Function spacesW 1,p,BV,N α, p characterized by the motions of ℝ n

Abstract

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.