On local Gevrey regularity for Gevrey vectors of subelliptic sums of squares -- an elementary proof of a sharp Gevrey Kotake-Narasimhan theorem

Abstract

We study the regularity of Gevrey vectors for H\"ormander operators $$ P = \sum_{j=1}^m X_j^2 + X_0 + c$$ where the $X_j$ are real vector fields and $c(x)$ is a smooth function, all in Gevrey class $G^{s}.$ The principal hypothesis is that $P$ satisfies the subelliptic estimate: for some $\varepsilon >0, \; \exists \,C$ such that $$\|v\|_\varepsilon^2 \leq C\left(|(Pv, v)| + \|v\|_0^2\right) \qquad \forall v\in C_0^\infty.$$ We prove directly (without the now familiar use of adding a variable $t$ and proving suitable hypoellipticity for $Q=-D_t^2-P$ and then, using the hypothesis on the iterates of $P$ on $u,$ constructiong a homogeneous solution $U$ for $Q$ whose trace on $t=0$ is just $u$) that for $s\geq 1,$\,$G^s(P,\Omega_0) \subset G^{s/\varepsilon}(\Omega_0);$ that is, $$\forall K\Subset \Omega_0, \;\exists C_K: \|P^j u\|_{L^2(K)}\leq C_K^{j+1} (2j)!^s, \;\forall j $$ $$\implies \forall K'\Subset \Omega_0, \;\exists \tilde C_{K'}:\,\|D^\ell u\|_{L^2(K')} \leq \tilde C_{K'}^{\ell+1} \ell!^{s/\epsilon}, \;\forall \ell.$$ 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. When $\epsilon =1,$ $P$ is elliptic and so we recover the original Kotake-Narasimhan theorem (\cite{KN1962}), which has been studied in many other classes, including ultradistributions (\cite{BJ}).