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European Mathematical Society Publishing House
2024-03-29 01:18:23
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https://www.ems-ph.org/meta/jmeta-stream.php?jrn=RMI&vol=34&iss=2&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
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European Mathematical Society Publishing House
Zuerich, Switzerland
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34
2018
2
Isoperimetric profiles and random walks on some permutation wreath products
Laurent
Saloff-Coste
Cornell University, Ithaca, USA
Tianyi
Zheng
University of California San Diego, La Jolla, USA
Isoperimetry, random walk, Schreier graphs
We study the isoperimetric profiles of certain families of finitely generated groups defined via marked Schreier graphs and permutation wreath products. The groups we study are among the “simplest” examples within a much larger class of groups, all defined via marked Schreier graphs and/or action on rooted trees, which includes such examples as the long range group, Grigorchuck group and the basillica group. The highly nonlinear structure of these groups make them both interesting and difficult to study. Because of the relative simplicity of the Schreier graphs that define the groups we study here (the key fact is that they contained very large regions that are “one dimensional”), we are able to obtain sharp explicit bounds on the L1 and L2 isoperimetric profiles of these groups. As usual, these sharp isoperimetric profile estimates provide sharp bounds on the probability of return of simple random walk. Nevertheless, within each of the families of groups we study there are also many cases for which the existing techniques appear inadequate and this leads to a variety of open problems.
Fourier analysis
Operator theory
481
540
10.4171/RMI/994
http://www.ems-ph.org/doi/10.4171/RMI/994
5
28
2018
Accessibility, Martin boundary and minimal thinness for Feller processes in metric measure spaces
Panki
Kim
Seoul National University, Republic of Korea
Renming
Song
University of Illinois at Urbana-Champaign, USA and Nankai University, Tianjin, China
Zoran
Vondraček
University of Zagreb, Croatia and University of Illinois, Urbana, USA
Martin boundary, Martin kernel, purely discontinuous Feller process, minimal thinness
In this paper we study the Martin boundary at infinity for a large class of purely discontinuous Feller processes in metric measure spaces. We show that if $\infty$ is accessible from an open set $D$, then there is only one Martin boundary point of $D$ associated with it, and this point is minimal. We also prove the analogous result for finite boundary points. As a consequence, we show that minimal thinness of a set is a local property.
Probability theory and stochastic processes
Potential theory
541
592
10.4171/RMI/995
http://www.ems-ph.org/doi/10.4171/RMI/995
5
28
2018
Solid hulls of weighted Banach spaces of entire functions
José
Bonet
Universidad Politecnia de Valencia, Spain
Jari
Taskinen
University of Helsinki, Finland
Weighted Banach spaces of entire functions, Taylor coefficients, solid hull, solid core
Given a continuous, radial, rapidly decreasing weight $v$ on the complex plane, we study the solid hull of its associated weighted space $H_v^\infty(\mathbb C)$ of all the entire functions $f$ such that $v|f|$ is bounded. The solid hull is found for a large class of weights satisfying the condition (B) of Lusky. Precise formulations are obtained for weights of the form $v(r)=$ exp$(-ar^p), a > 0, p > 0$. Applications to spaces of multipliers are included.
Functional analysis
Functions of a complex variable
593
608
10.4171/RMI/996
http://www.ems-ph.org/doi/10.4171/RMI/996
5
28
2018
Some remarks on the comparison principle in Kirchhoff equations
Giovany
Figueiredo
Universidade de Brasília, Brazil
Antonio
Suárez
Universidad de Sevilla, Spain
Kirchhoff equation, comparison principle, sub-supersolution method
In this paper we study the validity of the comparison principle and the sub-supersolution method for Kirchhoff type equations. We show that these principles do not work when the Kirchhoff function is increasing, contradicting some previous results. We give an alternative sub-supersolution method and apply it to some models.
Integral equations
Ordinary differential equations
Partial differential equations
609
620
10.4171/RMI/997
http://www.ems-ph.org/doi/10.4171/RMI/997
5
28
2018
Spectral permanence in a space with two norms
Hyeonbae
Kang
Inha University, Incheon, Republic of Korea
Mihai
Putinar
University of California, Santa Barbara, USA and Newcastle University, UK
Neumann–Poincaré operator, Lamé system, spectrum, finite section method
A generalization of a classical argument of Mark G. Krein leads us to the conclusion that the Neumann–Poincar´e operator associated to the Lamé system of linear elastostatics equations in two dimensions has the same spectrum on the Lebesgue space of the boundary as the more natural energy space. A similar result for the Neumann–Poincaré operator associated to the Laplace equation was stated by Poincaré and was proved rigorously a century ago by means of a symmetrization principle for non-selfadjoint operators. We develop the necessary theoretical framework underlying the spectral analysis of the Neumann–Poincaré operator, including also a discussion of spectral asymptotics of a Galerkin type approximation. Several examples from function theory of a complex variable and harmonic analysis are included.
Potential theory
Operator theory
Mechanics of deformable solids
Partial differential equations
621
635
10.4171/RMI/998
http://www.ems-ph.org/doi/10.4171/RMI/998
5
28
2018
The approximation property for spaces of Lipschitz functions with the bounded weak* topology
Antonio
Jiménez-Vargas
Universidad de Almería, Spain
Lipschitz spaces, approximation property, tensor product, epsilon product
Let $X$ be a pointed metric space and let Lip$_0(X)$ be the space of all scalar-valued Lipschitz functions on $X$ which vanish at the base point. We prove that Lip$_0(X)$ with the bounded weak* topology $\tau_{bw^*}$ has the approximation property if and only if the Lipschitz-free Banach space $\mathcal F(X)$ has the approximation property if and only if, for each Banach space $F$, each Lipschitz operator from $X$ into $F$ can be approximated by Lipschitz finite-rank operators within the unique locally convex topology $\gamma\tau_\gamma$ on Lip$_0(X,F)$ such that the Lipschitz transpose mapping $f\mapsto f^t$ is a topological isomorphism from Lip$_0(X,F),\gamma\tau_\gamma)$ to (Lip$_0(X),\tau_{bw^*})\epsilon F$.
Functional analysis
637
654
10.4171/RMI/999
http://www.ems-ph.org/doi/10.4171/RMI/999
5
28
2018
On the variance of the error term in the hyperbolic circle problem
Giacomo
Cherubini
University of Copenhagen, Denmark
Morten
Risager
University of Copenhagen, Denmark
Hyperbolic lattice points, Selberg’s pre-trace formula, fractional integration
Let $e(s)$ be the error term of the hyperbolic circle problem, and denote by $e_\alpha(s)$ the fractional integral to order $\alpha$ of $e(s)$. We prove that for any small $\alpha>0$ the asymptotic variance of $e_\alpha(s)$ is finite, and given by an explicit expression. Moreover, we prove that $e_\alpha(s)$ has a limiting distribution.
Number theory
Real functions
655
685
10.4171/RMI/1000
http://www.ems-ph.org/doi/10.4171/RMI/1000
5
28
2018
Parabolic Harnack inequality on fractal-type metric measure Dirichlet spaces
Janna
Lierl
University of Connecticut, Storrs, USA
Parabolic Harnack inequality, Moser iteration, weighted Poincaré inequality, heat kernel estimates, non-symmetric Dirichlet form, fractals
This paper proves the strong parabolic Harnack inequality for local weak solutions to the heat equation associated with time-dependent (nonsymmetric) bilinear forms. The underlying metric measure Dirichlet space is assumed to satisfy the volume doubling condition, the strong Poincaré inequality, and a cutoff Sobolev inequality. The metric is not required to be geodesic. Further results include a weighted Poincaré inequality, as well as upper and lower bounds for non-symmetric heat kernels.
Partial differential equations
Potential theory
Probability theory and stochastic processes
687
738
10.4171/RMI/1001
http://www.ems-ph.org/doi/10.4171/RMI/1001
5
28
2018
On the variation of maximal operators of convolution type II
Emanuel
Carneiro
Instituto de Matemática Pura e Aplicada, Rio de Janeiro, Brazil
Renan
Finder
Instituto de Matemática Pura e Aplicada, Rio de Janeiro, Brazil
Mateus
Sousa
Instituto de Matemática Pura e Aplicada, Rio de Janeiro, Brazil
Maximal functions, heat flow, Poisson kernel, Sobolev spaces, regularity, subharmonic, bounded variation, variation-diminishing, sphere
In this paper we establish that several maximal operators of convolution type, associated to elliptic and parabolic equations, are variation-diminishing. Our study considers maximal operators on the Euclidean space $\mathbb R^d$, on the torus $\mathbb T^d$ and on the sphere $\mathbb S^d$. The crucial regularity property that these maximal functions share is that they are subharmonic in the corresponding detachment sets.
Fourier analysis
Potential theory
Partial differential equations
Functional analysis
739
766
10.4171/RMI/1002
http://www.ems-ph.org/doi/10.4171/RMI/1002
5
28
2018
The Dirichlet boundary problem for second order parabolic operators satisfying a Carleson condition
Martin
Dindoš
Edinburgh University, UK
Sukjung
Hwang
Yonsei University, Seoul, Republic of Korea
Parabolic boundary value problem, Carleson condition, $L^p$ solvability
We establish $L^p$, $2\le p\le\infty$, solvability of the Dirichlet~boundary value problem for a parabolic equation $u_t- \mathrm {div}(A\nabla u) - \boldsymbol{B}\cdot\nabla u =0$ on time-varying domains with coefficient matrices $A=[a_{ij}]$ and $\boldsymbol{B} = [b_{i}]$ that satisfy a small Carleson condition. The results are sharp in the following sense. For a given value of $1 < p < \infty$ there exists operators that satisfy Carleson condition but fail to have $L^p$ solvability of the Dirichlet problem. Thus the assumption of smallness is sharp. Our results complements results of Hofmann, Lewis and Rivera-Noriega, where solvability of parabolic $L^p$ (for some large $p$) Dirichlet boundary value problem for coefficients that satisfy large Carleson condition was established. We also give a new (substantially shorter) proof of these results.
Partial differential equations
767
810
10.4171/RMI/1003
http://www.ems-ph.org/doi/10.4171/RMI/1003
5
28
2018
Norm convolution inequalities in Lebesgue spaces
Erlan
Nursultanov
Lomonosov Moscow State University, Kazakhstan Branch, Astana, Kazakhstan and RUDN University, Moscow, Russia
Sergey
Tikhonov
Centre de Recerca Matemática, Bellaterra, Spain, ICREA, Barcelona, Spain, and Universitat Autònoma de Barcelona, Spain
Nazerke
Tleukhanova
Gumilyov Eurasian National University, Astana, Kazakhstan
Convolution, Young–O’Neil inequality, oscillatory kernels
We obtain upper and similar lower estimates of the ($L_p, L_q$) norm for the convolution operator. The upper estimate improves on known convolution inequalities. The technique to obtain lower estimates is applied to study boundedness problems for oscillatory integrals.
Integral transforms, operational calculus
Fourier analysis
Operator theory
811
838
10.4171/RMI/1004
http://www.ems-ph.org/doi/10.4171/RMI/1004
5
28
2018
Smooth torus actions are described by a single vector field
Francisco Javier
Turiel
Universidad de Málaga, Spain
Antonio
Viruel
Universidad de Málaga, Spain
Torus, group action, vector field
Consider a smooth effective action of a torus $\mathbb T^n$ on a connected $C^{\infty}$-manifold $M$. Assume that $M$ is not a torus endowed with the natural action. Then we prove that there exists a complete vector field $X$ on $M$ such that the automorphism group of $X$ equals $\mathbb T^n \times \mathbb R$, where the factor $\mathbb R$ comes from the flow of $X$ and $\mathbb T^n$ is regarded as a subgroup of Diff$(M)$. Thus one may reconstruct the whole action of $\mathbb T^n$ from a single vector field.
Manifolds and cell complexes
Fourier analysis
Operator theory
839
852
10.4171/RMI/1005
http://www.ems-ph.org/doi/10.4171/RMI/1005
5
28
2018
Topological entropy of irregular sets
Luis
Barreira
Instituto Superior Técnico, Lisboa, Portugal
Jinjun
Li
Guangzhou University, China
Claudia
Valls
Instituto Superior Técnico, Lisboa, Portugal
Irregular sets, topological entropy, specification property
For expansive continuous maps with the specification property, we compute the topological entropy of the irregular set for the Birkhoff averages of a continuous function. This is the set of points for which the Birkhoff averages do not converge. The entropy is expressed in terms of a conditional variational principle. We also consider the general case of irregular sets obtained from ratios of Birkhoff averages of continuous functions. Moreover, we obtain a conditional variational principle for the topological entropy of the family of subsets of the irregular set formed by the points such that the set of accumulation points of the ratio of Birkhoff averages is a given interval. As nontrivial applications, we obtain conditional variational principles for the topological entropy of the level sets of local entropies, pointwise dimensions and Lyapunov exponents both on repellers and hyperbolic sets.
Dynamical systems and ergodic theory
Fourier analysis
853
878
10.4171/RMI/1006
http://www.ems-ph.org/doi/10.4171/RMI/1006
5
28
2018
The variance conjecture on hyperplane projections of the $\ell_p^n$ balls
David
Alonso
Universidad de Zaragoza, Spain
Jesús
Bastero
Universidad de Zaragoza, Spain
Variance conjecture, hyperplane projections, log-concave random vectors, convex bodies
We show that for any $1\leq p\leq\infty$, the family of random vectors uniformly distributed on hyperplane projections of the unit ball of $\ell_p^n$ verify the variance conjecture $$\mathrm {Var} |X|^2 \leq C \mathrm {max}_{\xi\in S^{n-1}}\mathbb E\langle X,\xi\rangle^2\, \mathbb E|X|^2,$$ where $C$ depends on $p$ but not on the dimension $n$ or the hyperplane. We will also show a general result relating the variance conjecture for a random vector uniformly distributed on an isotropic convex body and the variance conjecture for a random vector uniformly distributed on any Steiner symmetrization of it. As a consequence we will have that the class of random vectors uniformly distributed on any Steiner symmetrization of an $\ell_p^n$-ball verify the variance conjecture.
Functional analysis
Convex and discrete geometry
879
904
10.4171/RMI/1007
http://www.ems-ph.org/doi/10.4171/RMI/1007
5
28
2018
Algebraic connections vs. algebraic $\mathcal D$-modules: regularity conditions
Maurizio
Cailotto
Università di Padova, Italy
Luisa
Fiorot
Università di Padova, Italy
Gauss–Manin connection, $\mathcal D$-modules, De Rham cohomology
This paper is devoted to the comparison of the notions of regularity for algebraic connections and regularity for (holonomic) algebraic $\mathcal D$-modules.
Algebraic geometry
Several complex variables and analytic spaces
905
914
10.4171/RMI/1008
http://www.ems-ph.org/doi/10.4171/RMI/1008
5
28
2018
On a paper of Berestycki–Hamel–Rossi and its relations to the weak maximum principle at infinity, with applications
Marco
Magliaro
Universidade Federal do Ceará, Fortaleza, Brazil
Luciano
Mari
Scuola Normale Superiore, Pisa, Italy
Marco
Rigoli
Università di Milano, Italy
Weak maximum principle, Khas’minskii type conditions, Lichnerowicz equation, Ricci solitons
The aim of this paper is to study a new equivalent form of the weak maximum principle for a large class of differential operators on Riemannian manifolds. This new form has been inspired by the work of Berestycki, Hamel and Rossi for trace operators, and allows us to shed new light on it and to introduce a new sufficient bounded Khas’minskii type condition for its validity. We show its effectiveness by applying it to obtain some uniqueness results in a geometric setting.
Global analysis, analysis on manifolds
Partial differential equations
Differential geometry
915
936
10.4171/RMI/1009
http://www.ems-ph.org/doi/10.4171/RMI/1009
5
28
2018
The Loewner equation and Lipschitz graphs
Steffen
Rohde
University of Washington, Seattle, USA
Huy
Tran
Technische Universität Berlin, Germany
Michel
Zinsmeister
Université d'Orléans, France
Loewner differential equation
The proofs of continuity of Loewner traces in the stochastic and in the deterministic settings employ different techniques. In the former setting of the Schramm–Loewner evolution SLE, H¨older continuity of the conformal maps is shown by estimating the derivatives, whereas the latter setting uses the theory of quasiconformal maps. In this note, we adopt the former method to the deterministic setting and obtain a new and elementary proof that Hölder-1/2 driving functions with norm less than 4 generate simple arcs. We also give a sufficient condition for driving functions to generate curves that are graphs of Lipschitz functions.
Functions of a complex variable
Ordinary differential equations
937
948
10.4171/RMI/1010
http://www.ems-ph.org/doi/10.4171/RMI/1010
5
28
2018