A multiplier condition for hypoellipticity of complex vector fields with optimal loss of derivatives

Abstract

In Cz×Rt we consider the function g=g(z), set g1=∂zg, g11¯=∂z∂¯zg and define the operator Lg=∂z+ig1∂t. We discuss estimates with loss of derivatives, in the sense of Kohn, for the system (L¯g,fkLg) where (L¯g,Lg) is 12m subelliptic at 0 and f(0)=0, df(0)≠0. We prove estimates with a loss l=k−12m if the “multiplier” condition |f|≳|g11¯|12(m−1) is fulfilled. (For estimates without cut-off, subellipticity can be weakened to compactness and this results in a loss of l=k2(m−1).) For the choice (g,fk)=(|z|2m,z¯k) this result was obtained by Kohn and Bove–Derridj–Kohn–Tartakoff for m=1 and m≥1 respectively. Also, the loss l=k−12m was proven to be optimal. We show that it remains optimal for the model (g,fk)=(x2m,xk). Instead, for the model (g,fk)=(|z|2m,xk), in which the multiplier condition is violated, the loss is not lowered by the type and must be ≥k−12.