Seminario de Análisis y Aplicaciones UAM-ICMAT 2021

The aim of this talk is twofold. On one hand, we investigate some nonhomogeneous Gagliardo-Nirenberg-type inequalities. On the other hand, using the previously discussed inequalities, we analyse the standing waves for a fourth-order Schrödinger equation with mixed dispersion that minimize the associated energy when the \(L^2\)-norm (the mass) is kept fixed. Special attention will be paid to the method used to prove the non-homogeneous Gagliardo-Nirenberg-type inequalities. The talk is based on a joint work with Louis Jean-jean (Besançon, France), Rainer Mandel (Karlsruhe, Germany) and Mihai Mariş (Toulouse, France).

We study maximal operators \(\mathcal{M}^\mathcal{B}\) associated with various differentiation bases \(\mathcal{B}\) on the infinite-dimensional torus \(\mathbb{T}^\omega\). For the so-called Rubio de Francia basis \(\mathcal{R}\) the operator \(\mathcal{M}^\mathcal{R}\) is unbounded on \(L^p(\mathbb{T}^\omega)\) for every \(p \in [1, \infty)\). On the other hand, the operator determined by the restricted (dyadic) basis \(\mathcal{R}_0\) is of weak type \((1, 1)\), hence bounded on \(L^p(\mathbb{T}^\omega)\) for every \(p \in (1, \infty)\). We try to understand the interplay between the structure of a given basis \(\mathcal{B}\) and the behavior of \(\mathcal{M}^\mathcal{B}\). To this end, we look for intermediate bases \(\mathcal{R}_0 \subset \mathcal{\widetilde{R}} \subset \mathcal{R}\) which produce operators with more peculiar mapping properties. In particular, for given \(p \in (1, \infty)\) we construct \(\mathcal{\widetilde{R}}\) such that \(\mathcal{M}^{\mathcal{\widetilde{R}}}\) is bounded on \(L^p(\mathbb{T}^\omega)\) if and only if \(p \in (p_0, \infty]\).

We will present several results related to the Harmonic Analysis of functions defined on the infinite-dimensional torus \(\mathbb{T}^\omega\), which is the topological compact group consisting of the Cartesian product of countably infinite many copies of the one-dimensional torus, with its corresponding Haar measure. Such results include the study of absolutely divergent series, Calderón–Zygmund decomposition, and differentiation of integrals. Issues concerning maximal operators and theory of weights will be also introduced and further developed in a subsequent lecture by Dariusz Kosz.

In this talk we will discuss how certain kinds of uncertainty principles and interpolation formulas related to the Fourier transform are connected to sphere packing problems and talk about some recent developments on these fronts.

In this talk I will present some recent advances on Boundary Value Problems for the Laplace operator with rough boundary data in a bounded corkscrew domain in \(\mathbb{R}^{n+1}\) whose boundary is uniformly n-rectifiable and its measure theoretic boundary agrees with its topological boundary up to a set of n-dimensional Hausdorff measure zero. In particular, I will discuss the equivalence between solvability of the Dirichlet problem for the Laplacian with boundary data in \(L^{p'}\) and solvability of the regularity problem for the Laplacian with boundary data in an appropriate Sobolev space \(W^{1,p}\) , where \(p \in (1, 2 + \epsilon)\) and \(\frac{1}{p} + \frac{1}{p'} = 1\). As chord-arc domains satisfy the aforementioned geometric assumptions, our result answers a question posed by Carlos Kenig in 1991. This is joint work with Xavier Tolsa.

There many operators in Harmonic Analysis which can be described as an average of a family of operators \(\{T_j\}_j\) for which some boundedness properties are known. In particular, if \(T_j\) are uniformly bounded on \(L^p\) , then Minkowski integral inequality tells us that \(T\) also satisfies this property. But things change completely if the information that we have is that \(T_j\) are of weak type \((1, 1).\) However, under certain condition on the operators \(T_j\) , the weak type boundedness of \(T\) can be reached. This is a joint work with my student Sergi Baena.

We establish a compactness property for the difference between nonlinear and linear operators (or the Duhamel operator) related to the inhomogeneous nonlinear Schrödinger equation. The proof is based on a refined profile decomposition for the equation.

We introduce a new class of bilinear operators \(\{BC_a\}_a\) acting as a merger between two classical objects in harmonic analysis: the bilinear Hilbert transform and the linear Carleson-Stein-Wainger operator. The two opposing features (modulation invariance versus a kernel maximally modulated by the exponential of an imaginary monomial) of BC a require a simultaneous two-resolutions analysis and the use of a dilated time-frequency portrait. This is joint work with F. Bernicot, V. Lie, M. Vitturi.

We consider a quite general class of homogenization problems for singular integrals, initially motivated by random quasiconformal mappings. The talk is based on joint work with Steffen Rohde (U. Washington), Eero Saksman (U. Helsinki) and Terry Tao (UCLA).

Analysis on graphs studies the connections between geometrical or combinatorial properties of graphs and natural operators defined on them. In this talk, I will present a new geometrical construction leading to an infinite collection of families of graphs, where all the elements in each family are (finite) isospectral non-isomorphic graphs for the discrete magnetic Laplacian with standard weights. The construction is based on the notion of isospectral frames which, together with the s-partition of a natural number r, define the isospectral families of graphs by contraction of distinguished vertices. The isospectral frames have high symmetry and we use a spectral preorder of graphs studied in [2,3] to control the spectral spreading of the eigenvalues under elementary perturbations of the graph like vertex contraction and vertex virtualisation.
References:
[1] J.S. Fabila-Carrasco, F. Lledó and O. Post, A geometric construction of isospectral magnetic graphs, 2021 (in preparation).
[2] J.S. Fabila-Carrasco, F. Lledó and O. Post, Spectral preorder and perturbations of discrete weighted graphs. Math. Ann. (2020).
[3] J.S. Fabila-Carrasco, F. Lledó and O. Post, Spectral gaps and discrete magnetic Laplacians. Lin. Alg. Appl. 547 (2018) 183-216.

Picard's (little) theorem is one of the most striking results of classical Complex Analysis. Since Picard's original proof, a great variety of approaches, generalizations and surprising links with other areas have contributed to enrich the scope of Geometric Function Theory. Motivated by extensions of Picard's theorem to quasiregular mappings, J. Lewis obtained in 1994 the first purely real, harmonic proof of Picard's little theorem. In the talk we will report several variations on Picard’s theorem, with special emphasis on Lewis approach. In this direction, we will discuss a recent extension in terms of the range of harmonic maps in the plane.

In this talk I will explain a recent work, in collaboration with Albert Clop, where we provide a pointwise characterisation of nearly incompressible vector fields \(b : \mathbb{R}^n \to \mathbb{R}^n\) with \(|x| \log |x|\) growth at infinity for which \(\textrm{curl} b = Db − D^t b\) is bounded. In the plane we can go further and describe still in pointwise sense, the vector fields \(b : \mathbb{R}^2 \to \mathbb{R}^2\) for which \(|\textrm{div} b| + |\textrm{curl} b| \in L^\infty\) .

I will report on recent progress on the two weight problem for singular and fractional integral operators in \(\mathbb{R}^n\), in particular a two weight local Tb theorem in higher dimensions. Joint work with Christos Grigoriadis, Michalis Paparizos, Eric Sawyer, Chun-Yen Shen.

The notion of graph limits aims to provide a better understanding of large graphs by providing a limit object which is linked to a convergence notion for sequences of graphs. One of the problems that arise is the following: even when the convergent notions care about similar parameters, the different convergence notions either only work for certain families of graphs, or they trivialize for others. The best known example of this behaviour is the notion of left-convergence introduced by Lovasz et. al., and the Benjamini-Schramm convergence for bounded degree graphs: in both cases the convergence of the sequence involves the subgraph counts (we fix the subgraph to count, such as the triangle \(K_3\) , and let the parameter of the sequence grow). For the left-convergence, if the graphs \(G_n\) are not dense, then the limit trivializes; for the Benjamini-Schramm convergence, we can only consider sequences of graphs with bounded maximum degree. We present the following 'compactification' result: assuming the continuum hypothesis, there exists a set of positive sequences A (with the property that the quotient of every pair of sequences in \(A\) has a limit, possibly infinite), for which, for any subsequence \(b\) of positive numbers, there exist an \(a \in A\), a finite positive constant \(c\), and a subsubsequence \(d\) of \(b\) (indexed by \(I\)) such that \[ d_n/a_n \stackrel{n \to \infty, n \in I}{\longrightarrow} c \] (the limit of \(d\) along the subsequence is comparable to \(a\)). With this, we give a convergence notion that is the common generalization of the Benjamini-Schramm convergence and the left-convergence for graphs, and has the property that any sequence of graphs (with growing number of vertices) has a convergent subsequence.

SIO (Singular Integral Operator) theory and, Calderón-Zygmund theory specifically, developed starting from the '60s, provides a vast array of tools for dealing with operators that resemble the Hilbert transform \[ Hf(x) := \int_{\mathbb{R}} f(x-y) \frac{dy}{y} \] an ubiquitous operator in Complex Analysis, semi-linear PDEs, and many other branches of mathematics. Results valid for -valued functions were extended to Banach spaces-valued functions thanks to Bourgain's groundbreaking work on the deep relation between Banach space geometry and boundedness properties of vector-valued SIOs. Scalar-valued bounds for multilinear SIOs, like the bilinear Hilbert transform \[ BHT[f_1,f_2](x) = \int_{\mathbb{R}} f_1(x-t) f_2(x+t) \frac{dt}{t} \] are classic in time-frequency-scale analysis. Banach-space valued results have appeared only in the last couple of years. The well understood connections with Banach space geometry from linear theory are just starting to be investigated. Open questions and generalizations to non-commutative analysis abound and would come hand-in-hand with progress in understanding SIOs with worse singularities than of Calderón-Zygmund type that can often be realized as SIO-valued CZ operators.

Sea \(N\) un operador normal en un espacio de Hilbert. Hablaremos de la estructura espectral de una perturbación \(T = N + K\) del operador \(N\), donde \(K\) es un operador compacto suficientemente "suave". En particular, nos interesan la existencia de subespacios invariantes y si \(T\) es descomponible o no. El caso cuando la medida espectral µ de N es absolutamente continua respecto de la medida del área fue estudiado por el conferenciante en 1993; es un caso cuando \(\mu\) tiene "dimensión" dos. En este trabajo fue introducido un modelo cociente, definido en términos de ciertas clases vectoriales de Sobolev. El caso cuando \(\mu\) es discreta (en otras palabras, cuando \(N\) es diagonalizable) fue estudiado en una serie de trabajos de Ciprian Foias y sus coautores en 2007-2011. Podemos verlo como un caso de dimensión cero. En el trabajo que presentaremos, que es conjunto con Mihai Putinar, extendemos el modelo cociente a una amplia clase de medidas. Las medidas cuya "dimensión" es cercana a uno no quedan cubiertas por nuestro método.

We discuss some recent work related to weighted and unweighted Fourier restriction estimates for hyperbolic paraboloids in dimensions three and larger. In particular, we will discuss how one can prove linear estimates in higher dimensions by using bilinear restriction estimates due to S. Lee and A. Vargas. We will also discuss an application of this method to certain fractal restriction estimates for the hyperbolic paraboloid, which are joint with M.B Erdogan and T. Harris.

Let \(\Omega \subset \mathbb{R}^{n+1}, n \geq 2\), be a 1-sided NTA domain satisfying CDC. Let \(L_0 u = − \div(A_0 \nabla u), Lu = − \div(A\nabla u) \) be two real uniformly elliptic operators in \(\Omega\), and write \(\omega_{L_0} , \omega_L\) for the respective associated elliptic measures. We establish the equivalence between the following: (i) \(\omega_L \in A_\infty (ω_{L_0})\), (ii) \(L\) is \(L^p (ω_{L_0})\)-solvable, (iii) bounded null solutions of \(L\) satisfy Carleson measure estimates with respect to \(\omega_{L_0}\), (iv) \(S < N\) (i.e., the conical square function is controlled by the non-tangential maximal function) in \(L^q (\omega_{L_0})\) for any null solution of \(L\), and (v) \(L\) is BMO(\(\omega_{L_0})\)-solvable. This is a joint work with Ó. Dominguez, J.M. Martell, and P. Tradacete.

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.

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.

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.

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.

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.

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.

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.

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.)

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.

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).