- journal articles metadata
European Mathematical Society Publishing House
2024-03-29 08:25:16
15
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=RMI&vol=29&iss=4&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
4
Hardy spaces associated with different homogeneities and boundedness of composition operators
Yongsheng
Han
Auburn University, AUBURN, UNITED STATES
Chincheng
Lin
National Central University, CHUNG-LI, TAIWAN
Guozhen
Lu
Wayne State University, DETROIT, UNITED STATES
Zhuoping
Ruan
Nanjing University, NANJING, CHINA
Eric
Sawyer
McMaster University, HAMILTON, ONTARIO, CANADA
Hardy spaces, Calderón–Zygmund operators, discrete Calderón’s identity, almost orthogonality estimates, discrete Littlewood–Paley–Stein square functions
It is well known that standard Calderón–Zygmund singular integral operators with isotropic and nonisotropic homogeneities are bounded on the classical $H^p(\mathbb{R}^m)$ and nonisotropic $H^p_{h}(\mathbb{R}^m),$ respectively. In this paper, we develop a new Hardy space theory and prove that the composition of two Calderón–Zygmund singular integral operators with different homogeneities is bounded on this new Hardy space. Such a Hardy space has a multiparameter structure associated with the underlying mixed homogeneities arising from the two singular integral operators under consideration. The Calderón–Zygmund decomposition and an interpolation theorem hold on these new Hardy spaces.
Fourier analysis
1127
1157
10.4171/RMI/751
http://www.ems-ph.org/doi/10.4171/RMI/751
Pseudo-differential operators on fractals and other metric measure spaces
Marius
Ionescu
Colgate University, HAMILTON, UNITED STATES
Luke
Rogers
University of Connecticut, STORRS, UNITED STATES
Robert
Strichartz
Cornell University, ITHACA, UNITED STATES
Pseudo-differential operators, fractals, self-similar, elliptic, hypoelliptic, quasi-elliptic, Laplacian, metric measure space, sub-Gaussian heat kernel estimates, wavefront set, microlocal analysis
We define and study pseudo-differential operators on a class of fractals that include post-critically finite (p.c.f.) self-similar sets and Sierpiński carpets. Using sub-Gaussian estimates for the heat operator we prove that our operators have kernels that decay and, in the constant coefficient case, are smooth off the diagonal. Our analysis can be extended to products of fractals. While our results are applicable to a larger class of metric measure spaces with Laplacian, we use them to study elliptic, hypoelliptic, and quasi-elliptic operators on p.c.f. fractals, answering a few open questions posed in a series of recent papers. We extend our class of operators to include the so called Hörmander hypoelliptic operators and we initiate the study of wavefront sets and microlocal analysis on p.c.f. fractals.
Partial differential equations
Measure and integration
Functional analysis
Global analysis, analysis on manifolds
1159
1190
10.4171/RMI/752
http://www.ems-ph.org/doi/10.4171/RMI/752
On varieties with higher osculating defect
Pietro
De Poi
Università di Udine, UDINE, ITALY
Roberta
Di Gennaro
Università degli Studi di Napoli “Federico II”, NAPOLI, ITALY
Giovanna
Ilardi
Università degli Studi di Napoli “Federico II”, NAPOLI, ITALY
Algebraic varieties, osculating defects, higher fundamental forms, Laplace equations, scrolls
In this paper, using the method of moving frames, we generalise some of Terracini’s results on varieties with tangent defect. In particular, we characterise varieties with higher order osculating defect in terms of Jacobians of higher fundamental forms and moreover we characterise varieties with “small” higher fundamental forms as contained in scrolls.
Differential geometry
Algebraic geometry
Geometry
1191
1210
10.4171/RMI/753
http://www.ems-ph.org/doi/10.4171/RMI/753
Five squares in arithmetic progression over quadratic fields
Enrique
González-Jiménez
Universidad Autónoma de Madrid, MADRID, SPAIN
Xavier
Xarles
Universitat Autónoma de Barcelona, BARCELONA, SPAIN
Arithmetic progressions, squares, quadratic fields, elliptic curve Chabauty method, Mordel–Weil sieve
We provide several criteria to show over which quadratic number fields $\mathbb{Q}(\sqrt{D})$ there is a nonconstant arithmetic progression of five squares. This is carried out by translating the problem to the determination of when some genus five curves $C_D$ defined over $\mathbb{Q}$ have rational points, and then by using a Mordell–Weil sieve argument. Using an elliptic curve Chabauty-like method, we prove that, up to equivalence, the only nonconstant arithmetic progression of five squares over $–(\sqrt{409})$ is $7^2$, $13^2$, $17^2$, $409$, $ 23^2$. Furthermore, we provide an algorithm for constructing all the nonconstant arithmetic progressions of five squares over all quadratic fields. Finally, we state several problems and conjectures related to this problem.
Number theory
1211
1238
10.4171/RMI/754
http://www.ems-ph.org/doi/10.4171/RMI/754
On the boundedness of the Carleson operator near $L^1$
Victor
Lie
Purdue University, WEST LAFAYETTE, UNITED STATES
Time-frequency analysis, Carleson’s theorem
Based on the tile discretization elaborated in [14], we develop a Calderón–Zygmund type decomposition of the Carleson operator. As a consequence, through a unitary method that makes no use of extrapolation techniques, we recover previously known results regarding the largest rearrangement invariant space of functions with almost everywhere convergent Fourier series.
Fourier analysis
Operator theory
1239
1262
10.4171/RMI/755
http://www.ems-ph.org/doi/10.4171/RMI/755
Estimates for constant mean curvature graphs in $M\times\mathbb{R}$
José
Manzano
Universidad de Granada, GRANADA, SPAIN
Product manifolds, constant mean curvature, invariant surfaces, boundary curvature estimates, height estimates
We discuss some sharp estimates for a constant mean curvature graph $\Sigma$ in a Riemannian 3-manifold $M\times\mathbb{R}$ whose boundary $\partial\Sigma$ is contained in a slice $M\times\{t_0\}$ and satisfies a capillarity condition. We start by giving sharp lower bounds for the geodesic curvature of the boundary and improve these bounds when assuming additional restrictions on the maximum height attained by the graph in $M\times\mathbb{R}$. We also give a bound for the distance from an interior point to the boundary in terms of the height at that point, and characterize when these bounds are attained.
Differential geometry
Calculus of variations and optimal control; optimization
1263
1281
10.4171/RMI/756
http://www.ems-ph.org/doi/10.4171/RMI/756
On complete submanifolds with parallel mean curvature in product spaces
Dorel
Fetcu
Universitatea Tehnică “Gheorghe Asachi” din Iaşi, IAŞI, ROMANIA
Harold
Rosenberg
, RIO DE JANEIRO, BRAZIL
Submanifolds with parallel mean curvature vector field, Simons type equation
We prove a Simons type formula for submanifolds with parallel mean curvature vector field in product spaces of type $M^n(c)\times\mathbb{R}$, where $M^n(c)$ is a space form with constant sectional curvature $c$, and then we use it to characterize some of these submanifolds.
Differential geometry
1283
1306
10.4171/RMI/757
http://www.ems-ph.org/doi/10.4171/RMI/757
Some two-dimensional extensions of Bougerol’s identity in law for the exponential functional of linear Brownian motion
Jean
Bertoin
Universität Zürich, ZÜRICH, SWITZERLAND
Daniel
Dufresne
University of Melbourne, MELBOURNE, Victoria, AUSTRALIA
Marc
Yor
Université Paris VI, PARIS CEDEX 05, FRANCE
Brownian motion, exponential functional, Bougerol’s identity, local time, Bessel processes
We present a two-dimensional extension of an identity in distribution due to Bougerol [4] that involves the exponential functional of a linear Brownian motion. Even though this identity does not extend to the level of processes, we point out further striking relations in this direction.
Probability theory and stochastic processes
1307
1324
10.4171/RMI/758
http://www.ems-ph.org/doi/10.4171/RMI/758
Commutators, paraproducts and BMO in non-homogeneous martingale settings
Sergei
Treil
Brown University, PROVIDENCE, UNITED STATES
Paraproducts, commutators, BMO
In this paper we investigate the relations between (martingale) BMO spaces, paraproducts and commutators in non-homogeneous martingale settings. Some new, and one might add unexpected, results are obtained. Some alternative proof of known results are also presented.
Fourier analysis
1325
1372
10.4171/RMI/759
http://www.ems-ph.org/doi/10.4171/RMI/759
Normalisers of operator algebras and tensor product formulas
Martin
McGarvey
Queen's University Belfast, BELFAST, NORTHERN IRELAND, UNITED KINGDOM
Lina
Oliveira
Instituto Superior Técnico, LISBOA, PORTUGAL
Ivan
Todorov
Queen's University Belfast, BELFAST, NORTHERN IRELAND, UNITED KINGDOM
CSL algebra, masa-bimodule, nest algebra, normaliser
We establish a tensor product formula for bimodules over maximal abelian self-adjoint algebras and their supports. We use this formula to show that if $\mathcal{A}$ is the tensor product of finitely many continuous nest algebras, $\mathcal{B}$ is a CSL algebra and $\mathcal{A}$ and $\mathcal{B}$ have the same normaliser semigroup then either $\mathcal{A} =\mathcal{B}$ or $\mathcal{ A}^* = \mathcal{B}$. We show that the result does not hold without the assumption that the nests be continuous, answering in the negative a question previously raised in the literature.
Operator theory
Functional analysis
1373
1395
10.4171/RMI/760
http://www.ems-ph.org/doi/10.4171/RMI/760
Exponential growth of rank jumps for A-hypergeometric systems
María-Cruz
Fernández-Fernández
Universidad de Sevilla, SEVILLA, SPAIN
Hypergeometric, $D$-module, holonomic rank
The dimension of the space of holomorphic solutions at nonsingular points (also called the holonomic rank) of an $A$-hypergeometric system $M_A (\beta )$ is known to be bounded above by $ 2^{2d}\operatorname{vol}(A)$, where $d$ is the rank of the matrix $A$ and $\operatorname{vol}(A)$ is its normalized volume. This bound was thought to be much too large because it is exponential in $d$. Indeed, all the examples we have found in the literature satisfy $\operatorname{rank}(M_A (\beta ))1$.
Special functions
Commutative rings and algebras
Algebraic geometry
1397
1404
10.4171/RMI/761
http://www.ems-ph.org/doi/10.4171/RMI/761
Defining functions for unbounded $C^m$ domains
Phillip
Harrington
University of Arkansas, FAYETTEVILLE, UNITED STATES
Andrew
Raich
University of Arkansas, FAYETTEVILLE, UNITED STATES
Defining function, signed distance function, unbounded domains, uniformly $C^m$ defining function
For a domain $\Omega\subset\mathbb{R}^n$, we introduce the concept of a uniformly $C^m$ defining function. We characterize uniformly $C^m$ defining functions in terms of the signed distance function for the boundary and provide a large class of examples of unbounded domains with uniformly $C^m$ defining functions. Some of our results extend results from the bounded case.
Differential geometry
Several complex variables and analytic spaces
Global analysis, analysis on manifolds
1405
1420
10.4171/RMI/762
http://www.ems-ph.org/doi/10.4171/RMI/762
Ground states for pseudo-relativistic Hartree equations of critical type
Vittorio
Coti Zelati
Università degli Studi di Napoli Federico II, NAPOLI, ITALY
Margherita
Nolasco
Università degli Studi dell'Aquila, L'AQUILA, ITALY
Nonlinear Schrödinger equation, pseudo-relativistic Hartree approximation, solitary waves, ground states
We study the existence of ground state solutions for a class of nonlinear pseudo-relativistic Schrödinger equations with critical two-body interactions. Such equations are characterized by a nonlocal pseudo-differential operator closely related to the square root of the Laplacian. We investigate this problem using variational methods after transforming the problem to an elliptic equation with a nonlinear Neumann boundary conditions.
Partial differential equations
1421
1436
10.4171/RMI/763
http://www.ems-ph.org/doi/10.4171/RMI/763
A general form of the weak maximum principle and some applications
Guglielmo
Albanese
Università degli Studi di Milano, MILANO, ITALY
Luis
Alías
Universidad de Murcia, MURCIA, SPAIN
Marco
Rigoli
Università di Milano, MILANO, ITALY
Omori–Yau maximum principle, weak maximum principle, trace type operators, Riemannian manifolds
The aim of this paper is to introduce new forms of the weak and Omori–Yau maximum principles for linear operators, notably for trace type operators, and show their usefulness, for instance, in the context of PDEs and in the theory of hypersurfaces. In the final part of the paper we consider a large class of nonlinear operators and we show that our previous results can be appropriately generalized to this case.
Global analysis, analysis on manifolds
Differential geometry
1437
1476
10.4171/RMI/764
http://www.ems-ph.org/doi/10.4171/RMI/764
Optimal regularizing effect for scalar conservation laws
François
Golse
École Polytechnique, PARIS, FRANCE
Benoît
Perthame
Université Pierre et Marie Curie, PARIS CEDEX 05, FRANCE
Scalar conservation law, compensated compactness, regularizing effect, kinetic formulation
We investigate the regularity of bounded weak solutions of scalar conservation laws with uniformly convex flux in space dimension one, satisfying an entropy condition with entropy production term that is a signed Radon measure. We prove that all such solutions belong to the Besov space $B^{1/3,3}_{\infty,{\rm loc}}$. Since C. de Lellis and M. Westdickenberg [11] have proved the existence of such solutions that do not belong to $B^{s,p}_{q,{\rm loc}}$ if either $s>1/\max(p,3)$ or $s=1/3$ and $1\le q
Partial differential equations
Fluid mechanics
1477
1504
10.4171/RMI/765
http://www.ems-ph.org/doi/10.4171/RMI/765