Volume 17, Issue 3

We present a general technique to prove that a Markov process on a polish space E has continuous respectively càdlàg paths P x -a.s. for all x ∈ E provided this is the case under P μ  := ∫ P x μ ( dx  ) where μ is a measure with full support. As an application we consider the following SDE: d X t  = σ ( X t  ) d W t  + b ( X t  ) d t , for which we get a weak solution for every initial condition merely under very weak integrability conditions on σ and b.

We characterize the weights w such that ∫ 0 ∞ ƒ *( s ) p w ( s )  ds ≃ ∫ 0 ∞ ( ƒ **( s ) – ƒ *( s )) p w ( s )  ds . Our result generalizes a result due to Bennett–De Vore–Sharpley, where the usual Lorentz L p,q norm is replaced by an equivalent expression involving the functional ƒ  ** – ƒ  *. Sufficient conditions for the boundedness of maximal Calderón–Zygmund singular integral operators between classical Lorentz spaces are also given.

Denote by ƒ λ ( G  ) the number of free subgroups of index λm G , where m G is the least common multiple of the orders of the finite subgroups in G . The present paper develops a general theory for the p -divisibility of ƒ λ ( G  ), where p is a prime dividing m G . Among other things, we obtain an explicit combinatorial description of ƒ λ ( G  ) modulo p , leading to an optimal generalisation of Stothers’ explicit formula for the parity of ƒ λ (PSL 2 (ℤ)).

We study the classification of polynomial vector fields in two complex variables under the hypotheses that the singularities are isolated and the flow is complete. Normal forms are obtained for the case the generic orbit is diffeomorphic to ℂ. For the case the generic orbit is diffeomorphic to ℂ ∖ {0} and there is an affine singularity we classify the linear part of the vector field and prove the existence of entire linearization or first integral.

Using the relations between affine symmetric spaces and differentiable Bol loops we determine all 3-dimensional connected differentiable proper Bol loops having a non-solvable Lie group as the group topologically generated by their left translations.

By extending a technique used by Gutierrez and Wheeden[20], we obtain some sharp conditions for weighted Sobolev interpolation inequalities for weights satisfying a weighted Poincaré inequality.

Essential results from the theory of torus actions, initiated by P. A. Smith, are generalized to actions of pro-tori, i.e. compact connected abelian groups. We show that the fixed point set in a (rational cohomology) manifold, resp. sphere, is a rational cohomology manifold, resp. sphere, of even codimension. Borel’s dimension formula for the fixed spheres of codimension one subgroups is proved for actions of pro-tori on (cohomology) spheres. This yields a sharp upper bound for the group dimension. Finally, we describe some applications to actions of pro-tori on compact generalized polygons.

We prove a weighted version of Selberg’s orthogonality conjecture for automorphic L -functions attached to irreducible cuspidal representations of GL m over ℚ. Using this weighted orthogonality, we obtain the uniqueness of factorization of general L -functions. As a consequence, we prove that the L -function attached to an automorphic irreducible cuspidal representation of GL m over ℚ cannot be factored further.

The automorphism group of the random graph has a locally finite subgroup with the same orbits as the whole automorphism group on finite sequences of vertices.