Gevrey class

In mathematics, the Gevrey classes on a domain \Omega\subseteq \R^n, introduced by Maurice Gevrey, are spaces of functions 'between' the space of analytic functions C^\omega(\Omega) and the space of smooth (infinitely differentiable) functions C^\infty(\Omega). In particular, for \sigma \ge 1, the Gevrey class G^\sigma (\Omega), consists of those smooth functions g \in C^\infty(\Omega) such that for every compact subset K \Subset \Omega there exists a constant C, depending only on g, K, such that :\sup_{x \in K} |D^\alpha g(x)| \le C^{|\alpha|+1}|\alpha!|^\sigma \quad \forall \alpha \in \Z_{\geq 0}^n Where D^\alpha denotes the partial derivative of order \alpha (see multi-index notation).

When \sigma = 1, G^\sigma(\Omega) coincides with the class of analytic functions C^\omega(\Omega), but for \sigma 1 there are compactly supported functions in the class that are not identically zero (an impossibility in C^\omega). It is in this sense that they interpolate between C^\omega and C^\infty. The Gevrey classes find application in discussing the smoothness of solutions to certain partial differential equations: Gevrey originally formulated the definition while investigating the homogeneous heat equation, whose solutions are in G^2(\Omega).