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 Hormander operators $$\begin{aligned} P = \sum _{j=1}^m X_j^2 + X_0 + c \end{aligned}$$ 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 $$\begin{aligned} \Vert v\Vert _{\varepsilon }^2 \le C\left( |(Pv, v)| + \Vert v\Vert _0^2\right) \qquad \forall v\in C_0^\infty . \end{aligned}$$ 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, constructing a homogeneous solution U for Q whose trace on $$t=0$$ is just u) that for $$s\ge 1,$$ $$G^s(P,\Omega _0) \subset G^{s/\varepsilon }(\Omega _0);$$ that is, $$\begin{aligned}&\forall K\Subset \Omega _0, \;\exists C_K: \Vert P^j u\Vert _{L^2(K)}\le C_K^{j+1} (2j)!^s, \;\forall j\\&\quad \implies \forall K'\Subset \Omega _0, \;\exists \tilde{C}_{K'}:\,\Vert D^\ell u\Vert _{L^2(K')} \le \tilde{C}_{K'}^{\ell +1} \ell !^{s/\varepsilon }, \;\forall \ell . \end{aligned}$$ 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. When $$\varepsilon =1,$$ P is elliptic and so we recover the original Kotake–Narasimhan theorem (Kotake and Narasimhan in Bull Soc Math Fr 90(12):449–471, 1962), which has been studied in many other classes, including ultradifferentiable functions (Boiti and Journet in J Pseudo-Differ Oper Appl 8(2):297–317, 2017). We are indebted to M. Derridj for multiple conversations over the years.