  In a celebrated paper (Tokyo J. Math. 1984) K. Nishihara proved global
existence for Kirchhoff equations in a special class of initial data which lies
in between analytic functions and Gevrey spaces. This class was defined in
terms of Fourier components with weights satisfying suitable convexity and
integrability conditions.
  In this paper we extend this result by removing the convexity constraint, and
by replacing Nishihara's integrability condition with the simpler integrability
condition which appears in the usual characterization of quasi-analytic
functions.
  After the convexity assumptions have been removed, the resulting theory
reveals unexpected connections with some recent global existence results for
spectral-gap data.
