In a joint work with Adrián González-Pérez and Javier Parcet, we adapt a recently published proof of this result in order to deal with operators of weak Orlicz type. We partially solve an open problem related to convergence of martingales in von Neumann algebras.
Joint work with Adrián González-Pérez and Javier Parcet.
We consider harmonic Bergman spaces on radial trees and compute some onb and kernels. Assuming the tree to have bounded geometry we get sparse domination for the Bergman projector $P$, that implies boundedness results and provides weighted estimates for $P$. Allowing the tree to have unbounded geometry we deal with a nondoubling setting and prove a sparse domination-like result for $P$. Through a suitable dyadic system and a CZ decomposition, we get boundedness properties for integral operators
Joint work with J. Conde Alonso, F. De Mari, M. Monti and M. Vallarino.