TAILIEUCHUNG - Đề tài " Hypoellipticity and loss of derivatives "

Dedicated to Yum-Tong Siu for his 60th birthday. Abstract Let {X1 , . . . , Xp } be complex-valued vector fields in Rn and assume that they satisfy the bracket condition (. that their Lie algebra spans all vector fields). Our object is to study the operator E = Xi∗ Xi , where Xi∗ is the L2 adjoint of Xi . A result of H¨rmander is that when the Xi are real then E is o hypoelliptic and furthemore it is subelliptic (the restriction of a destribution u to an open set U is “smoother” then the restriction. | Annals of Mathematics Hypoellipticity and loss of derivatives By J. J. Kohn Annals of Mathematics 162 2005 943 986 Hypoellipticity and loss of derivatives By J. J. Kohn with an Appendix by Makhlouf Derridj and David S. Tartakoff Dedicated to Yum-Tong Siu for his 60th birthday. Abstract Let X . Xp be complex-valued vector fields in Rra and assume that they satisfy the bracket condition . that their Lie algebra spans all vector fields . Our object is to study the operator E X Xi where X is the L2 adjoint of Xi. A result of Hormander is that when the Xi 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 Xi are complex-valued if the bracket condition of order one is satisfied . if the Xi Xi Xj 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 and X2 such that the bracket condition of order k 1 is satisfied and we prove that the operator E X X1 X X2 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 DaEu are bounded on U whenever m and for some 3 with 3 m k 1 the restriction of u to U is not locally bounded. 1. Introduction We will be concerned with local C x hypoellipticity in the following sense. A linear differential operator operator E on Rra is hypoelliptic if whenever u is a distribution such that the restriction of Eu to an open set U c Rra is in C X U then the restriction of u to U is also in C X U . If E is hypoelliptic then it satisfies the following a priori estimates. Research was partially supported by NSF Grant DMS-9801626. 944 J. J. KOHN 1 Given open sets U U1 in Rn such that U c U c U c Rn a nonnegative integer p and a real number

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