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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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  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. 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). D. Geller, Analytic pseudodierential operators for Heisenberggroup and local solvability, Math. Notes 37, Princeton UniversityPress, Princeton, N.J. (1990). D.S. Tartako, Local analytic hypoellipticity for b on nondegenerateCauchy-Riemann manifolds, Proc. Nat. Acad.Sci. U.S.A. 75 (1978), 3027-3028. 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. 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. M. Christ and D. Geller, Counterexamples to analytic hypoellipticityfor domains of finite type, Ann. Math. 135 (1992),511-566. S.C. Chen and M.C. Shaw, Partial dierential equations inseveral complex variables, Studies in Advanced Math. 19(2001). 電子全文 推文當script無法執行時可按︰推文 網路書籤當script無法執行時可按︰網路書籤 推薦當script無法執行時可按︰推薦 評分當script無法執行時可按︰評分 引用網址當script無法執行時可按︰引用網址 轉寄當script無法執行時可按︰轉寄    top
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