Algèbres de Sobolev sur certains groupes nilpotents
Let \(N=V_ 1\oplus V_ 2\oplus...\oplus V_ m\) be a direct sum of real vector spaces such that \([V_ 1,V_ k]=V_{k+1}\), \(1\leq k\leq m-1\), \([V_ 1,V_ m]=0\) where [, ] is a Lie bracket operation. By Campbell- Hausdorff formula N is a nilpotent Lie group called stratified by \textit{G. B. Folland} [Ark. Mat. 13, 161-207 (1975; Zbl 0312.35026). See also \textit{G. B. Folland} and \textit{E. M. Stein}'s monograph ''Hardy spaces on homogeneous groups'' (1982; Zbl 0508.42025)]. Take \(\Delta\) a sublaplacian of N and let \(\rho =\sum^{m}_{i=1}i \dim V_ i\). It is shown in this paper that the non-isotropic Sobolev space \(S^ p_{\alpha}\) \((\alpha >0\), \(1<p<\infty)\) which is the domain of the fractional power \(\Delta^{\alpha /2}\) in \(L^ p(N)\) is a Banach algebra when \(\alpha >\rho /p\). Some results of spectral synthesis involving Bessel and Riesz capacities are given.
scientific article; zbMATH DE number 3916601