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 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
 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
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 [1] A. Nagel, Vector fields and nonisotropic metrics, in BeijingLectures in Harmonic Analysis, Ann. of Math. Studies 112,Princeton University Press, Princeton, N.J. (1986),241-306.[2] A. Nagel and E.M. Stein, Lectures on pseudo-dierential operators:Regularity theorems and applications to non-ellipticproblems, Math. Notes 34, Princeton University Press, Princeton,N.J. (1979).[3] D. Geller, Analytic pseudodierential operators for Heisenberggroup and local solvability, Math. Notes 37, Princeton UniversityPress, Princeton, N.J. (1990).[4] D.S. Tartako, Local analytic hypoellipticity for b on nondegenerateCauchy-Riemann manifolds, Proc. Nat. Acad.Sci. U.S.A. 75 (1978), 3027-3028.[5] F. Treves, Analytic hypoellipticity of a class of pseudodierentialoperators with double characteristics and applications tothe @-Neumann problem, Comm. in P.D.E. 3 (1978), 475-642.[6] G.B. Folland and E.M. Stein, Estimates for the @b complex andanalysis on the Heisenberg group, Comm. Pure and AppliedMath. 27 (1974), 429-522.[7] M. Christ and D. Geller, Counterexamples to analytic hypoellipticityfor domains of finite type, Ann. Math. 135 (1992),511-566.[8] S.C. Chen and M.C. Shaw, Partial dierential equations inseveral complex variables, Studies in Advanced Math. 19(2001).
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