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

• Published in: arXiv.org 2017-08
• Main Author: Tartakoff, David S
• Format: Article
• Language: English
• Subjects: Fields (mathematics); Hypotheses; Regularity; Theorems
• ISSN: 2331-8422

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}).