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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.

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
22E30 Analysis on real and complex Lie groups
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