- journal articles metadata
European Mathematical Society Publishing House
2024-03-29 06:49:17
35
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=RMI&vol=21&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)
21
2005
1
The Cauchy problem for viscous shallow water equations
Weike
Wang
Shanghai Jiao Tong University, SHANGHAI, CHINA
Chao-Jiang
Xu
Université de Rouen, MONT SAINT AIGNAN, FRANCE
Shallow water equation, Littlewood-Paley decomposition, global solution
In this paper we study the Cauchy problem for viscous shallow water equations. We work in the Sobolev spaces of index $s>2$ to obtain local solutions for any initial data, and global solutions for small initial data.
Partial differential equations
Fluid mechanics
General
1
24
10.4171/RMI/412
http://www.ems-ph.org/doi/10.4171/RMI/412
Codimension one symplectic foliations
Omegar
Calvo
CIMAT, GUANAJUATO - GTO, MEXICO
Vicente
Muñoz
Universidad Complutense de Madrid, MADRID, SPAIN
Francisco
Presas
Universidad Autónoma de Madrid, MADRID, SPAIN
Foliation, symplectic, asymptotically holomorphic
We define the concept of symplectic foliation on a symplectic manifold and provide a method of constructing many examples, by using asymptotically holomorphic techniques.
Differential geometry
Dynamical systems and ergodic theory
General
25
46
10.4171/RMI/413
http://www.ems-ph.org/doi/10.4171/RMI/413
Minimal Slant Submanifolds of the smallest dimension in $S$-manifolds
Alfonso
Carriazo
Universidad de Sevilla, SEVILLA, SPAIN
Luis
Fernández
Universidad de Sevilla, SEVILLA, SPAIN
María Belén
Hans-Uber
Universidad de Sevilla, SEVILLA, SPAIN
$S$-manifold, slant submanifold, minimal submanifold, smallest dimension
We study slant submanifolds of $S$-manifolds with the smallest dimension, specially minimal submanifolds and establish some relations between them and anti-invariant submanifolds in $S$-manifolds, similar to those ones proved by B.-Y. Chen for slant surfaces and totally real surfaces in Kaehler manifolds.
Differential geometry
General
47
66
10.4171/RMI/414
http://www.ems-ph.org/doi/10.4171/RMI/414
Weighted Sobolev-Lieb-Thirring inequalities
Kazuya
Tachizawa
Hokkaido University, SAPPORO, JAPAN
Sobolev-Lieb-Thirring inequalities, $\varphi$-transform, $A_p$-weights
We give a weighted version of the Sobolev-Lieb-Thirring inequality for suborthonormal functions. In the proof of our result we use $\varphi$-transform of Frazier-Jawerth.
Real functions
Fourier analysis
General
67
85
10.4171/RMI/415
http://www.ems-ph.org/doi/10.4171/RMI/415
Clifford and Harmonic Analysis on Cylinders and Tori
Soeren
Krausshar
Universiteit Gent, GENT, BELGIUM
John
Ryan
University of Arkansas, FAYETTEVILLE, UNITED STATES
Dirac operator, Clifford analysis, cotangent functions
Cotangent type functions in $\mathbb{R}^n$ are used to construct Cauchy kernels and Green kernels on the conformally flat manifolds $\mathbb{R}^n / \mathbb{Z}^k$ where $1\leq k\leq n$. Basic properties of these kernels are discussed including introducing a Cauchy formula, Green's formula, Cauchy transform, Poisson kernel, Szegö kernel and Bergman kernel for certain types of domains. Singular Cauchy integrals are also introduced as are associated Plemelj projection operators. These in turn are used to study Hardy spaces in this context. Also the analogues of Calderón-Zygmund type operators are introduced in this context, together with singular Clifford holomorphic, or monogenic, kernels defined on sector domains in the context of cylinders. Fundamental differences in the context of the $n$-torus arising from a double singularity for the generalized Cauchy kernel on the torus are also discussed.
Functions of a complex variable
Differential geometry
Global analysis, analysis on manifolds
General
87
110
10.4171/RMI/416
http://www.ems-ph.org/doi/10.4171/RMI/416
Resolution of a family of Galois embedding problems with cyclic kernel
Montserrat
Vela
Universitat Politècnica de Catalunya, BARCELONA, SPAIN
Galois embedding problems, generalized Clifford algebras
In this paper we compute the obstruction and the solutions of cyclic embedding problems given by $$ (E): \quad 0 \rightarrow \mathbb{Z}/n\mathbb{Z} \rightarrow E \rightarrow \Gamma=\mathbb{Z}/n\mathbb{Z} \times \stackrel{m)}{\cdots} \times \mathbb{Z}/n\mathbb{Z} \rightarrow 0 , $$ with $\mathbb{Z}/n\mathbb{Z}$ trivial $\Gamma$-modulo, finding adequate representations of $\Gamma$ in the automorphisms group of a generalized Clifford algebra.
Field theory and polynomials
Number theory
General
111
132
10.4171/RMI/417
http://www.ems-ph.org/doi/10.4171/RMI/417
A multiple set version of the $3k-3$ Theorem
Yahya ould
Hamidoune
Université Pierre et Marie Curie, PARIS, FRANCE
Alain
Plagne
Ecole Polytechnique, PALAISEAU CEDEX, FRANCE
$3k-3$ theorem, multiple set addition, $3k-4$ theorem, structure theory of set addition, Frobenius problem
In 1959, Freiman demonstrated his famous $3k-4$ Theorem which was to be a cornerstone in inverse additive number theory. This result was soon followed by a $3k-3$ Theorem, proved again by Freiman. This result describes the sets of integers $\mathcal{A}$ such that $| \mathcal{A}+\mathcal{A} | \leq 3 | \mathcal{A} | -3$. In the present paper, we prove a $3k-3$-like Theorem in the context of multiple set addition and describe, for any positive integer $j$, the sets of integers $\mathcal{A}$ such that the inequality $|j \mathcal{A} | \leq j(j+1)(| \mathcal{A} | -1)/2$ holds. Freiman's $3k-3$ Theorem is the special case $j=2$ of our result. This result implies, for example, the best known results on a function related to the Diophantine Frobenius number. Actually, our main theorem follows from a more general result on the border of $j\mathcal{A}$.
Number theory
General
133
161
10.4171/RMI/418
http://www.ems-ph.org/doi/10.4171/RMI/418
A note on the existence of $H$-bubbles via perturbation methods
Veronica
Felli
Università degli Studi di Milano-Bicocca, MILANO, ITALY
H-surfaces, nonlinear elliptic systems, perturbative methods
We study the problem of existence of surfaces in $\mathbb{R}^3$ parametrized on the sphere ${\mathbb S}^2$ with prescribed mean curvature $H$ in the perturbative case, i.e. for $H=H_0+\varepsilon H_1$, where $H_0$ is a nonzero constant, $H_1$ is a $C^2$ function and $\varepsilon$ is a small perturbation parameter.
Differential geometry
Partial differential equations
General
163
178
10.4171/RMI/419
http://www.ems-ph.org/doi/10.4171/RMI/419
Équation anisotrope de Navier-Stokes dans des espaces critiques
Marius
Paicu
Université Paris-Sud, ORSAY CEDEX, FRANCE
Navier-Stokes equation, critical spaces, anisotropic viscosity
We study the tridimensional Navier-Stokes equation when the value of the vertical viscosity is zero, in a critical space (invariant by the scaling). We shall prove local in time existence of the solution, respectively global in time when the initial data is small compared with the horizontal viscosity.
Partial differential equations
Fluid mechanics
General
179
235
10.4171/RMI/420
http://www.ems-ph.org/doi/10.4171/RMI/420
A quantitative version of Krein’s Theorem
M.
Fabian
Czech Academy of Sciences, PRAGUE 1, CZECH REPUBLIC
P.
Hájek
Czech Academy of Sciences, PRAGUE 1, CZECH REPUBLIC
Vicente
Montesinos
Universidad Politecnia de Valencia, VALENCIA, SPAIN
V.
Zizler
University of Alberta, EDMONTON, CANADA
Banach spaces, weak compactness, Krein’s Theorem
A quantitative version of Krein’s Theorem on convex hulls of weak compact sets is proved. Some applications to weakly compactly generated Banach spaces are given.
Functional analysis
General
237
248
10.4171/RMI/421
http://www.ems-ph.org/doi/10.4171/RMI/421
Quasinormal Families of Meromorphic Functions
Xuecheng
Pang
East China Normal University, SHANGHAI, CHINA
Shahar
Nevo
Bar-Ilan University, RAMAT-GAN, ISRAEL
Lawrence
Zalcman
Bar-Ilan University, RAMAT-GAN, ISRAEL
Quasinormal families, omitted values
Let $\mathcal{F}$ be a family of functions meromorphic on the plane domain $D$, all of whose zeros are multiple. Suppose that $f'(z)\ne 1$ for all $f\in \mathcal{F}$ and $z\in D.$ Then if $\mathcal{F}$ is quasinormal on $D$, it is quasinormal of order 1 there.
Functions of a complex variable
General
249
262
10.4171/RMI/422
http://www.ems-ph.org/doi/10.4171/RMI/422
Taille des valeurs de fonctions $L$ de carrés symétriques au bord de la bande critique
Emmanuel
Royer
Université de Montpellier II, MONTPELLIER CEDEX 5, FRANCE
Jie
Wu
Université Henri Poincaré, VANDOEUVRE LES NANCY, FRANCE
forme automorphe, carré symétrique, fonction $L$, valeur spéciale
For each weight $k$ and level $N$ square free and without small prime factors, we prove the existence of primitive forms $f_+$ and $f_-$ of weight $k$ and level $N$ such that $$ L(1,\sym^2f_+)\gg_{k}[\log\log(3N)]^{3} $$ and $$ L(1,\sym^2f_-)\ll_{k}[\log\log(3N)]^{-1}. $$ The result comes from a delicate study of the moments of $L(1,\sym^2 f)$. This study gives also results for squarefree levels but with small prime factors. It provides counterexamples to the equivalence between harmonic and natural means.
Number theory
General
263
312
10.4171/RMI/423
http://www.ems-ph.org/doi/10.4171/RMI/423
Interpolation and extrapolation of smooth functions by linear operators
Charles
Fefferman
Princeton University, PRINCETON, UNITED STATES
Whitney extension problem, linear operators
Let $C^{m , 1} ( \mathbb{R}^n)$ be the space of functions on $\mathbb{R}^n$ whose $m^{\sf th}$ derivatives are Lipschitz 1. For $E \subset \mathbb{R}^n$, let $C^{m , 1} (E)$ be the space of all restrictions to $E$ of functions in $C^{m,1} ( \mathbb{R}^n)$. We show that there exists a bounded linear operator $T: C^{m , 1} (E) \rightarrow C^{m , 1} ( \mathbb{R}^n)$ such that, for any $f \in C^{m , 1} ( E )$, we have $T f = f$ on $E$.
Calculus of variations and optimal control; optimization
Convex and discrete geometry
General
313
348
10.4171/RMI/424
http://www.ems-ph.org/doi/10.4171/RMI/424
2
A Simplified Proof of Desingularization and Applications
Ana María
Bravo
Universidad Autónoma de Madrid, MADRID, SPAIN
Santiago
Encinas
Universidad de Valladolid, VALLADOLID, SPAIN
Orlando
Villamayor U.
Universidad Autónoma de Madrid, MADRID, SPAIN
Resolution of singularities, desingularization
This paper contains a short and simplified proof of desingularization over fields of characteristic zero, together with various applications to other problems in algebraic geometry (among others, the study of the behavior of desingularization of families of embedded schemes, and a formulation of desingularization which is stronger than Hironaka's). Our proof avoids the use of the Hilbert-Samuel function and Hironaka's notion of normal flatness: First we define a procedure for principalization of ideals (i.e. a procedure to make an ideal invertible), and then we show that desingularization of a closed subscheme $X$ is achieved by using the procedure of principalization for the ideal ${\mathcal I}(X)$ associated to the embedded scheme $X$. The paper intends to be an introduction to the subject, focused on the motivation of ideas used in this new approach, and particularly on applications, some of which do not follow from Hironaka's proof.
Algebraic geometry
General
349
458
10.4171/RMI/425
http://www.ems-ph.org/doi/10.4171/RMI/425
Some Remarks on the Weak Maximum Principle
Marco
Rigoli
Università di Milano, MILANO, ITALY
Maura
Salvatori
Università di Milano, MILANO, ITALY
Marco
Vignati
Università di Milano, MILANO, ITALY
Maximum principles, volume growth
We obtain a maximum principle at infinity for solutions of a class of nonlinear singular elliptic differential inequalities on Riemannian manifolds under the sole geometrical assumptions of volume growth conditions. In the case of the Laplace-Beltrami operator we relate our results to stochastic completeness and parabolicity of the manifold.
Global analysis, analysis on manifolds
Differential geometry
General
459
481
10.4171/RMI/426
http://www.ems-ph.org/doi/10.4171/RMI/426
Dyadic BMO on the bidisk
Óscar
Blasco
Universidad de Valencia, VALENCIA, SPAIN
Sandra
Pott
Lund University, LUND, SWEDEN
BMO on the bidisk, Carleson measures, Haar multipliers
We give several new characterizations of the dual of the dyadic Hardy space $H^{1,d}(\mathbb{T}^2)$, the so-called dyadic BMO space in two variables and denoted ${\mathrm{BMO}}^{\mathit d}_{prod}}$. These include characterizations in terms of Haar multipliers, in terms of the ``symmetrised paraproduct'' $\Lambda_b$, in terms of the rectangular BMO norms of the iterated ``sweeps'', and in terms of nested commutators with dyadic martingale transforms. We further explore the connection between ${\mathrm{BMO}}^{\mathit d}_{prod}}$ and John-Nirenberg type inequalities, and study a scale of rectangular BMO spaces.
Fourier analysis
Operator theory
General
483
510
10.4171/RMI/427
http://www.ems-ph.org/doi/10.4171/RMI/427
Estimates of BMO type for singular integrals on spaces of homogeneous type and applications to hypoelliptic PDEs
Marco
Bramanti
Politecnico di Milano, MILANO, ITALY
Luca
Brandolini
Università di Bergamo, DALMINE, ITALY
BMO, hypoelliptic operators, singular integrals, spaces of homogeneous type
Let us consider the class of ``nonvariational uniformly hypoelliptic operators'': $$ Lu\equiv\sum_{i,j=1}^{q}a_{ij} (x) X_{i} X_{j} u $$ where: $X_1,X_2,\ldots,X_q$ is a system of H\"ormander vector fields in $\mathbb{R}^{n}$ ($n>q$), $\{a_{ij}\}$ is a $q\times q$ uniformly elliptic matrix, and the functions $a_{ij} (x)$ are continuous, with a suitable control on the modulus of continuity. We prove that: $$ \| X_{i} X_{j} u \|_{BMO(\Omega^{\prime})} \leq c \left\{\left\|Lu\right\|_{BMO(\Omega)} + \left\| u\right\|_{BMO(\Omega)} \right\} $$ for domains $\Omega^{\prime}\subset\subset\Omega$ that are regular in a suitable sense. Moreover, the space $BMO$ in the above estimate can be replaced with a scale of spaces of the kind studied by Spanne. To get this estimate, several results are proved, regarding singular and fractional integrals on general spaces of homogeneous type, in relation with function spaces of $BMO$ type.
Partial differential equations
Fourier analysis
Abstract harmonic analysis
General
511
556
10.4171/RMI/428
http://www.ems-ph.org/doi/10.4171/RMI/428
Extreme cases of weak type interpolation
Evgeniy
Pustylnik
Technion - Israel Institute of Technology, HAIFA, ISRAEL
Rearrangement invariant spaces, Boyd indices, weak interpolation
We consider quasilinear operators $T$ of {\it joint weak type} $(a,b;p,q)$ (in the sense of [Bennett, Sharpley: Interpolation of operators, Academic Press, 1988]) and study their properties on spaces $L_{\varphi,E}$ with the norm $\|\varphi(t)f^*(t) \|_{\tilde E}$, where $\tilde E$ is arbitrary rearrangement-invariant space with respect to the measure $dt/t$. A space $L_{\varphi,E}$ is said to be "close" to one of the endpoints of interpolation if the corresponding Boyd index of this space is equal to $1/a$ or to $1/p$. For all possible kinds of such ``closeness", we give sharp estimates for the function $\psi(t)$ so as to obtain that every $T:L_{\varphi,E}\to L_{\psi,E}$.
Functional analysis
General
557
576
10.4171/RMI/429
http://www.ems-ph.org/doi/10.4171/RMI/429
A Generalized Sharp Whitney Theorem for Jets
Charles
Fefferman
Princeton University, PRINCETON, UNITED STATES
Extension problems, Whitney convexity, Whitney $\omega$-convexity
Suppose that, for each point $x$ in a given subset $E \subset \mathbb{R}^n$, we are given an $m$-jet $f(x)$ and a convex, symmetric set $\sigma(x)$ of $m$-jets at $x$. We ask whether there exist a function $F \in C^{m , \omega} ( \mathbb{R}^n )$ and a finite constant $M$, such that the $m$-jet of $F$ at $x$ belongs to $f ( x ) + M \sigma ( x )$ for all $x \in E$. We give a necessary and sufficient condition for the existence of such $F , M$, provided each $\sigma(x)$ satisfies a condition that we call "Whitney $\omega$-convexity''.
Calculus of variations and optimal control; optimization
Convex and discrete geometry
General
577
688
10.4171/RMI/430
http://www.ems-ph.org/doi/10.4171/RMI/430
Corrigenda: $(n,2)$-sets have full Hausdorff dimension (Rev. Mat. Iberoamericana 20 (2004), no. 2, 381-393)
Themis
Mitsis
University of Crete, IRAKLION, GREECE
General
689
692
10.4171/RMI/431
http://www.ems-ph.org/doi/10.4171/RMI/431
Corrigenda: On the product theory of singular integrals (Rev. Mat. Iberoamericana 20 (2004), no. 2, 531-561)
Alexander
Nagel
University of Wisconsin, MADISON, UNITED STATES
Elias
Stein
Princeton University, PRINCETON, UNITED STATES
General
693
694
10.4171/RMI/432
http://www.ems-ph.org/doi/10.4171/RMI/432
3
Logarithmic derivative of the Euler $\Gamma$-function in Clifford analysis
Guy
Laville
Université de Caen, CAEN CEDEX, FRANCE
Louis
Randriamihamison
Institut National Polytechnique de Toulouse, TOULOUSE CEDEX, FRANCE
Non-commutative analysis, Clifford analysis, $\psi$-function, Euler constant, dilogarithm function
The logarithmic derivative of the $\Gamma$-function, namely the $\psi$-function, has numerous applications. We define analogous functions in a four dimensional space. We cut lattices and obtain Clifford-valued functions. These functions are holomorphic cliffordian and have similar properties as the $\psi$-function. These new functions show links between well-known constants: the Euler gamma constant and some generalisations, $\zeta^R(2)$, $\zeta^R(3)$. We get also the Riemann zeta function and the Epstein zeta functions.
Functions of a complex variable
Potential theory
Special functions
General
695
728
10.4171/RMI/433
http://www.ems-ph.org/doi/10.4171/RMI/433
Asymptotic windings over the trefoil knot
Jacques
Franchi
Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE
Trefoil knot, modular group, quasi-hyperbolic manifold, harmonic 1-forms, Brownian motion, geodesics, ergodic measures, geodesic flow, asymptotic laws
Consider the group $G:=PSL_2(\mathbb R)$ and its subgroups $\Gamma:= PSL_2(\mathbb{Z})$ and $\Gamma':= DSL_2(\mathbb{Z})$. $G/\Gamma$ is a canonical realization (up to an homeomorphism) of the complement $\mathbb S^3\setminus T$ of the trefoil knot $T$, and $G/\Gamma'$ is a canonical realization of the 6-fold branched cyclic cover of $\mathbb S^3\setminus T$, which has 3-dimensional cohomology of 1-forms. Putting natural left-invariant Riemannian metrics on $G$, it makes sense to ask which is the asymptotic homology performed by the Brownian motion in $G/\Gamma'$, describing thereby in an intrinsic way part of the asymptotic Brownian behavior in the fundamental group of the complement of the trefoil knot. A good basis of the cohomology of $ G/\Gamma'$, made of harmonic 1-forms, is calculated, and then the asymptotic Brownian behavior is obtained, by means of the joint asymptotic law of the integrals of the above basis along the Brownian paths. Finally the geodesics of $G$ are determined, a natural class of ergodic measures for the geodesic flow is exhibited, and the asymptotic geodesic behavior in $G/\Gamma'$ is calculated, by reduction to its Brownian analogue, though it is not precisely the same (counter to the hyperbolic case).
Global analysis, analysis on manifolds
Group theory and generalizations
Dynamical systems and ergodic theory
Probability theory and stochastic processes
729
770
10.4171/RMI/434
http://www.ems-ph.org/doi/10.4171/RMI/434
Potential Theory for Schrödinger operators on finite networks
Enrique
Bendito
Universitat Politècnica de Catalunya, BARCELONA, SPAIN
Ángeles
Carmona
Universitat Politècnica de Catalunya, BARCELONA, SPAIN
Andrés
Encinas
Universitat Politècnica de Catalunya, BARCELONA, SPAIN
Combinatorial Laplacian, Schrödinger operators, Dirichlet forms, Green kernel, Poisson kernel, Discrete Potential Theory, equilibrium measures, effective resistance
We aim here at analyzing the fundamental properties of positive semidefinite Schrödinger operators on networks. We show that such operators correspond to perturbations of the combinatorial Laplacian through 0-order terms that can be totally negative on a proper subset of the network. In addition, we prove that these discrete operators have analogous properties to the ones of elliptic second order operators on Riemannian manifolds, namely the monotonicity, the minimum principle, the variational treatment of Dirichlet problems and the condenser principle. Unlike the continuous case, a discrete Schrödinger operator can be interpreted as an integral operator and therefore a discrete Potential Theory with respect to its associated kernel can be built. We prove that the Schrödinger kernel satisfies enough principles to assure the existence of equilibrium measures for any proper subset. These measures are used to obtain systematic expressions of the Green and Poisson kernels associated with Dirichlet problems.
Potential theory
Ordinary differential equations
General
771
818
10.4171/RMI/435
http://www.ems-ph.org/doi/10.4171/RMI/435
Explicit spectral gap estimates for the linearized Boltzmann and Landau operators with hard potentials
Céline
Baranger
, CACHAN CEDEX, FRANCE
Clément
Mouhot
Université de Lyon, LYON CEDEX 07, FRANCE
Spectral gap, linearized Boltzmann operator, Landau linearized operator, geometrical properties, explicit, grazing collision limit, hard potentials
This paper deals with explicit spectral gap estimates for the linearized Boltzmann operator with hard potentials (and hard spheres). We prove that it can be reduced to the Maxwellian case, for which explicit estimates are already known. Such a method is constructive, does not rely on Weyl's Theorem and thus does not require Grad's splitting. The more physical idea of the proof is to use geometrical properties of the whole collision operator. In a second part, we use the fact that the Landau operator can be expressed as the limit of the Boltzmann operator as collisions become grazing in order to deduce explicit spectral gap estimates for the linearized Landau operator with hard potentials.
Fluid mechanics
Statistical mechanics, structure of matter
General
819
841
10.4171/RMI/436
http://www.ems-ph.org/doi/10.4171/RMI/436
The bidual of a tensor product of Banach spaces
Félix
Cabello Sánchez
Universidad de Extremadura, BADAJOZ, SPAIN
Ricardo
García
Universidad de Extremadura, BADAJOZ, SPAIN
Banach space, tensor product, dual space, infinite-dimensional holomorphy
This paper studies the relationship between the bidual of the (projective) tensor product of Banach spaces and the tensor product of their biduals.
Functional analysis
General
843
861
10.4171/RMI/437
http://www.ems-ph.org/doi/10.4171/RMI/437
Estimates in Besov spaces for transport and transport-diffusion equations with almost Lipschitz coefficients
Raphaël
Danchin
Université Paris 12 – Val de Marne, CRÉTEIL CEDEX, FRANCE
Transport equation, transport-diffusion equation, estimates in Besov spaces, almost Lipschitz vectorfield, loss of regularity
This paper aims at giving an overview of estimates in general Besov spaces for the Cauchy problem on $t=0$ related to the vector field $\partial_t+v\cdot\nabla$. The emphasis is on the conservation or loss of regularity for the initial data. When $\nabla v$ belongs to $L^1(0,T;L^\infty)$ (plus some convenient conditions depending on the functional space considered for the data), the initial regularity is preserved. On the other hand, if $\nabla v$ is slightly less regular (e.g. $\nabla v$ belongs to some limit space for which the embedding in $L^\infty$ fails), the regularity may coarsen with time. Different scenarios are possible going from linear to arbitrarily small loss of regularity. This latter result will be used in a forthcoming paper to prove global well-posedness for two-dimensional incompressible density-dependent viscous fluids (see [Danchin, R.: Local theory in critical spaces for compressible viscous and heat-conductive gases. Comm. Partial Differential Equations 26 (2001), 1183-1233, and Erratum, 27 (2002), 2531-2532.]). Besides, our techniques enable us to get estimates uniformly in $\nu\geq0$ when adding a diffusion term $-\nu\Delta u$ to the transport equation.
Partial differential equations
General
863
888
10.4171/RMI/438
http://www.ems-ph.org/doi/10.4171/RMI/438
Solution to the gradient problem of C. E. Weil
Zoltán
Buczolich
Eötvös Loránd University, BUDAPEST, HUNGARY
Gradient, Denjoy-Clarkson property, Lebesgue measure
In this paper we give a complete answer to the famous gradient problem of C. E. Weil. On an open set $G\subset \mathbb{R}^{2}$ we construct a differentiable function $f:G\to\mathbb{R}$ for which there exists an open set $\Omega_{1}\subset\mathbb{R}^{2}$ such that $\nabla f(\mathbf{p})\in \Omega_{1}$ for a $\mathbf{p}\in G$ but $\nabla f(\mathbf{q})\not\in\Omega_{1}$ for almost every $\mathbf{q}\in G$. This shows that the Denjoy-Clarkson property does not hold in higher dimensions.
Measure and integration
Dynamical systems and ergodic theory
General
889
910
10.4171/RMI/439
http://www.ems-ph.org/doi/10.4171/RMI/439
Fractional iteration in the disk algebra: prime ends and composition operators
Manuel
Contreras
Universidad de Sevilla, SEVILLA, SPAIN
Santiago
Díaz-Madrigal
Universidad de Sevilla, SEVILLA, SPAIN
Semigroups, disk algebra, prime end, starlike functions, spirallike functions
In this paper we characterize the semigroups of analytic functions in the unit disk which lead to semigroups of operators in the disk algebra. These characterizations involve analytic as well as geometric aspects of the iterates and they are strongly related to the classical theorem of Carath\'eodory about local connection and boundary behaviour of univalent functions.
Functions of a complex variable
Operator theory
General topology
General
911
928
10.4171/RMI/440
http://www.ems-ph.org/doi/10.4171/RMI/440
High order regularity for subelliptic operators on Lie groups of polynomial growth
Nick
Dungey
Macquarie University, SYDNEY, NSW, AUSTRALIA
Lie group, subelliptic operator, heat kernel, Riesz transform, regularity estimates
Let $G$ be a Lie group of polynomial volume growth, with Lie algebra $\mbox{\gothic g}$. Consider a second-order, right-invariant, subelliptic differential operator $H$ on $G$, and the associated semigroup $S_t = e^{-tH}$. We identify an ideal $\mbox{\gothic n}'$ of $\mbox{\gothic g}$ such that $H$ satisfies global regularity estimates for spatial derivatives of all orders, when the derivatives are taken in the direction of $\mbox{\gothic n}'$. The regularity is expressed as $L_2$ estimates for derivatives of the semigroup, and as Gaussian bounds for derivatives of the heat kernel. We obtain the boundedness in $L_p$, $1
Topological groups, Lie groups
Partial differential equations
Global analysis, analysis on manifolds
General
929
996
10.4171/RMI/441
http://www.ems-ph.org/doi/10.4171/RMI/441
Quasiconformal groups of compact type
Petra
Bonfert-Taylor
Wesleyan University, MIDDLETOWN, UNITED STATES
Gaven
Martin
Massey University, AUCKLAND, NEW ZEALAND
Quasiconformal groups, convergence groups, geometric finiteness, conical limit set
We establish that a quasiconformal group is of compact type if and only if its limits set is purely conical and find that the limit set of a quasiconformal group of compact type is uniformly perfect. A key tool is the result of Bowditch-Tukia on compact-type convergence groups. These results provide crucial tools for studying the deformations of quasiconformal groups and in establishing isomorphisms between such groups and conformal groups.
Manifolds and cell complexes
Group theory and generalizations
Functions of a complex variable
General
997
1012
10.4171/RMI/442
http://www.ems-ph.org/doi/10.4171/RMI/442
A note on lifting of Carnot groups
Andrea
Bonfiglioli
Università di Bologna, BOLOGNA, ITALY
Francesco
Uguzzoni
Università di Bologna, BOLOGNA, ITALY
Lifting of vector fields, Carnot groups, fundamental solutions, free groups
We prove that every homogeneous Carnot group can be lifted to a free homogeneous Carnot group. Though following the ideas of Rothschild and Stein, we give simple and self-contained arguments, providing a constructive proof, as shown in the examples.
Partial differential equations
Topological groups, Lie groups
Abstract harmonic analysis
General
1013
1035
10.4171/RMI/443
http://www.ems-ph.org/doi/10.4171/RMI/443
Approximation in law to the $d$-parameter fractional Brownian sheet based on the functional invariance principle
Xavier
Bardina
Universitat Autònoma de Barcelona, BELLATERRA, SPAIN
Carme
Florit
Universitat Autònoma de Barcelona, BELLATERRA, SPAIN
$d$-parameter fractional Brownian sheet, weak convergence, functional invariance principle.
We show a result of approximation in law of the $d$-parameter fractional Brownian sheet in the space of the continuous functions on $[0,T]^d$. The construction of these approximations is based on the functional invariance principle.
Probability theory and stochastic processes
General
1037
1052
10.4171/RMI/444
http://www.ems-ph.org/doi/10.4171/RMI/444
Generalized Hantzsche-Wendt flat manifolds
Juan
Rossetti
Universidad Nacional de Córdoba, CORDOBA, ARGENTINA
Andrzej
Szczepański
University of Gdansk, GDANSK, POLAND
Flat manifold, Bieberbach group, holonomy representation
We study the family of closed Riemannian $n$-manifolds with holonomy group isomorphic to $\mathbb{Z}_2^{n-1}$, which we call generalized Hantzsche-Wendt manifolds. We prove results on their structure, compute some invariants, and find relations between them, illustrated in a graph connecting the family.
Group theory and generalizations
Combinatorics
Differential geometry
Manifolds and cell complexes
1053
1070
10.4171/RMI/445
http://www.ems-ph.org/doi/10.4171/RMI/445
$L^p$ decay estimates for weighted oscillatory integral operators on $\mathbb{R}$
Malabika
Pramanik
California Institute of Technology, PASADENA, UNITED STATES
Chan Woo
Yang
Korea University, SEOUL, SOUTH KOREA
Oscillatory integral operators, decay rate, weight
In this paper, we formulate necessary conditions for decay rates of $L^p$ operator norms of weighted oscillatory integral operators on $\mathbb{R}$ and give sharp $L^2$ estimates and nearly sharp $L^p$ estimates.
Integral transforms, operational calculus
Partial differential equations
General
1071
1095
10.4171/RMI/446
http://www.ems-ph.org/doi/10.4171/RMI/446