Sobolev met Poincaré

There are several generalizations of the classical theory of Sobolev spaces as they are necessary for the applications to Carnot-Carathéodory spaces, subelliptic equations, quasiconformal mappings on Carnot groups and more general Loewner spaces, analysis on topological manifolds, potential theory on infinite graphs, analysis on fractals and the theory of Dirichlet forms. The aim of this paper is to present a unified approach to the theory of Sobolev spaces that covers applications to many of those areas. The variety of different areas of applications forces a very general setting. We are given a metric space X equipped with a doubling measure ¯. A generalization of a Sobolev function and its gradient is a pair u 2 L 1 loc (X), 0 g 2 L p (X) such that for every ball B ae X the Poincar'e-type inequality Z B ju \\Gamma uB j d¯ Cr `Z oeB g p d¯ ' 1=p holds, where r is the radius of B and oe 1, C ? 0 are fixed constants. Working in the above setting we show that basically...