Gain/Loss of derivatives for complex vector fields

TL;DR

This paper investigates derivative loss estimates for complex vector fields with subelliptic properties, establishing optimal bounds under certain conditions and analyzing models where the multiplier condition is violated.

Contribution

It provides new sharp estimates with derivative loss for systems involving complex vector fields, extending previous results to models with different structures and conditions.

Findings

Proves optimal derivative loss estimates under the multiplier condition.

Shows the loss cannot be improved in models violating the multiplier condition.

Extends known results to new model cases with different geometric structures.

Abstract

In $\C_z\times\R_t$ we consider the function $g=g(z)$, set $g_1=\di_z g$, $g_{1\bar 1}=\di_z\dib_zg$ and define the operator $L_g=\di_z+ig_1\di_t$. We discuss estimates with loss of derivatives, in the sense of Kohn, for the system $(\bar L_g,f^kL_g)$ where $(\bar L_g,L_g)$ is $\frac1{2m} $ subelliptic at 0 and $f(0)=0,\,\,df(0)\neq0$. We prove estimates with a loss $l=\frac{k-1}{2m} $ if the "multiplier" condition $|f|\simgeq |g_{1\bar 1}|^{\frac1{2(m-1)}}$ is fulfilled. (For estimates without cut-off, subellipticity can be weakened to compactness and this results in a loss of $l=\frac [{2(m-1)}$.) For the choice $(g,f^k)=(|z|^{2m},\bar z^k)$ this result was obtained by Kohn and Bove-Derridj-Kohn-Tartakoff for $m=1$ and $m\geq1$ respectively. Also, the loss $l=\frac{k-1}{2m}$ was proven to be optimal. We show that it remains optimal for the model $(g,f^k)=(x^{2m},x^k)$. Instead, for the model $(g,f^k)=(|z|^{2m},x^k)$, in which the multiplier condition is violated, the loss is not lowered by the type and must be $\geq \frac{k-1}2$.