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POINCARE INEQUALITY FOR ABSTRACT SPACES ALIREZA RANJBAR-MOTLAGHThe Poincare inequality is generalised to metric-measure spaces which support a strong version of the doubling condition. This generalises the Poincare inequality for manifolds whose Ricci curvature is bounded from below and metric-measure spaces which satisfy the measure contraction property.

Abstract. The purpose of this paper is to prove an embedding theorem for Sobolev type functions whose gradients are in a Lorentz space, in the framework of abstract metricmeasure spaces. We then apply this theorem to prove absolute continuity and differentiability of such functions.Introduction. In this article, we extend Morrey's embedding theorem (see for instance [GT, Thm. 7.17]) to Sobolev type functions whose generalized gradients are in a Lorentz space, when the underlying space is a metric-measure space. In fact, Morrey's theorem states that for any function f in the Sobolev space W 1,p (R n ), the following inequality is satisfied:

Abstract. The Poincaré inequality is extended to uniformly doubling metric-measure spaces which satisfy a version of the triangle comparison property. The proof is based on a generalization of the change of variables formula.1. Introduction. The purpose of this paper is to prove the Poincaré type inequality for metric-measure spaces, that is, metric spaces (X, d) with a measure µ (see [EG], [He] and [R] for the basic definitions). It is wellknown that the Poincaré inequality implies the Sobolev and isoperimetric inequalities in doubling and smooth spaces; see for instance [HK] and [F].Heinonen and Koskela [HeK] introduced the concept of upper gradient and they proved the Poincaré type inequality for abstract spaces. Bourdon and Pajot [BP] showed that there exist metrics on the boundary of some hyperbolic buildings such that the Poincaré type inequality is valid and the space is Ahlfors regular of non-integer dimension. Also, Laakso [L] constructed Ahlfors regular spaces of any given dimension greater than one such that the Poincaré type inequality is valid. Hanson and Heinonen [HH] constructed, for any integer n ≥ 2, a space with topological dimension n and Ahlfors regular of dimension n which supports the Poincaré type inequality, but has no manifold point.

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