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European Mathematical Society Publishing House
2024-03-29 16:08:26
13
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=RMI&vol=29&iss=3&update_since=2024-03-29
Revista Matemática Iberoamericana
Rev. Mat. Iberoamericana
RMI
0213-2230
2235-0616
General
10.4171/RMI
http://www.ems-ph.org/doi/10.4171/RMI
subscribers
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society (from 2012)
29
2013
3
The twisting representation of the L-function of a curve
Francesc
Fité
Universität Bielefeld, BIELEFELD, GERMANY
Joan-Carles
Lario
Universitat Politècnica de Catalunya, BARCELONA, SPAIN
Abelian varieties, genus 2 curves, L-functions, Artin representations
Let $C$ be a smooth projective curve defined over a number field and let $C'$ be a twist of $C$. In this article we relate the $\ell$-adic representations attached to the $\ell$-adic Tate modules of the Jacobians of $C$ and $C'$ through an Artin representation. This representation induces global relations between the local factors of the respective Hasse–Weil $L$-functions. We make these relations explicit in a particularly illustrative situation. For all but a finite number of $\overline{\mathbb{Q}}$-isomorphism classes of genus 2 curves defined over $\mathbb{Q}$ with $\operatorname{Aut}(C)\simeq D_8$ or $D_{12}$, we find a representative curve $C/\mathbb{Q}$ such that, for every isomorphism $\phi\colon C'\rightarrow C$ satisfying some mild condition, we are able to determine either the local factor $L_{ p}(C'/\mathbb{Q},T)$ or the product $L_{p}(C'/\mathbb{Q},T)\cdot L_{p}(C'/\mathbb{Q},-T)$ from the local factor $L_{p}(C/\mathbb{Q},T)$.
Number theory
Algebraic geometry
749
764
10.4171/RMI/738
http://www.ems-ph.org/doi/10.4171/RMI/738
Transversal multilinear Radon-like transforms: local and global estimates
Jonathan
Bennett
University of Birmingham, BIRMINGHAM, UNITED KINGDOM
Neal
Bez
University of Birmingham, EDGBASTON, UNITED KINGDOM
Susana
Gutiérrez
University of Birmingham, BIRMINGHAM, UNITED KINGDOM
Multilinear forms, convolution estimates, diffraction tomography
We prove local “$L^p$-improving” estimates for a class of multilinear Radon-like transforms satisfying a strong transversality hypothesis. As a consequence, we obtain sharp multilinear convolution estimates for measures supported on fully transversal submanifolds of Euclidean space of arbitrary dimension. Motivated by potential applications in diffraction tomography, we also prove global estimates for the same class of Radon-like transforms under a natural homogeneity assumption.
Integral transforms, operational calculus
765
788
10.4171/RMI/739
http://www.ems-ph.org/doi/10.4171/RMI/739
An operator inequality for weighted Bergman shift operators
Anders
Olofsson
Lund University, LUND, SWEDEN
Aron
Wennman
Royal Institute of Technology, STOCKHOLM, SWEDEN
Bergman shift operator, operator inequality, weighted shift operator
We prove an operator inequality for the Bergman shift operator on weighted Bergman spaces of analytic functions in the unit disc with weight function controlled by a curvature parameter $\alpha$ assuming nonnegative integer values. This generalizes results by Shimorin, Hedenmalm and Jakobsson concerning the cases $\alpha=0$ and $\alpha=1$. A naturally derived scale of Hilbert space operator inequalities is studied and shown to be relaxing as the parameter $\alpha>-1$ increases. Additional examples are provided in the form of weighted shift operators.
Operator theory
Functions of a complex variable
Functional analysis
789
808
10.4171/RMI/740
http://www.ems-ph.org/doi/10.4171/RMI/740
The automorphism group of Thompson’s group $F$: subgroups and metric properties
José
Burillo
Universitat Politècnica de Catalunya, CASTELLDEFELS, SPAIN
Sean
Cleary
The City College of CUNY, NEW YORK, UNITED STATES
Thompson’s group
We describe some of the geometric properties of the automorphism group $\operatorname {Aut}(F)$ of Thompson’s group $F$. We give realizations of $\operatorname {Aut}(F)$ geometrically via periodic tree pair diagrams, which lead to natural presentations and give effective methods for estimating the word length of elements. We study some natural subgroups of $\operatorname {Aut}(F)$ and their metric properties. In particular, we show that the subgroup of inner automorphisms of $F$ is at least quadratically distorted in $\operatorname {Aut}(F)$, whereas other subgroups of $\operatorname {Aut}(F)$ isomorphic to $F$ are undistorted.
Group theory and generalizations
809
828
10.4171/RMI/741
http://www.ems-ph.org/doi/10.4171/RMI/741
Bounds on the Walsh model for $M^{q,*}$ Carleson and related operators
Richard
Oberlin
Louisiana State University, BATON ROUGE, UNITED STATES
Return times theorem, Carleson’s theorem, multipliers, maximal-multipliers, variation-norm
We prove an extension of the Walsh-analog of the Carleson–Hunt theorem, where the $L^\infty$ norm defining the Carleson maximal operator has been replaced by an $L^q$ maximal-multiplier-norm. Additionally, we consider certain associated variation-norm estimates.
Fourier analysis
829
857
10.4171/RMI/742
http://www.ems-ph.org/doi/10.4171/RMI/742
Quasisymmetric Koebe uniformization
Sergei
Merenkov
University of Illinois at Urbana-Champaign, URBANA, UNITED STATES
Kevin
Wildrick
Universität Bern, BERN, SWITZERLAND
Quasiconformal mapping, metric spaces, uniformization
We study a quasisymmetric version of the classical Koebe uniformization theorem in the context of Ahlfors regular metric surfaces. We provide sufficient conditions for an Ahlfors 2-regular metric space $X$ homeomorphic to a domain in the standard 2-sphere $\mathbb{S}^2$ to be quasisymmetrically equivalent to a circle domain in $\mathbb{S}^2$. We also give an example showing the sharpness of these conditions.
Functions of a complex variable
859
910
10.4171/RMI/743
http://www.ems-ph.org/doi/10.4171/RMI/743
Initial boundary value problems for the two-component shallow water systems
Kai
Yan
Sun Yat-sen University, GUANGZHOU, GUANGDONG, CHINA
Zhaoyang
Yin
Sun Yat-sen University, GUANGZHOU, GUANGDONG, CHINA
Two-component Camassa–Holm system, modified two-component value problems, local well-possedness, Besov spaces, blow-up and global existence, global weak solutions
In this paper we study initial boundary value problems of three types of two-component shallow water systems on the half line subject to homogeneous Dirichlet boundary conditions. We first prove local well-possedness of the two-component Camassa–Holm system, the modified two-component Camassa–Holm system, and the two-component Degasperis–Procesi system in the Besov spaces. Then, we are able to specify certain conditions on the initial data which on the one hand guarantee global existence and on the other hand produce solutions with a finite lifespan. Moreover, in the case of finite time singularities we are able to describe the precise blow-up scenario for breaking waves. Finally we investigate global weak solutions for the two-component Camassa–Holm system and the modified two-component Camassa–Holm system on the half line, respectively. Our approach is based on sharp extension results for functions on the half line and several symmetry preserving properties of the systems under discussion.
Partial differential equations
911
938
10.4171/RMI/744
http://www.ems-ph.org/doi/10.4171/RMI/744
Monodromy zeta function formula for embedded $\mathbf{Q}$-resolutions
Jorge
Martín-Morales
Academia General Militar, ZARAGOZA, SPAIN
Quotient singularity, weighted blow-up, embedded $\mathbf{Q}$-resolution, monodromy zeta function
In previous work we have introduced the notion of an embedded $\mathbf{Q}$-resolution, which essentially consists in allowing the final ambient space to contain abelian quotient singularities. Here we give a generalization to this setting of N. A’Campo’s formula for the monodromy zeta function of a singularity. Some examples of its application are shown.
Several complex variables and analytic spaces
939
967
10.4171/RMI/745
http://www.ems-ph.org/doi/10.4171/RMI/745
Density of Lipschitz functions and equivalence of weak gradients in metric measure spaces
Luigi
Ambrosio
Scuola Normale Superiore, PISA, ITALY
Nicola
Gigli
Université de Nice-Sophia Antipolis, NICE CEDEX 02, FRANCE
Giuseppe
Savaré
Università di Pavia, PAVIA, ITALY
Weak upper gradients, Sobolev functions, optimal transport
We compare several notions of weak (modulus of) gradients in metric measure spaces and prove their equivalence. Using tools from optimal transportation theory we prove density in energy of Lipschitz maps independently of doubling and Poincaré assumptions on the metric measure space.
Functional analysis
Global analysis, analysis on manifolds
969
996
10.4171/RMI/746
http://www.ems-ph.org/doi/10.4171/RMI/746
Infinitely many nonradial solutions for the Hénon equation with critical growth
Juncheng
Wei
University of British Columbia, VANCOUVER, CANADA
Shusen
Yan
University of New England, ARMIDALE, AUSTRALIA
Hénon's equation, infinitely many solutions, critical Sobolev exponent, reduction method
We consider the following Hénon equation with critical growth: \[ (*) \begin{cases} - \Delta u = |y|^\alpha \, u^{\frac{N+2}{N-2}},\; u>0, & y\in B_1(0) , \\ u=0, &\text{on } \partial B_1(0), \end{cases} \] where $ \alpha>0$ is a positive constant, $ B_1(0)$ is the unit ball in $\mathbb{R}^N$, and $N\ge 4$. Ni [9] proved the existence of a radial solution and Serra [12] proved the existence of a nonradial solution for $\alpha$ large and $N \geq 4$. In this paper, we show the existence of a nonradial solution for any $\alpha>0$ and $N \geq 4$. Furthermore, we prove that equation (*) has infinitely many nonradial solutions, whose energy can be made arbitrarily large.
Partial differential equations
Operator theory
997
1020
10.4171/RMI/747
http://www.ems-ph.org/doi/10.4171/RMI/747
Single annulus $L^p$ estimates for Hilbert transforms along vector fields
Michael
Bateman
University of California Los Angeles, LOS ANGELES, UNITED STATES
Carleson’s theorem, time-frequency analysis, Stein’s conjecture, Zygmund’s conjecture, differentiation of vector fields, Hilbert transform in direction of a vector field
We prove $L^p$ estimates, $p\in (1,\infty)$, on the Hilbert transform along a one variable vector field acting on functions with frequency support in an annulus. Estimates when $p>2$ were proved by Lacey and Li. This paper also contains key technical ingredients for a companion paper with Christoph Thiele in which $L^p$ estimates are established for the full Hilbert transform. The operators considered here are singular integral variants of maximal operators arising in the study of planar differentiation problems.
Fourier analysis
1021
1069
10.4171/RMI/748
http://www.ems-ph.org/doi/10.4171/RMI/748
On the Riemann surface type of random planar maps
James
Gill
Saint Louis University, ST. LOUIS, UNITED STATES
Steffen
Rohde
University of Washington, SEATTLE, UNITED STATES
We show that the (random) Riemann surfaces of the Angel–Schramm uniform infinite planar triangulation and of Sheffield’s infinite necklace construction are both parabolic. In other words, Brownian motion on these surfaces is recurrent. We obtain this result as a corollary to a more general theorem on subsequential distributional limits of random unbiased disc triangulations, following work of Benjamini and Schramm.
Functions of a complex variable
Probability theory and stochastic processes
1071
1090
10.4171/RMI/749
http://www.ems-ph.org/doi/10.4171/RMI/749
Lewy–Stampacchia type estimates for variational inequalities driven by (non)local operators
Raffaella
Servadei
Università della Calabria, COSENZA, ITALY
Enrico
Valdinoci
Università degli Studi di Milano, MILANO, ITALY
Variational inequalities, integrodifferential operators, fractional Laplacian
The purpose of this paper is to derive some Lewy–Stampacchia estimates in some cases of interest, such as the ones driven by non-local operators. Since we will perform an abstract approach to the problem, this will provide, as a byproduct, Lewy–Stampacchia estimates in more classical cases as well. In particular, we can recover the known estimates for the standard Laplacian, the $p$-Laplacian, and the Laplacian in the Heisenberg group. In the non-local framework we prove a Lewy–Stampacchia estimate for a general integrodifferential operator and, as a particular case, for the fractional Laplacian. As far as we know, the abstract framework and the results in the non-local setting are new.
Partial differential equations
Calculus of variations and optimal control; optimization
1091
1126
10.4171/RMI/750
http://www.ems-ph.org/doi/10.4171/RMI/750