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Lipschitz spaces on compact Lie groups. (English) Zbl 0639.43003

A compact connected Lie group G (which for simplicity is assumed to be simply connected) is considered. A function \(f\in L^ p(G)\) is said to belong to the Lipschitz space \(\Lambda_{\alpha}^{p,q}(G)\) \((\alpha >0\), \(1\leq p,q\leq \infty)\) if \(\| f\|_{p,q;\alpha}=\| f\|_ p+(\int^{1}_{0}(t^{-\alpha}\omega_ k(t,f;L^ p))^ qdt/^ t)^ q\) is finite (if \(q<\infty\); usual modifications if \(q=\infty)\). Here k is any integer \(>\alpha\) and \(\omega_ k(t,f;L^ p)\) denotes the k-th order \(L^ p\) modulus of smoothness. Some characterizations of \(\Lambda_{\alpha}^{p,q}\) are given. In particular, if \(f\in L^ p(G),\) set \[ L_{\alpha}^{p,q}(f)=\| f\|_ p+\sum_{| I| =k}(\int^{1}_{0}(t^{(k-\alpha)/^ 2}\| X\quad If(.,t)\|_ p)^ qdt/t)^{1/q}<\infty \] (if \(q<\infty\); usual modifications if \(q=\infty)\) where \(f(.,t)=f*W_ t\), \(W_ t\) being the Weierstrass kernel on G, and \(X^ I\) is any differential monomial of order k. The main result is the following Theorem. \(f\in \Lambda_{\alpha}^{p,q}(G)\) if and only if \(L_{\alpha}^{p,q}(f)<\infty\). Moreover \(L_{\alpha}^{p,q}\) and \(\|.\|_{p,q;\alpha}\) are equivalent norms.
Reviewer: S.Meda

MSC:

43A15 \(L^p\)-spaces and other function spaces on groups, semigroups, etc.
22E30 Analysis on real and complex Lie groups
43A75 Harmonic analysis on specific compact groups
42B99 Harmonic analysis in several variables
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References:

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