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Function spaces on Lie groups, the Riemannian approach. (English) Zbl 0587.46036
Let k be an appropriate $$C^{\infty}$$ function in $$R^ n$$ with compact support. Let $$f\in {\mathcal S}'(R^ n)$$ and let $k(t,f)(x)=\int_{R^ n}k(| y|)f(x+ty)dy,\quad x\in R^ n,\quad t>0,$ be some means. The smoothness of f can be measured via the behaviour of these means with respect to some $$L_ p(R^ n)-L_ q(R_+^ 1)$$-quasi-norms, $$0<p\leq \infty$$, $$0<q\leq \infty$$. In this way two scales of spaces $$B^ s_{pq}(R^ n)$$ and $$F^ s_{pq}(R^ n)$$ can be treated which cover Sobolev spaces, Besov-Lipschitz spaces, Bessel-potential spaces, Hardy spaces, Zygmund classes, Hölder spaces etc. Very roughly, the aim of the paper is to extend this approach from $$R^ n$$ to a separable Lie group G. The above means must be replaced by $k(t,f)(x)=\int_{{\mathfrak g}}k(| x|)f(x\cdot \exp tx)dx,\quad x\in G,\quad 0<t<r,$ where $${\mathfrak g}$$ stands for the corresponding Lie algebra and $$| X|$$ comes from some Riemannian metric.

##### MSC:
 4.6e+36 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 2.2e+31 Analysis on real and complex Lie groups
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