Volume 22, Issue 5

We present abstract inhomogeneous Strichartz estimates for dispersive operators, extending previous work by M. Keel and T. Tao on the one hand, and generalising results of D. Foschi, M. Vilela, M. Nakamura and T. Ozawa on the other hand. It is shown that these abstract estimates imply new inhomogeneous Strichartz estimates for the wave equation and some Schrödinger equations involving potentials.

Let G be a finite group and let H be a subgroup of G . H is said to be semipermutable in G if H permutes with every subgroup K of G with (| H |, | K |) = 1. A number of new characterizations of finite solvable BT-groups are given, where a BT-group is one in which semipermutability is a transitive relation.

As it is well known, there is an element in the family of classical Lie algebras which presents more symmetries than the others. We are speaking, of course, about the algebra which, together with the usual two-fold symmetry which is common to the whole family, presents an order three symmetry which makes of it an “exceptional” classical Lie algebra. In this work we describe the fine group gradings induced by outer automorphisms of the Lie algebra . The qualitative approach used here emphasizes some algebraic aspects, as triality, and it is devoted to gradings involving the outer automorphisms of order three. The computation of the remaining fine group gradings, the ones induced by inner automorphisms and outer ones of order two, follows the same pattern as the classical case with n ≥ 5 and so it has been previously considered in the literature.

In this paper, we continue the analysis of [Scarabotti, Tolli, Proc. London Math. Soc.: 2009] on finite homogeneous spaces whose associated permutation representation decomposes with multiplicity. We extend the theory of Gelfand–Tsetlin bases to permutation representations. Then we study several concrete examples on the symmetric groups, generalizing the Gelfand pair of the Johnson scheme. We also extend part of the Okounkov–Vershik theory to the Young permutation module M a . In particular we constuct explicit Gelfand–Tsetlin bases for the representation S n –1,1 . We also give an explicit Gelfand–Tsetlin decomposition for the permutation module associated with a three-parts partitions, using James reformulation of the Young rule by means of intertwining operators (Radon transforms). Several statistical applications, refining previous work by Diaconis, are given. Finally, the spectrum of several invariant operators is determined.

We prove that if a vector-function ƒ belongs to the Morrey space , with , n ≥ 3, N ≥ 2, λ ∈ [0, n – 2], then there exists a very weak solution u of the system such that Du belongs to the space for any , provided the matrix of coefficients ( A ij ) has L ∞ ∩ VMO entries.

The amplitude of a Differential Graded module M is defined in terms of its homology H( M ) by amp M = sup{ i | H i ( M ) ≠ 0} – inf{ i | H i ( M ) ≠ 0}. The main result of this paper is that if R is a sufficiently nice local chain Differential Graded algebra with amp R < ∞, then each compact Differential Graded R -module M with non-zero homology satisfies amp M ≥ amp R .

We fully classify all Lagrangian submanifolds of a complex Grassmannian which are an orbit of a compact group of isometries and have positive Euler characteristic.

We prove an analogue of Kostant's convexity theorem for thick affine buildings and give an application for groups with affine BN -pairs. Recall that there are two natural retractions of the affine building onto a fixed apartment A : The retraction r centered at an alcove in A and the retraction ρ centered at a chamber in the spherical building at infinity. We prove that for each special vertex x ∈ A the set ρ ( r –1 ( W .x )) is a certain convex hull of W .x . The proof can be reduced to a statement about Coxeter complexes and heavily relies on a character formula for highest weight representations of algebraic groups.

Given a fibration over the circle, we relate the eigenspace decomposition of the algebraic monodromy, the homological finiteness properties of the fiber, and the formality properties of the total space. In the process, we prove a more general result about iterated group extensions. As an application, we obtain new criteria for formality of spaces, and 1-formality of groups, illustrated by bundle constructions and various examples from low-dimensional topology and singularity theory.

We treat as a potential operator of order α , and discuss the size of the sets where it becomes +∞, where ƒ is non-negative and has compact support. We express the results in terms of Hausdorff capacity (measure).

Let R be a ring. A left R -module M (resp. right R -module N ) is called singly injective (resp. singly flat) if Ext 1 ( F / C, M ) = 0 (resp. Tor 1 ( N, F / C ) = 0) for any cyclic submodule C of any finitely generated free left R -module F . It is shown that R is a left pseudo-coherentring if and only if any direct product of singly flat right R -modules is singly flat if and only if any direct limit of singly injective left R -modules is singly injective if and only if every right R -module has a singly flat preenvelope; and every left R -module over a left pseudo-coherent ring R has a singly injective cover. Some applications are also given.

Let Γ be a Fuchsian group acting on the hyperbolic plane ℍ. Any eigenfunction of the Laplacian, any automorphic form on an hyperbolic surface ℍ/ Γ induce a distribution on the boundary ∂ ℍ. This distribution is the derivative of a certain order of a fonction F on ∂ ℍ : the derivative of order 1 in the case of bounded eigenfunctions, the derivative of order in the case of cuspidal automorphic forms of weight k . For cuspidal eigenfunctions (resp. for cuspidal automorphic forms) the optimal Hölder exponent of F at a point ξ ∈ ∂ ℍ can be computed exactly when ℍ/ Γ has finite volume : this exponent depends only on the eigenvalue of the function (resp. on the weight of the automorphic form) and on the fact that the ray oξ be recurrent or not ; in particular F is not differentiable at any point, except possibly at the cusps, depending on the eigenvalue (resp. on the weight). For regular but non-cuspidal automorphic forms, there is a continuous family of possibilities for the modulus of continuity. We study in details the case of automorphic forms of weight (like the Jacobi theta function, whose associate function F on ∂ ℍ has imaginary part the Riemann function : using properties of the geodesic flow such as the Khintchine-Sullivan theorem, we show that at almost all ξ a modulus of continuity is , for any є > 0.