Inégalités de Sobolev produit sur les groupes de Lie nilpotents. (Product Sobolev inequalities on nilpotent Lie groups)
This is a continuation of studies by \textit{N. Th. Varopoulos} on analysis on Lie groups [J. Funct. Anal. 66, 406-431 (1986; Zbl 0595.22008); ibid. 76, 346-410 (1988; Zbl 0634.22008)], and by the author [C. R. Acad. Sci., Paris, Sér. I 305, 295-297 (1987; Zbl 0625.43001)]. Here and in the following reviews, G denotes a real, nilpotent, connected Lie group and \(X_ 1,...,X_ k\) left-invariant vector-fields on G which, with bracket formation, generate the Lie algebra of G. \(\| \cdot \|_ p\) denotes the \(L^ p\)-norm with respect to right Haar measure. The author searches for the class of positive exponents w(i) such that a product inequality of Sobolev type, \[ \| f\|_{n/(n-1)}\leq C\prod^{k}_{i=1}\| X_ if\|_ 1^{w(i)}, \] should hold for all \(f\in D_ 0^{\infty}(G)\). Necessary and sufficient conditions are found when G is simply-connected. If G is not simply-connected, but if its unique maximal compact subgroup is a circle, then necessary and sufficient conditions are also found.
scientific article; zbMATH DE number 4142345
scientific article