The sharp Gevrey Kotake–Narasimhan theorem with an elementary proof

We study the regularity of Gevrey vectors for Hörmander operators\[P={\sum \limits_{j=1}^{m}}{X_{j}^{2}}\mathrm{+}{X_{0}}\mathrm{+}c\]where the $X_j$ are real, smooth vector fields and $C_x$ is a smooth function, all in Gevrey class $G^s$. $P$ is assumed to satisfy a subelliptic estimate in an open set $\Omega_0$: for some $\varepsilon \gt 0$ there exists a constant $C$ such that\[\lVert v{\| _{\varepsilon }^{2}}\leq C\left(|(Pv,v)|\mathrm{+}\| v{\| _{0}^{2}}\right)\qquad \forall v \in {C_{0}^{\mathrm{\infty }}}({\Omega _{0}}).\]We prove directly that for $s \gt 1 , G^s (P,\Omega_0) \subset G^{s / \varepsilon} (\Omega_0)$, i.e.,\[\begin{aligned}&{}\forall K \Subset \Omega _{0}, \;\exists C_{K}: \|P^{j} u\|_{L^{2}(K)} \leq C_{K}^{j+1} (2j)!^{s}, \;\forall j \\\implies{}&{} \forall K'\Subset \Omega _{0}, \;\exists \tilde{C}_{K'}:\,\|D^{\ell }u\|_{L^{2}(K')} \leq \tilde{C}_{K'}^{\ell +1} \ell !^{s/\varepsilon }, \;\forall \ell .\end{aligned}\]In other words, Gevrey growth of derivatives of $u$ as measured by iterates of $P$ yields Gevrey regularity for $u$ in a larger Gevrey class dictated by the size of $\varepsilon$ in the a priori estimate.

When $\varepsilon=1$, $P$ is elliptic and so we recover the original Kotake-Narasimhan theorem ([9]), which has been studied in many other classes, including the class of ultradifferentiable functions ([1]).

Our result has appeared previously ([1]) but with a proof that one colleague referred to as ‘incomplete’, perhaps recalling their initial reaction that our approach would be ‘very long’ if written out in all detail. We have chosen to come up with a less ‘detailed’ but more intuitive proof, in the last section, that should leave no doubt of the complete adequacy of this approach.

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Received 30 September 2020

Accepted 9 March 2021

Published 13 May 2022