Sobolev inequalities on Lie groups and symmetric spaces (Q910874)

Let G be a connected real Lie group and m(g) the modular function relating left and right Haar measure. Put \(\nabla f=(X_ 1f,...,X_ kf)\). The problem studied here is to find necessary and sufficient conditions on various indices in order that an inequality of Sobolev type (or some related inequality), \[ \| m^{-1/n}f\|_{np/(n-p)}\leq C\| \nabla f\|_ p\quad (n>p\geq 1), \] should hold for all \(f\in C_ 0^{\infty}(G)\). Amongst these indices, the so-called spectral gap associated to the vector-fields \(X_ j\) plays an essential rôle.