On functional linear partial differential equations in Gevrey spaces of holomorphic functions.

Stéphane Malek

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We consider $n$ -tuples of commuting operators $a = a_{1}, ..., a_{n}$ on a Banach space with real spectra. The holomorphic functional calculus for $a$ is extended to algebras of ultra-differentiable functions on $ℝ^{n}$ , depending on the growth of $∥ exp (i a \cdot t) ∥$ , $t \in ℝ^{n}$ , when $| t | \to \infty$ . In the non-quasi-analytic case we use the usual Fourier transform, whereas for the quasi-analytic case we introduce a variant of the FBI transform, adapted to ultradifferentiable classes.