This paper is motivated by the behavior of the heat diffusion kernel p(t)(x) on a general unimodular Lie group. Indeed. contrary to what happens in R(n), the P-t(x) on a general Lie group is behaving like t(-delta(t)/2) for two possibly distinct integers delta(t), one for t tending to 0 and another for t tending to infinity, namely d and D. This forces us to consider a natural generalization of Lorentz spaces with different indices at ''zero'' and at ''infinity.'' (C) 1996 Academic Press, Inc.
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