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

Abstract

Let {X 1 , . . ., X p } be complex-valued vector fields in 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 = 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 ≥ 1 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 α Eu are bounded on U whenever |α| ≤ m and for some β, with |β| = mk + 1, the restriction of D β u to U is not locally bounded.

Locations

• Annals of Mathematics - View - PDF