an:03888306
Zbl 0557.43007
Rubenthaler, Hubert
La surjectivit?? de l'application moyenne pour les espaces pr??homog??nes
EN
J. Funct. Anal. 60, 80-94 (1985).
00149952
1985
j
43A85 46F10
mean map; homogeneous polynomial; theorem of Borel; asymptotic developments
Let f be a homogeneous polynomial on \({\mathbb{R}}^ n\). For C a connected component of \(\{f(x)\neq 0\}\) and s a complex number one considers the integral \(Z(\phi,s)=\int_{C}\phi (x)| f(x)|^ sdx,\) where \(\phi\) is a test function. This integral converges for Re s\(>0\) and admits a meromorphic continuation. The author studies the distributions occurring as coefficients in the Laurent developments of the function \(s\mapsto Z(\phi,s)\) at its poles. He proves a generalization of a theorem of Borel, which corresponds to the special case \(f(x)=x\), \(n=1\), and says: for any sequence \(c_ k\) of complex numbers there exists a \(C^{\infty}\) function \(\phi\) on \({\mathbb{R}}\) such that \(\phi^{(k)}(0)=c_ k.\) Let \(M_{\phi}(t)\) be the integral of \(\phi\) on \(\{f(x)=t\}\cap C.\) As a corollary of the previous result, since Z(\(\phi\),s) is the Mellin transform of \(M_{\phi}(t)\), the space of the functions \(M_{\phi}\) can be described in terms of asymptotic developments at 0.
J.Faraut