We deal with self-affine fractals in 1R2. We examine the notion of affine dimension of a fractal proposed in [26]. To this end, we introduce a generalized affine Hausdorif dimension related to a family of Borel sets. Among other results, we prove that for a suitable class of self-affine fractals (which includes all the so-called general Sierpiñski carpets), under the "open set condition", the affine dimension of the fractal coincides-up to a constant-not only with its Hausdorif dimension arising from a non-isotropic distance D9 in lR2 , but also with the generalized affine Hausdorif dimension related to the family of all balls in (1R 2 , Do). We conclude the paper with a comparison between this assertion and results already known in the literature.

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We characterize the Sobolev spaces WI'P(R ~) (p e]l, + ~ ]), the space BV(R ~) of functions of bounded variation, the Nikol'skij spaces N ~, P (R ~) = B~ (R n) (p E [1, + :c [, )~ e ]0, 1[) by means of the group of all motions--i.e., orientation preserving isometries--of R ~ (these spaces are usually defined or characterized by means of differences of functions, involving only the translations). Moreover, we show that the characterizations we present here give rise to norms equivalent to the usual ones. This is done also with the goal of extending the results from R n to noncompact homogeneous manifolds, in particular in connection with applications to PDE's.