- journal articles metadata
European Mathematical Society Publishing House
2024-03-28 23:31:49
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https://www.ems-ph.org/meta/jmeta-stream.php?jrn=RMI&vol=26&update_since=2024-03-28
Revista Matemática Iberoamericana
Rev. Mat. Iberoamericana
RMI
0213-2230
2235-0616
General
10.4171/RMI
http://www.ems-ph.org/doi/10.4171/RMI
subscribers
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society (from 2012)
26
2010
1
Threshold solutions for the focusing 3D cubic Schrödinger equation
Thomas
Duyckaerts
Institut Galilée, Université Paris 13, VILLETANEUSE, FRANCE
Svetlana
Roudenko
Arizona State University, TEMPE, UNITED STATES
Nonlinear Schrödinger equation, scattering, profile decomposition, blow-up
We study the focusing 3d cubic NLS equation with $H^1$ data at the mass-energy threshold, namely, when $M[u_0]E[u_0]{=}M[Q]E[Q]$. In earlier works of Holmer-Roudenko and Duyckaerts-Holmer-Roudenko, the behavior of solutions (i.e., scattering and blow up in finite time) was classified when $M[u_0]E[u_0] < M[Q]E[Q]$. In this paper, we first exhibit 3 special solutions: $e^{it} Q$ and $Q^\pm$, where $Q$ is the ground state, $Q^\pm$ exponentially approach the ground state solution in the positive time direction, $Q^+$ has finite time blow up and $Q^-$ scatters in the negative time direction. Secondly, we classify solutions at this threshold and obtain that up to $\dot{H}^{1/2}$ symmetries, they behave exactly as the above three special solutions, or scatter and blow up in both time directions as the solutions below the mass-energy threshold. These results are obtained by studying the spectral properties of the linearized Schrödinger operator in this mass-supercritical case, establishing relevant modulational stability and careful analysis of the exponentially decaying solutions to the linearized equation.
Partial differential equations
General
1
56
10.4171/RMI/592
http://www.ems-ph.org/doi/10.4171/RMI/592
Valiron’s construction in higher dimension
Filippo
Bracci
Università di Roma “Tor Vergata”, ROMA, ITALY
Graziano
Gentili
Università degli Studi di Firenze, FIRENZE, ITALY
Pietro
Poggi-Corradini
Kansas State University, MANHATTAN, UNITED STATES
Linearization, dynamics of holomorphic self-maps, intertwining maps, iteration theory, hyperbolic maps.
We consider holomorphic self-maps $\varphi$ of the unit ball $\mathbb B^N$ in $\mathbb C^N$ ($N=1,2,3,\dots$). In the one-dimensional case, when $\varphi$ has no fixed points in $\mathbb D\defeq \mathbb B^1$ and is of hyperbolic type, there is a classical renormalization procedure due to Valiron which allows to semi-linearize the map $\varphi$, and therefore, in this case, the dynamical properties of $\varphi$ are well understood. In what follows, we generalize the classical Valiron construction to higher dimensions under some weak assumptions on $\varphi$ at its Denjoy-Wolff point. As a result, we construct a semi-conjugation $\sigma$, which maps the ball into the right half-plane of $\mathbb C$, and solves the functional equation $\sigma\circ \varphi=\lambda \sigma$, where $\lambda > 1$ is the (inverse of the) boundary dilation coefficient at the Denjoy-Wolff point of $\varphi$.
Several complex variables and analytic spaces
Functions of a complex variable
General
57
76
10.4171/RMI/593
http://www.ems-ph.org/doi/10.4171/RMI/593
Hölder exponents of arbitrary functions
Antoine
Ayache
Université Lille 1, VILLENEUVE D'ASQ, FRANCE
Stéphane
Jaffard
Université Paris Est, CRÉTEIL CEDEX, FRANCE
Hölder regularity, Hölder exponents, wavelets.
The functional class of Hölder exponents of continuous function has been completely characterized by P. Andersson, K. Daoudi, S. Jaffard, J. Lévy Véhel andY. Meyer [1, 2, 6, 9]; these authors have shown that this class exactly corresponds to that of the lower limits of the sequences of nonnegative continuous functions. The problem of determining whether or not the H¨older exponents of discontinuous (and even unbounded) functions can belong to a larger class remained open during the last decade. The main goal of our article is to show that this is not the case: the latter H¨older exponents can also be expressed as lower limits of sequences of continuous functions. Our proof mainly relies on a “wavelet-leader” reformulation of a nice characterization of pointwise Hölder regularity due to P. Anderson.
Real functions
Fourier analysis
Numerical analysis
General
77
89
10.4171/RMI/594
http://www.ems-ph.org/doi/10.4171/RMI/594
Estimates for the X-ray transform restricted to 2-manifolds
M. Burak
Erdoğan
University of Illinois, URBANA, UNITED STATES
Richard
Oberlin
Louisiana State University, BATON ROUGE, UNITED STATES
Radon, X-ray, transform, mixed-norm
We prove almost sharp mixed-norm estimates for the X-ray transform restricted to lines whose directions lie on certain well-curved two dimensional manifolds.
Fourier analysis
General
91
114
10.4171/RMI/595
http://www.ems-ph.org/doi/10.4171/RMI/595
On the Conley decomposition of Mather sets
Patrick
Bernard
Université de Paris Dauphine, PARIS CEDEX 16, FRANCE
Semi-continuity of the Aubry set, minimizing measures, chain transitivity
In the context of Mather’s theory of Lagrangian systems, we study the decomposition in chain-transitive classes of the Mather invariant sets. As an application, we prove, under appropriate hypotheses, the semi-continuity of the so-called Aubry set as a function of the Lagrangian.
Dynamical systems and ergodic theory
Calculus of variations and optimal control; optimization
General
115
132
10.4171/RMI/596
http://www.ems-ph.org/doi/10.4171/RMI/596
Vector-valued distributions and Hardy’s uncertainty principle for operators
Michael
Cowling
University of New South Wales, SYDNEY, NSW, AUSTRALIA
Bruno
Demange
Université de Grenoble I, SAINT-MARTIN D'HERES, FRANCE
M.
Sundari
Chennai Mathematical Institute, SIRUSERI, INDIA
Uncertainty principle, linear operators, Hardy’s theorem
Suppose that $f$ is a function on $\mathbb{R}^n$ such that $\exp(a |\cdot|^2) f$ and $\exp(b |\cdot|^2) \hat f$ are bounded, where $a,b > 0$. Hardy's Uncertainty Principle asserts that if $ab > \pi^2$, then $f = 0$, while if $ab = \pi^2$, then $f = c\exp(-a|\cdot|^2)$. In this paper, we generalise this uncertainty principle to vector-valued functions, and hence to operators. The principle for operators can be formulated loosely by saying that the kernel of an operator cannot be localised near the diagonal if the spectrum is also localised.
Fourier analysis
General
133
146
10.4171/RMI/597
http://www.ems-ph.org/doi/10.4171/RMI/597
Lebesgue points and the fundamental convergence theorem for superharmonic functions on metric spaces
Anders
Björn
Linköping University, LINKÖPING, SWEDEN
Jana
Björn
Linköping University, LINKÖPING, SWEDEN
Mikko
Parviainen
Helsinki University of Technology, HELSINKI UNIVERSITY OF TECHNOL, FINLAND
$mathcal{A}$-harmonic, fundamental convergence theorem, Lebesgue point, metric space, Newtonian function, nonlinear, p-harmonic, quasicontinuous, Sobolev function, superharmonic, superminimizer, supersolution, weak upper gradient
We prove the nonlinear fundamental convergence theorem for superharmonic functions on metric measure spaces. Our proof seems to be new even in the Euclidean setting. The proof uses direct methods in the calculus of variations and, in particular, avoids advanced tools from potential theory. We also provide a new proof for the fact that a Newtonian function has Lebesgue points outside a set of capacity zero, and give a sharp result on when superharmonic functions have $L^q$-Lebesgue points everywhere.
Potential theory
Partial differential equations
General
147
174
10.4171/RMI/598
http://www.ems-ph.org/doi/10.4171/RMI/598
Real analytic parameter dependence of solutions of differential equations
Paweł
Domański
Adam Mickiewicz University, POZNAN, POLAND
Analytic dependence on parameters, linear partial differential operator, convolution operator, linear partial differential equation with constant coefficients, injective tensor product, surjectivity of tensorized operators, space of distributions, currents, space of ultradistributions in the sense of Beurling, functor Proj1, PLS-space, locally convex space, vector valu
We consider the problem of real analytic parameter dependence of solutions of the linear partial differential equation $P(D)u=f$, i.e., the question if for every family $(f\sb\lambda)\subseteq \mathscr{D}'(\Omega)$ of distributions depending in a real analytic way on $\lambda\in U$, $U$ a real analytic manifold, there is a family of solutions $(u\sb\lambda)\subseteq \dio$ also depending analytically on $\lambda$ such that $$ P(D)u\sb\lambda=f\sb\lambda\qquad \text{for every $\lambda\in U$}, $$ where $\Omega\subseteq \mathbb{R}\sp d$ an open set. For general surjective variable coefficients operators or operators acting on currents over a smooth manifold we give a solution in terms of an abstract ``Hadamard three circle property'' for the kernel of the operator. The obtained condition is evaluated giving the full solution (usually in terms of the symbol) for operators with constant coefficients and open (convex) $\Omega\subseteq\mathbb{R}\sp d$ if $P(D)$ is of one of the following types: 1) elliptic, 2) hypoelliptic, 3) homogeneous, 4) of two variables, 5) of order two or 6) if $P(D)$ is the system of Cauchy-Riemann equations. An analogous problem is solved for convolution operators of one variable. In all enumerated cases, it follows that the solution is in the affirmative if and only if $P(D)$ has a linear continuous right inverse which shows a striking difference comparing with analogous smooth or holomorphic parameter dependence problems. The paper contains the whole theory working also for operators on Beurling ultradistributions $\mathscr{D}'\sb{(\omega)}$. We prove a characterization of surjectivity of tensor products of general surjective linear operators on a wide class of spaces containing most of the natural spaces of classical analysis.
Partial differential equations
Functional analysis
General
175
238
10.4171/RMI/599
http://www.ems-ph.org/doi/10.4171/RMI/599
Taylor Formula on step two Carnot Groups
Gabriella
Arena
Università degli Studi di Catania, CATANIA, ITALY
Andrea
Caruso
Università degli Studi di Catania, CATANIA, ITALY
Antonio
Causa
Università degli Studi di Catania, CATANIA, ITALY
Carnot groups, Taylor formulas
In the setting of step two Carnot groups we give an explicit representation of Taylor polynomial in terms of a suitable basis of the real vector space of left invariant differential operators, acting pointwisely on monomials like the ordinary Euclidean iterated derivations.
Topological groups, Lie groups
Real functions
General
239
259
10.4171/RMI/600
http://www.ems-ph.org/doi/10.4171/RMI/600
Wellposedness and regularity of solutions of an aggregation equation
Dong
Li
University of Iowa, IOWA CITY, UNITED STATES
José
Rodrigo
University of Warwick, COVENTRY, UNITED KINGDOM
Aggregation equations, well-posedness, higher regularity
We consider an aggregation equation in $\mathbb R^d$, $d\ge 2$ with fractional dissipation: $u_t + \nabla\cdot(u \nabla K*u)=-\nu \Lambda^\gamma u $, where $\nu\ge 0$, $0 < \gamma\le 2$ and $K(x)=e^{-|x|}$. In the supercritical case, $0 < \gamma < 1$, we obtain new local wellposedness results and smoothing properties of solutions. In the critical case, $\gamma=1$, we prove the global wellposedness for initial data having a small $L_x^1$ norm. In the subcritical case, $\gamma > 1$, we prove global wellposedness and smoothing of solutions with general $L_x^1$ initial data.
Partial differential equations
General
261
294
10.4171/RMI/601
http://www.ems-ph.org/doi/10.4171/RMI/601
Exploding solutions for a nonlocal quadratic evolution problem
Dong
Li
University of Iowa, IOWA CITY, UNITED STATES
José
Rodrigo
University of Warwick, COVENTRY, UNITED KINGDOM
Xiaoyi
Zhang
University of Iowa, IOWA CITY, UNITED STATES
Nonlinear parabolic equation, fractional diffusion, chemotaxis
We consider a nonlinear parabolic equation with fractional diffusion which arises from modelling chemotaxis in bacteria. We prove the wellposedness, continuation criteria and smoothness of local solutions. In the repulsive case we prove global wellposedness in Sobolev spaces. Finally in the attractive case, we prove that for a class of smooth initial data the $L_x^\infty$-norm of the corresponding solution blows up in finite time. This solves a problem left open by Biler and Woyczy\'nski [Biler, P. and Woyczy\'Nski, W.A.: Global and exploding solutions for nonlocal quadratic evolution problems. SIAM J. Appl. Math. {\bf 59} (1999), no. 3, 845-869].
Partial differential equations
General
295
332
10.4171/RMI/602
http://www.ems-ph.org/doi/10.4171/RMI/602
Measure of submanifolds in the Engel group
Enrico
Le Donne
ETH Zürich, ZÜRICH, SWITZERLAND
Valentino
Magnani
Università di Pisa, PISA, ITALY
Engel group, submanifolds, Hausdorff measure
We find all intrinsic measures of $C^{1,1}$ smooth submanifolds in the Engel group, showing that they are equivalent to the corresponding $d$-dimensional spherical Hausdorff measure restricted to the submanifold. The integer $d$ is the degree of the submanifold. These results follow from a different approach to negligibility, based on a blow-up technique.
Global analysis, analysis on manifolds
Topological groups, Lie groups
Differential geometry
General
333
346
10.4171/RMI/603
http://www.ems-ph.org/doi/10.4171/RMI/603
Singular integrals in nonhomogeneous spaces: $L^2$ and $L^p$ continuity from Hölder estimates
Marco
Bramanti
Politecnico di Milano, MILANO, ITALY
Singular integrals, nonhomogeneous spaces, $L^p$ spaces, Hölder spaces
We present a result of $L^p$ continuity of singular integrals of Calderón-Zygmund type in the context of bounded nonhomogeneous spaces, well suited to be applied to problems of a priori estimates for partial differential equations. First, an easy and selfcontained proof of $L^2$ continuity is got by means of $C^{\alpha}$ continuity, thanks to an abstract theorem of Krein. Then $L^p$ continuity is derived adapting known results by Nazarov-Treil-Volberg about singular integrals in nonhomogeneous spaces.
Fourier analysis
Operator theory
General
347
366
10.4171/RMI/604
http://www.ems-ph.org/doi/10.4171/RMI/604
2
Convergence of metric graphs and energy forms
Atsushi
Kasue
Kanazawa University, KANAZAWA, JAPAN
Network, resistance form, resistance metric, Gromov-Hausdorff convergence, Γ-convergence, harmonic function of finite Dirichlet sum, Kuramochi compactification
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.
Potential theory
Differential geometry
Probability theory and stochastic processes
General
367
448
10.4171/RMI/605
http://www.ems-ph.org/doi/10.4171/RMI/605
The Howe dual pair in Hermitean Clifford analysis
Fred
Brackx
Ghent University, GENT, BELGIUM
Hennie
De Schepper
Ghent University, GENT, BELGIUM
David
Eelbode
Universiteit Antwerpen, ANTWERPEN, BELGIUM
Vladimír
Souček
Charles University, PRAHA 8, CZECH REPUBLIC
Hermitean Clifford analysis, Howe dual pair
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.
Topological groups, Lie groups
Linear and multilinear algebra; matrix theory
Functions of a complex variable
General
449
479
10.4171/RMI/606
http://www.ems-ph.org/doi/10.4171/RMI/606
Riesz transforms on forms and $L^p$-Hodge decomposition on complete Riemannian manifolds
Xiang-Dong
Li
Chinese Academy of Sciences, BEIJING, CHINA
Hodge decomposition, martingale transforms, Riesz transforms, Weitzenböck curvature
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.
Differential geometry
Global analysis, analysis on manifolds
General
481
528
10.4171/RMI/607
http://www.ems-ph.org/doi/10.4171/RMI/607
On the cluster size distribution for percolation on some general graphs
Antar
Bandyopadhyay
Indian Statistical Institute, NEW DELHI, INDIA
Jeffrey
Steif
Chalmers University of Technology, GOTHENBURG, SWEDEN
Ádám
Timár
Universität Bonn, BONN, GERMANY
Amenability, Cayley graphs, cluster size distribution, exponential decay, percolation, sub-exponential decay
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.
Probability theory and stochastic processes
Statistical mechanics, structure of matter
General
529
550
10.4171/RMI/608
http://www.ems-ph.org/doi/10.4171/RMI/608
A priori Hölder estimate, parabolic Harnack principle and heat kernel estimates for diffusions with jumps
Zhen-Qing
Chen
University of Washington, SEATTLE, UNITED STATES
Takashi
Kumagai
Kyoto University, KYOTO, JAPAN
Symmetric Markov process, pseudo-differential operator, diffusion process, jump process, Lévy system, hitting probability, parabolic function, a priori Hölder estimate, parabolic Harnack inequality, transition density, heat kernel estimates
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.
Probability theory and stochastic processes
Potential theory
Operator theory
General
551
589
10.4171/RMI/609
http://www.ems-ph.org/doi/10.4171/RMI/609
End-point estimates and multi-parameter paraproducts on higher dimensional tori
John
Workman
, ALEXANDRIA, UNITED STATES
Paraproduct, polydisc
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.
Fourier analysis
General
591
610
10.4171/RMI/610
http://www.ems-ph.org/doi/10.4171/RMI/610
Socle theory for Leavitt path algebras of arbitrary graphs
Gonzalo
Aranda Pino
Universidad de Málaga, MÁLAGA, SPAIN
Dolores
Martín Barquero
Universidad de Málaga, MÁLAGA, SPAIN
Cándido
Martín González
Universidad de Málaga, MÁLAGA, SPAIN
Mercedes
Siles Molina
Universidad de Málaga, MÁLAGA, SPAIN
Leavitt path algebra, graph C*-algebra, socle, arbitrary graph, minimal left ideal
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.
Associative rings and algebras
General
611
638
10.4171/RMI/611
http://www.ems-ph.org/doi/10.4171/RMI/611
Lowest uniformizations of closed Riemann orbifolds
Rubén
Hidalgo
Universidad Técnica Federico Santa María, VALPARAÍSO, CHILE
Orbifolds, Schottky groups, Kleinian groups
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.
Functions of a complex variable
General
639
649
10.4171/RMI/612
http://www.ems-ph.org/doi/10.4171/RMI/612
Bernstein-Heinz-Chern results in calibrated manifolds
Guanghan
Li
Hubei University, WUHAN, CHINA
Isabel
Salavessa
Instituto Superior Técnico, LISBOA, PORTUGAL
Calibrated geometry, parallel mean curvature, Heinz-inequality, Bernstein
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.
Differential geometry
Global analysis, analysis on manifolds
General
651
692
10.4171/RMI/613
http://www.ems-ph.org/doi/10.4171/RMI/613
Toeplitz operators on Bergman spaces with locally integrable symbols
Jari
Taskinen
University of Helsinki, HELSINKI, FINLAND
Jani
Virtanen
University of Helsinki, HELSINKI, FINLAND
Toeplitz operators, Bergman spaces, boundedness, compactness, Fredholm properties
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.
Operator theory
General
693
706
10.4171/RMI/614
http://www.ems-ph.org/doi/10.4171/RMI/614
A convolution estimate for two-dimensional hypersurfaces
Ioan
Bejenaru
University of Chicago, CHICAGO, UNITED STATES
Sebastian
Herr
University of California, BERKELEY, UNITED STATES
Daniel
Tataru
University of California, BERKELEY, UNITED STATES
Transversality, hypersurface, convolution, $L^2$ estimate, induction on scales
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.
Fourier analysis
Operator theory
General
707
728
10.4171/RMI/615
http://www.ems-ph.org/doi/10.4171/RMI/615
3
Topological and analytical properties of Sobolev bundles. II. Higher dimensional cases
Takeshi
Isobe
Tokyo Institute of Technology, TOKYO, JAPAN
Sobolev bundle, topology of bundle, Yang-Mills functional, variational problem
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.
Functional analysis
Manifolds and cell complexes
Global analysis, analysis on manifolds
General
729
798
10.4171/RMI/616
http://www.ems-ph.org/doi/10.4171/RMI/616
Contact properties of codimension 2 submanifolds with flat normal bundle
Juan José
Nuño-Ballesteros
Universitat de València, BURJASSOT (VALENCIA), SPAIN
M.C.
Romero-Fuster
Universitat de València, BURJASSOT (VALENCIA), SPAIN
Asymptotic directions, ν-principal curvature foliation, umbilicity, sphericity, normal curvature
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.
Differential geometry
Global analysis, analysis on manifolds
General
799
824
10.4171/RMI/617
http://www.ems-ph.org/doi/10.4171/RMI/617
Le Théorème du symbole total d'un opérateur différentiel $p$-adique
Zoghman
Mebkhout
Université Paris 7 Denis Diderot, PARIS CEDEX 05, FRANCE
Luis
Narváez Macarro
Universidad de Sevilla, SEVILLA, SPAIN
Affinoid algebra, Dwork-Monsky-Washnitzer algebra, †-scheme, †-adic differential operator
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.
Algebraic geometry
General
825
859
10.4171/RMI/618
http://www.ems-ph.org/doi/10.4171/RMI/618
The $(L^1,L^1)$ bilinear Hardy-Littlewood function and Furstenberg averages
Idris
Assani
University of North Carolina at Chapel Hill, CHAPEL HILL, UNITED STATES
Zoltán
Buczolich
Eötvös Loránd University, BUDAPEST, HUNGARY
Furstenberg averages, bilinear Hardy–Littlewood maximal function
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 $\frac{1}{p}+\frac{1}{q} \leq 1$.
Dynamical systems and ergodic theory
Measure and integration
General
861
890
10.4171/RMI/619
http://www.ems-ph.org/doi/10.4171/RMI/619
Aronson-Bénilan type estimate and the optimal Hölder continuity of weak solutions for the 1-D degenerate Keller-Segel systems
Yoshie
Sugiyama
Tsuda University, TOKYO, JAPAN
Parabolic system of degenerate type, Keller-Segel, porous medium, Aronson- Bénilan estimate, interface, optimal Hölder continuity
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 $t$.
Partial differential equations
General
891
913
10.4171/RMI/620
http://www.ems-ph.org/doi/10.4171/RMI/620
Well-posedness in critical spaces for the compressible Navier-Stokes equations with density dependent viscosities
Qionglei
Chen
Institute of Applied Physics and Computational Mathematics, BEIJING, HAIDIAN DISTRICT, CHINA
Changxing
Miao
Institute of Applied Physics and Computational Mathematics, BEIJING, CHINA
Zhifei
Zhang
Peking University, BEIJING, CHINA
Compressible Navier-Stokes equations, Besov spaces, Bony’s paraproduct, Fourier localization
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.
Partial differential equations
General
915
946
10.4171/RMI/621
http://www.ems-ph.org/doi/10.4171/RMI/621
On some maximal multipliers in $L^p$
Ciprian
Demeter
Indiana University, BLOOMINGTON, UNITED STATES
Maximal multipliers, phase space projections
We extend an $L^2$ maximal multiplier result of Bourgain to all $L^p$ spaces, $1 < p < \infty$.
Fourier analysis
General
947
964
10.4171/RMI/622
http://www.ems-ph.org/doi/10.4171/RMI/622
Overdetermined problems in unbounded domains with Lipschitz singularities
Alberto
Farina
Université Picardie Jules Verne, AMIENS, FRANCE
Enrico
Valdinoci
Università di Roma Tor Vergata, ROMA, ITALY
Elliptic partial differential equations, rigidity results, nonexistence of solutions
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.
Partial differential equations
General
965
974
10.4171/RMI/623
http://www.ems-ph.org/doi/10.4171/RMI/623
Loewner chains in the unit disk
Manuel
Contreras
Universidad de Sevilla, SEVILLA, SPAIN
Santiago
Díaz-Madrigal
Universidad de Sevilla, SEVILLA, SPAIN
Pavel
Gumenyuk
University of Bergen, BERGEN, NORWAY
Loewner chains, evolution families
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.
Functions of a complex variable
Ordinary differential equations
General
975
1012
10.4171/RMI/624
http://www.ems-ph.org/doi/10.4171/RMI/624
Elliptic equations in the plane satisfying a Carleson measure condition
Martin
Dindoš
Edinburgh University, EDINBURGH, UNITED KINGDOM
David
Rule
Heriot-Watt University, EDINBURGH, UNITED KINGDOM
Elliptic equations, Carleson measure condition, Neumann problem, regularity problem, distributional inequalities, inhomogeneous equation
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].
Partial differential equations
General
1013
1034
10.4171/RMI/625
http://www.ems-ph.org/doi/10.4171/RMI/625
A counterexample for the geometric traveling salesman problem in the Heisenberg group
Nicolas
Juillet
Université de Grenoble I, SAINT-MARTIN D'HERES, FRANCE
Heisenberg group, Carnot-Carathéodory metric, rectifiable curve, Traveling Salesman Problem
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.
Measure and integration
General
1035
1056
10.4171/RMI/626
http://www.ems-ph.org/doi/10.4171/RMI/626
Maps from Riemannian manifolds into non-degenerate Euclidean cones
Luciano
Mari
Università di Milano, MILANO, ITALY
Marco
Rigoli
Università di Milano, MILANO, ITALY
Maximum principles, harmonic maps, isometric immersion, Riemannian manifold
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 $\mathbb{R}^{2m-1}$.
Differential geometry
Partial differential equations
General
1057
1074
10.4171/RMI/627
http://www.ems-ph.org/doi/10.4171/RMI/627
The $C^m$ Norm of a Function with Prescribed Jets I
Charles
Fefferman
Princeton University, PRINCETON, UNITED STATES
Whitney extension theorem, optimal $C^m$ norm
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.
Calculus of variations and optimal control; optimization
Convex and discrete geometry
General
1075
1098
10.4171/RMI/628
http://www.ems-ph.org/doi/10.4171/RMI/628