Gevrey Hypoellipticity for Non-subelliptic Operators

In J. J. Kohn’s recent paper [5] the operator ∂ ∂ − iz|z|2(m−1) ∂z ∂t was introduced and shown to be hypoelliptic, yet to lose k−1 derivatives in m L2 Sobolev norms. Christ [3] showed that the addition of one more variable destroyed hypoellipticity altogether. Here we show that this operator with ∂2 an Oleinik-type singularity Em,k + |z|2(p−1) ∂s2 is Gevrey hypoelliptic Gs for 2m any s ≥ p−k , (2m > p > k). A related result is that for the ‘real’ version, ∂ ∂ with X = ∂x − ixq−1 ∂t , Em,k = Lm Lm + Lm |z|2k Lm , L= ∂2 ∂2 ∗ = X X + (xk X)∗ (xk X) + x2(p−1) 2 ∂s2 ∂s q is Gevrey hypoelliptic Gs for any s ≥ p−k , (q > p > k), although the method of proof is different, and that the result is sharp. The situation is reminiscent of the Baouendi-Goulaouic example, in which adding a new variable to a Grushin type analytic hypoelliptic operator de- stroys the analyticity and drops the regularity to G2 yet prefacing that new second derivative by a power of the first variable to obtain an Oleinik-type operator improves the Gevrey index substantially (cf. [1])