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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 u003e0, ; exists ,C) such that $$begin{aligned} Vert vVert _{varepsilon }^2 le Cleft( |(Pv, v)| + Vert vVert _0^2right) qquad forall vin C_0^infty . end{aligned}$$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 (sge 1,) (G^s(P,Omega _0) subset G^{s/varepsilon }(Omega _0);) that is, $$begin{aligned}u0026forall KSubset Omega _0, ;exists C_K: Vert P^j uVert _{L^2(K)}le C_K^{j+1} (2j)!^s, ;forall ju0026quad implies forall Ku0027Subset Omega _0, ;exists tilde{C}_{Ku0027}:,Vert D^ell uVert _{L^2(Ku0027)} le tilde{C}_{Ku0027}^{ell +1} ell !^{s/varepsilon }, ;forall ell . end{aligned}$$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.

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