Seminario de Análisis y Aplicaciones UAM-ICMAT 2019

Maximal directional singular integrals are defined by considering a one dimensional singular integral operator acting along a line in the Euclidean space, and then studying the maximal value as the line changes through a set of directions. Unlike the case of maximal directional averages, when considering singular integrals we are forced to admit only finite sets of directions in order to get \(L^p\) boundedness. In this talk we will talk about the case when the set of directions is finite and lacunary, which gives us optimal \(L^p\) bounds depending on the number of directions.

For \(f \in L^p(\mathbb{R}^2)\), let \(A_r f(x)\) be the mean of \(f\) over the circle of radius \(r\) centered at \(x\). For a set \(E \subset [1, 2]\) one can ask about \(L^p \to L^q\) estimates for the maximal function \(\sup_{r\in E} |A_r f(x)|\). How does the \((p, q)\)-parameter range for such estimates depend on the set \(E\)? We discuss results from two recent papers, one with Theresa Anderson, Kevin Hughes and Joris Roos, and one with Joris Roos.

Given a measure space \(X,\mu\) and a measurable function \(p(\cdot) : X \to [1,\infty]\), the variable Lebesgue space \(L^{p(\cdot)}(X)\) consists of all measurable functions \(f\) such that for some \(\lambda > 0\), $$ \rho(f/\lambda) = \int_{X_*} \left(\frac{|f(x)|}{\lambda}\right)^{p(x)} d\mu + \lambda^{-1} \|f\|_{L^{\infty}(X_\infty)} < \infty, $$ where \(X_* = \{x \in X : p(x) < \infty\}\) and \(X_\infty = \{x \in X : p(x) = \infty\}\). This space is a Banach function space with the norm $$ \|f\|_{L^{p(\cdot)}(X)} = \inf \{\lambda > 0 : \rho(f/\lambda) \leq 1\}. $$ These spaces, particularly when \(X = \mathbb{R}^n\) and \(\mu\) is Lebesgue measure, have been extensively studied for the past 20 years. It is well-known that if $$ p_+ = \textrm{ess} \sup_{x \in X} p(x) < \infty, $$ then these spaces have many properties analogous to the classical Lebesgue spaces. In particular, the dual space \(L^{p(\cdot)}(X)^*\) is isomorphic to \(L^{p'(\cdot)}(X)\), where the exponent is defined pointwise by $$ \frac{1}{p(x)} + \frac{1}{p'(x)} = 1. $$ This is no longer true if \(p_+ = \infty\), even if \(p(x) < \infty\) everywhere, and it has been a long standing problem to characterize the dual space in this case. I will discuss recent progress on this problem, both for general measure spaces and for the special case of the discrete sequence spaces \(\ell^{p(\cdot)}\), where \(X = \mathbb{N}\) and \(\mu\) is the discrete counting measure. We give a direct sum decomposition of the dual space as \(L^{p'(\cdot)}(X)\oplus L^{p(\cdot)}_{\textrm{germ}}(X)\), where \(L^{p(\cdot)}_{\textrm{germ}}(X)\), the germ space, intuitively consists of functions that "live" where the exponent function is unbounded. We give a number of properties of the germ space, particularly in the case of sequence spaces. This talk is joint work with José Conde-Alonso, Jesús Ocariz, and Alex Amenta.

A classical result in the theory of entire functions of exponential type, Shannon’s interpolation formula predicates that, given a function whose Fourier transform vanishes outside the interval \([−\frac{1}{2}, \frac{1}{2}]\), it is possible to recover it from its values at the integers. More specifically, it holds, in suitable sense of convergence, that $$ f(x) = \sum_{n \in \mathbb{Z}} f(n) \frac{\sin(\pi(x - n))}{\pi(x - n)} . $$ This formula is unfortunately unavailable for arbitrary Schwartz functions on the real line, but a recent result of Radchenko and Viazovska provides us with an explicit construction of an interpolation basis for even Schwartz functions. It states, in a nutshell, that we can recover explicitly the function given its values at the square roots of integers. We will discuss a bit these results and explore the question of determining which pairs of sets \((A, B)\) satisfy that, if a Schwartz function \(f\) vanishes on \(A\) and its Fourier transform vanishes on \(B\), then \(f \equiv 0\), with particular interest in the cases where \(A = \{\pm n^\alpha\}_{n\in\mathbb{N}}\) and \(B = \{\pm n^\beta\}_{n\in\mathbb{N}}\) are sets of powers of integers.

There are many problems arising from geometry than can be treat from harmonic analysis. For example, problems about the size of sets containing certain geometric configurations are related to the boundedness of maximal operators. In this talk, we will present a discretized maximal operator associated to averaging over (neighborhoods of) squares in the plane and, more generally, \(k-\)skeletons in \(\mathbb{R}^n\). These results are motivated by, and partially extend, recent results on sets that contain a scaled \(k-\)skeleton of the unit cube with center in every point of \(\mathbb{R}^n\).

The classical obstacle problem is a free boundary problem that appears naturally in the study of the Stefan problem, the Frostman equilibrium measure, the Helle-Shaw flow, the Dam problem, or the pricing of American options. Caffarelli obtained in the 1970’s a fundamental breakthrough: he gave a robust sufficient condition that implies the local smoothness of the free boundary. Complementarily, in the last years we worked towards obtaining a more complete understanding of singularities. This has lead us to proving, in dimensions three and four, a conjecture of Schaeffer which asserts that for generic boundary data there are no singularities (although one can construct examples of boundary data and obstacles for which the singular set is as large as the regular set!).

The Calderón's problem is to decide whether the conductivity of a body can be uniquely recovered from measurements of potential and current at the boundary. In this talk I will introduce the problem and the main ideas behind the method of Complex Geometrical Optics (CGO) solution. Finally, I will show how bilinear estimates come into the method, and what is the extension of Tao's bilinear theorem we need.

En esta charla presentaremos los principales resultados de un artículo en conjunto con el Profesor Gustavo Garrigós de la Universidad de Murcia. En una primera instancia se obtienen fórmulas explícitas del valor medio para las soluciones de las ecuaciones de difusión del calor asociadas a los operadores de Ornstein-Uhlenbeck y de Hermite. A partir de estos resultados, como es usual en teoría potencial, se obtienen numerosas propiedades relevantes para las soluciones de las ecuaciones mencionadas anteriormente, tales como principios del máximo, propagación infinita, teoremas de unicidad, desigualdades de tipo Harnack, entre otros.

In this talk we will discuss bases in Banach lattices, and how they can be used to measure (non)-embeddability of a Banach space into a lattice. We will give several characterizations of basic sequences that respect the lattice structure, and discuss some of the more unexpected corollaries. Most of the talk will focus on bibases, but I will also comment on existence of non-negative bases in Hilbert space.

In this talk we will discuss the extension of the regularity theory for solutions of elliptic PDE in divergence form \(Lu = −\text{div}(A \cdot \nabla u)\) with merely bounded coefficients in unbounded domains to operators of the form \(Lu = −\text{div}(A \cdot \nabla u + bu) − c \cdot \nabla u − du\), in an open set \(\Omega \subset \mathbb{R}^n , n \geq 3\), with possibly infinite Lebesgue measure. We assume that the \(n \times n\) matrix \(A\) is uniformly elliptic with real, merely bounded n and possibly non-symmetric coefficients, and either \(b, c \in L_{\text{loc}}^{n,\infty}(\Omega)\) and \(d \in L_{\text{loc}}^{\frac{n}{2},\infty}(\Omega)\), or \(|b|^2 , |c|^2 , |d| \in K_{\text{loc}}(\Omega)\), where \(K_{\text{loc}}(\Omega)\) stands for the local Stummel-Kato class. Let \(K_{\text{Dini},2}(\Omega)\) be a variant of \(K(\Omega)\) satisfying a Carleson-Dini-type condition. We develop a De Giorgi/Nash/Moser theory and also prove a Wiener-type criterion for boundary regularity. Assuming global conditions on the coefficients, we show that the variational Dirichlet problem is well-posed and, assuming \(−\text{div} c + d \leq 0\), we construct the Green's function associated with \(L\) satisfying quantitative estimates. Under the additional hypothesis \(|b + c|^2 ∈ K'(\Omega)\), we show that it satisfies global pointwise bounds and also construct the Green's function associated with the formal adjoint operator of \(L\). An important feature of our results is that all the estimates are scale invariant and independent of \(\Omega\), while we do not assume smallness of the norms of the coefficients or coercivity of the associated bilinear form.

The theory of singular integral operators originally initiated by by A.P. Calderón and A. Zygmund in the 1950's continues to expand, with geometric questions taking center stage at the moment. From the seminal work of G. David and S. Semmes in the 1990's we now know that uniform rectifiability is the most natural geometric conditions ensuring boundedness on Lp spaces. For a variety of purposes it is of interest to understand not just when the norm of such singular integral operators is finite but rather when the said norm is actually small. This lecture elaborates on the specific analytic and geometric features which determine the latter aspect. One of the main results identifies the largest class of singular integral operators (dubbed "generalized double layers") which have small norms as mappings on Lebesgue, Sobolev, Hardy, BMO, and Hölder spaces considered on sufficiently flat "surfaces". This is based, in part, on joint work with Juan José Mar´n, José Maréa Martell, Dorina Mitrea, and Irina Mitrea.

In this talk I will discuss boundary value problems for second-order elliptic constant (complex) coefficient systems in open subsets of the Euclidean ambient satisfying certain geometric properties, best expressed in the language of Geometric Measure Theory. One of the most prominent features is the smallness of the BMO semi-norm of the outward unit normal, which should be thought as some scale-invariant demand of flatness (this does not force the boundary to be regular as, in fact, this may contain spiral points, for example). In this geometric environment we are going to formulate both Dirichlet and Neumann Problems, and prove existence, uniqueness, estimates, and integral representation formulas for their respective solutions. This relies on work in collaboration with Juan José Mar´n, José Maréa Martell, Irina Mitrea, and Marius Mitrea.

En este seminario trataremos varios problemas que se ubican en la intersección del análisis armónico, las ecuaciones en derivadas parciales y la teoría geométrica de la medida. Estudiaremos cómo la geometría de un dominio en \(\mathbb{R}^n\) influye en las propiedades de acotación de ciertos operadores definidos en su frontera y las aplicaciones de este hecho a los problemas de valor en la frontera para sistemas elípticos de segundo orden, homogéneos, con coeficientes complejos constantes. Más específicamente, el comportamiento del vector normal unitario exterior es la característica geométrica clave que nos permitirá acotar operadores integrales (como las transformadas de Riesz o los potenciales de capa) en ciertos espacios de funciones. A su vez, éste es un paso fundamental para estudiar problemas de valor en la frontera en dominios SKT no acotados. En la dirección contraria, de estos operadores extraemos información acerca de la geometría del dominio. También trataremos el caso del semiespacio superior, estableciendo resultados para el problema de Dirichlet con dato en la frontera en espacios generalizados de Hölder y Morrey-Campanato. Además, mostraremos un teorema de tipo Fatou y una fórmula de representación integral de Poisson para soluciones en el semiespacio superior. Seminario previo a la defensa de la tesis doctoral.

Bilinear restriction (or extension) estimates for free waves give an efficient way to exploit both transversality and curvature, and have a number of applications in harmonic analysis and PDE. We give an overview of these estimates, as well as some recent extensions of the bilinear theory from free waves to the adapted function spaces \(U^p\) . These extensions give a way to connect the bi bilinear restriction theory with the global well-posedness problem for dispersive PDE. In particular, they can be used to give another solution to the division problem for the wave maps equation.

El lenguaje de semigrupos es una herramienta general y unificadora que resulta útil para formular y analizar propiedades de operadores fraccionarios y para obtener resultados de regularidad en algunos espacios funcionales relacionados. En esta charla mostraremos algunos problemas relacionados con la acotación de operadores elípticos y parabólicos fraccionarios en espacios de Lebesgue y Hölder, donde esta técnica clarifica definiciones, facilita los cálculos y permite probar algunos resultados que desde otros puntos de vista involucraría estimaciones mucho más complejas o incluso podrían no resolverse. Presentación previa a la defensa de la tesis doctoral.

Dado un espacio de Banach \(\mathbb{X}\) sobre \(\mathbb{C}\) y una base \(\mathcal{B} = (e_j)_{j=1}^\infty\), uno de los problemas fundamentales en Teoría de Aproximación es como representar un elemento \(f = \sum_{n=1}^\infty a_n \in \mathbb{X}\) por sumas finitas de \(m\) términos: $$ \mathcal{T}_m(f) = \sum_{n \in A} b_n e_n $$ para un conjunto \(A\) adecuado y unos escalares \(b_n\) que en principio no tienen porque coincidir con los coeficientes de \(f\). La colección \((\mathcal{T}_m(f))_m\) se llama algoritmo de aproximación de \(m\)-términos. En esta charla introduciremos dos algoritmos aproximación, que son el Thresholding Greedy Algorithm y el Thresholding Chebyshev Greedy Algorithm. Para ambos algoritmos, estudiaremos su eficiencia a través del llamado parámetro de Lebesgue y, además, veremos que condiciones ha de satisfacer la base \(\mathcal{B}\) para garantizar ciertos tipos de convergencia. Presentación previa a la defensa de la tesis doctoral.

En la charla se presentará la conjetura de Kannan-Lovász-Simonovits sobre el salto espectral, en su vertiente funcional e isoperimétrica. Se relacionará con otras conjeturas que forman la parte más importante del análisis geométrico asintótico. Tambien se darán las mejores estimaciones conocidas actualmente que son debidas a Eldan, Lee y Vempala.

For a domain \(\Omega\) in space-time \(\mathbb{R}^{n+1}\), quantitative, scale-invariant absolute continuity (more precisely, the weak-\(A_\infty\) property) of caloric measure with respect to a natural version of "surface" measure on the "quasi-lateral boundary" (a subset of the parabolic boundary which is simply the usual lateral boundary for cylinders and Lip(1,1/2) domains), is equivalent to the solvability of a suitable version of the initial-Dirichlet problem with lateral (more precisely "quasi-lateral") data in some \(L^p\) space, \(p < \infty\). We establish two criteria for the weak-\(A_\infty\) property to hold. The first, based on changing the pole of the parabolic measure without a change of pole formula, extends elliptic results of Bennewitz and Lewis to the parabolic setting. The second, based on extrapolation from the endpoint, extends an elliptic result of Dindos, Kenig and Pipher. This is joint work with A. Genschaw.

The classical Minkowski problem consists in finding a convex polyhedron from data consisting of normals to their faces and their surface areas. In the smooth case, the corresponding problem for convex bodies is to find the convex body given the Gauss curvature of its boundary, as a function of the unit normal. The proof consists of three parts: existence, uniqueness and regularity. In this talk, we study a Minkowski problem for certain measure, called p-capacitary surface area measure, associated to a compact convex set with nonempty interior and its p−harmonic capacitary function. We will discuss existence, uniqueness, and regularity of this problem under this setting and see connections with the Brunn-Minkowski inequality and Monge-Ampere equation.

Which subspaces \(X_p \subset L^p(\mathbb{R}^d)\) give good initial data for the wave equation \(\partial_t^2 u = \Delta u\), in the sense that \(u(0, .) \in X_p , \partial_t u(0, .) = 0\) implies \(u(t, .) \in L^p\) ? A classical answer is \(X_p = W^{(d−1)|1/p−1/2|,p}\). Its weakness is that, while \(u(t, .) \in L^p\), one does not have \(u(t, .) \in X_p\) , i.e. one looses derivatives. In this talk, we consider a new Hardy space that contains \(W^{(d−1)|1/p−1/2|,p}\), but is invariant under the action of the wave group. More generally, we show that this space is invariant under the action of a large class of Fourier Integral Operators. This allows us to recover and extend (to symbols that are less regular, and not compactly supported in space) a celebrated result of Seeger-Sogge-Stein on the \(L^p\) boundedness of FIO. More generally we set up a Hardy space theory that is meant to do for hyperbolic equations what the usual Hardy space theory does for elliptic and parabolic equations. This is possible because FIOs lifted to function spaces over phase space through appropriate wave packet transforms turn out to have a diffusive behaviour (not in space but in phase space, and with respect to an appropriate metric). In this talk, we discuss this perspective, the deep ideas from microlocal analysis and harmonic analysis that it builds upon, and its potential. This is joint work with Andrew Hassell and Jan Rozendaal (ANU).

The usual one third trick allows to reduce problems involving general cubes to a countable family. Moreover, this covering lemma uses only dyadic cubes, which allows to use nice martingale properties in harmonic analysis problems. In this talk, we consider alternatives to this technique in spaces equipped with nonhomogeneous measures. This entails additional difficulties which forces us to consider martingale filtrations that are not regular. The dyadic covering that we find can be used to clarify the relationship between martingale BMO spaces and the most natural BMO space in this setting.

In this talk I will present some recent results on the sharp form of a mixed norm Fourier extension inequality first proposed by Luis Vega in his Ph.D. thesis (Madrid - 1988). This problem has interesting connections to questions of independent interest in the theory of special functions. The talk will be accessible to a broad audience, with a minimal background in Analysis. This is based on a joint work with D. Oliveira e Silva (Birmingham) and M. Sousa (Munich).

The quantum vortices of a superfluid are described as nodal lines of a solution to the time-dependent Gross-Pitaevskii equation. Experiments in Lab and extensive numerical computations show that quantum vortices cross, each of them breaking into two parts and exchanging part of itself for part of the other. This phenomenon, known as quantum vortex reconnection, occurs even though the superfluid does not lose its smoothness. This usually leads to a change of topology of the quantum vortices. In this talk I will show that, given any initial and final congurations of quantum vortices (which do not need to be topologically equivalent) and any conceivable way of reconnecting them (that is, of transforming one into the other), there is a Schwartz initial datum whose associated solution is smooth and realizes this specific vortex reconnection scenario. Key for the proof of this result is a new global approximation property of the linear Schrodinger equation. It ensures that a function that satisfies the Schrödinger equation in a spacetime set satisfying certain mild topological properties, can be approximated, in a suitable norm, by a global solution defined by a Schwartz initial datum. This is based on joint work with Alberto Enciso.

Given a metric measure space \((X, d, \mu)\), its doubling constant is given by $$ C_\mu = \sup_{x \in X, r > 0} \frac{\mu(B(x, 2r))}{\mu(B(x, r))} , $$ where \(B(x, r)\) denotes the open ball of radius \(r\) centered at \(x\). Clearly, \(C_\mu \geq 1\), and in the case X reduces to a singleton \(C_\mu = 1\). At first, one might think that for a metric space with more than one point, the constant \(C_\mu\) could be very close to one. However, we will actually see that in general \(C_\mu \geq 2\). The talk is based on a joint work with Javier Soria (Barcelona).

In the context of von Neumann algebras bilateral almost uniform convergence plays the role of almost everywhere convergence in measure spaces. It is known, after the work for Junge and Xu, that there is a maximal ergodic theorem in the \(L_p\)-spaces of von Neumann algebras that holds true and implies bilateral almost everywhere convergence. One substantial difference between that theorem and the classical one being that the optimal constants grow like \((p - 1) - 2\) - as opposed to \((p - 1) - 1\) -. This has profound implications, for instance the optimal extrapolation space for maximal ergodic inequalities is \(L \log^2 L\) - as opposed to \(L \log L\) -. Here, we are going to present new results that give almost uniform con- vergence for maximals in several indices for operators in \(L \log^2 L\). Our results generalize a technique of Jessen, Marcinkiewicz and Zygmund. After that, we will discuss aplication to the free group algebra. This is joint work with Jose Conde-Alonso and Javier Parcet.

In the fifties Bishop proposed the family \(T_\alpha f(x) = xf(\{x+\alpha\})\) acting on \(L^2 [0,1)\) as a possible source of operators without invariant subspaces. Years later Davie showed that \(T_\alpha\) actually has invariant subspaces whenever \(\alpha\) is not a Liouville number (so for almost all \(\alpha\)). In this talk I will speak about recent work with F. Chamizo, E. Gallardo and M. Monsalve in which we extend Davie's method to some Liouville numbers \(\alpha\) and show that these techniques cannot work for every \(\alpha\).

The uniformity norms, introduced by Gowers in his famous work on Szemerédi's theorem, have become a central tool in arithmetic combinatorics, especially to count linear configurations in subsets of compact abelian groups, and more generally to analyze averages of functions over such configurations. An important result related to the Gowers norms is the so-called inverse theorem, proved for functions on finite intervals of integers by Green, Tao and Ziegler. This theorem tells us essentially that such a function has large Gowers norm of order \(k+1\) only if the function correlates with a specific type of function, called a nilsequence, that is generated by a rotation on a nilmanifold of step \(k\). I shall discuss recent joint work with Balázs Szegedy in which, building up on the theory of nilspaces initiated by Camarena and Szegedy, we obtain in particular a generalization of the Green-Tao-Ziegler inverse theorem, valid for functions on compact abelian groups and also on nilmanifolds.

Consider a local solution \(u ∈ W_{\textrm{loc}}^{1,2}\) to an inhomogeneous elliptic partial differential equation in divergence form \(\textrm{div} (A\nabla u) = f\) where \(A\) is a uniformly elliptic matrix with measurable coefficient and \(f\) is a source term in a suitable \(L^p\) space. Classical results in regularity theory tell that when the source term \(f\) is slightly better than what is required for the existence of a solution as above, the regularity of the solution itself is also better than what was assumed a priori. This is traditionally seen as a consequence of Gehring's lemma about open-ended property of reverse Hölder classes. In this talk, I discuss a functional analytic point of view on the topic with special focus on extensions to parabolic and fractional PDEs.

Let \(\mathcal{L}\) be a smooth second-order real differential operator in divergence form on a manifold of dimension \(n\). Under a bracket-generating condition, we show that the ranges of validity of spectral multiplier estimates of Mihlin-Hörmander type and wave propagator estimates of Miyachi–Peral type for \(\mathcal{L}\) cannot be wider than the corresponding ranges for the Laplace operator on \(\mathbb{R}^n\). The result applies to all sub-Laplacians on Carnot groups and more general sub-Riemannian manifolds, without restrictions on the step. The proof hinges on a Fourier integral representation for the wave propagator associated with \(\mathcal{L}\) and nondegeneracy properties of the sub-Riemannian geodesic flow. This is a joint work with Alessio Martini and Sebastiano Nicolussi Golo.

In harmonic analysis terms, Lafforgue/de la Salle rigidity theorem for \(SL_n(\mathbb{R})\) implies that Fourier summability fails in \(L_p\) when p is large enough in terms of the rank n − 1. It refines older celebrated results by Harish-Chandra, Cowling or Haagerup, and spotlights the dramatic difference between abelian and semisimple harmonic analysis. We shall present the first sufficient condition for \(L_p\)-boundedness of Fourier multipliers in this context, which is reminiscent of the Hörmander-Mikhlin criterion, but substantially and necessarily different to accommodate rigidity. Next, we shall introduce a major strengthening of the rigidity theorem and link it with Bochner-Riesz summability problems. Emphasis will be put on the harmonic analysis aspects of both of these results. Joint work with Éric Ricard and Mikael de la Salle.

En esta charla estudiaremos el problema de perturbación de operadores elípticos en dominios irregulares. Dados dos operadores \(L_0 = − \textrm{div}(A_0 \nabla\cdot)\) y \(L = − \textrm{div}(A\nabla\cdot)\), buscamos condiciones en la discrepancia entre \(A_0\) y \(A\) que nos permitan transferir buenas propiedades de un operador a otro, como el hecho de que la medida elı́ptica pertenezca a la clase \(A_\infty\). Extendemos el resultado de Fefferman-Kenig-Pipher (1991) a dominios del tipo 1-sided CAD, estableciendo que si la discrepancia entre las matrices satisface una condición tipo medida de Carleson entonces la pertenencia a \(A_\infty\) de la medida elíptica de uno de los operadores implica la misma propiedad para el otro. Para probar este resultado presentamos dos métodos que son distintos del usado por Fefferman-Kenig-Pipher. Uno de ellos usa la técnica denominada "extrapolación para medidas de Carleson". El segundo, que explicaremos en detalle y que es válido para operadores no simétricos, pasa por estudiar la propiedad de que todas las soluciones acotadas de un operador satisfagan estimaciones de tipo medida de Carleson.

We study the problem of dominating the dyadic strong maximal function by (1,1)-type sparse forms based on rectangles with sides parallel to the axes, and show that such domination is impossible. Our proof relies on an explicit construction of a pair of maximally separated point sets with respect to an appropriately defined notion of distance.

En esta charla estudiaremos un modelo de formación de opinión, donde los agentes están caracterizados, además de por su opinión respecto a cierto tema, por su capacidad de convencer al otro y su facilidad o reticencia a cambiar de opinión. Demostraremos que en presencia de agentes firmes o testarudos (que no modifican su opinión), hay convergencia al consenso, situado en un valor que es una media de las opiniones iniciales ponderada por la capacidad de convicción de los individuos, pero unicamente entre la población testaruda. En otras palabras, los agentes firmes determinan la opinión final de la población. Más aún, hallamos la tasa de convergencia al consenso y observamos que, a mayor número de agentes testarudos, menor es el tiempo de convergencia.