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Extension criteria for homogeneous Sobolev spaces of functions of one variable Rev. Mat. Iberoam. (IF 1.142) Pub Date : 2020-08-26 Pavel Shvartsman
For each $p > 1$ and each positive integer $m$, we give intrinsic characterizations of the restriction of the homogeneous Sobolev space $L_{p}^{m}(\mathbb{R})$ to an arbitrary closed subset $E$ of the real line. We show that the classical one-dimensional Whitney extension operator is "universal" for the scale of $L_{p}^{m}(\mathbb{R})$ spaces in the following sense: For every $p\in(1,\infty]$, it provides
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Non-cutoff Boltzmann equation with polynomial decay perturbations Rev. Mat. Iberoam. (IF 1.142) Pub Date : 2020-08-25 Ricardo Alonso; Yoshinori Morimoto; Weiran Sun; Tong Yang
The Boltzmann equation without the angular cutoff is considered when the initial data is a small perturbation of a global Maxwellian and decays algebraically in the velocity variable. We obtain a well-posedness theory in the perturbative framework for both mild and strong angular singularities. The three main ingredients in the proof are the moment propagation, the spectral gap of the linearized operator
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A characterization of Krull monoids for which sets of lengths are (almost) arithmetical progressions Rev. Mat. Iberoam. (IF 1.142) Pub Date : 2020-08-20 Alfred Geroldinger; Wolfgang Alexander Schmid
Let $H$ be a Krull monoid with finite class group $G$ and suppose that every class contains a prime divisor. Then sets of lengths in $H$ have a well-defined structure which depends only on the class group $G$. With methods from additive combinatorics we establish a characterization of those class groups $G$ guaranteeing that all sets of lengths are (almost) arithmetical progressions.
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A Kakeya maximal function estimate in four dimensions using planebrushes Rev. Mat. Iberoam. (IF 1.142) Pub Date : 2020-08-20 Nets Hawk Katz; Joshua Zahl
We obtain an improved Kakeya maximal function estimate in $\mathbb R^4$ using a new geometric argument called the planebrush. A planebrush is a higher dimensional analogue of Wolff’s hairbrush, which gives effective control on the size of Besicovitch sets when the lines through a typical point concentrate into a plane. When Besicovitch sets do not have this property, the existing trilinear estimates
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The cubic Schrödinger regime of the Landau–Lifshitz equation with a strong easy-axis anisotropy Rev. Mat. Iberoam. (IF 1.142) Pub Date : 2020-08-05 André de Laire; Philippe Gravejat
We pursue our work on the asymptotic regimes of the Landau–Lifshitz equation. We put the focus on the cubic Schrödinger equation, which is known to describe the dynamics of uniaxial ferromagnets in a regime of strong easy-axis anisotropy. In any dimension, we rigorously prove this claim for solutions with sufficient regularity. In this regime, we additionally classify the one-dimensional solitons of
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Structure of globally hyperbolic spacetimes-with-timelike-boundary Rev. Mat. Iberoam. (IF 1.142) Pub Date : 2020-07-28 Luis Aké Hau; José Luis Flores Dorado; Miguel Sánchez Caja
Globally hyperbolic spacetimes-with-timelike-boundary $(\overline{M} = M \cup \partial M, g)$ are the natural class of spacetimes where regular boundary conditions (eventually asymptotic, if $\partial M$ is obtained by means of a conformal embedding) can be posed. $\partial M$ represents the naked singularities and can be identified with a part of the intrinsic causal boundary. Apart from general properties
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Sub-Riemannian structures do not satisfy Riemannian Brunn–Minkowski inequalities Rev. Mat. Iberoam. (IF 1.142) Pub Date : 2020-07-27 Nicolas Juillet
We prove that no Brunn–Minkowski inequality from the Riemannian theories of curvature-dimension and optimal transportation can be satisfied by a strictly sub-Riemannian structure. Our proof relies on the same method as for the Heisenberg group together with new investigations by Agrachev, Barilari and Rizzi on ample normal geodesics of sub-Riemannian structures and the geodesic dimension attached to
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Highest weight modules for affine Lie superalgebras Rev. Mat. Iberoam. (IF 1.142) Pub Date : 2020-07-22 Lucas Calixto; Vyacheslav Futorny
We describe Borel and parabolic subalgebras of affine Lie superalgebras and study the Verma type modules associated to such subalgebras. We give necessary and sufficient conditions under which these modules are simple.
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Spiders’ webs of doughnuts Rev. Mat. Iberoam. (IF 1.142) Pub Date : 2020-07-22 Alastair Fletcher; Daniel Stoertz
If $f\colon \mathbb{R}^3 \to \mathbb{R}^3$ is a uniformly quasiregular mapping with Julia set $J(f)$, a genus $g$ Cantor set, for $g\geq 1$, then for any linearizer $L$ at any repelling periodic point of $f$, the fast escaping set $A(L)$ consists of a spiders' web structure containing embedded genus $g$ tori on any sufficiently large scale. In other words, $A(L)$ contains a spiders' web of doughnuts
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Embedding the Heisenberg group into a bounded-dimensional Euclidean space with optimal distortion Rev. Mat. Iberoam. (IF 1.142) Pub Date : 2020-07-03 Terence Tao
Let $H := \Big(\begin{smallmatrix} 1 & \mathbb{R} & \mathbb{R} \\ 0 & 1 & \mathbb{R} \\ 0 & 0 & 1 \end{smallmatrix}\Big)$ denote the Heisenberg group with the usual Carnot–Carathéodory metric $d$. It is known (since the work of Pansu and Semmes) that the metric space $(H,d)$ cannot be embedded in a bilipschitz fashion into a Hilbert space; however, from a general theorem of Assouad, for any $0 < \varepsilon
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A coordinate-free proof of the finiteness principle for Whitney’s extension problem Rev. Mat. Iberoam. (IF 1.142) Pub Date : 2020-04-14 Jacob Carruth; Abraham Frei-Pearson; Arie Israel; Bo'az Klartag
We present a coordinate-free version of Fefferman’s solution of Whitney’s extension problem in the space $C^{m−1,1}(\mathbb R^n)$. While the original argument relies on an elaborate induction on collections of partial derivatives, our proof uses the language of ideals and translation-invariant subspaces in the ring of polynomials. We emphasize the role of compactness in the proof, first in the familiar
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Lipschitz regularity for orthotropic functionals with nonstandard growth conditions Rev. Mat. Iberoam. (IF 1.142) Pub Date : 2020-04-07 Pierre Bousquet; Lorenzo Brasco
We consider a model convex functional with orthotropic structure and super-quadratic nonstandard growth conditions. We prove that bounded local minimizers are locally Lipschitz, with no restrictions on the ratio between the highest and the lowest growth rates.
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Small local action of singular integrals on spaces of non-homogeneous type Rev. Mat. Iberoam. (IF 1.142) Pub Date : 2020-04-07 Benjamin Jaye; Tomás Merchán
Fix $d\geq 2$ and $s\in (0,d)$. In this paper we introduce a notion called small local action associated to a singular integral operator, which is a necessary condition for the existence of the principal value integral. Our goal is to understand the geometric properties of a measure for which an associated singular integral has small local action. We revisit Mattila's theory of symmetric measures and
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Critical weak-$L^p$ differentiability of singular integrals Rev. Mat. Iberoam. (IF 1.142) Pub Date : 2020-04-03 Luigi Ambrosio; Augusto C. Ponce; Rémy Rodiac
We establish that for every function $u \in L^1_{\rm loc}(\Omega)$ whose distributional Laplacian $\Delta u$ is a signed Borel measure in an open set $\Omega$ in $\mathbb{R}^{N}$, the distributional gradient $\nabla u$ is differentiable almost everywhere in $\Omega$ with respect to the weak-$L^{N/(N-1)}$ Marcinkiewicz norm. We show in addition that the absolutely continuous part of $\Delta u$ with
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Discrete dynamics and differentiable stacks Rev. Mat. Iberoam. (IF 1.142) Pub Date : 2020-04-03 Alejandro Cabrera; Matias L. del Hoyo; Enrique Pujals
In this paper we relate the study of actions of discrete groups over connected manifolds to that of their orbit spaces seen as differentiable stacks. We show that the orbit stack of a discrete dynamical system on a simply connected manifold encodes the dynamics up to conjugation and inversion. We also prove a generalization of this result for arbitrary discrete groups and non-simply connected manifolds
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Two weight inequalities for positive operators: doubling cubes Rev. Mat. Iberoam. (IF 1.142) Pub Date : 2020-04-03 Wei Chen; Michael T. Lacey
For the maximal operator $M$ on $\mathbb{R}^{d}$, and $ 1 < p,\rho < \infty$, there is a finite constant $D=D _{p, \rho }$ so that this holds. For all weights $w,\sigma$ on $\mathbb{R}^{d}$, the operator $M(\sigma \cdot)$ is bounded from $L^{p}(\sigma )\to L^{p}(w)$ if and only if the pair of weights $(w,\sigma)$ satisfy the two weight $A_{p}$ condition, and this testing inequality holds: $$\int_{Q}
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On measures that improve $L^q$ dimension under convolution Rev. Mat. Iberoam. (IF 1.142) Pub Date : 2020-03-26 Eino Rossi; Pablo Shmerkin
The $L^q$ dimensions, for $1 < q < \infty$, quantify the degree of smoothness of a measure. We study the following problem on the real line: when does the $L^q$ dimension improve under convolution? This can be seen as a variant of the well-known $L^p$-improving property. Our main result asserts that uniformly perfect measures (which include Ahlfors-regular measures as a proper subset) have the property
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On the two-systole of real projective spaces Rev. Mat. Iberoam. (IF 1.142) Pub Date : 2020-03-20 Lucas Ambrozio; Rafael Montezuma
We establish an integral-geometric formula for minimal two-spheres inside homogeneous three-spheres, and use it to provide a characterisation of each homogeneous metric on the three-dimensional real projective space as the unique metric with the largest possible two-systole among metrics with the same volume in its conformal class.
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Newton–Okounkov bodies of exceptional curve valuations Rev. Mat. Iberoam. (IF 1.142) Pub Date : 2020-03-20 Carlos Galindo; Julio José Moyano-Fernández; Francisco Monserrat; Matthias Nickel
We prove that the Newton–Okounkov body associated to the flag $E_{\bullet}:= \{ X=X_r \supset E_r \supset \{q\} \}$, defined by the surface $X$ and the exceptional divisor $E_r$ given by any divisorial valuation of the complex projective plane $\mathbb{P}^2$, with respect to the pull-back of the line-bundle $\mathcal{O}_{\mathbb{P}^2} (1)$ is either a triangle or a quadrilateral, characterizing when
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Supports and extreme points in Lipschitz-free spaces Rev. Mat. Iberoam. (IF 1.142) Pub Date : 2020-03-18 Ramón J. Aliaga; Eva Pernecká
For a complete metric space $M$, we prove that the finitely supported extreme points of the unit ball of the Lipschitz-free space $\mathcal{F}{M}$ are precisely the elementary molecules $(\delta(p)-\delta(q))/d(p,q)$ defined by pairs of points $p,q$ in $M$ such that the triangle inequality $d(p,q) < d(p,r)+d(q,r)$ is strict for any $r\in M$ different from $p$ and $q$. To this end, we show that the
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On a problem of Sárközy and Sós for multivariate linear forms Rev. Mat. Iberoam. (IF 1.142) Pub Date : 2020-03-18 Juanjo Rué; Christoph Spiegel
We prove that for pairwise co-prime numbers $k_1,\dots,k_d \geq 2$ there does not exist any infinite set of positive integers $\mathcal{A}$ such that the representation function $r_{\mathcal{A}}(n) = \# \{ (a_1, \dots, a_d) {\in} \mathcal{A}^d : k_1 a_1 + \cdots + k_d a_d = n \}$ becomes constant for $n$ large enough. This result is a particular case of our main theorem, which poses a further step
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Elliptic fibrations on K3 surfaces with a non-symplectic involution fixing rational curves and a curve of positive genus Rev. Mat. Iberoam. (IF 1.142) Pub Date : 2020-03-18 Alice Garbagnati; Cecília Salgado
In this paper we complete the classification of the elliptic fibrations on K3 surfaces which admit a non-symplectic involution acting trivially on the Néron–Severi group. We use the geometric method introduced by Oguiso and moreover we provide a geometric construction of the fibrations classified. If the non-symplectic involution fixes at least one curve of genus 1, we relate all the elliptic fibrations
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On the factorization of iterated polynomials Rev. Mat. Iberoam. (IF 1.142) Pub Date : 2020-03-16 Lucas Reis
Let $\mathbb{F}_q$ be the finite field with $q$ elements and $f, g\in \mathbb{F}_q[x]$ be polynomials of degree at least one. This paper deals with the asymptotic growth of certain arithmetic functions associated to the factorization of the iterated polynomials $f(g^{(n)}(x))$ over $\mathbb{F}_q$, such as the largest degree of an irreducible factor and the number of irreducible factors. In particular
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On a Pólya functional for rhombi, isosceles triangles, and thinning convex sets Rev. Mat. Iberoam. (IF 1.142) Pub Date : 2020-03-16 Michiel van den Berg; Vincenzo Ferone; Carlo Nitsch; Cristina Trombetti
Let $\Omega$ be an open convex set in $\mathbb{R}^m$ with finite width, and with boundary $\partial \Omega$. Let $v_{\Omega}$ be the torsion function for $\Omega$, i.e., the solution of $-\Delta v=1, v|_{\partial\Omega}=0$. An upper bound is obtained for the product of $\Vert v_{\Omega}\Vert_{L^{\infty}(\Omega)}\lambda(\Omega)$, where $\lambda(\Omega)$ is the bottom of the spectrum of the Dirichlet
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Complex orthogonal geometric structures of dimension three Rev. Mat. Iberoam. (IF 1.142) Pub Date : 2020-03-16 Mayra Méndez
A complex orthogonal (geometric) structure on a complex manifold is a geometric structure locally modelled on a non-degenerate quadric. One of the first examples of such a structure on a compact manifold of dimension three was constructed by Guillot. In this paper, we show that the same manifold carries a family of uniformizable complex orthogonal (geometric) structures which includes Guillot’s structure;
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Poincaré profiles of groups and spaces Rev. Mat. Iberoam. (IF 1.142) Pub Date : 2020-03-02 David Hume; John M. Mackay; Romain Tessera
We introduce a spectrum of monotone coarse invariants for metric measure spaces called Poincaré profiles. The two extremes of this spectrum determine the growth of the space, and the separation profile as defined by Benjamini–Schramm–Timár. In this paper we focus on properties of the Poincaré profiles of groups with polynomial growth, and of hyperbolic spaces, where we deduce a connection between these
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Stein–Weiss inequalities with the fractional Poisson kernel Rev. Mat. Iberoam. (IF 1.142) Pub Date : 2020-02-25 Lu Chen; Zhao Liu; Guozhen Lu; Chunxia Tao
In this paper, we establish the following Stein–Weiss inequality with the fractional Poisson kernel: \begin{align*} (\star)\qquad \int_{\mathbb{R}^n_{+}} \int_{\partial\mathbb{R}^n_{+}} |\xi|^{-\alpha} f(\xi) &\,P(x,\xi,\gamma)\, g(x)\, |x|^{-\beta}\, d\xi\, dx \\ &\leq C_{n,\alpha,\beta,p,q'}\, \|g\|_{L^{q'}(\mathbb{R}^n_{+})}\, \|f\|_{L^p(\partial \mathbb{R}^{n}_{+})}, \end{align*} where $$ P(x,\xi
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Gauge theory and $\mathrm G_2$-geometry on Calabi–Yau links Rev. Mat. Iberoam. (IF 1.142) Pub Date : 2020-02-21 Omegar Calvo-Andrade; Lázaro O. Rodríguez Díaz; Henrique N. Sá Earp
The 7-dimensional link $K$ of a weighted homogeneous hypersurface on the round 9-sphere in $\mathbb{C}^5$ has a nontrivial null Sasakian structure which is contact Calabi–Yau, in many cases. It admits a canonical co-calibrated $\mathrm G_2$-structure $\varphi$ induced by the Calabi–Yau 3-orbifold basic geometry. We distinguish these pairs $(K,\varphi)$ by the Crowley–Nordström $\mathbb{Z}_{48}$-valued
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Uniqueness and stability of the saddle-shaped solution to the fractional Allen–Cahn equation Rev. Mat. Iberoam. (IF 1.142) Pub Date : 2020-02-21 Juan Carlos Felipe-Navarro; Tomás Sanz-Perela
In this paper we prove the uniqueness of the saddle-shaped solution $u\colon \mathbb{R}^{2m} \to \mathbb{R}$ to the semilinear nonlocal elliptic equation $(-\Delta)^\gamma u = f(u)$ in $\mathbb{R}^{2m}$, where $\gamma \in (0,1)$ and $f$ is of Allen–Cahn type. Moreover, we prove that this solution is stable if $2m\geq 14$. As a consequence of this result and the connection of the problem with nonlocal
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A limiting free boundary problem for a degenerate operator in Orlicz–Sobolev spaces Rev. Mat. Iberoam. (IF 1.142) Pub Date : 2020-02-19 Jefferson Abrantes Santos; Sergio H. Monari Soares
A free boundary optimization problem involving the $\Phi$-Laplacian in Orlicz–Sobolev spaces is considered for the case where $\Phi$ does not satisfy the natural conditions introduced by Lieberman. A minimizer $u\Phi$ having non-degeneracy at the free boundary is proved to exist and some important consequences are established, namely, the Lipschitz regularity of $u\Phi$ along the free boundary, that
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The Radó–Kneser–Choquet theorem for $p$-harmonic mappings between Riemannian surfaces Rev. Mat. Iberoam. (IF 1.142) Pub Date : 2020-02-19 Tomasz Adamowicz; Jarmo Jääskeläinen; Aleksis Koski
In the planar setting, the Radó–Kneser–Choquet theorem states that a harmonic map from the unit disk onto a Jordan domain bounded by a convex curve is a diffeomorphism provided that the boundary mapping is a homeomorphism. We prove the injectivity criterion of Radó–Kneser–Choquet for $p$-harmonic mappings between Riemannian surfaces. In our proof of the injectivity criterion we approximate the $p$-harmonic
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Kähler manifolds with geodesic holomorphic gradients Rev. Mat. Iberoam. (IF 1.142) Pub Date : 2020-02-17 Andrzej Derdzinski; Paolo Piccione
We prove a dichotomy theorem about compact Kähler manifolds admitting nontrivial real-holomorphic geodesic gradient vector fields, which has the following consequence: either such a manifold satisfies an additional integrability condition, or through every zero of the real-holomorphic geodesic gradient there passes an uncountable family of totally geodesic, holomorphically immersed complex projective
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Discrepancy for convex bodies with isolated flat points Rev. Mat. Iberoam. (IF 1.142) Pub Date : 2020-02-14 Luca Brandolini; Leonardo Colzani; Bianca Gariboldi; Giacomo Gigante; Giancarlo Travaglini
We consider the discrepancy of the integer lattice with respect to the collection of all translated copies of a dilated convex body having a finite number of flat, possibly non-smooth, points in its boundary. We estimate the $L^p$ norm of the discrepancy with respect to the translation variable, as the dilation parameter goes to infinity. If there is a single flat point with normal in a rational direction
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Sharp Adams–Moser–Trudinger type inequalities in the hyperbolic space Rev. Mat. Iberoam. (IF 1.142) Pub Date : 2020-02-14 Quốc Anh Ngô; Van Hoang Nguyen
The purpose of this paper is to establish some Adams–Moser–Trudinger inequalities, which are the borderline cases of the Sobolev embedding, in the hyperbolic space $\mathbb H^n$. First, we prove a sharp Adams’ inequality of order two with the exact growth condition in $\mathbb H^n$. Then we use it to derive a sharp Adams-type inequality and an Adachi–Tanakat-ype inequality. We also prove a sharp Adams-type
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Regularity estimates in weighted Morrey spaces for quasilinear elliptic equations Rev. Mat. Iberoam. (IF 1.142) Pub Date : 2020-02-12 Giuseppe Di Fazio; Truyen Nguyen
We study regularity for solutions of quasilinear elliptic equations of the form $\mathrm {div}\mathbf{A}(x,u,\nabla u)=\mathrm {div}\mathbf{F}$ in bounded domains in $\mathbb{R}^n$. The vector field $\mathbf{A}$ is assumed to be continuous in $u$, and its growth in $\nabla u$ is like that of the $p$-Laplace operator. We establish interior gradient estimates in weighted Morrey spaces for weak solutions
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On the rate of convergence of semigroups of holomorphic functions at the Denjoy–Wolff point Rev. Mat. Iberoam. (IF 1.142) Pub Date : 2020-02-12 Dimitrios Betsakos; Manuel D. Contreras; Santiago Díaz-Madrigal
Let $\{\phi_t\}$ be a semigroup of holomorphic self-maps of the unit disc $\mathbb{D}$ with Denjoy–Wolff point $\tau\in \partial\mathbb{D}$. We study the rate of convergence of the semigroup to $\tau$, that is, given $z\in \overline{\mathbb{D}}$, we discuss the behavior of $|\phi_{t}(z)-\tau|$ as $t$ goes to $+\infty$.
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Sidon set systems Rev. Mat. Iberoam. (IF 1.142) Pub Date : 2020-02-11 Javier Cilleruelo; Oriol Serra; Maximilian Wötzel
A family $\mathcal{A}$ of $k$-subsets of $\{1,2,\dots, N\}$ is a Sidon system if the sumsets $A+B$, $A,B\in \mathcal{A}$ are pairwise distinct. We show that the largest cardinality $F_k(N)$ of a Sidon system of $k$-subsets of $[N]$ satisfies $F_k(N)\le {N-1\choose k-1}+N-k$ and the asymptotic lower bound $F_k(N)=\Omega_k(N^{k-1})$. More precise bounds on $F_k(N)$ are obtained for $k\le 3$. We also
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Teichmüller space of circle diffeomorphisms with Hölder continuous derivatives Rev. Mat. Iberoam. (IF 1.142) Pub Date : 2020-02-10 Katsuhiko Matsuzaki
Based on the quasiconformal theory of the universal Teichmüller space, we introduce the Teichmüller space of diffeomorphisms of the unit circle with $\alpha$-Hölder continuous derivatives as a subspace of the universal Teichmüller space. We characterize such a diffeomorphism quantitatively in terms of the complex dilatation of its quasiconformal extension and the Schwarzian derivative given by the
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Regularity of the singular set in a two-phase problem for harmonic measure with Hölder data Rev. Mat. Iberoam. (IF 1.142) Pub Date : 2020-02-10 Matthew Badger; Max Engelstein; Tatiana Toro
In non-variational two-phase free boundary problems for harmonic measure, we examine how the relationship between the interior and exterior harmonic measures of a domain $\Omega \subset \mathbb R^n$ influences the geometry of its boundary. This type of free boundary problem was initially studied by Kenig and Toro in 2006, and was further examined in a series of separate and joint investigations by
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Ground state solutions for quasilinear Schrödinger equations with variable potential and superlinear reaction Rev. Mat. Iberoam. (IF 1.142) Pub Date : 2020-02-10 Sitong Chen; Vicenţiu D. Rădulescu; Xianhua Tang; Binlin Zhang
This paper is concerned with the following quasilinear Schrödinger equation: $$-\Delta u+V(x)u-\frac{1}{2}\Delta (u^2)u= g(u), \quad x\in \mathbb{R}^N,$$ where $N\ge 3$, $V\in \mathcal{C}(\mathbb R^N,[0,\infty))$ and $g\in \mathcal{C}(\mathbb{R}, \mathbb{R})$ is superlinear at infinity. By using variational and some new analytic techniques, we prove the above problem admits ground state solutions under
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On Landis’ conjecture in the plane for some equations with sign-changing potentials Rev. Mat. Iberoam. (IF 1.142) Pub Date : 2020-02-10 Blair Davey
In this article, we investigate the quantitative unique continuation properties of real-valued solutions to elliptic equations in the plane. Under a general set of assumptions on the operator, we establish quantitative forms of Landis’ conjecture. Of note, we prove a version of Landis’ conjecture for solutions to $−\Delta u + Vu = 0$, where $V$ is a bounded function whose negative part exhibits polynomial
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An abundance of simple left braces with abelian multiplicative Sylow subgroups Rev. Mat. Iberoam. (IF 1.142) Pub Date : 2020-01-16 Ferran Cedó; Eric Jespers; Jan Okniński
Braces were introduced by Rump to study involutive non-degenerate set-theoretic solutions of the Yang–Baxter equation. A constructive method for producing all such finite solutions from a description of all finite left braces has been recently discovered. It is thus a fundamental problem to construct and classify all simple left braces, as they can be considered as building blocks for the general theory
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The Pohozaev identity for the anisotropic $p$-Laplacian and estimates of the torsion function Rev. Mat. Iberoam. (IF 1.142) Pub Date : 2020-01-16 Qiaoling Wang; Changyu Xia
In this paper we prove a Pohozaev identity for the weighted anisotropic $p$-Laplace operator. As an application of the identity, we deduce the nonexistence of nontrivial solutions of the Dirichlet problem for the weighted anisotropic $p$-Laplacian in star-shaped domains of $\mathbb R^n$. We also provide an upper bound estimate for the first Dirichet eigenvalue of the anisotropic $p$-Laplacian on bounded
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Semilinear elliptic equations with Hardy potential and gradient nonlinearity Rev. Mat. Iberoam. (IF 1.142) Pub Date : 2020-01-13 Konstantinos Gkikas; Phuoc-Tai Nguyen
Let $\Omega \subset {\mathbb R}^N$ ($N \geq 3$) be a $C^2$ bounded domain, and let $\delta$ be the distance to $\partial \Omega$. In this paper, we study positive solutions of the equation $(\star)\ -L_\mu u+ g(|\nabla u|) = 0$ in $\Omega$, where $L_\mu=\Delta + {\mu}/{\delta^2} $, $\mu \in (0,{1}/{4}]$ and $g$ is a continuous, nondecreasing function on ${\mathbb R}_+$. We prove that if $g$ satisfies
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The Chirka–Lindelöf and Fatou theorems for $\overline\partial_J$-subsolutions Rev. Mat. Iberoam. (IF 1.142) Pub Date : 2020-01-10 Alexandre Sukhov
This paper studies boundary properties of bounded functions with bounded $\overline\partial_J$ differential on strictly pseudoconvex domains in an almost complex manifold.
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On scattering for the cubic defocusing nonlinear Schrödinger equation on the waveguide $\mathbb R^2 \times \mathbb T$ Rev. Mat. Iberoam. (IF 1.142) Pub Date : 2020-01-10 Xing Cheng; Zihua Guo; Kailong Yang; Lifeng Zhao
In this article, we will show the scattering of the cubic defocusing nonlinear Schrödinger equation on the waveguide $\mathbb{R}^2\times \mathbb{T}$ in $H^1$. We first establish the linear profile decomposition in $H^{1}(\mathbb{R}^2 \times \mathbb{T})$ motivated by the linear profile decomposition of the mass-critical Schrödinger equation in $L^2(\mathbb{R}^2)$. Then by using the solution of the cubic
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Dynamical aspects of the generalized Schrödinger problem via Otto calculus – A heuristic point of view Rev. Mat. Iberoam. (IF 1.142) Pub Date : 2020-01-10 Ivan Gentil; Christian Léonard; Luigia Ripani
The defining equation $$(\ast)\qquad \dot \omega_t=-F'(\omega_t)$$ of a gradient flow is kinetic in essence. This article explores some dynamical (rather than kinetic) features of gradient flows (i) by embedding equation $(\ast)$ into the family of slowed down gradient flow equations: $\dot \omega^{\varepsilon}_t=- \varepsilon F'( \omega ^{ \varepsilon}_t)$, where $\varepsilon > 0$, and (ii) by considering
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On differential polynomial rings over nil algebras Rev. Mat. Iberoam. (IF 1.142) Pub Date : 2020-01-07 Mikhail Chebotar; Wen-Fong Ke
Let $R$ be a nil algebra over a field of characteristic 0, and let $\delta$ be a derivation of $R$. Then the differential polynomial ring $R[X, \delta]$ cannot be mapped onto a unital simple ring homomorphically.
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On continuation properties after blow-up time for $L^2$-critical gKdV equations Rev. Mat. Iberoam. (IF 1.142) Pub Date : 2020-01-07 Yang Lan
In this paper, we consider a blow-up solution $u(t)$ (close to the soliton manifold) to the $L^2$-critical gKdV equation $\partial_tu+(u_{xx}+u^5)_x=0$, with finite blow-up time $T < +\infty$. We expect to construct a natural extension of $u(t)$ after the blow-up time. To do this, we consider the solution $u_{\gamma}(t)$ to the saturated $L^2$-critical gKdV equation $\partial_tu+(u_{xx}+u^5-\gamma
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On a class of nonlinear Schrödinger–Poisson systems involving a nonradial charge density Rev. Mat. Iberoam. (IF 1.142) Pub Date : 2020-01-07 Carlo Mercuri; Teresa Megan Tyler
In the spirit of the classical work of P. H. Rabinowitz on nonlinear Schrödinger equations, we prove existence of mountain-pass solutions and least energy solutions to the nonlinear Schrödinger–Poisson system $$\left\{\begin{array}{lll} - \Delta u+ u + \rho (x) \phi u = |u|^{p-1} u, &x\in \mathbb{R}^3, \\ -\Delta \phi=\rho(x) u^2, & x\in \mathbb{R}^3, \end{array}\right.$$ under different assumptions
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Almost everywhere divergence of spherical harmonic expansions and equivalence of summation methods Rev. Mat. Iberoam. (IF 1.142) Pub Date : 2020-01-07 Xianghong Chen; Dashan Fan; Juan Zhang
We show that there exists an integrable function on the $n$-sphere $(n \geq 2)$, whose Cesàro $(C, (n − 1)/2)$ means with respect to the spherical harmonic expansion diverge unboundedly almost everywhere. This extends results of Stein (1961) for flat tori and complements the work of Taibleson (1985) for spheres. We also study relations among Cesàro, Riesz, and Bochner–Riesz means.
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On the cohomology class of fiber-bunched cocycles on semi simple Lie groups Rev. Mat. Iberoam. (IF 1.142) Pub Date : 2020-01-03 Paulo Varandas
We study the cohomological equation associated to linear cocycles on semi simple Lie groups $\mathcal G$ over hyperbolic dynamics. We give sufficient conditions for the solution of the cohomological equation of fiber-bunched cocycles to be unique and for the Hölder conjugacy class of the cocycle to coincide with $C^\nu(M,\mathcal G)$. In particular, we prove that there exists an open and dense subset
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Genus one Lefschetz fibrations on disk cotangent bundles of surfaces Rev. Mat. Iberoam. (IF 1.142) Pub Date : 2020-01-03 Burak Ozbagci
We describe a Lefschetz fibration of genus one on the disk cotangent bundle of any closed orientable surface $\Sigma$. As a corollary, we obtain an explicit genus one open book decomposition adapted to the canonical contact structure on the unit cotangent bundle of $\Sigma$.
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A global well-posedness result for the Rosensweig system of ferrofluids Rev. Mat. Iberoam. (IF 1.142) Pub Date : 2020-01-03 Francesco De Anna; Stefano Scrobogna
In this paper we study a Bloch–Torrey regularization of the Rosensweig system for ferrofluids. The scope of this paper is twofold. First of all, we investigate the existence and uniqueness of Leray–Hopf solutions of this model in the whole space $\mathbb R^2$. In the second part of this paper we investigate both the long-time behavior of weak solutions and the propagation of Sobolev regularities in
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On compactness of commutators of multiplication and bilinear pseudodifferential operators and a new subspace of BMO Rev. Mat. Iberoam. (IF 1.142) Pub Date : 2020-01-03 Rodolfo H. Torres; Qingying Xue
It is known that the compactness of the commutators of point-wise multiplication with bilinear homogeneous Calderón–Zygmund operators acting on product of Lebesgue spaces is characterized by the multiplying function being in the space CMO. This space is the closure in BMO of its subspace of smooth functions with compact support. It is shown in this work that for bilinear Calderón–Zygmund operators
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On the failure of the Hörmander multiplier theorem in a limiting case Rev. Mat. Iberoam. (IF 1.142) Pub Date : 2019-12-17 Lenka Slavíková
We discuss the Hörmander multiplier theorem for $L^p$ boundedness of Fourier multipliers in which the multiplier belongs to a fractional Sobolev space with smoothness $s$. We show that this theorem does not hold in the limiting case $|1/p-1/2|=s/n$.
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Almgren’s frequency formula for an extension problem related to the anisotropic fractional Laplacian Rev. Mat. Iberoam. (IF 1.142) Pub Date : 2019-12-17 Raimundo Leitão
We consider the anisotropic version of an extension problem studied by Caffarelli and Silvestre. We present the anisotropic fractional Laplacian and prove the Almgren’s frequency formula obtained by Caffarelli and Silvestre for the anisotropic case.
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Overdetermined problems and constant mean curvature surfaces in cones Rev. Mat. Iberoam. (IF 1.142) Pub Date : 2019-12-04 Filomena Pacella; Giulio Tralli
We consider a partially overdetermined problem in a sector-like domain $\Omega$ in a cone $\Sigma$ in $\mathbb{R}^N$, $N\geq 2$, and prove a rigidity result of Serrin type by showing that the existence of a solution implies that $\Omega$ is a spherical sector, under a convexity assumption on the cone. We also consider the related question of characterizing constant mean curvature compact surfaces $\Gamma$
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Zariski K3 surfaces Rev. Mat. Iberoam. (IF 1.142) Pub Date : 2019-11-11 Toshiyuki Katsura; Matthias Schütt
We construct Zariski K3 surfaces of Artin invariant 1, 2 and 3 in many characteristics. In particular, we prove that any supersingular Kummer surface is Zariski if $p\not\equiv 1$ mod 12. Our methods combine different approaches such as quotients by the group scheme $\alpha_p$, Kummer surfaces, and automorphisms of hyperelliptic curves.
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Index and first Betti number of $f$-minimal hypersurfaces and self-shrinkers Rev. Mat. Iberoam. (IF 1.142) Pub Date : 2019-10-21 Debora Impera; Michele Rimoldi; Alessandro Savo
We study the Morse index of self-shrinkers for the mean curvature flow and, more generally, of $f$-minimal hypersurfaces in a weighted Euclidean space endowed with a convex weight. When the hypersurface is compact, we show that the index is bounded from below by an affine function of its first Betti number. When the first Betti number is large, this improves index estimates known in literature. In
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