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 Hörmander 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 $\epsilon >0,\exists C$ such that $$\Vert v{\Vert }_{\epsilon }^2\le C\left(|(Pv,v)|+\Vert v{\Vert }_0^2\right)\forall v\in C_0^{\infty }.$$ 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 $s\ge 1,$\,$G^s(P,{\Omega }_0)\subset G^{s/\epsilon }({\Omega }_0);$ that is, $$\forall K⋐{\Omega }_0,\exists C_K:\Vert P^{j}u{\Vert }_{L^2(K)}\le C_K^{j+1}(2j){!}^s,\forall j$$ $$⟹\forall K^{\prime }⋐{\Omega }_0,\exists {\tilde{C}}_{K^{\prime }}:\Vert D^{\ell }u{\Vert }_{L^2(K^{\prime })}\le {\tilde{C}}_{K^{\prime }}^{\ell +1}\ell {!}^{s/\epsilon },\forall \ell .$$ 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}).

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