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

[1] A. Nagel, Vector fields and nonisotropic metrics, in Beijing
Lectures 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-elliptic
problems, Math. Notes 34, Princeton University Press, Princeton,
N.J. (1979).
[3] D. Geller, Analytic pseudodierential operators for Heisenberg
group and local solvability, Math. Notes 37, Princeton University
Press, Princeton, N.J. (1990).
[4] D.S. Tartako, Local analytic hypoellipticity for b on nondegenerate
Cauchy-Riemann manifolds, Proc. Nat. Acad.
Sci. U.S.A. 75 (1978), 3027-3028.
[5] F. Treves, Analytic hypoellipticity of a class of pseudodierential
operators with double characteristics and applications to
the @-Neumann problem, Comm. in P.D.E. 3 (1978), 475-642.
[6] G.B. Folland and E.M. Stein, Estimates for the @b complex and
analysis on the Heisenberg group, Comm. Pure and Applied
Math. 27 (1974), 429-522.
[7] M. Christ and D. Geller, Counterexamples to analytic hypoellipticity
for domains of finite type, Ann. Math. 135 (1992),
511-566.
[8] S.C. Chen and M.C. Shaw, Partial dierential equations in
several complex variables, Studies in Advanced Math. 19
(2001).