Sur les transformées de Riesz dans le cas du Laplacien avec drift
Noël
Lohoué;
Sami
Mustapha
2139-2147
Abstract: We prove $L^p$ estimates for Riesz transforms with drift.
Hardy inequalities with optimal constants and remainder terms
Filippo
Gazzola;
Hans-Christoph
Grunau;
Enzo
Mitidieri
2149-2168
Abstract: We show that the classical Hardy inequalities with optimal constants in the Sobolev spaces $W_0^{1,p}$ and in higher-order Sobolev spaces on a bounded domain $\Omega\subset\mathbb{R} ^n$ can be refined by adding remainder terms which involve $L^p$ norms. In the higher-order case further $L^p$ norms with lower-order singular weights arise. The case $1<p<2$ being more involved requires a different technique and is developed only in the space $W_0^{1,p}$.
A unified approach to improved $L^p$ Hardy inequalities with best constants
G.
Barbatis;
S.
Filippas;
A.
Tertikas
2169-2196
Abstract: We present a unified approach to improved $L^p$ Hardy inequalities in $\mathbf{R}^N$. We consider Hardy potentials that involve either the distance from a point, or the distance from the boundary, or even the intermediate case where the distance is taken from a surface of codimension $1<k<N$. In our main result, we add to the right hand side of the classical Hardy inequality a weighted $L^p$ norm with optimal weight and best constant. We also prove nonhomogeneous improved Hardy inequalities, where the right hand side involves weighted $L^q$ norms, $q \neq p$.
Luzin gaps
Ilijas
Farah
2197-2239
Abstract: We isolate a class of $F_{\sigma\delta}$ ideals on $\mathbb{N}$ that includes all analytic P-ideals and all $F_\sigma$ ideals, and introduce `Luzin gaps' in their quotients. A dichotomy for Luzin gaps allows us to freeze gaps, and prove some gap preservation results. Most importantly, under PFA all isomorphisms between quotient algebras over these ideals have continuous liftings. This gives a partial confirmation to the author's rigidity conjecture for quotients $\mathcal{P}(\mathbb{N} )/\mathcal{I}$. We also prove that the ideals $\operatorname{NWD}(\mathbb{Q} )$ and $\operatorname{NULL}(\mathbb{Q} )$have the Radon-Nikodým property, and (using OCA$_\infty$) a uniformization result for $\mathcal{K}$-coherent families of continuous partial functions.
Generic integral manifolds for weight two period domains
James
A.
Carlson;
Domingo
Toledo
2241-2249
Abstract: We define the notion of a generic integral element for the Griffiths distribution on a weight two period domain, draw the analogy with the classical contact distribution, and then show how to explicitly construct an infinite-dimensional family of integral manifolds tangent to a given element.
Maximum norms of random sums and transient pattern formation
Thomas
Wanner
2251-2279
Abstract: Many interesting and complicated patterns in the applied sciences are formed through transient pattern formation processes. In this paper we concentrate on the phenomenon of spinodal decomposition in metal alloys as described by the Cahn-Hilliard equation. This model depends on a small parameter, and one is generally interested in establishing sharp lower bounds on the amplitudes of the patterns as the parameter approaches zero. Recent results on spinodal decomposition have produced such lower bounds. Unfortunately, for higher-dimensional base domains these bounds are orders of magnitude smaller than what one would expect from simulations and experiments. The bounds exhibit a dependence on the dimension of the domain, which from a theoretical point of view seemed unavoidable, but which could not be observed in practice. In this paper we resolve this apparent paradox. By employing probabilistic methods, we can improve the lower bounds for certain domains and remove the dimension dependence. We thereby obtain optimal results which close the gap between analytical methods and numerical observations, and provide more insight into the nature of the decomposition process. We also indicate how our results can be adapted to other situations.
Parametrized $\diamondsuit$ principles
Justin
Tatch
Moore;
Michael
Hrusák;
Mirna
Dzamonja
2281-2306
Abstract: We will present a collection of guessing principles which have a similar relationship to $\diamondsuit$ as cardinal invariants of the continuum have to ${CH}$. The purpose is to provide a means for systematically analyzing $\diamondsuit$ and its consequences. It also provides for a unified approach for understanding the status of a number of consequences of ${CH}$ and $\diamondsuit$in models such as those of Laver, Miller, and Sacks.
On the adjunction mapping of very ample vector bundles of corank one
Antonio
Lanteri;
Marino
Palleschi;
Andrew
J.
Sommese
2307-2324
Abstract: Let $\mathcal{E}$ be a very ample vector bundle of rank $n-1$ over a smooth complex projective variety $X$ of dimension $n\geq 3$. The structure of $(X,\mathcal{E})$ being known when $\kappa (K_{X} + \det \mathcal{E}) \leq 0$, we investigate the structure of the adjunction mapping when $0 < \kappa (K_{X} + \det \mathcal{E}) < n$.
Units in some families of algebraic number fields
L.
Ya.
Vulakh
2325-2348
Abstract: Multi-dimensional continued fractions associated with $GL_n({\mathbf Z})$ are introduced and applied to find systems of fundamental units in some families of totally real fields and fields with signature (2,1).
Young wall realization of crystal bases for classical Lie algebras
Seok-Jin
Kang;
Jeong-Ah
Kim;
Hyeonmi
Lee;
Dong-Uy
Shin
2349-2378
Abstract: In this paper, we give a new realization of crystal bases for finite-dimensional irreducible modules over classical Lie algebras. The basis vectors are parameterized by certain Young walls lying between highest weight and lowest weight vectors.
Phase transitions in phylogeny
Elchanan
Mossel
2379-2404
Abstract: We apply the theory of Markov random fields on trees to derive a phase transition in the number of samples needed in order to reconstruct phylogenies. We consider the Cavender-Farris-Neyman model of evolution on trees, where all the inner nodes have degree at least $3$, and the net transition on each edge is bounded by $\epsilon$. Motivated by a conjecture by M. Steel, we show that if $2 (1 - 2 \epsilon)^2 > 1$, then for balanced trees, the topology of the underlying tree, having $n$ leaves, can be reconstructed from $O(\log n)$ samples (characters) at the leaves. On the other hand, we show that if $2 (1 - 2 \epsilon)^2 < 1$, then there exist topologies which require at least $n^{\Omega(1)}$ samples for reconstruction. Our results are the first rigorous results to establish the role of phase transitions for Markov random fields on trees, as studied in probability, statistical physics and information theory, for the study of phylogenies in mathematical biology.
Stable representatives for symmetric automorphisms of groups and the general form of the Scott conjecture
Mihalis
Sykiotis
2405-2441
Abstract: Let $G$ be a group acting on a tree $X$ such that all edge stabilizers are finite. We extend Bestvina-Handel's theory of train tracks for automorphisms of free groups to automorphisms of $G$ which permute vertex stabilizers. Using this extension we show that there is an upper bound depending only on $G$ for the complexity of the graph of groups decomposition of the fixed subgroups of such automorphisms of $G$.
Dialgebra cohomology as a G-algebra
Anita
Majumdar;
Goutam
Mukherjee
2443-2457
Abstract: It is well known that the Hochschild cohomology $H^*(A,A)$ of an associative algebra $A$ admits a G-algebra structure. In this paper we show that the dialgebra cohomology $HY^*(D,D)$ of an associative dialgebra $D$ has a similar structure, which is induced from a homotopy G-algebra structure on the dialgebra cochain complex $CY^*(D,D)$.
Norms of linear-fractional composition operators
P.
S.
Bourdon;
E.
E.
Fry;
C.
Hammond;
C.
H.
Spofford
2459-2480
Abstract: We obtain a representation for the norm of the composition operator $C_\phi$ on the Hardy space $H^2$ whenever $\phi$ is a linear-fractional mapping of the form $\phi(z) = b/(cz +d)$. The representation shows that, for such mappings $\phi$, the norm of $C_\phi$ always exceeds the essential norm of $C_\phi$. Moreover, it shows that a formula obtained by Cowen for the norms of composition operators induced by mappings of the form $\phi(z) = sz +t$ has no natural generalization that would yield the norms of all linear-fractional composition operators. For rational numbers $s$ and $t$, Cowen's formula yields an algebraic number as the norm; we show, e.g., that the norm of $C_{1/(2-z)}$ is a transcendental number. Our principal results are based on a process that allows us to associate with each non-compact linear-fractional composition operator $C_\phi$, for which $\Vert C_\phi\Vert> \Vert C_\phi\Vert _e$, an equation whose maximum (real) solution is $\Vert C_\phi\Vert^2$. Our work answers a number of questions in the literature; for example, we settle an issue raised by Cowen and MacCluer concerning co-hyponormality of a certain family of composition operators.
Presentations of noneffective orbifolds
Andre
Henriques;
David
S.
Metzler
2481-2499
Abstract: It is well known that an effective orbifold $M$ (one for which the local stabilizer groups act effectively) can be presented as a quotient of a smooth manifold $P$ by a locally free action of a compact Lie group $K$. We use the language of groupoids to provide a partial answer to the question of whether a noneffective orbifold can be so presented. We also note some connections to stacks and gerbes.
The length of harmonic forms on a compact Riemannian manifold
Paul-Andi
Nagy;
Constantin
Vernicos
2501-2513
Abstract: We study $(n+1)$-dimensional Riemannian manifolds with harmonic forms of constant length and first Betti number equal to $n$ showing that they are $2$-step nilmanifolds with some special metrics. We also characterize, in terms of properties on the product of harmonic forms, the left-invariant metrics among them. This allows us to clarify the case of equality in the stable isosytolic inequalities in that setting. We also discuss other values of the Betti number.
Automorphisms of subfactors from commuting squares
Anne
Louise
Svendsen
2515-2543
Abstract: We study an infinite series of irreducible, hyperfinite subfactors, which are obtained from an initial commuting square by iterating Jones' basic construction. They were constructed by Haagerup and Schou and have $A_{\infty}$as principal graphs, which means that their standard invariant is ``trivial''. We use certain symmetries of the initial commuting squares to construct explicitly non-trivial outer automorphisms of these subfactors. These automorphisms capture information about the subfactors which is not contained in the standard invariant.
Poincaré's closed geodesic on a convex surface
Wilhelm
P. A.
Klingenberg
2545-2556
Abstract: We present a new proof for the existence of a simple closed geodesic on a convex surface $M$. This result is due originally to Poincaré. The proof uses the ${2k}$-dimensional Riemannian manifold ${_{k}\Lambda M} = \hbox {(briefly)} \Lambda$ of piecewise geodesic closed curves on $M$ with a fixed number $k$ of corners, $k$ chosen sufficiently large. In $\Lambda$ we consider a submanifold $\overset{\approx }{\Lambda }_{0}$ formed by those elements of $\Lambda$ which are simple regular and divide $M$ into two parts of equal total curvature $2\pi$. The main burden of the proof is to show that the energy integral $E$, restricted to $\overset{\approx }{\Lambda }_{0}$, assumes its infimum. At the end we give some indications of how our methods yield a new proof also for the existence of three simple closed geodesics on $M$.