Revista Matemática Iberoamericana Articles (Project Euclid)

Convergence of metric graphs and energy forms

Thu, 05 Aug 2010 15:41 EDT

Atsushi Kasue

Source: Rev. Mat. Iberoamericana, Volume 26, Number 2, 367--448.

Abstract:
In this paper, we begin with clarifying spaces obtained as limits of sequences of finite networks from an analytic point of view, and we discuss convergence of finite networks with respect to the topology of both the Gromov-Hausdorff distance and variational convergence called $\Gamma$-convergence. Relevantly to convergence of finite networks to infinite ones, we investigate the space of harmonic functions of finite Dirichlet sums on infinite networks and their Kuramochi compactifications.

The Howe dual pair in Hermitean Clifford analysis

Thu, 05 Aug 2010 15:41 EDT

Fred Brackx , Hennie De Schepper , David Eelbode , Vladimir Souček

Source: Rev. Mat. Iberoamericana, Volume 26, Number 2, 449--479.

Abstract:
Clifford analysis offers a higher dimensional function theory studying the null solutions of the rotation invariant, vector valued, first order Dirac operator $\partial$. In the more recent branch Hermitean Clifford analysis, this rotational invariance has been broken by introducing a complex structure $J$ on Euclidean space and a corresponding second Dirac operator $\partial_J$, leading to the system of equations $\partial f = 0 = \partial_J f$ expressing so-called Hermitean monogenicity. The invariance of this system is reduced to the unitary group. In this paper we show that this choice of equations is fully justified. Indeed, constructing the Howe dual for the action of the unitary group on the space of all spinor valued polynomials, the generators of the resulting Lie superalgebra reveal the natural set of equations to be considered in this context, which exactly coincide with the chosen ones.

Riesz transforms on forms and $L^p$-Hodge decomposition on complete Riemannian manifolds

Thu, 05 Aug 2010 15:41 EDT

Xiang-Dong Li

Source: Rev. Mat. Iberoamericana, Volume 26, Number 2, 481--528.

Abstract:
In this paper we prove the Strong $L^p$-stability of the heat semigroup generated by the Hodge Laplacian on complete Riemannian manifolds with non-negative Weitzenböck curvature. Based on a probabilistic representation formula, we obtain an explicit upper bound of the $L^p$-norm of the Riesz transforms on forms on complete Riemannian manifolds with suitable curvature conditions. Moreover, we establish the Weak $L^p$-Hodge decomposition theorem on complete Riemannian manifolds with non-negative Weitzenböck curvature.

On the cluster size distribution for percolation on some general graphs

Thu, 05 Aug 2010 15:41 EDT

Antar Bandyopadhyay , Jeffrey Steif , Ádám Timár

Source: Rev. Mat. Iberoamericana, Volume 26, Number 2, 529--550.

Abstract:
We show that for any Cayley graph, the probability (at any $p$) that the cluster of the origin has size $n$ decays at a well-defined exponential rate (possibly 0). For general graphs, we relate this rate being positive in the supercritical regime with the amenability/nonamenability of the underlying graph.

A priori Hölder estimate, parabolic Harnack principle and heat kernel estimates for diffusions with jumps

Thu, 05 Aug 2010 15:41 EDT

Zhen-Qing Chen , Takashi Kumagai

Source: Rev. Mat. Iberoamericana, Volume 26, Number 2, 551--589.

Abstract:
In this paper, we consider the following type of non-local (pseudo-differential) operators $\mathcal{L}$ on $\mathbb{R}^d$: \begin{align*} \mathcal{L} u(x) = \frac{1}{2} \sum_{i, j=1}^d \frac{\partial}{\partial x_i} \Big(a_{ij}(x) \frac{\partial u(x)}{\partial x_j}\Big) \\+ \lim_{\varepsilon \downarrow 0} \int_{\{y\in \mathbb{R}^d: |y-x|>\varepsilon\}} (u(y)-u(x)) J(x, y) dy, \end{align*} where $A(x)=(a_{ij}(x))_{1\leq i,j\leq d}$ is a measurable $d\times d$ matrix-valued function on $\mathbb{R}^d$ that is uniformly elliptic and bounded and $J$ is a symmetric measurable non-trivial non-negative kernel on $\mathbb{R}^d \times \mathbb{R}^d$ satisfying certain conditions. Corresponding to $\mathcal{L}$ is a symmetric strong Markov process $X$ on $\mathbb{R}^d$ that has both the diffusion component and pure jump component. We establish a priori Hölder estimate for bounded parabolic functions of $\mathcal{L}$ and parabolic Harnack principle for positive parabolic functions of $\mathcal{L}$. Moreover, two-sided sharp heat kernel estimates are derived for such operator $\mathcal{L}$ and jump-diffusion $X$. In particular, our results apply to the mixture of symmetric diffusion of uniformly elliptic divergence form operator and mixed stable-like processes on $\mathbb{R}^d$. To establish these results, we employ methods from both probability theory and analysis.

End-point estimates and multi-parameter paraproducts on higher dimensional tori

Thu, 05 Aug 2010 15:41 EDT

John T. Workman

Source: Rev. Mat. Iberoamericana, Volume 26, Number 2, 591--610.

Abstract:
Analogues of multi-parameter multiplier operators on $\mathbb{R}^d$ are defined on the torus $\mathbb{T}^d$. It is shown that these operators satisfy the classical Coifman-Meyer theorem. In addition, $L(\log L)^n$ end-point estimates are proved.

Socle theory for Leavitt path algebras of arbitrary graphs

Thu, 05 Aug 2010 15:41 EDT

Gonzalo Aranda Pino , Dolores Martín Barquero , Cándido Martín González , Mercedes Siles Molina

Source: Rev. Mat. Iberoamericana, Volume 26, Number 2, 611--638.

Abstract:
The main aim of the paper is to give a socle theory for Leavitt path algebras of arbitrary graphs. We use both the desingularization process and combinatorial methods to study Morita invariant properties concerning the socle and to characterize it, respectively. Leavitt path algebras with nonzero socle are described as those which have line points, and it is shown that the line points generate the socle of a Leavitt path algebra. A concrete description of the socle of a Leavitt path algebra is obtained: it is a direct sum of matrix rings (of finite or infinite size) over the base field. New proofs of the Graded Uniqueness and of the Cuntz-Krieger Uniqueness Theorems are given, by using very different means.

Lowest uniformizations of closed Riemann orbifolds

Thu, 05 Aug 2010 15:41 EDT

Rubén A. Hidalgo

Source: Rev. Mat. Iberoamericana, Volume 26, Number 2, 639--649.

Abstract:
A Kleinian group containing a Schottky group as a finite index subgroup is called a Schottky extension group. If $\Omega$ is the region of discontinuity of a Schottky extension group $K$, then the quotient $\Omega/K$ is a closed Riemann orbifold; called a Schottky orbifold. Closed Riemann surfaces are examples of Schottky orbifolds as a consequence of the Retrosection Theorem. Necessary and sufficient conditions for a Riemann orbifold to be a Schottky orbifold are due to M. Reni and B. Zimmermann in terms of the signature of the orbifold. It is well known that the lowest uniformizations of a closed Riemann surface are exactly those for which the Deck group is a Schottky group. In this paper we extend such a result to the class of Schottky orbifolds, that is, we prove that the lowest uniformizations of a Schottky orbifold are exactly those for which the Deck group is a Schottky extension group.

Bernstein-Heinz-Chern results in calibrated manifolds

Thu, 05 Aug 2010 15:41 EDT

Guanghan Li , Isabel M. C. Salavessa

Source: Rev. Mat. Iberoamericana, Volume 26, Number 2, 651--692.

Abstract:
Given a calibrated Riemannian manifold $\overline{M}$ with parallel calibration $\Omega$ of rank $m$ and $M$ an orientable m-submanifold with parallel mean curvature $H$, we prove that if $\cos\theta$ is bounded away from zero, where $\theta$ is the $\Omega$-angle of $M$, and if $M$ has zero Cheeger constant, then $M$ is minimal. In the particular case $M$ is complete with $Ricci^M\geq 0$ we may replace the boundedness condition on $\cos\theta$ by $\cos\theta\geq Cr^{-\beta}$, when $r\rightarrow+\infty$, where $0 < \beta < 1$ and $C > 0$ are constants and $r$ is the distance function to a point in $M$. Our proof is surprisingly simple and extends to a very large class of submanifolds in calibrated manifolds, in a unified way, the problem started by Heinz and Chern of estimating the mean curvature of graphic hypersurfaces in Euclidean spaces. It is based on an estimation of $\|H\|$ in terms of $\cos\theta$ and an isoperimetric inequality. In a similar way, we also give some conditions to conclude $M$ is totally geodesic. We study some particular cases.

Toeplitz operators on Bergman spaces with locally integrable symbols

Thu, 05 Aug 2010 15:41 EDT

Jari Taskinen , Jani Virtanen

Source: Rev. Mat. Iberoamericana, Volume 26, Number 2, 693--706.

Abstract:
We study the boundedness of Toeplitz operators $T_a$ with locally integrable symbols on Bergman spaces $A^p(\mathbb{D})$, $1 < p < \infty$. Our main result gives a sufficient condition for the boundedness of $T_a$ in terms of some ``averages'' (related to hyperbolic rectangles) of its symbol. If the averages satisfy an ${o}$-type condition on the boundary of $\mathbb{D}$, we show that the corresponding Toeplitz operator is compact on $A^p$. Both conditions coincide with the known necessary conditions in the case of nonnegative symbols and $p=2$. We also show that Toeplitz operators with symbols of vanishing mean oscillation are Fredholm on $A^p$ provided that the averages are bounded away from zero, and derive an index formula for these operators.

A convolution estimate for two-dimensional hypersurfaces

Thu, 05 Aug 2010 15:41 EDT

Ioan Bejenaru , Sebastian Herr , Daniel Tataru

Source: Rev. Mat. Iberoamericana, Volume 26, Number 2, 707--728.

Abstract:
Given three transversal and sufficiently regular hypersurfaces in $\mathbb{R}^3$ it follows from work of Bennett-Carbery-Wright that the convolution of two $L^2$ functions supported of the first and second hypersurface, respectively, can be restricted to an $L^2$ function on the third hypersurface, which can be considered as a nonlinear version of the Loomis-Whitney inequality. We generalize this result to a class of $C^{1,\beta}$ hypersurfaces in $\mathbb{R}^3$, under scaleable assumptions. The resulting uniform $L^2$ estimate has applications to nonlinear dispersive equations.

Topological and analytical properties of Sobolev bundles. II. Higher dimensional cases

Fri, 27 Aug 2010 08:57 EDT

Takeshi Isobe

Source: Rev. Mat. Iberoamericana, Volume 26, Number 3, 729--798.

Abstract:
We define various classes of Sobolev bundles and connections and study their topological and analytical properties. We show that certain kinds of topologies (which depend on the classes) are well-defined for such bundles and they are stable with respect to the natural Sobolev topologies. We also extend the classical Chern-Weil theory for such classes of bundles and connections. Applications related to variational problems for the Yang-Mills functional are also given.

Contact properties of codimension 2 submanifolds with flat normal bundle

Fri, 27 Aug 2010 08:57 EDT

J. J. Nuño-Ballesteros , M. C. Romero-Fuster

Source: Rev. Mat. Iberoamericana, Volume 26, Number 3, 799--824.

Abstract:
Given an immersed submanifold $M^n\subset\mathbb{R}^{n+2}$, we characterize the vanishing of the normal curvature $R_D$ at a point $p \in M$ in terms of the behaviour of the asymptotic directions and the curvature locus at $p$. We relate the affine properties of codimension 2 submanifolds with flat normal bundle with the conformal properties of hypersurfaces in Euclidean space. We also characterize the semiumbilical, hypespherical and conformally flat submanifolds of codimension 2 in terms of their curvature loci.

Le Théorème du symbole total d'un opérateur différentiel $p$-adique

Fri, 27 Aug 2010 08:57 EDT

Zoghman Mebkhout , Luis Narváez Macarro

Source: Rev. Mat. Iberoamericana, Volume 26, Number 3, 825--859.

Abstract:
Let ${\mathcal X}^\dagger$ be a smooth $\dagger$-scheme (in the sense of Meredith) over a complete discrete valuation ring $(V, {\mathfrak m})$ of unequal characteristics $(0,p)$ and let ${\mathcal D}^\dagger_{{\mathcal X}^\dagger/V}$ be the sheaf of $V$-linear endomorphisms of ${\mathcal O}_{{\mathcal X}^\dagger}$ whose reduction modulo ${\mathfrak m}^s$ is a linear differential operator of order bounded by an affine function in $s$. In this paper we prove that locally there is an ${\mathcal O}_{{\mathcal X}^\dagger}$-isomorphism between the sections of ${\mathcal D}^\dagger_{{\mathcal X}^\dagger/V}$ and the overconvergent total symbols, and we deduce a cohomological triviality property.

The $(L^1,L^1)$ bilinear Hardy-Littlewood function and Furstenberg averages

Fri, 27 Aug 2010 08:57 EDT

Idris Assani , Zoltán Buczolich

Source: Rev. Mat. Iberoamericana, Volume 26, Number 3, 861--890.

Abstract:
Let $(X,\mathcal{B}, \mu, T)$ be an ergodic dynamical system on a non-atomic finite measure space. Consider the maximal function $$ R^* : (f, g) \in L^1 \times L^1 \rightarrow R^*(f, g)(x) = \sup_{n} \frac{f(T^n x) g(T^{2n} x)}{n}. $$ We show that there exist $f$ and $g$ such that $R^*(f, g)(x)$ is not finite almost everywhere. Two consequences are derived. The bilinear Hardy-Littlewood maximal function fails to be a.e. finite for all functions $(f, g)\in L^1\times L^1$. The Furstenberg averages do not converge for all pairs of $(L^1,L^1)$ functions, while by a result of J. Bourgain these averages converge for all pairs of $(L^p,L^q)$ functions with $\frac{1}{p}+\frac{1}{q} \leq 1$.

Aronson-Bénilan type estimate and the optimal Hölder continuity of weak solutions for the 1-D degenerate Keller-Segel systems

Fri, 27 Aug 2010 08:57 EDT

Yoshie Sugiyama

Source: Rev. Mat. Iberoamericana, Volume 26, Number 3, 891--913.

Abstract:
We consider the Keller-Segel system of degenerate type (KS)$_m$ with $m > 1$ below. We establish a uniform estimate of $\partial_x^2 u^{m-1}$ from below. The corresponding estimate to the porous medium equation is well-known as an Aronson-Bénilan type. We apply our estimate to prove the optimal Hölder continuity of weak solutions of (KS)$_m$. In addition, we find that the set $D(t):=\{ x \in \mathbb{R}; u(x,t) > 0\}$ of positive region to the solution $u$ is monotonically non-decreasing with respect to $t$.

Well-posedness in critical spaces for the compressible Navier-Stokes equations with density dependent viscosities

Fri, 27 Aug 2010 08:57 EDT

Qionglei Chen , Changxing Miao , Zhifei Zhang

Source: Rev. Mat. Iberoamericana, Volume 26, Number 3, 915--946.

Abstract:
In this paper, we prove the local well-posedness in critical Besov spaces for the compressible Navier-Stokes equations with density dependent viscosities under the assumption that the initial density is bounded away from zero.

On some maximal multipliers in $L^p$

Fri, 27 Aug 2010 08:57 EDT

Ciprian Demeter

Source: Rev. Mat. Iberoamericana, Volume 26, Number 3, 947--964.

Abstract:
We extend an $L^2$ maximal multiplier result of Bourgain to all $L^p$ spaces, $1 < p < \infty$.

Overdetermined problems in unbounded domains with Lipschitz singularities

Fri, 27 Aug 2010 08:57 EDT

Alberto Farina , Enrico Valdinoci

Source: Rev. Mat. Iberoamericana, Volume 26, Number 3, 965--974.

Abstract:
We study the overdetermined problem $$ \left\{ \begin{array}{cc} \Delta u + f(u) = 0 & \mbox{ in $\Omega$,} \\ u = 0 & \mbox{ on $\partial\Omega$,} \\ \partial_\nu u = c & \mbox{ on $\Gamma$,} \end{array} \right. $$ where $\Omega$ is a locally Lipschitz epigraph, that is $C^3$ on $\Gamma\subseteq\partial\Omega$, with $\partial\Omega\setminus\Gamma$ consisting in nonaccumulating, countably many points. We provide a geometric inequality that allows us to deduce geometric properties of the sets $\Omega$ for which monotone solutions exist. In particular, if $\mathcal{C} \in \mathbb{R}^n$ is a cone and either $n=2$ or $n=3$ and $f \ge 0$, then there exists no solution of $$ \left\{ \begin{array}{cc} \Delta u + f(u) = 0 & \mbox{ in $\mathcal{C}$,} \\ u > 0 & \mbox{ in $\mathcal{C}$,} \\ u = 0 & \mbox{ on $\partial\mathcal{C}$,} \\ \partial_\nu u = c & \mbox{ on $\partial\mathcal{C} \setminus \{0\}$.} \end{array} \right. $$ This answers a question raised by Juan Luis Vázquez.

Loewner chains in the unit disk

Fri, 27 Aug 2010 08:57 EDT

Manuel D. Contreras , Santiago Díaz-Madrigal , Pavel Gumenyuk

Source: Rev. Mat. Iberoamericana, Volume 26, Number 3, 975--1012.

Abstract:
In this paper we introduce a general version of the notion of Loewner chains which comes from the new and unified treatment, given in [Bracci, F., Contreras, M.D. and Díaz-Madrigal, S.: Evolution families and the Loewner equation I: the unit disk. To appear in J. Reine Angew. Math.] of the radial and chordal variant of the Loewner differential equation, which is of special interest in geometric function theory as well as for various developments it has given rise to, including the famous Schramm-Loewner evolution. In this very general setting, we establish a deep correspondence between these chains and the evolution families introduced in [Bracci, F., Contreras, M.D. and Díaz-Madrigal, S.: Evolution families and the Loewner equation I: the unit disk. To appear in J. Reine Angew. Math.]. Among other things, we show that, up to a Riemann map, such a correspondence is one-to-one. In a similar way as in the classical Loewner theory, we also prove that these chains are solutions of a certain partial differential equation which resembles (and includes as a very particular case) the classical Loewner-Kufarev PDE.

Elliptic equations in the plane satisfying a Carleson measure condition

Fri, 27 Aug 2010 08:57 EDT

Martin Dindoš , David J. Rule

Source: Rev. Mat. Iberoamericana, Volume 26, Number 3, 1013--1034.

Abstract:
In this paper we settle (in dimension $n=2$) the open question whether for a divergence form equation $\div (A\nabla u) = 0$ with coefficients satisfying certain minimal smoothness assumption (a Carleson measure condition), the $L^p$ Neumann and Dirichlet regularity problems are solvable for some values of $p\in (1,\infty)$. The related question for the $L^p$ Dirichlet problem was settled (in any dimension) in 2001 by Kenig and Pipher [Kenig, C.E. and Pipher, J.: The Dirichlet problem for elliptic equations with drift terms. Publ. Mat. 45 (2001), no. 1, 199-217].

A counterexample for the geometric traveling salesman problem in the Heisenberg group

Fri, 27 Aug 2010 08:57 EDT

Nicolas Juillet

Source: Rev. Mat. Iberoamericana, Volume 26, Number 3, 1035--1056.

Abstract:
We are interested in characterizing the compact sets of the Heisenberg group that are contained in a curve of finite length. Ferrari, Franchi and Pajot recently gave a sufficient condition for those sets, adapting a necessary and sufficient condition due to P. Jones in the Euclidean setting. We prove that this condition is not necessary.

Maps from Riemannian manifolds into non-degenerate Euclidean cones

Fri, 27 Aug 2010 08:57 EDT

Luciano Mari , Marco Rigoli

Source: Rev. Mat. Iberoamericana, Volume 26, Number 3, 1057--1074.

Abstract:
Let $M$ be a connected, non-compact $m$-dimensional Riemannian manifold. In this paper we consider smooth maps $\varphi: M \rightarrow \mathbb{R}^n$ with images inside a non-degenerate cone. Under quite general assumptions on $M$, we provide a lower bound for the width of the cone in terms of the energy and the tension of $\varphi$ and a metric parameter. As a side product, we recover some well known results concerning harmonic maps, minimal immersions and Kähler submanifolds. In case $\varphi$ is an isometric immersion, we also show that, if $M$ is sufficiently well-behaved and has non-positive sectional curvature, $\varphi(M)$ cannot be contained into a non-degenerate cone of $\mathbb{R}^{2m-1}$.

The $C^m$ Norm of a Function with Prescribed Jets I

Fri, 27 Aug 2010 08:57 EDT

Charles Fefferman

Source: Rev. Mat. Iberoamericana, Volume 26, Number 3, 1075--1098.

Abstract:
We prove a variant of the classical Whitney extension theorem, in which the $C^m$-norm of the extending function is controlled up to a given, small percentage error.

Global existence for the primitive equations with small anisotropic viscosity

Fri, 04 Feb 2011 09:13 EST

Frédéric Charve , Van-Sang Ngo

Source: Rev. Mat. Iberoamericana, Volume 27, Number 1, 1--38.

Abstract:
In this paper, we consider the primitive equations with zero vertical viscosity, zero vertical thermal diffusivity, and the horizontal viscosity and horizontal thermal diffusivity of size $\varepsilon^\alpha$ where $0 < \alpha < \alpha_0$. We prove the global existence of a unique strong solution for large data provided that the Rossby number is small enough (the rotation and the vertical stratification are large).

Le théorème du symbole total d'un opérateur différentiel $p$-adique d'échelon $h\geq0$

Fri, 04 Feb 2011 09:13 EST

Zoghman Mebkhout

Source: Rev. Mat. Iberoamericana, Volume 27, Number 1, 39--92.

Abstract:
In this article we prove the total symbol theorem for the $p$-adic differential operators of degree $h\geq 0$ for the echelon filtration and the noetherianity of the ring of the $p$-adic differential operators of degree $h\geq 0$ for the echelon filtration over a $\dagger$-adic affine smooth scheme small enough.

Isoperimetry for spherically symmetric log-concave probability measures

Fri, 04 Feb 2011 09:13 EST

Nolwen Huet

Source: Rev. Mat. Iberoamericana, Volume 27, Number 1, 93--122.

Abstract:
We prove an isoperimetric inequality for probability measures $\mu$ on $\mathbb{R}^n$ with density proportional to $\exp(-\phi(\lambda |x|))$, where $|x|$ is the euclidean norm on $\mathbb{R}^n$ and $\phi$ is a non-decreasing convex function. It applies in particular when $\phi(x)=x^\alpha$ with $\alpha \ge 1$. Under mild assumptions on $\phi$, the inequality is dimension-free if $\lambda$ is chosen such that the covariance of $\mu$ is the identity.

Universal objects in categories of reproducing kernels

Fri, 04 Feb 2011 09:13 EST

Daniel Beltiţă , José E. Galé

Source: Rev. Mat. Iberoamericana, Volume 27, Number 1, 123--179.

Abstract:
We continue our earlier investigation on generalized reproducing kernels, in connection with the complex geometry of $C^*$- algebra representations, by looking at them as the objects of an appropriate category. Thus the correspondence between reproducing $(-*)$-kernels and the associated Hilbert spaces of sections of vector bundles is made into a functor. We construct reproducing $(-*)$-kernels with universality properties with respect to the operation of pull-back. We show how completely positive maps can be regarded as pull-backs of universal ones linked to the tautological bundle over the Grassmann manifold of the Hilbert space $\ell^2(\mathbb{N})$.

Tropical plane geometric constructions: a transfer technique in Tropical Geometry

Fri, 04 Feb 2011 09:13 EST

Luis Felipe Tabera

Source: Rev. Mat. Iberoamericana, Volume 27, Number 1, 181--232.

Abstract:
The notion of geometric construction is introduced. This notion allows to compare incidence configurations both lying in the algebraic and the tropical plane. We provide sufficient conditions in a geometric construction to ensure that there is always an algebraic counterpart related by tropicalization. We also present some results to detect if this algebraic counterpart cannot exist. With these tools, geometric constructions are applied to transfer classical theorems to the tropical framework, we provide a notion of "constructible incidence theorem" and then several tropical versions of classical theorems are proved such as the converse of Pascal's, Fano's or Cayley-Bacharach theorems.

Regularity for solutions of the total variation denoising problem

Fri, 04 Feb 2011 09:13 EST

Vicent Caselles , Antonin Chambolle , Matteo Novaga

Source: Rev. Mat. Iberoamericana, Volume 27, Number 1, 233--252.

Abstract:
The main purpose of this paper is to prove a local Hölder regularity result for the solutions of the total variation based denoising problem assuming that the datum is locally Hölder continuous. We also prove a global estimate on the modulus of continuity of the solution in convex domains of $\mathbb{R}^N$ and some extensions of this result for the total variation minimization flow.

Cluster solutions for the Schrödinger-Poisson-Slater problem around a local minimum of the potential

Fri, 04 Feb 2011 09:13 EST

David Ruiz , Giusi Vaira

Source: Rev. Mat. Iberoamericana, Volume 27, Number 1, 253--271.

Abstract:
In this paper we consider the system in $\mathbb{R}^3$ \begin{equation} \left\{ \begin{array}{l} -\varepsilon^2 \Delta u + V(x)u + \phi(x)u = u^p, \\ -\Delta \phi = u^2, \end{array} \right. \end{equation} for $p\in (1,5)$. We prove the existence of multi-bump solutions whose bumps concentrate around a local minimum of the potential $V(x)$. We point out that such solutions do not exist in the framework of the usual Nonlinear Schrödinger Equation.

Construction of multi-soliton solutions for the $L^2$-supercritical gKdV and NLS equations

Fri, 04 Feb 2011 09:13 EST

Raphaël Côte , Yvan Martel , Frank Merle

Source: Rev. Mat. Iberoamericana, Volume 27, Number 1, 273--302.

Abstract:
Multi-soliton solutions, i.e. solutions behaving as the sum of $N$ given solitons as $t \to +\infty$, were constructed for the $L^2$ critical and subcritical (NLS) and (gKdV) equations in previous works (see [Merle, F.: Construction of solutions with exactly $k$ blow-up points for the Schrödinger equation with critical nonlinearity. Comm. Math. Phys. 129 (1990), no. 2, 223-240], [Martel, Y.: Asymptotic $N$-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations. Amer. J. Math. 127 (2005), no. 5, 1103-1140] and [Martel, Y. and Merle, F.: Multi solitary waves for nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 23 (2006), 849-864]). In this paper, we extend the construction of multi-soliton solutions to the $L^2$ supercritical case both for (gKdV) and (NLS) equations, using a topological argument to control the direction of instability.

Constant curvature foliations in asymptotically hyperbolic spaces

Fri, 04 Feb 2011 09:13 EST

Rafe Mazzeo , Frank Pacard

Source: Rev. Mat. Iberoamericana, Volume 27, Number 1, 303--333.

Abstract:
Let $(M,g)$ be an asymptotically hyperbolic manifold with a smooth conformal compactification. We establish a general correspondence between semilinear elliptic equations of scalar curvature type on $\partial M$ and Weingarten foliations in some neighbourhood of infinity in $M$. We focus mostly on foliations where each leaf has constant mean curvature, though our results apply equally well to foliations where the leaves have constant $\sigma_k$-curvature. In particular, we prove the existence of a unique foliation near infinity in any quasi-Fuchsian 3-manifold by surfaces with constant Gauss curvature. There is a subtle interplay between the precise terms in the expansion for $g$ and various properties of the foliation. Unlike other recent works in this area, by Rigger ([The foliation of asymptotically hyperbolic manifolds by surfaces of constant mean curvature (including the evolution equations and estimates). Manuscripta Math. 113 (2004), 403-421]) and Neves-Tian ([Existence and uniqueness of constant mean curvature foliation of asymptotically hyperbolic 3-manifolds. Geom. Funct. Anal. 19 (2009), no.3, 910-942], [Existence and uniqueness of constant mean curvature foliation of asymptotically hyperbolic 3-manifolds. II. J. Reine Angew. Math. 641 (2010), 69-93]), we work in the context of conformally compact spaces, which are more general than perturbations of the AdS-Schwarzschild space, but we do assume a nondegeneracy condition.

Strong $A_\infty$-weights are $A_\infty$-weights on metric spaces

Fri, 04 Feb 2011 09:13 EST

Riikka Korte , Outi Elina Kansanen

Source: Rev. Mat. Iberoamericana, Volume 27, Number 1, 335--354.

Abstract:
We prove that every strong $A_\infty$-weight is a Muckenhoupt weight in Ahlfors-regular metric measure spaces that support a Poincaré inequality. We also explore the relations between various definitions for $A_\infty$-weights in this setting, since some of these characterizations are needed in the proof of the main result.

The Jet of an Interpolant on a Finite Set

Fri, 04 Feb 2011 09:13 EST

Charles Fefferman , Arie Israel

Source: Rev. Mat. Iberoamericana, Volume 27, Number 1, 355--360.

Abstract:
We study functions $F \in C^m (\mathbb{R}^n)$ having norm less than a given constant $M$, and agreeing with a given function $f$ on a finite set $E$. Let $\Gamma_f (S,M)$ denote the convex set formed by taking the $(m-1)$-jets of all such $F$ at a given finite set $S \subset \mathbb{R}^n$. We provide an efficient algorithm to compute a convex polyhedron $\tilde{\Gamma}_f (S,M)$, such that $$ \Gamma_f (S,cM) \subset \tilde{\Gamma}_f (S,M) \subset \Gamma_f (S,CM), $ where $c$ and $C$ depend only on $m$ and $n$.

$L^2$ boundedness for maximal commutators with rough variable kernels

Fri, 10 Jun 2011 09:37 EDT

Yanping Chen , Yong Ding , Ran Li

Source: Rev. Mat. Iberoamericana, Volume 27, Number 2, 361--391.

Abstract:
For $b\in BMO(\mathbb{R}^n)$ and $k\in\mathbb{N}$, the $k$-th order maximal commutator of the singular integral operator $T$ with rough variable kernels is defined by $$ T^{\ast}_{b,k}f(x) = \sup_{\varepsilon > 0} \biggl| \int_{|x-y| > \varepsilon} \frac{\Omega(x,x-y)}{|x-y|^n} (b(x)-b(y))^{k} f(y) dy \biggl|. $$ In this paper the authors prove that the $k$-th order maximal commutator $T^{\ast}_{b,k}$ is a bounded operator on $L^2(\mathbb{R}^n)$ if $\Omega$ satisfies the same conditions given by Calderón and Zygmund. Moreover, the $L^2$-boundedness of the $k$-th order commutator of the rough maximal operator $M_\Omega$ with variable kernel, which is defined by $$ M_{\Omega;b,k}f(x) = \sup_{r > 0} \dfrac{1}{r^{n}} \int_{|x-y| < r} |\Omega(x,x-y)| |b(x)-b(y)|^{k} |f(y)| dy, $$ is also given here. These results obtained in this paper are substantial improvement and extension of some known results.

A new hypoelliptic operator on almost CR manifolds

Fri, 10 Jun 2011 09:37 EDT

Raphaël Ponge

Source: Rev. Mat. Iberoamericana, Volume 27, Number 2, 393--414.

Abstract:
The aim of this paper is to present the construction, out of the Kohn-Rossi complex, of a new hypoelliptic operator $Q_L$ on almost CR manifolds equipped with a real structure. The operator acts on all $(p,q)$-forms, but when restricted to $(p,0)$-forms and $(p,n)$-forms it is a sum of squares up to sign factor and lower order terms. Therefore, only a finite type condition condition is needed to have hypoellipticity on those forms. However, outside these forms $Q_L$ may fail to be hypoelliptic, as it is shown in the example of the Heisenberg group $\mathbb{H}^{5}$.

Coefficient multipliers on Banach spaces of analytic functions

Fri, 10 Jun 2011 09:37 EDT

Óscar Blasco , Miroslav Pavlović

Source: Rev. Mat. Iberoamericana, Volume 27, Number 2, 415--447.

Abstract:
Motivated by an old paper of Wells [J. London Math. Soc. {\bf 2} (1970), 549-556] we define the space $X\otimes Y$, where $X$ and $Y$ are "homogeneous" Banach spaces of analytic functions on the unit disk $\mathbb{D}$, by the requirement that $f$ can be represented as $f=\sum_{j=0}^\infty g_n * h_n$, with $g_n\in X$, $h_n\in Y$ and $\sum_{n=1}^\infty \|g_n\|_X \|h_n\|_Y < \infty$. We show that this construction is closely related to coefficient multipliers. For example, we prove the formula $((X\otimes Y),Z)=(X,(Y,Z))$, where $(U,V)$ denotes the space of multipliers from $U$ to $V$, and as a special case $(X\otimes Y)^*=(X,Y^*)$, where $U^*=(U,H^\infty)$. We determine $H^1\otimes X$ for a class of spaces that contains $H^p$ and $\ell^p$ $(1\le p\le 2)$, and use this together with the above formulas to give quick proofs of some important results on multipliers due to Hardy and Littlewood, Zygmund and Stein, and others.

The $\varepsilon$-strategy in variational analysis: illustration with the closed convexification of a function

Fri, 10 Jun 2011 09:37 EDT

Jean-Baptiste Hiriart-Urruty , Marco A. López , Michel Volle

Source: Rev. Mat. Iberoamericana, Volume 27, Number 2, 449--474.

Abstract:
In this work, we concentrate our interest and efforts on general variational (or optimization) problems which do not have solutions necessarily, but which do have approximate solutions (or solutions within $\varepsilon > 0$). We shall see how to recover all the (exact) minimizers of the relaxed version of the original problem (by closed-convexification of the objective function) in terms of the $\varepsilon $-minimizers of the original problem. Applications to two approximation problems in a Hilbert space setting will be shown.

Quantitative uniqueness for second order elliptic operators with strongly singular coefficients

Fri, 10 Jun 2011 09:37 EDT

Ching-Lung Lin , Gen Nakamura , Jenn-Nan Wang

Source: Rev. Mat. Iberoamericana, Volume 27, Number 2, 475--491.

Abstract:
In this paper we study the local behavior of a solution to second order elliptic operators with sharp singular coefficients in lower order terms. One of the main results is the bound on the vanishing order of the solution, which is a quantitative estimate of the strong unique continuation property. Our proof relies on Carleman estimates with carefully chosen phases. A key strategy in the proof is to derive doubling inequalities via three-sphere inequalities. Our method can also be applied to certain elliptic systems with similar singular coefficients.

High-dimensional Menger-type curvatures. Part I: Geometric multipoles and multiscale inequalities

Fri, 10 Jun 2011 09:37 EDT

Gilad Lerman , J. Tyler Whitehouse

Source: Rev. Mat. Iberoamericana, Volume 27, Number 2, 493--555.

Abstract:
We define discrete and continuous Menger-type curvatures. The discrete curvature scales the volume of a $(d+1)$-simplex in a real separable Hilbert space $H$, whereas the continuous curvature integrates the square of the discrete one according to products of a given measure (or its restriction to balls). The essence of this paper is to establish an upper bound on the continuous Menger-type curvature of an Ahlfors regular measure $\mu$ on $H$ in terms of the Jones-type flatness of $\mu$ (which adds up scaled errors of approximations of $\mu$ by $d$-planes at different scales and locations). As a consequence of this result we obtain that uniformly rectifiable measures satisfy a Carleson-type estimate in terms of the Menger-type curvature. Our strategy combines discrete and integral multiscale inequalities for the polar sine with the "geometric multipoles" construction, which is a multiway analog of the well-known method of fast multipoles.

Pseudo-localisation of singular integrals in $L^p$

Fri, 10 Jun 2011 09:37 EDT

Tuomas P. Hytönen

Source: Rev. Mat. Iberoamericana, Volume 27, Number 2, 557--584.

Abstract:
As a step in developing a non-commutative Calderón-Zygmund theory, J. Parcet (J. Funct. Anal. {\bf 256} (2009), no. 2, 509-593) established a new pseudo-localisation principle for classical singular integrals, showing that $Tf$ has small $L^2$ norm outside a set which only depends on $f\in L^2$ but not on the arbitrary normalised Calderón-Zygmund operator $T$. Parcet also asked if a similar result holds true in $L^p$ for $p\in(1,\infty)$. This is answered in the affirmative in the present paper. The proof, which is based on martingale techniques, even somewhat improves on the original $L^2$ result.

A proof of hypoellipticity for Kohn's operator via FBI

Fri, 10 Jun 2011 09:37 EDT

Gregorio Chinni

Source: Rev. Mat. Iberoamericana, Volume 27, Number 2, 585--604.

Abstract:
A new proof of both analytic and $C^{\infty} $ hypoellipticity of Kohn's operator is given using FBI techniques introduced by J. Sjöstrand. The same proof allows us to obtain both kind of hypoellipticity at the same time.

Submetries vs. submersions

Fri, 10 Jun 2011 09:37 EDT

Luis Guijarro , Gerard Walschap

Source: Rev. Mat. Iberoamericana, Volume 27, Number 2, 605--619.

Abstract:
We study submetries between Alexandrov spaces and show how some of the usual features of Riemannian submersions fail due to the lack of smoothness.

The center of a Leavitt path algebra

Fri, 10 Jun 2011 09:37 EDT

Gonzalo Aranda Pino , Kathi Crow

Source: Rev. Mat. Iberoamericana, Volume 27, Number 2, 621--644.

Abstract:
In this paper the center of a Leavitt path algebra is computed for a wide range of situations. A basis as a $K$-vector space is found for $Z(L(E))$ when $L(E)$ enjoys some finiteness condition such as being artinian, semisimple, noetherian and locally noetherian. The main result of the paper states that a simple Leavitt path algebra $L(E)$ is central (i.e. the center reduces to the base field $K$) when $L(E)$ is unital and has zero center otherwise. Finally, this result is extended, under some mild conditions, to the case of exchange Leavitt path algebras.

Multi-parameter Carnot-Carathéodory balls and the theorem of Frobenius

Fri, 10 Jun 2011 09:37 EDT

Brian Street

Source: Rev. Mat. Iberoamericana, Volume 27, Number 2, 645--732.

Abstract:
We study multi-parameter Carnot-Carathéodory balls, generalizing results due to Nagel, Stein and Wainger in the single parameter setting. The main technical result is seen as a uniform version of the theorem of Frobenius. In addition, we study maximal functions associated to certain multi-parameter families of Carnot-Carathéodory balls.

Finiteness of endomorphism algebras of CM modular abelian varieties

Tue, 09 Aug 2011 12:19 EDT

Josep González

Source: Rev. Mat. Iberoamericana, Volume 27, Number 3, 733--750.

Abstract:
Let $A_f$ be the abelian variety attached by Shimura to a normalized newform $f\in S_2(\Gamma_1(N))^{\operatorname{new}}$. We prove that for any integer $n > 1$ the set of pairs of endomorphism algebras $\big( \operatorname{End}_{\overline{\mathbb{Q}}}(A_f) \otimes \mathbb{Q}, \operatorname{End}_\mathbb{Q}(A_f) \otimes \mathbb{Q} \big)$ obtained from all normalized newforms $f$ with complex multiplication such that $\dim A_f=n$ is finite. We determine that this set has exactly 83 pairs for the particular case $n=2$ and show all of them. We also discuss a conjecture related to the finiteness of the set of number fields $\operatorname{End}_\mathbb{Q}(A_f) \otimes \mathbb{Q}$ for the non-CM case.

Partial regularity for subquadratic parabolic systems by $\mathcal{A}$-caloric approximation

Tue, 09 Aug 2011 12:19 EDT

Christoph Scheven

Source: Rev. Mat. Iberoamericana, Volume 27, Number 3, 751--801.

Abstract:
We establish a partial regularity result for weak solutions of nonsingular parabolic systems with subquadratic growth of the type $$ \partial_t u - \mathrm{div} a(x,t,u,Du) = B(x,t,u,Du), $$ where the structure function $a$ satisfies ellipticity and growth conditions with growth rate $\frac{2n}{n+2} < p < 2$. We prove Hölder continuity of the spatial gradient of solutions away from a negligible set. The proof is based on a variant of a harmonic type approximation lemma adapted to parabolic systems with subquadratic growth.

Regularity, local behavior and partial uniqueness for self-similar profiles of Smoluchowski's coagulation equation

Tue, 09 Aug 2011 12:19 EDT

José A. Cañizo , Stéphane Mischler

Source: Rev. Mat. Iberoamericana, Volume 27, Number 3, 803--839.

Abstract:
We consider Smoluchowski's equation with a homogeneous kernel of the form $a(x,y) = x^\alpha y ^\beta + x^\beta y^\alpha$ with $-1 < \alpha \leq \beta < 1$ and $\lambda := \alpha + \beta \in (-1,1)$. We first show that self-similar solutions of this equation are infinitely differentiable and prove sharp results on the behavior of self-similar profiles at $y = 0$ in the case $\alpha < 0$. We also give some partial uniqueness results for self-similar profiles: in the case $\alpha = 0$ we prove that two profiles with the same mass and moment of order $\lambda$ are necessarily equal, while in the case $\alpha < 0$ we prove that two profiles with the same moments of order $\alpha$ and $\beta$, and which are asymptotic at $y = 0$, are equal. Our methods include a new representation of the coagulation operator, and estimates of its regularity using derivatives of fractional order.

Harmonic polynomials and tangent measures of harmonic measure

Tue, 09 Aug 2011 12:19 EDT

Matthew Badger

Source: Rev. Mat. Iberoamericana, Volume 27, Number 3, 841--870.

Abstract:
We show that on an NTA domain if each tangent measure to harmonic measure at a point is a polynomial harmonic measure then the associated polynomials are homogeneous. Geometric information for solutions of a two-phase free boundary problem studied by Kenig and Toro is derived.

Auslander bounds and homological conjectures

Tue, 09 Aug 2011 12:19 EDT

Jiaqun Wei

Source: Rev. Mat. Iberoamericana, Volume 27, Number 3, 871--884.

Abstract:
Inspired by recent works on rings satisfying Auslander's conjecture, we study invariants, called Auslander bounds, and prove that they have strong relations to some homological conjectures.

Nonnegative solutions of the heat equation on rotationally symmetric Riemannian manifolds and semismall perturbations

Tue, 09 Aug 2011 12:19 EDT

Minoru Murata

Source: Rev. Mat. Iberoamericana, Volume 27, Number 3, 885--907.

Abstract:
Let $M$ be a rotationally symmetric Riemannian manifold, and $\Delta$ be the Laplace-Beltrami operator on $M$. We establish a necessary and sufficient condition for the constant function 1 to be a semismall perturbation of $-\Delta +1$ on $M$, and give optimal sufficient conditions for uniqueness of nonnegative solutions of the Cauchy problem to the heat equation. As an application, we determine the structure of all nonnegative solutions to the heat equation on $M\times(0,T)$.

Steiner and Schwarz symmetrization in warped products and fiber bundles with density

Tue, 09 Aug 2011 12:19 EDT

Frank Morgan , Sean Howe , Nate Harman

Source: Rev. Mat. Iberoamericana, Volume 27, Number 3, 909--918.

Abstract:
We provide very general symmetrization theorems in arbitrary dimension and codimension, in products, warped products, and certain fiber bundles such as lens spaces, including Steiner, Schwarz, and spherical symmetrization and admitting density.

On the interplay between Lorentzian Causality and Finsler metrics of Randers type

Tue, 09 Aug 2011 12:19 EDT

Erasmo Caponio , Miguel Ángel Javaloyes , Miguel Sánchez

Source: Rev. Mat. Iberoamericana, Volume 27, Number 3, 919--952.

Abstract:
We obtain some results in both Lorentz and Finsler geometries, by using a correspondence between the conformal structure (Causality) of standard stationary spacetimes on $M = \mathbb{R} \times S$ and Randers metrics on $S$. In particular: (1) For stationary spacetimes: we give a simple characterization of when $\mathbb{R} \times S$ is causally continuous or globally hyperbolic (including in the latter case, when $S$ is a Cauchy hypersurface), in terms of an associated Randers metric. Consequences for the computability of Cauchy developments are also derived. (2) For Finsler geometry: Causality suggests that the role of completeness in many results of Riemannian Geometry (geodesic connectedness by minimizing geodesics, Bonnet-Myers, Synge theorems) is played by the compactness of symmetrized closed balls in Finslerian Geometry. Moreover, under this condition we show that for any Randers metric $R$ there exists another Randers metric $\tilde R$ with the same pregeodesics and geodesically complete. Even more, results on the differentiability of Cauchy horizons in spacetimes yield consequences for the differentiability of the Randers distance to a subset, and vice versa.

Geometric-arithmetic averaging of dyadic weights

Tue, 09 Aug 2011 12:19 EDT

Jill Pipher , Lesley A. Ward , Xiao Xiao

Source: Rev. Mat. Iberoamericana, Volume 27, Number 3, 953--976.

Abstract:
The theory of Muckenhoupt's weight functions arises in many areas of analysis, for example in connection with bounds for singular integrals and maximal functions on weighted spaces. We prove that a certain averaging process gives a method for constructing $A_p$ weights from a measurably varying family of dyadic $A_p$ weights. This averaging process is suggested by the relationship between the $A_p$ weight class and the space of functions of bounded mean oscillation. The same averaging process also constructs weights satisfying reverse Hölder ($RH_p$) conditions from families of dyadic $RH_p$ weights, and extends to the polydisc as well.

Closed ideals of $A^\infty$ and a famous problem of Grothendieck

Tue, 09 Aug 2011 12:19 EDT

S. R. Patel

Source: Rev. Mat. Iberoamericana, Volume 27, Number 3, 977--995.

Abstract:
Using Fréchet algebraic technique, we show the existence of a nuclear Fréchet space without basis, thus providing yet another proof (of a different flavor) of a negative answer to a well known problem of Grothendieck from 1955. Using Fefferman's construction (which is based on complex-variable technique) of a $C^\infty$-function on the unit circle with certain properties, we give much simpler, transparent, and "natural" examples of restriction spaces without bases of nuclear Fréchet spaces of $C^\infty$-functions; these latter spaces, being classical objects of study, have attracted some attention because of their relevance to the theories of PDE and complex dynamical systems, and harmonic analysis. In particular, the restriction space $A^\infty(E)$, being a quotient algebra of the algebra $A^\infty(\Gamma)$, is the central one to other examples; the algebras $A^\infty$ had played a crucial role in solving a well-known problem of Kahane and Katznelson in the negative.

Sub-Riemannian geometry of parallelizable spheres

Tue, 09 Aug 2011 12:19 EDT

Mauricio Godoy Molina , Irina Markina

Source: Rev. Mat. Iberoamericana, Volume 27, Number 3, 997--1022.

Abstract:
The first aim of the present paper is to compare various sub-Riemannian structures over the three dimensional sphere $S^3$ originating from different constructions. Namely, we describe the sub-Riemannian geometry of $S^3$ arising through its right action as a Lie group over itself, the one inherited from the natural complex structure of the open unit ball in $\mathbb{C}^2$ and the geometry that appears when it is considered as a principal $S^1$-bundle via the Hopf map. The main result of this comparison is that in fact those three structures coincide. We present two bracket generating distributions for the seven dimensional sphere $S^7$ of step 2 with ranks 6 and 4. The second one yields to a sub-Riemannian structure for $S^7$ that is not widely present in the literature until now. One of the distributions can be obtained by considering the CR geometry of $S^7$ inherited from the natural complex structure of the open unit ball in $\mathbb{C}^4$. The other one originates from the quaternionic analogous of the Hopf map.

Product kernels adapted to curves in the space

Tue, 09 Aug 2011 12:19 EDT

Valentina Casarino , Paolo Ciatti , Silvia Secco

Source: Rev. Mat. Iberoamericana, Volume 27, Number 3, 1023--1057.

Abstract:
We establish $L^p$-boundedness for a class of operators that are given by convolution with product kernels adapted to curves in the space. The $L^p$ bounds follow from the decomposition of the adapted kernel into a sum of two kernels with singularities concentrated respectively on a coordinate plane and along the curve. The proof of the $L^p$-estimates for the two corresponding operators involves Fourier analysis techniques and some algebraic tools, namely the Bernstein-Sato polynomials. As an application, we show that these bounds can be exploited in the study of $L^p-L^q$ estimates for analytic families of fractional operators along curves in the space.

$L^p$ improving multilinear Radon-like transforms

Tue, 09 Aug 2011 12:19 EDT

Betsy Stovall

Source: Rev. Mat. Iberoamericana, Volume 27, Number 3, 1059--1085.

Abstract:
We characterize (up to endpoints) the $k$-tuples $(p_1,\ldots,p_k)$ for which certain $k$-linear generalized Radon transforms map the product $L^{p_1} \times \cdots \times L^{p_k}$ boundedly into $\mathbb{R}$. This generalizes a result of Tao and Wright.