Hypoellipticity and loss of derivatives (with an Appendix by Makhlouf Derridj and David S. Tartakoff)

Abstract

Let $\{X_1,\dots,X_p\}$ be complex-valued vector fields in $\mathbb R^n$ and assume that they satisfy the bracket condition (i.e. that their Lie algebra spans all vector fields). Our object is to study the operator $E=\sum X_i^*X_i$, where $X_i^*$ is the $L_2$ adjoint of $X_i$. A result of Hörmander is that when the $X_i$ are real then $E$ is hypoelliptic and furthemore it is subelliptic (the restriction of a destribution $u$ to an open set $U$ is “smoother” then the restriction of $Eu$ to $U$). When the $X_i$ are complex-valued if the bracket condition of order one is satisfied (i.e. if the $\{X_i,[X_i,X_j]\}$ span), then we prove that the operator $E$ is still subelliptic. This is no longer true if brackets of higher order are needed to span. For each $k\ge1$ we give an example of two complex-valued vector fields, $X_1$ and $X_2$, such that the bracket condition of order $k+1$ is satisfied and we prove that the operator $E=X_1^*X_1+X_2^*X_2$ is hypoelliptic but that it is not subelliptic. In fact it “loses” $k$ derivatives in the sense that, for each $m$, there exists a distribution $u$ whose restriction to an open set $U$ has the property that the $D^\alpha Eu$ are bounded on $U$ whenever $|\alpha|\le m$ and for some $\beta$, with $|\beta|=m-k+1$, the restriction of $D^\beta u$ to $U$ is not locally bounded.