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[[alternative]]On hypoellipticity of the Cauchy Riemann operator on weakly pseudoconvex CR manifolds

By [[author]]李光祥 and 李光祥

Abstract

[[abstract]]On hypoellipticity of the @b operator on weakly pseudoconvex CR manifold Let D Cn, n 2, be a CR manifold with smooth boundary, and let r be a smooth defining function for D. Hence, the set {Lk = @r @zn @ @zk ? @r @zk @ @zn | k = 1, 2, · · · , n ? 1} forms a global basis for the space of tangential (1,0) vector fields on the boundary bD. If D is strongly pseudoconvex, then bD is strongly pseudoconvex CR manifold. For example, we consider the Siegel upper half space = {(z0, zn) 2 Cn | Imzn > |z0|2} Cn. The set {Lk = @ @zk + 2izk @ @zn | k = 1, · · · , n ? 1} forms a global basis for the space of tangential (1,0) vector fields on the boundary b . If we choose T = ?2i @ @t , then the Levi matrix is the identity matrix. Moreover, the surface b is a strictly pseudoconvex CR manifold. As coordinates for the surface we use Hn = Cn?1 × R 3 (z0, t) 7! (z0, t + i|z0|2); the vector fields pull back to Zk = @ @zk + izk @ @t . The Heisenberg group Hn is a strictly pseudoconvex CR manifold with type (1,0) vector fields spanned by Z1, . . . ,Zn?1. Then we can get b = @b@ b+@ b@b is hypoelliptic on Hn for (0, q)-forms when 1 q n?1. But hypoellipticity of @ b does not always hold on a pseudoconvex CR manifold M which is not strongly pseudoconvex. For example, we consider the domain D = {(z1, z2) 2 C2 | Imz2 > [ReZ1]m,m 4 is even}. Set M to be the boundary bD, and the tangential (1,0) vector field on M is Z = @ @z1 + im 2 xm?1 1 @ @t , where x = Rez1 and z2 = t + is. Let S((z, t); (w, s)) be the Szeg¨o projection from L2(C × R) onto the kernel of Z. Define the distribution K(z, t) = S((z, t); (0, 0)). Then we can prove that K is not analytic away from 0. In the case M = {(z1, z2, z3) 2 C3 | Imz3 = [Rez1]m +|z2|2,m 4 is even}, the tangential (1,0) vector fields are spanned by Z1 = @ @z1 +im 2 xm?1 1 @ @t , and Z2 = @ @z2 +iz2 @ @t . Similarly, the Szeg¨o projection S is the orthogonal projection from L2 onto {f 2 L2 | Z1f = Z2 = 0}. Let J(z1, z2, t) = S((z1, z2, t), (0, 0, 0)). Then we can prove that J is not analytic away from 0, too. Now, we consider M = {(z1, z2) 2 C2 | Imz2 = xm,m 4 is even}. We prove the failure of @b to be analytic hypoelliptic on M directly. We examine f(x) = e2(x+xm) Rx ?1 e?4(s+sm)ds , and define f(x + iy, t) = Z 0 ?1 e?2ite?2i||1/myf(||1/mx) d . A calculation shows @b@ bf = 0, but @ bf(0 ? i, t) is not analytic at t = 0. 1

Topics: hypoelliptic, pseudoconvex, CR manifold, [[classification]]37
Year: 2010
OAI identifier: oai:ir.lib.ntnu.edu.tw:309250000Q/17717
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