Revista Matemática Iberoamericana Articles (Project Euclid)
http://projecteuclid.org/euclid.rmi
The latest articles from Revista Matemática Iberoamericana on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2010 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Thu, 05 Aug 2010 15:41 EDTFri, 10 Jun 2011 09:37 EDThttp://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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Convergence of metric graphs and energy forms
http://projecteuclid.org/euclid.rmi/1275671307
<strong>
Atsushi
Kasue
</strong><p><strong>Source: </strong>Rev. Mat. Iberoamericana, Volume 26, Number 2, 367--448.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.rmi/1275671307_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDT
The Howe dual pair in Hermitean Clifford analysis
http://projecteuclid.org/euclid.rmi/1275671308
<strong>
Fred
Brackx
</strong>, <strong>
Hennie
De Schepper
</strong>, <strong>
David
Eelbode
</strong>, <strong>
Vladimir
Souček
</strong><p><strong>Source: </strong>Rev. Mat. Iberoamericana, Volume 26, Number 2, 449--479.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.rmi/1275671308_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDT
Riesz transforms on forms and $L^p$-Hodge decomposition on
complete Riemannian manifolds
http://projecteuclid.org/euclid.rmi/1275671309
<strong>
Xiang-Dong
Li
</strong><p><strong>Source: </strong>Rev. Mat. Iberoamericana, Volume 26, Number 2, 481--528.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.rmi/1275671309_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDT
On the cluster size distribution for percolation on some general graphs
http://projecteuclid.org/euclid.rmi/1275671310
<strong>
Antar
Bandyopadhyay
</strong>, <strong>
Jeffrey
Steif
</strong>, <strong>
Ádám
Timár
</strong><p><strong>Source: </strong>Rev. Mat. Iberoamericana, Volume 26, Number 2, 529--550.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.rmi/1275671310_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDT
A priori Hölder estimate, parabolic Harnack principle and heat kernel
estimates for diffusions with jumps
http://projecteuclid.org/euclid.rmi/1275671311
<strong>
Zhen-Qing
Chen
</strong>, <strong>
Takashi
Kumagai
</strong><p><strong>Source: </strong>Rev. Mat. Iberoamericana, Volume 26, Number 2, 551--589.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.rmi/1275671311_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDT
End-point estimates and multi-parameter paraproducts on higher dimensional tori
http://projecteuclid.org/euclid.rmi/1275671312
<strong>
John T.
Workman
</strong><p><strong>Source: </strong>Rev. Mat. Iberoamericana, Volume 26, Number 2, 591--610.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.rmi/1275671312_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDT
Socle theory for Leavitt path algebras of arbitrary graphs
http://projecteuclid.org/euclid.rmi/1275671313
<strong>
Gonzalo
Aranda Pino
</strong>, <strong>
Dolores
Martín Barquero
</strong>, <strong>
Cándido
Martín González
</strong>, <strong>
Mercedes
Siles Molina
</strong><p><strong>Source: </strong>Rev. Mat. Iberoamericana, Volume 26, Number 2, 611--638.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.rmi/1275671313_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDT
Lowest uniformizations of closed Riemann orbifolds
http://projecteuclid.org/euclid.rmi/1275671314
<strong>
Rubén A.
Hidalgo
</strong><p><strong>Source: </strong>Rev. Mat. Iberoamericana, Volume 26, Number 2, 639--649.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.rmi/1275671314_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDT
Bernstein-Heinz-Chern results in calibrated manifolds
http://projecteuclid.org/euclid.rmi/1275671315
<strong>
Guanghan
Li
</strong>, <strong>
Isabel M. C.
Salavessa
</strong><p><strong>Source: </strong>Rev. Mat. Iberoamericana, Volume 26, Number 2, 651--692.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.rmi/1275671315_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDT
Toeplitz operators on Bergman spaces with locally integrable symbols
http://projecteuclid.org/euclid.rmi/1275671316
<strong>
Jari
Taskinen
</strong>, <strong>
Jani
Virtanen
</strong><p><strong>Source: </strong>Rev. Mat. Iberoamericana, Volume 26, Number 2, 693--706.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.rmi/1275671316_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDT
A convolution estimate for two-dimensional hypersurfaces
http://projecteuclid.org/euclid.rmi/1275671317
<strong>
Ioan
Bejenaru
</strong>, <strong>
Sebastian
Herr
</strong>, <strong>
Daniel
Tataru
</strong><p><strong>Source: </strong>Rev. Mat. Iberoamericana, Volume 26, Number 2, 707--728.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.rmi/1275671317_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDT
Topological and analytical properties of Sobolev bundles. II. Higher dimensional cases
http://projecteuclid.org/euclid.rmi/1282913821<strong>
Takeshi
Isobe
</strong><p><strong>Source: </strong>Rev. Mat. Iberoamericana, Volume 26, Number 3, 729--798.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.rmi/1282913821_Fri, 27 Aug 2010 08:57 EDTFri, 27 Aug 2010 08:57 EDT
Contact properties of codimension 2 submanifolds with flat normal bundle
http://projecteuclid.org/euclid.rmi/1282913822<strong>
J. J.
Nuño-Ballesteros
</strong>, <strong>
M. C.
Romero-Fuster
</strong><p><strong>Source: </strong>Rev. Mat. Iberoamericana, Volume 26, Number 3, 799--824.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.rmi/1282913822_Fri, 27 Aug 2010 08:57 EDTFri, 27 Aug 2010 08:57 EDT
Le Théorème du symbole total d'un opérateur différentiel $p$-adique
http://projecteuclid.org/euclid.rmi/1282913823<strong>
Zoghman
Mebkhout
</strong>, <strong>
Luis
Narváez Macarro
</strong><p><strong>Source: </strong>Rev. Mat. Iberoamericana, Volume 26, Number 3, 825--859.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.rmi/1282913823_Fri, 27 Aug 2010 08:57 EDTFri, 27 Aug 2010 08:57 EDT
The $(L^1,L^1)$ bilinear Hardy-Littlewood function and Furstenberg averages
http://projecteuclid.org/euclid.rmi/1282913824<strong>
Idris
Assani
</strong>, <strong>
Zoltán
Buczolich
</strong><p><strong>Source: </strong>Rev. Mat. Iberoamericana, Volume 26, Number 3, 861--890.</p><p><strong>Abstract:</strong><br/>
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
$\frac{1}{p}+\frac{1}{q} \leq 1$.
</p>projecteuclid.org/euclid.rmi/1282913824_Fri, 27 Aug 2010 08:57 EDTFri, 27 Aug 2010 08:57 EDT
Aronson-Bénilan type estimate and the optimal Hölder continuity of
weak solutions for the 1-D degenerate Keller-Segel systems
http://projecteuclid.org/euclid.rmi/1282913825<strong>
Yoshie
Sugiyama
</strong><p><strong>Source: </strong>Rev. Mat. Iberoamericana, Volume 26, Number 3, 891--913.</p><p><strong>Abstract:</strong><br/>
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 $t$.
</p>projecteuclid.org/euclid.rmi/1282913825_Fri, 27 Aug 2010 08:57 EDTFri, 27 Aug 2010 08:57 EDT
Well-posedness in critical spaces for the compressible Navier-Stokes equations
with density dependent viscosities
http://projecteuclid.org/euclid.rmi/1282913826<strong>
Qionglei
Chen
</strong>, <strong>
Changxing
Miao
</strong>, <strong>
Zhifei
Zhang
</strong><p><strong>Source: </strong>Rev. Mat. Iberoamericana, Volume 26, Number 3, 915--946.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.rmi/1282913826_Fri, 27 Aug 2010 08:57 EDTFri, 27 Aug 2010 08:57 EDT
On some maximal multipliers in $L^p$
http://projecteuclid.org/euclid.rmi/1282913827<strong>
Ciprian
Demeter
</strong><p><strong>Source: </strong>Rev. Mat. Iberoamericana, Volume 26, Number 3, 947--964.</p><p><strong>Abstract:</strong><br/>
We extend an $L^2$ maximal multiplier result of Bourgain to all
$L^p$ spaces, $1 < p < \infty$.
</p>projecteuclid.org/euclid.rmi/1282913827_Fri, 27 Aug 2010 08:57 EDTFri, 27 Aug 2010 08:57 EDT
Overdetermined problems in unbounded domains with Lipschitz singularities
http://projecteuclid.org/euclid.rmi/1282913828<strong>
Alberto
Farina
</strong>, <strong>
Enrico
Valdinoci
</strong><p><strong>Source: </strong>Rev. Mat. Iberoamericana, Volume 26, Number 3, 965--974.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.rmi/1282913828_Fri, 27 Aug 2010 08:57 EDTFri, 27 Aug 2010 08:57 EDT
Loewner chains in the unit disk
http://projecteuclid.org/euclid.rmi/1282913829<strong>
Manuel D.
Contreras
</strong>, <strong>
Santiago
Díaz-Madrigal
</strong>, <strong>
Pavel
Gumenyuk
</strong><p><strong>Source: </strong>Rev. Mat. Iberoamericana, Volume 26, Number 3, 975--1012.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.rmi/1282913829_Fri, 27 Aug 2010 08:57 EDTFri, 27 Aug 2010 08:57 EDT
Elliptic equations in the plane satisfying a Carleson measure condition
http://projecteuclid.org/euclid.rmi/1282913830<strong>
Martin
Dindoš
</strong>, <strong>
David J.
Rule
</strong><p><strong>Source: </strong>Rev. Mat. Iberoamericana, Volume 26, Number 3, 1013--1034.</p><p><strong>Abstract:</strong><br/>
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].
</p>projecteuclid.org/euclid.rmi/1282913830_Fri, 27 Aug 2010 08:57 EDTFri, 27 Aug 2010 08:57 EDT
A counterexample for the geometric traveling salesman problem in the Heisenberg group
http://projecteuclid.org/euclid.rmi/1282913831<strong>
Nicolas
Juillet
</strong><p><strong>Source: </strong>Rev. Mat. Iberoamericana, Volume 26, Number 3, 1035--1056.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.rmi/1282913831_Fri, 27 Aug 2010 08:57 EDTFri, 27 Aug 2010 08:57 EDT
Maps from Riemannian manifolds into non-degenerate Euclidean cones
http://projecteuclid.org/euclid.rmi/1282913832<strong>
Luciano
Mari
</strong>, <strong>
Marco
Rigoli
</strong><p><strong>Source: </strong>Rev. Mat. Iberoamericana, Volume 26, Number 3, 1057--1074.</p><p><strong>Abstract:</strong><br/>
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
$\mathbb{R}^{2m-1}$.
</p>projecteuclid.org/euclid.rmi/1282913832_Fri, 27 Aug 2010 08:57 EDTFri, 27 Aug 2010 08:57 EDT
The $C^m$ Norm of a Function with Prescribed Jets I
http://projecteuclid.org/euclid.rmi/1282913833<strong>
Charles
Fefferman
</strong><p><strong>Source: </strong>Rev. Mat. Iberoamericana, Volume 26, Number 3, 1075--1098.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.rmi/1282913833_Fri, 27 Aug 2010 08:57 EDTFri, 27 Aug 2010 08:57 EDT
Global existence for the primitive equations with small anisotropic viscosity
http://projecteuclid.org/euclid.rmi/1296828828<strong>
Frédéric
Charve
</strong>, <strong>
Van-Sang
Ngo
</strong><p><strong>Source: </strong>Rev. Mat. Iberoamericana, Volume 27, Number 1, 1--38.</p><p><strong>Abstract:</strong><br/>
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).
</p>projecteuclid.org/euclid.rmi/1296828828_Fri, 04 Feb 2011 09:13 ESTFri, 04 Feb 2011 09:13 EST
Le théorème du symbole total d'un opérateur différentiel
$p$-adique d'échelon $h\geq0$
http://projecteuclid.org/euclid.rmi/1296828829<strong>
Zoghman
Mebkhout
</strong><p><strong>Source: </strong>Rev. Mat. Iberoamericana, Volume 27, Number 1, 39--92.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.rmi/1296828829_Fri, 04 Feb 2011 09:13 ESTFri, 04 Feb 2011 09:13 EST
Isoperimetry for spherically symmetric log-concave probability measures
http://projecteuclid.org/euclid.rmi/1296828830<strong>
Nolwen
Huet
</strong><p><strong>Source: </strong>Rev. Mat. Iberoamericana, Volume 27, Number 1, 93--122.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.rmi/1296828830_Fri, 04 Feb 2011 09:13 ESTFri, 04 Feb 2011 09:13 EST
Universal objects in categories of reproducing kernels
http://projecteuclid.org/euclid.rmi/1296828831<strong>
Daniel
Beltiţă
</strong>, <strong>
José E.
Galé
</strong><p><strong>Source: </strong>Rev. Mat. Iberoamericana, Volume 27, Number 1, 123--179.</p><p><strong>Abstract:</strong><br/>
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 $\ell^2(\mathbb{N})$.
</p>projecteuclid.org/euclid.rmi/1296828831_Fri, 04 Feb 2011 09:13 ESTFri, 04 Feb 2011 09:13 EST
Tropical plane geometric constructions: a transfer technique in Tropical Geometry
http://projecteuclid.org/euclid.rmi/1296828832<strong>
Luis Felipe
Tabera
</strong><p><strong>Source: </strong>Rev. Mat. Iberoamericana, Volume 27, Number 1, 181--232.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.rmi/1296828832_Fri, 04 Feb 2011 09:13 ESTFri, 04 Feb 2011 09:13 EST
Regularity for solutions of the total variation denoising problem
http://projecteuclid.org/euclid.rmi/1296828833<strong>
Vicent
Caselles
</strong>, <strong>
Antonin
Chambolle
</strong>, <strong>
Matteo
Novaga
</strong><p><strong>Source: </strong>Rev. Mat. Iberoamericana, Volume 27, Number 1, 233--252.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.rmi/1296828833_Fri, 04 Feb 2011 09:13 ESTFri, 04 Feb 2011 09:13 EST
Cluster solutions for the Schrödinger-Poisson-Slater problem around a local
minimum of the potential
http://projecteuclid.org/euclid.rmi/1296828834<strong>
David
Ruiz
</strong>, <strong>
Giusi
Vaira
</strong><p><strong>Source: </strong>Rev. Mat. Iberoamericana, Volume 27, Number 1, 253--271.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.rmi/1296828834_Fri, 04 Feb 2011 09:13 ESTFri, 04 Feb 2011 09:13 EST
Construction of multi-soliton solutions for the $L^2$-supercritical
gKdV and NLS equations
http://projecteuclid.org/euclid.rmi/1296828835<strong>
Raphaël
Côte
</strong>, <strong>
Yvan
Martel
</strong>, <strong>
Frank
Merle
</strong><p><strong>Source: </strong>Rev. Mat. Iberoamericana, Volume 27, Number 1, 273--302.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.rmi/1296828835_Fri, 04 Feb 2011 09:13 ESTFri, 04 Feb 2011 09:13 EST
Constant curvature foliations in asymptotically hyperbolic spaces
http://projecteuclid.org/euclid.rmi/1296828836<strong>
Rafe
Mazzeo
</strong>, <strong>
Frank
Pacard
</strong><p><strong>Source: </strong>Rev. Mat. Iberoamericana, Volume 27, Number 1, 303--333.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.rmi/1296828836_Fri, 04 Feb 2011 09:13 ESTFri, 04 Feb 2011 09:13 EST
Strong $A_\infty$-weights are $A_\infty$-weights on metric spaces
http://projecteuclid.org/euclid.rmi/1296828837<strong>
Riikka
Korte
</strong>, <strong>
Outi Elina
Kansanen
</strong><p><strong>Source: </strong>Rev. Mat. Iberoamericana, Volume 27, Number 1, 335--354.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.rmi/1296828837_Fri, 04 Feb 2011 09:13 ESTFri, 04 Feb 2011 09:13 EST
The Jet of an Interpolant on a Finite Set
http://projecteuclid.org/euclid.rmi/1296828838<strong>
Charles
Fefferman
</strong>, <strong>
Arie
Israel
</strong><p><strong>Source: </strong>Rev. Mat. Iberoamericana, Volume 27, Number 1, 355--360.</p><p><strong>Abstract:</strong><br/>
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
$$
\Gamma_f (S,cM) \subset \tilde{\Gamma}_f (S,M) \subset \Gamma_f (S,CM),
$
where $c$ and $C$ depend only on $m$ and $n$.
</p>projecteuclid.org/euclid.rmi/1296828838_Fri, 04 Feb 2011 09:13 ESTFri, 04 Feb 2011 09:13 EST
$L^2$ boundedness for maximal commutators with rough variable kernels
http://projecteuclid.org/euclid.rmi/1307713031<strong>
Yanping
Chen
</strong>, <strong>
Yong
Ding
</strong>, <strong>
Ran
Li
</strong><p><strong>Source: </strong>Rev. Mat. Iberoamericana, Volume 27, Number 2, 361--391.</p><p><strong>Abstract:</strong><br/>
For $b\in BMO(\mathbb{R}^n)$ and $k\in\mathbb{N}$, the $k$-th order maximal commutator
of the singular integral operator $T$ with rough variable kernels is defined by
$$
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.
</p>projecteuclid.org/euclid.rmi/1307713031_Fri, 10 Jun 2011 09:37 EDTFri, 10 Jun 2011 09:37 EDT
A new hypoelliptic operator on almost CR manifolds
http://projecteuclid.org/euclid.rmi/1307713032<strong>
Raphaël
Ponge
</strong><p><strong>Source: </strong>Rev. Mat. Iberoamericana, Volume 27, Number 2, 393--414.</p><p><strong>Abstract:</strong><br/>
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}$.
</p>projecteuclid.org/euclid.rmi/1307713032_Fri, 10 Jun 2011 09:37 EDTFri, 10 Jun 2011 09:37 EDT
Coefficient multipliers on Banach spaces of analytic functions
http://projecteuclid.org/euclid.rmi/1307713033<strong>
Óscar
Blasco
</strong>, <strong>
Miroslav
Pavlović
</strong><p><strong>Source: </strong>Rev. Mat. Iberoamericana, Volume 27, Number 2, 415--447.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.rmi/1307713033_Fri, 10 Jun 2011 09:37 EDTFri, 10 Jun 2011 09:37 EDT
The $\varepsilon$-strategy in variational analysis: illustration with the closed convexification of a function
http://projecteuclid.org/euclid.rmi/1307713034<strong>
Jean-Baptiste
Hiriart-Urruty
</strong>, <strong>
Marco A.
López
</strong>, <strong>
Michel
Volle
</strong><p><strong>Source: </strong>Rev. Mat. Iberoamericana, Volume 27, Number 2, 449--474.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.rmi/1307713034_Fri, 10 Jun 2011 09:37 EDTFri, 10 Jun 2011 09:37 EDT
Quantitative uniqueness for second order elliptic operators with strongly singular coefficients
http://projecteuclid.org/euclid.rmi/1307713035<strong>
Ching-Lung
Lin
</strong>, <strong>
Gen
Nakamura
</strong>, <strong>
Jenn-Nan
Wang
</strong><p><strong>Source: </strong>Rev. Mat. Iberoamericana, Volume 27, Number 2, 475--491.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.rmi/1307713035_Fri, 10 Jun 2011 09:37 EDTFri, 10 Jun 2011 09:37 EDT
High-dimensional Menger-type curvatures. Part I: Geometric multipoles and multiscale inequalities
http://projecteuclid.org/euclid.rmi/1307713036<strong>
Gilad
Lerman
</strong>, <strong>
J. Tyler
Whitehouse
</strong><p><strong>Source: </strong>Rev. Mat. Iberoamericana, Volume 27, Number 2, 493--555.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.rmi/1307713036_Fri, 10 Jun 2011 09:37 EDTFri, 10 Jun 2011 09:37 EDT
Pseudo-localisation of singular integrals in $L^p$
http://projecteuclid.org/euclid.rmi/1307713037<strong>
Tuomas P.
Hytönen
</strong><p><strong>Source: </strong>Rev. Mat. Iberoamericana, Volume 27, Number 2, 557--584.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.rmi/1307713037_Fri, 10 Jun 2011 09:37 EDTFri, 10 Jun 2011 09:37 EDT
A proof of hypoellipticity for Kohn's operator via FBI
http://projecteuclid.org/euclid.rmi/1307713038<strong>
Gregorio
Chinni
</strong><p><strong>Source: </strong>Rev. Mat. Iberoamericana, Volume 27, Number 2, 585--604.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.rmi/1307713038_Fri, 10 Jun 2011 09:37 EDTFri, 10 Jun 2011 09:37 EDT
Submetries vs. submersions
http://projecteuclid.org/euclid.rmi/1307713039<strong>
Luis
Guijarro
</strong>, <strong>
Gerard
Walschap
</strong><p><strong>Source: </strong>Rev. Mat. Iberoamericana, Volume 27, Number 2, 605--619.</p><p><strong>Abstract:</strong><br/>
We study submetries between Alexandrov spaces and show how some of the usual
features of Riemannian submersions fail due to the lack of smoothness.
</p>projecteuclid.org/euclid.rmi/1307713039_Fri, 10 Jun 2011 09:37 EDTFri, 10 Jun 2011 09:37 EDT
The center of a Leavitt path algebra
http://projecteuclid.org/euclid.rmi/1307713040<strong>
Gonzalo
Aranda Pino
</strong>, <strong>
Kathi
Crow
</strong><p><strong>Source: </strong>Rev. Mat. Iberoamericana, Volume 27, Number 2, 621--644.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.rmi/1307713040_Fri, 10 Jun 2011 09:37 EDTFri, 10 Jun 2011 09:37 EDT
Multi-parameter Carnot-Carathéodory balls and the theorem of Frobenius
http://projecteuclid.org/euclid.rmi/1307713041<strong>
Brian
Street
</strong><p><strong>Source: </strong>Rev. Mat. Iberoamericana, Volume 27, Number 2, 645--732.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.rmi/1307713041_Fri, 10 Jun 2011 09:37 EDTFri, 10 Jun 2011 09:37 EDT
Finiteness of endomorphism algebras of CM modular abelian varieties
http://projecteuclid.org/euclid.rmi/1312906776<strong>
Josep
González
</strong><p><strong>Source: </strong>Rev. Mat. Iberoamericana, Volume 27, Number 3, 733--750.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.rmi/1312906776_Tue, 09 Aug 2011 12:19 EDTTue, 09 Aug 2011 12:19 EDT
Partial regularity for subquadratic parabolic systems by $\mathcal{A}$-caloric approximation
http://projecteuclid.org/euclid.rmi/1312906777<strong>
Christoph
Scheven
</strong><p><strong>Source: </strong>Rev. Mat. Iberoamericana, Volume 27, Number 3, 751--801.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.rmi/1312906777_Tue, 09 Aug 2011 12:19 EDTTue, 09 Aug 2011 12:19 EDT
Regularity, local behavior and partial uniqueness for self-similar profiles of
Smoluchowski's coagulation equation
http://projecteuclid.org/euclid.rmi/1312906778<strong>
José A.
Cañizo
</strong>, <strong>
Stéphane
Mischler
</strong><p><strong>Source: </strong>Rev. Mat. Iberoamericana, Volume 27, Number 3, 803--839.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.rmi/1312906778_Tue, 09 Aug 2011 12:19 EDTTue, 09 Aug 2011 12:19 EDT
Harmonic polynomials and tangent measures of harmonic measure
http://projecteuclid.org/euclid.rmi/1312906779<strong>
Matthew
Badger
</strong><p><strong>Source: </strong>Rev. Mat. Iberoamericana, Volume 27, Number 3, 841--870.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.rmi/1312906779_Tue, 09 Aug 2011 12:19 EDTTue, 09 Aug 2011 12:19 EDT
Auslander bounds and homological conjectures
http://projecteuclid.org/euclid.rmi/1312906780<strong>
Jiaqun
Wei
</strong><p><strong>Source: </strong>Rev. Mat. Iberoamericana, Volume 27, Number 3, 871--884.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.rmi/1312906780_Tue, 09 Aug 2011 12:19 EDTTue, 09 Aug 2011 12:19 EDT
Nonnegative solutions of the heat equation on rotationally symmetric Riemannian
manifolds and semismall perturbations
http://projecteuclid.org/euclid.rmi/1312906781<strong>
Minoru
Murata
</strong><p><strong>Source: </strong>Rev. Mat. Iberoamericana, Volume 27, Number 3, 885--907.</p><p><strong>Abstract:</strong><br/>
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 $M\times(0,T)$.
</p>projecteuclid.org/euclid.rmi/1312906781_Tue, 09 Aug 2011 12:19 EDTTue, 09 Aug 2011 12:19 EDT
Steiner and Schwarz symmetrization in warped products and fiber bundles with density
http://projecteuclid.org/euclid.rmi/1312906782<strong>
Frank
Morgan
</strong>, <strong>
Sean
Howe
</strong>, <strong>
Nate
Harman
</strong><p><strong>Source: </strong>Rev. Mat. Iberoamericana, Volume 27, Number 3, 909--918.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.rmi/1312906782_Tue, 09 Aug 2011 12:19 EDTTue, 09 Aug 2011 12:19 EDT
On the interplay between Lorentzian Causality and Finsler metrics of Randers type
http://projecteuclid.org/euclid.rmi/1312906783<strong>
Erasmo
Caponio
</strong>, <strong>
Miguel Ángel
Javaloyes
</strong>, <strong>
Miguel
Sánchez
</strong><p><strong>Source: </strong>Rev. Mat. Iberoamericana, Volume 27, Number 3, 919--952.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.rmi/1312906783_Tue, 09 Aug 2011 12:19 EDTTue, 09 Aug 2011 12:19 EDT
Geometric-arithmetic averaging of dyadic weights
http://projecteuclid.org/euclid.rmi/1312906784<strong>
Jill
Pipher
</strong>, <strong>
Lesley A.
Ward
</strong>, <strong>
Xiao
Xiao
</strong><p><strong>Source: </strong>Rev. Mat. Iberoamericana, Volume 27, Number 3, 953--976.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.rmi/1312906784_Tue, 09 Aug 2011 12:19 EDTTue, 09 Aug 2011 12:19 EDT
Closed ideals of $A^\infty$ and a famous problem of Grothendieck
http://projecteuclid.org/euclid.rmi/1312906785<strong>
S. R.
Patel
</strong><p><strong>Source: </strong>Rev. Mat. Iberoamericana, Volume 27, Number 3, 977--995.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.rmi/1312906785_Tue, 09 Aug 2011 12:19 EDTTue, 09 Aug 2011 12:19 EDT
Sub-Riemannian geometry of parallelizable spheres
http://projecteuclid.org/euclid.rmi/1312906786<strong>
Mauricio
Godoy Molina
</strong>, <strong>
Irina
Markina
</strong><p><strong>Source: </strong>Rev. Mat. Iberoamericana, Volume 27, Number 3, 997--1022.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.rmi/1312906786_Tue, 09 Aug 2011 12:19 EDTTue, 09 Aug 2011 12:19 EDT
Product kernels adapted to curves in the space
http://projecteuclid.org/euclid.rmi/1312906787<strong>
Valentina
Casarino
</strong>, <strong>
Paolo
Ciatti
</strong>, <strong>
Silvia
Secco
</strong><p><strong>Source: </strong>Rev. Mat. Iberoamericana, Volume 27, Number 3, 1023--1057.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.rmi/1312906787_Tue, 09 Aug 2011 12:19 EDTTue, 09 Aug 2011 12:19 EDT
$L^p$ improving multilinear Radon-like transforms
http://projecteuclid.org/euclid.rmi/1312906788<strong>
Betsy
Stovall
</strong><p><strong>Source: </strong>Rev. Mat. Iberoamericana, Volume 27, Number 3, 1059--1085.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.rmi/1312906788_Tue, 09 Aug 2011 12:19 EDTTue, 09 Aug 2011 12:19 EDT