Regularity results for $\bar\partial_b$ on CR-manifolds of hypersurface type

Abstract

We introduce a class of embedded CR manifolds satisfying a geometric condition that we call weak Y(q). For such manifolds, we show that dbar-b has closed range on L2L^2 and that the complex Green operator is continuous on L2L^2. Our methods involves building a weighted norm from a microlocal decomposition. We also prove that at any Sobolev level there is a weight such that the complex Green operator inverting the weighted Kohn Laplacian is continuous. Thus, we can solve the dbar-b equation in C∞C^\infty. Comment: 23 pages

Full text

arXiv:0912.2823v1 [math.CV] 15 Dec 2009
REGULARITY RESULTS FOR ¯
∂bON CR-MANIFOLDS OF HYPERSURFACE
TYPE
PHILLIP S. HARRINGTON AND ANDREW RAICH
1. Introduction and Results
In this article, we introduce a class of embedded CR manifolds satisfying a geometric condition
that we call weak Y(q). For such manifolds, we show that ¯
∂bhas closed range on L2and that the
complex Green operator is continuous on L2. Our methods involves building a weighted norm from
a microlocal decomposition. We also prove that at any Sobolev level there is a weight such that
the complex Green operator inverting the weighted Kohn Laplacian is continuous. Thus, we can
solve the ¯
∂b-equation in C∞.
Let M2n−1⊂CNbe a C∞compact, orientable CR-manifold, N≥n. We say that Mis of
hypersurface type if the CR-dimension of Mis n−1, so that the complex tangent bundle of M
splits into a complex subbundle of dimension n−1, the conjugate of the complex subbundle, and
one totally real direction. When the de Rham complex on Mis restricted to the complex subbundle,
we obtain the ¯
∂bcomplex.
When Mis the boundary of a pseudoconvex domain, closed range for ¯
∂bwas obtained in [Sha85],
[Koh86], and [BS86]. This work was extended to pseudoconvex manifolds of hypersurface type by
Nicoara in [Nic06]. When the domain is not pseudoconvex, there is a condition Y(q) which is
known to imply subelliptic estimates for the complex Green operator acting on (0, q) forms (see
[FK72] or [CS01] for details on Y(q)). In this article, we will adapt the microlocal analysis used in
[Nic06, Rai] to obtain closed range results for ¯
∂bon manifolds satisfying weak Y(q).
When Mis a CR-manifold of hypersurface type, the tangent space of Mcan be spanned by
(1,0) vector fields L1,...,Ln−1, their conjugates, and a totally imaginary vector field Tspanning
the remaining direction. If ¯
∂b
∗denotes the Hilbert space adjoint of ¯
∂bwith respect to the L2inner
product on M, we have a basic identity for (0, q) forms φof the form

¯
∂bφ
2+
¯
∂b
∗φ
2=X
J∈Iq
n−1
X
j=1 
¯
LjφJ
2+X
I∈Iq−1
n−1
X
j,k=1
Re(cjk T φj I , φkI ) + ···
where cjk denotes the Levi-form of Min local coordinates (see for example the proof of Theorem
8.3.5 in [CS01]) and Iqis the set of increasing q-tuples. The difficulty in using the basic identity to
prove regularity estimates for ¯
∂brests in controlling the Re(cjk T φjI , φk I ) terms. When Msatisfies
Y(q), integration by parts can be performed on the gradient term in such a way that

¯
∂bφ
2+
¯
∂b
∗φ
2≥C(X
J∈Iq
n−1
X
j=1 
¯
LjφJ
2+X
J∈Iq
n−1
X
j=1
kLjφJk2) + ··· .
Using H¨ormander’s classic result on sums of squares [H¨or67], this can be used to estimate kφk1/2. On
manifolds where the Levi-form degenerates, it may still be possible to choose good local coordinates
The first author is partially supported by NSF grant DMS-1002332 second author is partially supported by NSF
grant DMS-0855822.
1

so that with a suitable integration by parts, there is the estimate

¯
∂bφ
2+
¯
∂b
∗φ
2≥X
J∈Iq
n−1
X
j=m+1 
¯
LjφJ
2+X
J∈Iq
m
X
j=1
kLjφJk2+··· .
for some integer m. Unfortunately, since such an estimate no longer bounds all of the Ljand ¯
Lj
derivatives, it is not possible to control kφk1/2. Hence, a weight function is needed to provide some
positivity in the L2-norm. The key idea in [Nic06, Rai] is to microlocalize and decompose a form
φinto pieces whose Fourier transform is supported on specific regions. The authors then build a
weighted norm based on the decomposition. In this weighted L2-space, the cjkTterms are under
control and a basic estimate holds. If the weight function is t|z|2, then Nicoara proves that ¯
∂bhas
closed range in L2and in Hs, and if the weight function is obtained from property (Pq), then Raich
shows that the complex Green operator is compact on Hs(M) for all s≥0.
It is already known through an integration by parts argument (see the work of Ahn, Baracco and
Zampieri [ABZ06] or Zampieri [Zam08]) that local regularity estimates hold on a class of domains
where the Levi-form has degeneracies and mixed signature (known as q-pseudoconvex domains).
Our method is to apply microlocal analysis to the integration by parts argument used in the q-
pseudoconvex case to obtain a more general sufficient condition for (global) L2and Sobolev space
estimates.
Our main results are the following.
Theorem 1.1. Let M2n−1be a C∞compact, orientable weakly Y(q)CR-manifold embedded in
CN,N≥nand 1≤q≤n−2. Then the following hold:
(i) The operators ¯
∂b:L2
0,q(M)→L2
0,q+1(M)and ¯
∂b:L2
0,q−1(M)→L2
0,q(M)have closed range;
(ii) The operators ¯
∂b
∗:L2
0,q+1(M)→L2
0,q(M)and ¯
∂b
∗:L2
0,q(M)→L2
0,q−1(M)have closed range;
(iii) The Kohn Laplacian defined by b=¯
∂b¯
∂b
∗+¯
∂b
∗¯
∂bhas closed range on L2
0,q(M);
(iv) The complex Green operator Gqis continuous on L2
0,q(M);
(v) The canonical solution operators for ¯
∂b,¯
∂b
∗Gq:L2
0,q(M)→L2
0,q−1(M)and
Gq¯
∂∗
b,t :L2
0,q+1(M)→L2
0,q(M), are continuous;
(vi) The canonical solution operators for ¯
∂b
∗,¯
∂bGq:L2
0,q(M)→L2
0,q+1(M)and
Gq¯
∂b:L2
0,q−1(M)→L2
0,q(M), are continuous;
(vii) The space of harmonic forms Hq(M), defined to be the (0, q)-forms annihilated by ¯
∂band
¯
∂b
∗is finite dimensional;
(viii) If ˜q=qor q+ 1 and α∈L2
0,˜q(M)so that ¯
∂bα= 0, then there exists u∈L2
0,˜q−1(M)so that
¯
∂bu=α;
(ix) The Szeg¨o projections Sq=I−¯
∂b
∗¯
∂bGqand Sq−1=I−¯
∂b
∗Gq¯
∂bare continuous on L2
0,q(M)
and L2
0,q−1(M), respectively.
These results will be obtained by studying a family of weighted operators with respect to a norm
|kφ|ktdefined in terms of the weights et|z|2and e−t|z|2and the microlocal decomposition of φ. For
such operators, we will also be able to obtain Sobolev space estimates, as follows:
Theorem 1.2. Let M2n−1be a C∞compact, orientable weakly Y(q)CR-manifold embedded in
CN,N≥n. For s≥0there exists Ts≥0so that the following hold:
(i) The operators ¯
∂b:L2
0,q(M)→L2
0,q+1(M)and ¯
∂b:L2
0,q−1(M)→L2
0,q(M)have closed range
with respect to |k· |kt. Additionally, for any s > 0if t≥Ts, then ¯
∂b:Hs
0,q(M)→Hs
0,q+1(M)
and ¯
∂b:Hs
0,q−1(M)→Hs
0,q(M)have closed range;
2

(ii) The operators ¯
∂∗
b,t :L2
0,q+1(M)→L2
0,q(M)and ¯
∂∗
b,t :L2
0,q(M)→L2
0,q−1(M)have closed
range with respect to |k · |kt. Additionally, if t≥Ts, then ¯
∂∗
b,t :Hs
0,q+1(M)→Hs
0,q(M)and
¯
∂∗
b,t :Hs
0,q(M)→Hs
0,q−1(M)have closed range;
(iii) The Kohn Laplacian defined by b,t =¯
∂b¯
∂∗
b,t +¯
∂∗
b,t ¯
∂bhas closed range on L2
0,q(M)(with
respect to |k · |kt) and also on Hs
0,q(M)if t≥Ts;
(iv) The space of harmonic forms Hq
t(M), defined to be the (0, q)-forms annihilated by ¯
∂band
¯
∂∗
b,t is finite dimensional;
(v) The complex Green operator Gq,t is continuous on L2
0,q(M)(with respect to |k · |kt) and also
on Hs
0,q(M)if t≥Ts;
(vi) The canonical solution operators for ¯
∂b,¯
∂∗
b,tGq,t :L2
0,q(M)→L2
0,q−1(M)and Gq,t ¯
∂∗
b,t :
L2
0,q+1(M)→L2
0,q(M)are continuous (with respect to |k · |kt). Additionally,
¯
∂∗
b,tGq,t :Hs
0,q(M)→Hs
0,q−1(M)and Gq,t ¯
∂∗
b,t :Hs
0,q+1(M)→Hs
0,q(M)are continuous if
t≥Ts.
(vii) The canonical solution operators for ¯
∂∗
b,t,¯
∂bGq,t :L2
0,q(M)→L2
0,q+1(M)and Gq,t ¯
∂b:
L2
0,q−1(M)→L2
0,q(M)are continuous (with respect to |k · |kt). Additionally,
¯
∂bGq,t :Hs
0,q(M)→Hs
0,q+1(M)and Gq,t ¯
∂b:Hs
0,q−1(M)→Hs
0,q(M)are continuous if
t≥Ts.
(viii) The Szeg¨o projections Sq,t =I−¯
∂∗
b,t ¯
∂bGq,t and Sq−1,t =I−¯
∂∗
b,tGq,t ¯
∂bare continuous on
L2
0,q(M)and L2
0,q−1(M), respectively and with respect to |k · |kt. Additionally, if t≥Ts, then
Sq,t and Sq−1,t are continuous on Hs
0,q and Hs
0,q−1, respectively.
(ix) If ˜q=qor q+ 1 and α∈Hs
0,q(M)so that ¯
∂bα= 0 and α⊥ H˜q
t(with respect to |k · |kt),
then there exists u∈Hs
0,˜q−1(M)so that
¯
∂bu=α;
(x) If ˜q=qor q+ 1 and α∈C∞
0,˜q(M)satisfies ¯
∂bα= 0 and α⊥ H˜q
t(with respect to h·,·it),
then there exists u∈C∞
0,˜q−1(M)so that
¯
∂bu=α.
Remark 1.3.We will see below that the proof of Theorem 1.1 follows from Theorem 1.2 and the
fact that the weighted and unweighted norms are equivalent. We will see in the proof of the main
theorem that the constants improve as t→ ∞. In particular, we will show that kϕk2
t≤AtQb,t(ϕ, ϕ)
where At→0 as t→ ∞. A (weak) consequence is that if the weight is strong enough, ¯
∂and ¯
∂b
∗
have closed range in weighted L2with a constant that does not depend on the weight. In the
unweighted case, this means the constants may be quite large. For a more quantitative discussion,
see Remark 7.1 below.
Additionally, our results hold for any abstract CR-manifold for which a q-compatible function
exists. q-compatible functions are defined in Definition 2.7. They play the analogous role here of
CR-plurisubharmonic functions in [Nic06, Rai].
In Section 2, we introduce the notion of weak Y(q) manifolds and q-compatible functions. In
Section 3, we set up the microlocal analysis and build the weighted norm. Additionally, we compute
¯
∂band ¯
∂b
∗in local coordinates. In Section 4, we adapt the microlocal analysis in [Nic06, Rai] and
prove a basic estimate: Proposition 4.1. In Section 5, we use the basic estimate to begin the study
of the regularity theory for ¯
∂b, and we prove Theorems 1.2 and 1.1 in Sections 6 and 7, respectively.
2. Definitions and Notation
2.1. CR manifolds and ¯
∂b.
3

Definition 2.1. Let M⊂CNbe a C∞manifold of real dimension 2n−1, n≥2. Mis called a CR-
manifold of hypersurface type if Mis equipped with a subbundle T1,0(M) of the complexified
tangent bundle CT M =T M ⊗Cso that
(i) dimCT1,0(M) = n−1;
(ii) T1,0(M)∩T0,1(M) = {0}where T0,1(M) = T1,0(M);
(iii) T1,0(M) satisfies the following integrability condition: if L1, L2are smooth sections of
T1,0(M), then so is the commutator [L1, L2].
Since Mis a submanifold of CN, we can generate T1,0
z(M) for z∈Mfrom the induced CR-
structure on Mas follows: set T1,0
z(M) = T1,0
z(CN)∩Tz(M)⊗C(under the natural inclusions).
Since the complex dimension of T1,0
z(M) is n−1 for all z∈M, we can let T1,0(M) = Sz∈MT1,0
z(M).
Observe that conditions (ii) and (iii) are automatically satisfied in this case.
For the remainder of this article, M2n−1is a smooth, orientable CR-manifold of hypersurface
type embedded in CNfor some N≥n. Let Λ0,q(M) be the bundle of (0, q)-forms on M, i.e.,
Λ0,q(M) = Vq(T0,1(M)∗). Denote the C∞sections of Λ0,q(M) by C∞
0,q(M).
We construct ¯
∂busing the fact that M⊂CN. There is a Hermitian inner product on Λ0,q (M)
given by
(ϕ, ψ) = ZM
hϕ, ψixdV,
where dV is the volume element on Mand hϕ, ψixis the induced inner product on Λ0,q(M). This
metric is compatible with the induced CR-structure, i.e., the vector spaces T1,0
z(M) and T0,1
z(M)
are orthogonal under the inner product. The involution condition (iii) of Definition 2.1 means that
¯
∂bcan be defined as the restriction of the de Rham exterior derivative dto Λ(0,q)(M). The inner
product gives rise to an L2-norm k·k0, and we also denote the closure of ¯
∂bin this norm by ¯
∂b(by
an abuse of notation). In this way, ¯
∂b:L2
0,q(M)→L2
0,q+1(M) is a well-defined, closed, densely
defined operator, and we define ¯
∂b
∗:L2
0,q+1(M)→L2
0,q(M) to be the L2-adjoint of ¯
∂b. The Kohn
Laplacian b:L2
0,q(M)→L2
0,q(M) is defined as
b=¯
∂b
∗¯
∂b+¯
∂b¯
∂b
∗.
2.2. The Levi form and eigenvalue conditions. The induced CR-structure has a local or-
thonormal basis L1,...,Ln−1for the (1,0)-vector fields in a neighborhood Uof each point x∈M.
Let ω1,...,ωn−1be the dual basis of (1,0)-forms that satisfy hωj, Lki=δj k. Then ¯
L1,..., ¯
Ln−1
is a local orthonormal basis for the (0,1)-vector fields with dual basis ¯ω1,..., ¯ωn−1in U. Also,
T(U) is spanned by L1,...,Ln−1,¯
L1,..., ¯
Ln−1, and an additional vector field Ttaken to be purely
imaginary (so ¯
T=−T). Let γbe the purely imaginary global 1-form on Mthat annihilates
T1,0(M)⊕T0,1(M) and is normalized so that hγ, T i=−1.
Definition 2.2. The Levi form at a point x∈Mis the Hermitian form given by hdγx, L ∧¯
L′i
where L, L′∈T1,0
x(U), Ua neighborhood of x∈M.
Definition 2.3. We call Mweakly pseudoconvex if there exists a form γsuch that the Levi
form is positive semi-definite at all x∈Mand strictly pseudoconvex if there is a form γsuch
that the Levi form is positive definite at all x∈M.
The following two (standard) definitions are taken from Chen and Shaw [CS01].
Definition 2.4. Let Mbe an oriented CR-manfiold of real dimension 2n−1 with n≥2. Mis
said to satisfy condition Z(q), 1 ≤q≤n−1, if the Levi form associated with Mhas at least
n−qpositive eigenvalues or at least q+ 1 negative eigenvalues at every boundary point. Mis said
to satisfy condition Y(q), 1 ≤q≤n−1 if the Levi form has at least either max{n−q, q + 1}
4

eigenvalues of the same sign of min{n−q, q + 1}pairs of eigenvalues of opposite signs at every
point on M.
Note that Y(q) is equivalent to Z(q) and Z(n−1−q). The necessity of the symmetric requirements
for ¯
∂bat levels qand n−1−qstems from the duality between (0, q)-forms and (0, n −1−q)-forms
(see [FK72] or [RS08] for details).
Z(q) and Y(q) are classical conditions and natural extensions of strict pseudoconvexity. We
wish, however, for an extension of weak pseudoconvexity. Let P∈Mand Ube a special boundary
neighborhood. Then there exists an orthonormal basis L1,...,Ln−1of T1,0(U). By the Cartan
formula (see [Bog91], p.14),
hdγ, Lj∧¯
Lki=−hγ, [Lj,¯
Lk]i.
If
[Lj,¯
Lk] = cjk Tmod T1,0(U)⊕T0,1(U),
then hdγ, Lj∧¯
Lki=cjk . For this reason, the matrix (cjk )1≤j,k≤n−1is called the Levi form with
respect to L1,...,Ln−1.
By weakening the definition of Z(q), we obtain:
Definition 2.5. Let Mbe a smooth, compact, oriented CR-manifold of hypersurface type of real
dimension 2n−1. We say Msatisfies Z(q) weakly at P if there exists
(i) a special boundary neighborhood U⊂Mcontaining P;
(ii) an integer m=m(U)6=q;
(iii) an orthonormal basis L1,...,Ln−1of T1,0(U) so that µ1+···+µq−(c11 +···+cmm )≥0
on U, where µ1,...,µn−1are the eigenvalues of the Levi form in increasing order.
We say that Mis weakly Z(q) if Mis Z(q) weakly at Pfor all P∈Mand the condition m > q
or m < q is independent of U⊂M. As above, Msatisfies Y(q) weakly at P if Msatisfies Z(q)
weakly at Pand Z(n−1−q) weakly at P.
To see that Definition 2.5 generalizes condition Z(q), choose coordinates diagonalizing cjk at
Pso that cjj |P=µj. If the Levi-form has at least n−qpositive eigenvalues, then µq>0, so
we can let m=q−1 and obtain µ1+···+µq−(c11 +···+cmm) = µq>0 at P. If the Levi-
form has at least q+ 1 negative eigenvalues, then µq+1 <0, so we can let m=q+ 1 and obtain
µ1+···+µq−(c11 +···+cmm ) = −µq+1 >0 at P. In either case, the sum is strictly positive at
P, so the estimate extends to a neighborhood U.
The preceding argument also shows that weak-Z(q) is satisfied by domains where the Levi-form
is locally diagonalizable and has at least n−qnon-negative eigenvalues or q+ 1 non-positive
eigenvalues. However, diagonalizability is not necessary. Consider the hypersurface in C5defined
by ρ(z) = Im z5+|z3|2+|z4|2+ (Re z1)(|z1|2−2|z2|2). Under the coordinates Lj=∂
∂zj−2i∂ρ
∂zj
∂
∂z5
and T= 2i∂
∂z5+ 2i∂
∂¯z5the Levi-form looks like




2 Re z1−z20 0
−¯z2−2 Re z10 0
0 0 1 0
0 0 0 1



.
We can compute the eigenvalues of this matrix in increasing order as
n−p4(Re z1)2+|z2|2,p4(Re z1)2+|z2|2,1,1o.
Since the corresponding eigenvectors are discontinuous at P= 0, the Levi-form can not be diag-
onalized in a neighborhood of P= 0. In fact, we can not even continuously separate the positive
and negative eigenspaces. Let q= 2 and m= 0. The sum of the two smallest eigenvalues is zero,
so this domain satisfies weak Z(2), which is equivalent to weak Y(2) when n= 5.
5

The signature of the Levi-form may also change locally. If we let ρ(z) = Im z5+|z2|2+|z3|2+
|z4|2+ Re((z1)2¯z1) with Ljand Tas before, then we have a diagonal Levi-form with eigenvalues
{2 Re(z1),1,1,1}. When Re(z1)>0, we have four positive eigenvalues. When Re(z1)<0, we
have three positive and one negative eigenvalues. Note that since we always have at least three
positive eigenvalues, this satisfies the standard definition of Y(2). From the standpoint of weak
Z(2), we can take m= 0 and obtain µ1+µ2= 2 Re(z1) + 1 >0 near P, or we can take m= 1 and
obtain µ1+µ2−c11 = (2 Re(z1) + 1) −2 Re(z1) = 1 >0, so either value of mmay work. Hence,
the appropriate value of mneed not be constant on M. However, since we disallow m=q, the
condition m < q or m > q must be global.
If we can choose m < q independent of the local neighborhood U, then weak Z(q) agrees
with (q−1)-pseudoconvexity (see [Zam08] for the definition on boundaries of domains and further
references, or [ABZ06] for generic CR submanifolds). If Msatisfies weak Z(1) for a choice of m= 0,
then Mis simply a weakly pseudoconvex CR-manifold of hypersurface type.
Remark 2.6.For a CR-manifold Mthat satisfies weak Y(q), the mthat corresponds to Z(q) has
no relation to the mthat corresponds to Z(n−1−q). To emphasize this, we may use mqfor the
integer-valued function on Mthat corresponds to weak Z(q) and similarly for mn−1−qfor weak
Z(n−1−q).
2.3. q-compatible functions. Let Iq={J= (j1,...,jq)∈Nq: 1 ≤j1<··· < jq≤n−1}.
Let λbe a function defined near Mand define the 2-form
(1) Θλ=1
2∂b¯
∂bλ−¯
∂b∂bλ+1
2ν(λ)dγ.
where νis the real normal to M. We will sometimes consider Θλto be the matrix Θλ= (Θλ
jk ).
Definition 2.7. Let Mbe a smooth, compact, oriented CR-manifold of hypersurface type of real
dimension 2n−1 satisfying Z(q) weakly at some point P∈M. Let λbe a smooth function
near M. We say λis q-compatible with Mat Pif there exists a special boundary neighborhood
U⊂Mcontaining P, an integer mq=mq(U) from weak Z(q), an orthonormal basis L1,...,Ln−1
of T1,0(U), and a constant Bλ>0 satisfying
(i) µ1+···+µq−(c11 +···+cmm)≥0 on U, where µ1,...,µn−1are the eigenvalues of the
Levi form in increasing order.
(ii) b1+···+bq−(Θ11 +···+ Θmm)≥Bλon Uif m < q, where b1, . . . , bn−1are the eigenvalues
of Θ in increasing order.
(iii) bn−q+···+bn−1−(Θ11 +···+ Θmm )≤ −Bλon Uif m > q.
We call Bλthe positivity constant of λ. Observe that if Mis pseudoconvex, Msatisfies Definition
2.5 for any 1 ≤q≤n−1 and any orthonormal basis L1,...,Ln−1by selecting m= 0. Hence,
plurisubharmonic functions will be q-compatible with pseudoconvex domains for any 1 ≤q≤n−1.
Remark 2.8.If λ=|z|2then Proposition 3.1 below proves that Θ = ∂¯
∂when tested against
complex tangent vectors of M. Tested against such vectors, Θ|z|2=I. Since this is diagonal and
all of the eigenvalue of Iare 1, b1+··· +bq−(Θ11 +··· + Θmm ) = q−m≥1 if q > m and
bn−q+··· +bn−1−(Θ11 +··· + Θmm) = q−m≤ −1 if q < m. Hence, λ=|z|2is always a
q-compatible function on Mwith positivity constant 1.
Remark 2.9.Without the requirement that {L1,...,Ln−1}are orthonormal, λ=|z|2may not
be a q-compatible function for all values of m6=q. For a given choice of non-orthonormal local
coordinates, we can always define a local function which is q-compatible for all allowable qand m,
but there is no guarantee that such local functions could be made global. Hence, if we remove the
restriction that the local coordinates in Definition 2.7 are orthonormal, we must also assume the
existence of a global function which is q-compatible for all allowable choices of qand m.
6

Remark 2.10.We note that if for every Bλ>0 there exists a q-compatible function λsatisfying
0≤λ≤1 with positivity constant Bλ, then the methods of [Rai] can be incorporated into our
current paper to show that the complex Green operator is compact. Such a condition is analogous
to Catlin’s Property (P) [Cat84].
In this article, constants with no subscripts may depend on n,N,Mbut not any relevant q-
compatible function. Those constants will be denoted with an appropriate subscript. The constant
Awill be reserved for the constant in the construction of the pseudodifferential operator in Section
3.
3. Computations in Local Coordinates
3.1. Local coordinates and CR-plurisubharmonicity. The following result is proved in [Rai].
Proposition 3.1. Let M2n−1be a smooth, orientable CR-manifold of hypersurface type embedded
in CNfor some N≥n. If λis a smooth function near M,L∈T1,0(M), and νis the real part of
the complex normal to M, then on M
1
2∂¯
∂λ −¯
∂∂λ, L ∧¯
L−1
2∂b¯
∂bλ−¯
∂b∂bλ, L ∧¯
L=1
2ν{λ}hdγ, L ∧¯
Li
3.2. Pseudodifferential Operators. We follow the setup for the microlocal analysis in [Rai].
Since Mis compact, there exists a finite cover {Uν}νso each Uνhas a special boundary system
and can be parameterized by a hypersurface in Cn(Uνmay be shrunk as necessary). To set up
the microlocal analysis, we need to define the appropriate pseudodifferential operators on each Uν.
Let ξ= (ξ1,...,ξ2n−2, ξ2n−1) = (ξ′, ξ2n−1) be the coordinates in Fourier space so that ξ′is dual to
the part of T(M) in the maximal complex subspace (i.e., T1,0(M)⊕T0,1(M)) and ξ2n−1is dual to
the totally real part of T(M), i.e.,the “bad” direction T. Define
C+={ξ:ξ2n−1≥1
2|ξ′|and |ξ| ≥ 1};
C−={ξ:−ξ∈ C+};
C0={ξ:−3
4|ξ′| ≤ ξ2n−1≤3
4|ξ′|} ∪ {ξ:|ξ| ≤ 1}.
Note that C+and C−are disjoint, but both intersect C0nontrivially. Next, we define functions on
{|ξ|:|ξ|2= 1}. Let
ψ+(ξ) = 1 when ξ2n−1≥3
4|ξ′|and supp ψ+⊂ {ξ:ξ2n−1≥1
2|ξ′|};
ψ−(ξ) = ψ+(−ξ);
ψ0(ξ) satisfies ψ0(ξ)2= 1 −ψ+(ξ)2−ψ−(ξ)2.
Extend ψ+,ψ−, and ψ0homogeneously outside of the unit ball, i.e., if |ξ| ≥ 1, then
ψ+(ξ) = ψ+(ξ/|ξ|), ψ−(ξ) = ψ−(ξ/|ξ|),and ψ0(ξ) = ψ0(ξ/|ξ|).
Also, extend ψ+,ψ−, and ψ0smoothly inside the unit ball so that (ψ+)2+ (ψ−)2+ (ψ0)2= 1.
Finally, for a fixed constant A > 0 to be chosen later, define for any t > 0
ψ+
t(ξ) = ψ(ξ/(tA)), ψ−
t(ξ) = ψ−(ξ/(tA)),and ψ0
t(ξ) = ψ0(ξ/(tA)).
Next, let Ψ+
t, Ψ−
t, and Ψ0be the pseudodifferential operators of order zero with symbols ψ+
t,ψ−
t,
and ψ0
t, respectively. The equality (ψ+
t)2+ (ψ−
t)2+ (ψ0
t)2= 1 implies that
(Ψ+
t)∗Ψ+
t+ (Ψ0
t)∗Ψ0
t+ (Ψ−
t)∗Ψ−
t=Id.
7

We will also have use for pseudodifferential operators that “dominate” a given pseudodifferential
operator. Let ψbe cut-off function and ˜
ψbe another cut-off function so that ˜
ψ|supp ψ≡1. If Ψ
and ˜
Ψ are pseudodifferential operators with symbols ψand ˜
ψ, respectively, then we say that ˜
Ψ
dominates Ψ.
For each Uν, we can define Ψ+
t, Ψ−
t, and Ψ0
tto act on functions or forms supported in Uν, so let
Ψ+
ν,t, Ψ−
ν,t, and Ψ0
ν,t be the pseudodifferential operators of order zero defined on Uν, and let C+
ν,C−
ν,
and C0
νbe the regions of ξ-space dual to Uνon which the symbol of each of those pseudodifferential
operators is supported. Then it follows that:
(Ψ+
ν,t)∗Ψ+
ν,t + (Ψ0
ν,t)∗Ψ0
ν,t + (Ψ−
ν,t)∗Ψ−
ν,t =Id.
Let ˜
Ψ+
µ,t and ˜
Ψ−
µ,t be pseudodifferential operators that dominate Ψ+
µ,t and Ψ−
µ,t, respectively (where
Ψ+
µ,t and Ψ−
µ,t are defined on some Uµ). If ˜
C+
µand ˜
C−
µare the supports of ˜
Ψ+
µ,t and ˜
Ψ−
µ,t, respectively,
then we can choose {Uµ},˜
ψ+
µ,t, and ˜
ψ−
µ,t so that the following result holds.
Lemma 3.2. Let Mbe a compact, orientable, embedded CR-manifold. There is a finite open
covering {Uµ}µof Mso that if Uµ, Uν∈ {Uµ}have nonempty intersection, then there exists a
diffeomorphism ϑbetween Uνand Uµwith Jacobian Jϑso that:
(i) t
Jϑ(˜
C+
µ)∩ C−
ν=∅and C+
ν∩t
Jϑ(˜
C−
µ) = ∅where t
Jϑis the inverse of the transpose of Jϑ;
(ii) Let ϑ
Ψ+
µ,t,ϑ
Ψ−
µ,t, and ϑ
Ψ0
µ,t be the transfers of Ψ+
µ,t,Ψ−
µ,t, and Ψ0
µ,t, respectively via ϑ. Then
on {ξ:ξ2n−1≥4
5|ξ′|and |ξ| ≥ (1 + ǫ)tA}, the principal symbol of ϑ
Ψ+
µ,t is identically 1, on
{ξ:ξ2n−1≤ −4
5|ξ′|and |ξ| ≥ (1 + ǫ)tA}, the principal symbol of ϑ
Ψ−
µ,t is identically 1, and
on {ξ:−1
3ξ2n−1≥1
3|ξ′|and |ξ| ≥ (1 + ǫ)tA}, the principal symbol of ϑ
Ψ0
µ,t is identically 1,
where ǫ > 0can be very small;
(iii) Let ϑ˜
Ψ+
µ,t,ϑ˜
Ψ−
µ,t be the transfers via ϑof ˜
Ψ+
µ,t and ˜
Ψ−
µ,t, respectively. Then the principal
symbol of ϑ˜
Ψ+
µ,t is identically 1 on C+
νand the principal symbol of ϑ˜
Ψ−
µ,t is identically 1 on
C−
ν;
(iv) ˜
C+
µ∩˜
C−
µ=∅.
We will suppress the left superscript ϑas it should be clear from the context which pseudodif-
ferential operator must be transferred. The proof of this lemma is contained in Lemma 4.3 and its
subsequent discussion in [Nic06].
If Pis any of the operators Ψ+
µ,t, Ψ−
µ,t, or Ψ0
µ,t, then it is immediate that
(2) Dα
ξσ(P) = 1
|t|αqα(x, ξ)
for |α| ≥ 0, where q(x, ξ) is bounded independently of t.
3.3. Norms. We have a volume form dV on M, and we define the following inner products and
norms on functions (with their natural generalizations to forms). Let λbe a smooth function
defined near M. We define
(φ, ϕ)λ=ZM
φ¯ϕ e−λdV, and kϕk2
λ= (ϕ, ϕ)λ
In particular, (φ, ϕ)0=RMφ¯ϕ dV and kϕk2
0= (ϕ, ϕ)0are the standard (unweighted) L2inner
product and norm. If ϕ=PJ∈IqϕJ¯ωJ, then we use the common shorthand kϕk=PJ∈IqkϕJk
where k · k represents any norm of ϕ.
We also need a norm that is well-suited for the microlocal arguments. Let λ+and λ−be smooth
functions defined near M. Let {ζν}be a partition of unity subordinate to the covering {Uν}
8

satisfying Pνζ2
ν= 1. Also, for each ν, let ˜
ζνbe a cutoff function that dominates ζνso that
supp ˜
ζν⊂Uν. Then we define the global inner product and norm as follows:
hφ, ϕiλ+,λ−=hφ, ϕi±=X
νh(˜
ζνΨ+
ν,tζνφν,˜
ζνΨ+
ν,tζνϕν)λ+
+ (˜
ζνΨ0
ν,tζνφν,˜
ζνΨ0
ν,tζνϕν)0+ (˜
ζνΨ−
ν,tζνφν,˜
ζνΨ−
ν,tζνϕν)λ−i
and
|kϕ|k2
λ+,λ−=|kϕ|k2
±=X
νhk˜
ζνΨ+
ν,tζνϕνk2
λ++k˜
ζνΨ0
ν,tζνϕνk2
0+k˜
ζνΨ−
ν,tζνϕνk2
λ−i,
where ϕνis the form ϕexpressed in the local coordinates on Uν. The superscript νwill often be
omitted.
For a form ϕsupported on M, the Sobolev norm of order sis given by the following:
kϕk2
s=X
ν
k˜
ζνΛsζνϕνk2
0
where Λ is defined to be the pseudodifferential operator with symbol (1 + |ξ|2)1/2.
In [Rai], it is shown that there exist constants c±and C±so that
(3) c±kϕk2
0≤ |kϕ|k2
λ+,λ−≤C±kϕk2
0
where c±and C±depend on maxM{|λ+|+|λ−|} (assuming tA ≥1). Additionally, there exists an
invertible self-adjoint operator H±so that (φ, ϕ)0=hφ, H±ϕi±.
3.4. ¯
∂band its adjoints. If fis a function on M, in local coordinates,
¯
∂bf=
n−1
X
j=1
¯
Ljf¯ωj,
while if ϕis a (0, q)-form, there exist functions mJ
Kso that
¯
∂bϕ=X
J∈Iq
K∈Iq+1
n−1
X
j=1
ǫjJ
K¯
LjϕJ¯ωK+X
J∈Iq
K∈Iq+1
ϕJmJ
K¯ωK
where ǫjJ
Kis 0 if {j} ∪ J6=Kas sets and is the sign of the permutation that reorders jJ as K. We
also define
ϕjI =X
J∈Iq
ǫjI
JϕJ
(in this case, |I|=q−1 and |J|=q). Let ¯
L∗
jbe the adjoint of ¯
Ljin (·,·)0,¯
L∗,λ
jbe the adjoint of
¯
Ljin (·,·)λ. We define ¯
∂b
∗and ¯
∂∗,λ
bin L2(M) and L2(M, e−λ), respectively. In this paper, λstands
for λ+or λ−and we will abbreviate ¯
∂∗,λ+
bby ¯
∂∗,+
band similarly for ¯
∂∗,−
b,¯
L∗,+,¯
L∗,−, etc.
On a (0, q)-form ϕ, we have (for some functions fj∈C∞(U))
¯
∂b
∗ϕ=X
I∈Iq−1
n−1
X
j=1
¯
L∗
jϕjI ¯ωI+X
I∈Iq−1
J∈Iq
mI
JϕJ¯ωI
=−X
I∈Iq−1
n−1
X
j=1 LjϕjI +fjϕj I ¯ωI+X
I∈Iq−1
J∈Iq
mI
JϕJ¯ωI
9

¯
∂∗,λ
bϕ=X
I∈Iq−1
n−1
X
j=1
¯
L∗,λ
jϕjI ¯ωI+X
I∈Iq−1
mI
JϕJ¯ωI
(4)
=−X
I∈Iq−1
n−1
X
j=1 LjϕjI −LjλϕjI +fjϕjI ¯ωI+X
I∈Iq−1
J∈Iq
mI
JϕJ¯ωI
Consequently, we see that
¯
∂∗,λ
b=¯
∂b
∗−[¯
∂b
∗, λ],
and both adjoints have the same domain. Finally, let ¯
∂∗
b,±be the adjoint of ¯
∂bwith respect to
h· ,·i±.
The computations proving Lemma 4.8 and Lemma 4.9 and equation (4.4) in [Nic06] can be
applied here with only a change of notation, so we have the following two results, recorded here as
Lemma 3.3 and Lemma 3.4. The meaning of the results is that ¯
∂∗
b,±acts like ¯
∂∗,+
bfor forms whose
support is basically C+and ¯
∂∗,−
bon forms whose support is basically C−.
Lemma 3.3. On smooth (0, q)-forms,
¯
∂∗
b,±=¯
∂b
∗−X
µ
ζ2
µ˜
Ψ+
µ,t[¯
∂b
∗, λ+] + X
µ
ζ2
µ˜
Ψ−
µ,t[¯
∂b
∗, λ−]
+X
µ˜
ζµ[˜
ζµΨ+
µ,tζµ,¯
∂b]∗˜
ζµΨ+
µ,tζµ+ζµ(Ψ+
µ,t)∗˜
ζµ[¯
∂∗,+
b,˜
ζµΨ+
µ,tζµ]˜
ζµ
+˜
ζµ[˜
ζµΨ−
µ,tζµ,¯
∂b]∗˜
ζµΨ−
µ,tζµ+ζµ(Ψ+
µ,t)∗˜
ζµ[¯
∂∗,−
b,˜
ζµΨ−
µ,tζµ]˜
ζµ+EA,
where the error term EAis a sum of order zero terms and “lower order” terms. Also, the symbol
of EAis supported in C0
µfor each µ.
We are now ready to define the energy forms that we use. Let
Qb,±(φ, ϕ) = h¯
∂bφ, ¯
∂bϕi±+h¯
∂∗
b,±φ, ¯
∂∗
b,±ϕi±
Qb,+(φ, ϕ) = ( ¯
∂bφ, ¯
∂bϕ)λ++ ( ¯
∂∗,+
bφ, ¯
∂∗,+
bϕ)λ+
Qb,0(φ, ϕ) = ( ¯
∂bφ, ¯
∂bϕ)0+ ( ¯
∂b
∗φ, ¯
∂b
∗ϕ)0
Qb,−(φ, ϕ) = ( ¯
∂bφ, ¯
∂bϕ)λ−+ ( ¯
∂∗,−
bφ, ¯
∂∗,−
bϕ)λ−.
Lemma 3.4. If ϕis a smooth (0, q)-form on M, then there exist constants K, K±and K′with
K≥1so that
(5)
KQb,±(ϕ, ϕ) + KtX
ν
k˜
ζν˜
Ψ0
ν,tζνϕνk2
0+K′kϕk2
0+Ot(kϕk2
−1)≥X
νhQb,+(˜
ζνΨ+
ν,tζνϕν,˜
ζνΨ+
ν,tζνϕν)
+Qb,0(˜
ζνΨ0
ν,tζνϕν,˜
ζνΨ0
ν,tζνϕν) + Qb,−(˜
ζνΨ−
ν,tζνϕν,˜
ζνΨ−
ν,tζνϕν)i
Kand K′do not depend on t, λ−or λ+.
Also, since ¯
∂∗,λ
b=¯
∂b
∗+ “lower order” and Ψλ
µ,t satisfies (2), commuting ¯
∂∗,λ
bby Ψλ
µ,t creates error
terms of order 0 that do not depend on tor λ, although lower order terms that may depend on t
and λ.
10

4. The Basic Estimate
The goal of this section is to prove a basic estimate for smooth forms on M.
Proposition 4.1. Let M⊂CNbe a compact, orientable CR-manifold of hypersurface type of
dimension 2n−1and 1≤q≤n−2. Assume that Madmits functions λ1and λ2where λ1is a
q-compatible function and λ2is an (n−1−q)-compatible function with positivity constants Bλ+
and Bλ−, respectively. Let ϕ∈Dom( ¯
∂b)∩Dom( ¯
∂b
∗). Set
λ+=(tλ1if mq< q
−tλ1if mq> q
and
λ−=(−tλ2if mn−1−q< n −1−q
tλ2if mn−1−q> n −1−q.
There exist constants K,K±, and K′
±where Kdoes not depend on λ+and λ−so that
tB±|kϕ|k2
±≤KQb,±(ϕ, ϕ) + K|kϕ|k2
±+K±X
νX
J∈Iq
k˜
ζν˜
Ψ0
ν,tζνϕν
Jk2
0+K′
±kϕk2
−1.
The constant B±= min{Bλ+, Bλ−}.
For Theorem 1.1, we will use λ1=λ2=|z|2.
4.1. Local Estimates. The crucial multilinear algebra that we need is contained in the following
lemma from Straube [Str]:
Lemma 4.2. Let B= (bjk)1≤j,k≤nbe a Hermitian matrix and 1≤q≤n. The following are
equivalent:
(i) If u∈Λ(0,q), then X
K∈Iq−1
n
X
j,k=1
bjk uj K ukK ≥M|u|2.
(ii) The sum of any qeigenvalues of Bis at least M.
(iii)
q
X
s=1
n
X
j,k=1
bjk ts
jts
k≥Mwhenever t1,...,tqare orthonormal in Cn.
We work on a fixed U=Uν. On this neighborhood, as above, there exists an orthonormal basis
of vector fields L1,...,Ln,¯
L1,..., ¯
Lnso that
(6) [Lj,¯
Lk] = cjk T+
n−1
X
ℓ=1
(dℓ
jk Lℓ−¯
dℓ
kj ¯
Lℓ)
if 1 ≤j, k ≤n−1, and T=Ln−¯
Ln. Note that cjk are the coefficients of the Levi form. Recall
that ¯
L∗,+,¯
L∗, and ¯
L∗,−are the adjoints of ¯
Lin (·,·)λ+, (·,·)0, and (·,·)λ−, respectively. From (4),
we see that
¯
L∗,λ
j=−Lj+Ljλ−fj
and plugging this into (6), we have
(7) [¯
L∗,λ
j,¯
Lk] = −cjk T+
n−1
X
ℓ=1 dℓ
jk (¯
L∗,λ
ℓ−Lℓλ+fℓ) + ¯
dℓ
kj ¯
Lℓ−¯
LkLjλ+¯
Lkfj.
Because of Lemma 3.4, we may turn our attention to the the quadratic
Qb,λ(ϕ, ϕ) = ( ¯
∂bϕ, ¯
∂bϕ)λ+ ( ¯
∂∗,λ
bϕ, ¯
∂∗,λ
bϕ)λ.
11

We introduce the error term
E(ϕ)≤C
kϕk2
λ+
n−1
X
j=1
|(h¯
Ljϕ, ϕ)λ|
=C
kϕk2
λ+
n−1
X
j=1
|(˜
h¯
L∗,λ
jϕ, ϕ)λ|

where the operators ¯
Ljand ¯
L∗,λ
jact componentwise, Cis a constant independent of ϕand λ, and
hand ˜
hare bounded functions that are independent of t,A,λ+,λ−, and the other quantities that
are carefully minding. Recall the definition that ϕj K =PJ∈IqǫjK
JϕJ. As in the proof of Lemma
4.2 in [Rai], we compute that for smooth ϕsupported in a special boundary neighborhood,
Qb,λ(ϕ, ϕ) = X
J∈Iq
n−1
X
j=1
k¯
LjϕJk2
λ+X
I∈Iq−1
n−1
X
j,k=1
Re cjk T ϕj I , ϕkI λ+E(ϕ)
+X
I∈Iq−1
n−1
X
j,k=1 (1
2(¯
LjLkλ+Lj¯
Lkλ)ϕjI , ϕkI λ+1
2
n−1
X
ℓ=1 (dℓ
jk Lℓλ+dℓ
jk ¯
Lℓλ)ϕjI , ϕkI λ).(8)
The weak Z(q)-hypothesis suggests that we ought to integrate by parts to take advantage of the
positivity/negativity conditions. By (7) and integration by parts, we have
(9)
k¯
LjϕJk2
λ− k ¯
L∗,λ
jϕJk2
λ=−Re(cjj T ϕJ, ϕJ)−
n−1
X
ℓ=1
Re dℓ
jj (Lℓλ)ϕJ, ϕJ−Re(( ¯
LjLjλ)ϕJ, ϕJ) + E(ϕ).
Consequently, we can use (7) and (9) to obtain
Qb,λ(ϕ, ϕ) = X
J∈Iqnm
X
j=1
k¯
L∗,λ
jϕJk2
λ+
n−1
X
j=m+1
k¯
LjϕJk2
λo+E(ϕ)
+X
I∈Iq−1
n−1
X
j,k=1
Re cjk T ϕj I , ϕkI λ−X
J∈Iq
m
X
j=1
Re cjj T ϕJ, ϕJλ
+X
I∈Iq−1
n−1
X
j,k=1 (1
2(¯
LjLkλ+Lj¯
Lkλ)ϕjI , ϕkI λ+1
2
n−1
X
ℓ=1 (dℓ
jk Lℓλ+dℓ
jk ¯
Lℓλ)ϕjI , ϕkI λ)
(10)
−X
J∈Iq
m
X
j=1 (1
2(¯
LjLjλ+Lj¯
Ljλ)ϕJ, ϕJλ+1
2
n−1
X
ℓ=1 (dℓ
jj Lℓλ+dℓ
jj ¯
Lℓλ)ϕJ, ϕJλ).
We are now in a position to control the “bad” direction terms. Recall the following consequence
of the sharp G˚arding inequality from [Rai].
Proposition 4.3. Let Rbe a first order pseudodifferential operator such that σ(R)≥κwhere κ
is some positive constant and (hjk)a hermitian matrix (that does not depend on ξ). Then there
exists a constant Csuch that if the sum of any qeigenvalues of (hjk)is nonnegative, then
Re nX
I∈Iq−1
n−1
X
j,k=1 hjkRujI , uk I o≥κRe X
I∈Iq−1
n−1
X
j,k=1 hjkuj I , ukI −Ckuk2,
12

and if the the sum of any collection of (n−1−q)eigenvalues of (hjk )is nonnegative, then
Re nX
J∈Iq
n−1
X
j=1 hjj RuJ, uJ−X
I∈Iq−1
n−1
X
j,k=1 hjkRujI , uk I o
≥κRe nX
J∈Iq
n−1
X
j=1 hjj uJ, uJ−X
I∈Iq−1
n−1
X
j,k=1 hjkuj I , ukI o−Ckuk2.
Note that (hjk ) may be a matrix-valued function in zbut may not depend on ξ.
The following lemma is the analog of Lemma 4.6 in [Rai].
Lemma 4.4. Let Mbe as in Theorem 1.2 and ϕa(0, q)-form supported on Uso that up to a
smooth term ˆϕis supported in C+. Let
(h+
jk ) = (cj k )−δjk
1
q
m
X
ℓ=1
cℓℓ.
Then
Re nX
I∈Iq−1
n−1
X
j,k=1 h+
jk T ϕj I , ϕkI λo
≥tA Re nX
I∈Iq−1
n−1
X
j,k=1 h+
jk ϕj I , ϕkI λo−O(kϕk2
λ)−Ot(k˜
ζν˜
Ψ0
tϕk2
0).
where the constant in O(kϕk2
λ)does not depend on t.
Proof. Observe that the eigenvalues of (h+
jk ) are µj−1
qPm
ℓ=1 cℓℓ, so the smallest possible sum of
any qeigenvalues of (h+
jk ) is
µ1+···+µq−
m
X
ℓ=1
cℓℓ ≥0.
With this inequality in hand, we employ the argument of Proposition 4.6 from [Rai] with the
following changes. First, we replace cjk with h+
jk . Also, we replace the Awith tA (for example, the
sentence “By construction, ξ2n−1≥Ain C+. . . ” gets replaced by “By construction, ξ2n−1≥tA in
C+. . . ”). 
Observe that
(11) X
I∈Iq−1
n−1
X
j,k=1
Re cjk T ϕj I , ϕkI λ−X
J∈Iq
m
X
j=1
Re cjj T ϕJ, ϕJλ=
Re nX
I∈Iq−1
n−1
X
j,k=1 h+
jk T ϕj I , ϕkI λo.
Now that we can eliminate the Tterms, we turn to controlling the remaining terms.
Proposition 4.5. Let ϕ∈Dom( ¯
∂b)∩Dom( ¯
∂b
∗)be a (0, q)-form supported in U. Assume that λ
is a q-compatible function with positivity constant Bλ+. If m < q, choose λ+=tλ and if m > q,
choose λ+=−tλ. Then there exists a constant Cthat is independent of Bλ+so that
Qb,+(˜
ζΨ+
tϕ, ˜
ζΨ+
tϕ) + Ck˜
ζΨ+
tϕk2
λ++Ot(k˜
ζ˜
Ψ0
tϕk2
0)≥tBλ+k˜
ζΨ+
tϕk2
λ+.
13

Proof. Let
s+
jk =1
2(¯
LkLjλ++Lj¯
Lkλ+) + 1
2
n−1
X
ℓ=1
(dℓ
jk Lℓλ++dℓ
kj ¯
Lℓλ+)
and
r+
jk =s+
jk −1
qδjk
m
X
ℓ=1
sℓℓ
In this case (10) can be rewritten as
Qb,+(φ, φ) = X
J∈Iqnm
X
j=1
k¯
L∗,+
jφJk2
λ++
n−1
X
j=m+1
k¯
LjφJk2
λ+o+E(ϕ)
+X
I∈Iq−1
n−1
X
j,k=1
Re (r+
jk +h+
jk T)φj I , φkI λ+.
As noted in [Nic06, Rai], one can check that if L=Pn−1
j=1 ξjLj(where ξjis constant), then
D1
2∂b¯
∂bλ+−¯
∂b∂bλ+, L ∧¯
LE=
n−1
X
j,k=1
s+
jk ξj¯
ξk.
This means that s+
jk = Θ+
jk −1
2ν(λ+)cjk . Thus, if
Γλ+
jk = Θλ+
jk −1
qδjk
m
X
ℓ=1
Θλ+
ℓℓ
then
Qb,+(φ, φ) = X
J∈Iqnm
X
j=1
k¯
L∗,+
jφJk2
λ++
n−1
X
j=m+1
k¯
LjφJk2
λ+o+E(ϕ)
+X
I∈Iq−1
n−1
X
j,k=1
Re Γλ+
jk +h+
jk (T−1
2ν(λ+))φjI , φkI λ+.
Next, we replace φwith ˜
ζΨ+
tϕ. Since supp ˜
ζ⊂U′, and the Fourier transform of ˜
ζΨ+
tϕis
supported in C+up to a smooth term, we can use Lemma 4.4 to control the Tterms. Therefore,
from (10) and the form of E(ϕ), we have that
Qb,+(˜
ζΨ+
tϕ, ˜
ζΨ+
tϕ)≥(1 −ǫ)X
J∈Iqnm
X
j=1
k¯
L∗,+
j˜
ζΨ+
tϕJk2
λ++
n−1
X
j=m+1
k¯
Lj˜
ζΨ+
tϕJk2
λ+o
+X
I∈Iq−1
n−1
X
j,k=1
Re Γλ+
jk +h+
jk (tA −1
2ν(λ+))˜
ζΨ+
tϕjI ,˜
ζΨ+
tϕkI λ+
−O(k˜
ζΨ+
tϕk2
0)−Ot(k˜
ζν˜
Ψ0
tϕk2
0).
If we choose A≥1
2|ν(λ)|, then tA −1
2ν(λ+)≥0. Since the sum of any qeigenvalues of (h+
jk ) is
nonnegative, these terms are strictly positive. If m < q, then the sum of any qeigenvalues of Γλ+
is the sum of qeigenvalues of tΘλminus the sum of the first mdiagonal terms of tΘλ. If m > q,
the sum of any qeigenvalues of Γλ+is the sum of the first mdiagonal terms of tΘλminus the sum
14

of qeigenvalues of of tΘλ. In either case, by the q-compatibility of λ, we know that this sum is at
least tBλ+where Bλ+is the positivity constant of λ. By Lemma 4.2, this means that
Qb,+(˜
ζΨ+
tϕ, ˜
ζΨ+
tϕ) + Ck˜
ζΨ+
tϕk2
0+Ot(k˜
ζν˜
Ψ0
tϕk2
0)≥tBλ+k˜
ζΨ+
tϕk2
λ+.

Observe that the statement of Proposition 4.5 is independent of the choice of local coordinates
L1,...,Ln−1and m6=q. Hence, to handle the terms with support in C−, we may choose new local
coordinates and a new value of mso that Definitions 2.5 and 2.7 hold with (n−1−q) in place of
q. We again integrate (8) by parts and compute
Qb,λ(ϕ, ϕ) = X
J∈Iqnm
X
j=1
k¯
LjϕJk2
λ+
n−1
X
j=m+1
k¯
L∗,λ
jϕJk2
λo+E(ϕ)
+X
I∈Iq−1
n−1
X
j,k=1
Re cjk T ϕj I , ϕkI λ−X
J∈Iq
n−1
X
j=m+1
Re cjj T ϕJ, ϕJλ
+X
I∈Iq−1
n−1
X
j,k=1 (1
2(¯
LjLkλ+Lj¯
Lkλ)ϕjI , ϕkI λ+1
2
n−1
X
ℓ=1 (dℓ
jk Lℓλ+dℓ
jk ¯
Lℓλ)ϕjI , ϕkI λ)
(12)
−X
J∈Iq
n−1
X
j=m+1 (1
2(¯
LjLjλ+Lj¯
Ljλ)ϕJ, ϕJλ+1
2
n−1
X
ℓ=1 (dℓ
jj Lℓλ+dℓ
jj ¯
Lℓλ)ϕJ, ϕJλ)
By the argument of Lemma 4.4, we can also establish the following:
Lemma 4.6. Let Mbe as in Theorem 1.2 and ϕbe a (0, q)-form supported on Uso that up to a
smooth term, ˆϕis supported in C−. Let
(h−
jk ) = (cjk)−δjk
1
n−1−q
m
X
ℓ=1
cℓℓ.
Then
X
J∈Iq
n−1
X
j=1 h−
jj (−T)ϕJ, ϕJλ−X
I∈Iq−1
n−1
X
j,k=1 h−
jk (−T)ϕj I , ϕkI λ
≥tAX
J∈Iq
n−1
X
j=1 h−
jj ϕJ, ϕJλ−X
I∈Iq−1
n−1
X
j,k=1 h−
jk ϕj I , ϕkI λ+O(kϕk2
λ) + Ot(k˜
ζν˜
Ψ0
tϕk2
0).
In a similar fashion to (11), we have the equality
(13) X
J∈Iq
n−1
X
j=m+1
Re cjj T ϕJ, ϕJλ−X
I∈Iq−1
n−1
X
j,k=1
Re cjk T ϕj I , ϕkI λ
= Re nX
J∈Iq
n−1
X
j=1 h−
jj T ϕJ, ϕJλ−X
I∈Iq−1
n−1
X
j,k=1 h−
jk T ϕj I , ϕkI λo.
Applying these to the proof of Proposition 4.5, we obtain
Proposition 4.7. Let ϕ∈Dom( ¯
∂b)∩Dom( ¯
∂b
∗)be a (0, q)-form supported in U. Assume that λis
an (n−1−q)-compatible function with positivity constant Bλ−. If m > n −1−q, choose λ−=tλ
15

and if m < n −1−q, choose λ−=−tλ. Then there exists a constant Cthat is independent of Bλ−
so that
Qb,−(˜
ζΨ−
tϕ, ˜
ζΨ−
tϕ) + Ck˜
ζΨ−
tϕk2
λ−+Ot(k˜
ζ˜
Ψ0
tϕk2
0)≥tBλ−k˜
ζΨ−
tϕk2
λ−.
We are now ready to prove the basic estimate, Proposition 4.1.
Proof (Proposition 4.1). From (5), there exist constants K,K±so that
KQb,±(ϕ, ϕ) + K±X
ν
k˜
ζν˜
Ψ0
ν,tζνϕνk2
0+K′kϕk2
0+O±(kϕk2
−1)
≥X
νhQb,+(˜
ζνΨ+
ν,tζνϕν,˜
ζνΨ+
ν,tζνϕν) + Qb,−(˜
ζνΨ−
ν,tζνϕν,˜
ζνΨ−
ν,tζνϕν)i.
From Proposition 4.5 and Proposition 4.7 it follows that by increasing the size of K,K±, and K′
KQb,±(ϕ, ϕ) + K±X
ν
k˜
ζν˜
Ψ0
ν,tζνϕνk2
0+K′kϕk2
0+O±(kϕk2
−1)≥tB±kϕk2
0
where B±= min{Bλ−, Bλ+}.
4.2. A Sobolev estimate in the “elliptic directions”. For forms whose Fourier transforms
are supported up to a smooth term in C0, we have better estimates. The following results are in
[Nic06, Rai].
Lemma 4.8. Let ϕbe a (0,1)-form supported in Uνfor some νsuch that up to a smooth term, ˆϕ
is supported in ˜
C0
ν. There exist positive constants C > 1and C1>0so that
CQb,±(ϕ, H±ϕ) + C1kϕk2
0≥ kϕk2
1.
The proof in [Nic06] also holds at level (0, q).
We can use Lemma 4.8 to control terms of the form k˜
ζνΨ0
ν,tζνϕνk2
0.
Proposition 4.9. For any ǫ > 0, there exists Cǫ,±>0so that
k˜
ζνΨ0
ν,tζνϕνk2
0≤ǫQb,±(ϕν, ϕν) + Cǫ,±kϕνk2
−1.
See [Rai] for a proof of this proposition.
5. Regularity Theory for ¯
∂b
5.1. Closed range for b,±.For 1 ≤q≤n−2, let
Hq
±={ϕ∈Dom( ¯
∂b)∩Dom( ¯
∂b
∗) : ¯
∂bϕ= 0,¯
∂∗
b,±ϕ= 0}
={ϕ∈Dom( ¯
∂b)∩Dom( ¯
∂b
∗) : Qb,±(ϕ, ϕ) = 0}
be the space of ±-harmonic (0, q)-forms.
Lemma 5.1. Let M2n−1be a smooth, embedded CR-manifold of hypersurface type that admits a
q-compatible function λ+and an (n−1−q)-compatible function λ−. If t > 0is suitably large and
1≤q≤n−2, then
(i) Hq
±is finite dimensional;
(ii) There exists Cthat does not depend on λ+and λ−so that for all (0, q)-forms ϕ∈Dom( ¯
∂b)∩
Dom( ¯
∂b
∗)satisfying ϕ⊥ Hq
±(with respect to h·,·i±) we have
(14) |kϕ|k2
±≤CQb,±(ϕ, ϕ).
16

Proof. For ϕ∈ H±, we can use Proposition 4.1 with tsuitably large (to absorb terms) so that
tB±|kϕ|k2
±≤C±X
ν
k˜
ζνΨ0
ν,tζµϕνk2
0+kϕk2
−1.
Also, by Proposition 4.9,
X
ν
k˜
ζνΨ0
ν,tζµϕνk2
0≤C±kϕk2
−1.
since Qb,±(ϕ, ϕ) = 0. The unit ball in H±∩L2(M) is compact, and hence finite dimensional.
Assume that (14) fails. Then there exists ϕk⊥ H±with |kϕk|k±= 1 so that
(15) |kϕk|k2
±≥kQb,±(ϕk, ϕk).
For ksuitably large, we can use Proposition 4.1 and the above argument to absorb Qb,±(ϕk, ϕk)
by B±|kϕk|k±to get:
(16) |kϕk|k2
±≤C±kϕkk2
−1.
Since L2(M) is compact in H−1(M), there exists a subsequence ϕkjthat converges in H−1(M).
However, (16) forces ϕkjto converge in L2(M) as well. Although the norm (Qb,±(·,·) + |k · |k2
±)1/2
dominates the L2(M)-norm, (15) applied to ϕjkshows that ϕjkconverges in the (Qb,±(·,·)+|k·|k2
±)1/2
norm as well. The limit ϕsatisfies |kϕ|k±= 1 and ϕ⊥ H±. However, a consequence of (15) is that
ϕ∈ H±. This is a contradiction and (14) holds. 
Let
⊥Hq
±={ϕ∈L2
0,q(M) : hϕ, φi±= 0,for all φ∈ Hq
±}.
On ⊥Hq
±, define
b,±=¯
∂b¯
∂∗
b,±+¯
∂∗
b,±¯
∂b.
Since ¯
∂∗
b,±=H±¯
∂b
∗+ [ ¯
∂b
∗, H±], Dom( ¯
∂∗
b,±) = Dom( ¯
∂b
∗). This causes
Dom(b,±) = {ϕ∈L2
0,q(M) : ϕ∈Dom( ¯
∂b)∩Dom( ¯
∂b
∗),¯
∂bϕ∈Dom( ¯
∂b
∗),and ¯
∂b
∗ϕ∈Dom( ¯
∂b)}.
6. Proof of Theorem 1.2.
6.1. Closed range in L2.From Remark 2.8, we know that |z|2is a q-compatible functions with
a positivity constant of 1. Thus, for suitably large t, the space of harmonic (0, q)-forms Hq
t:= Hq
±
is finite dimensional. Moreover, if we use h·,·itfor h·,·i±and Qb,t for Qb,±, then for ϕ⊥ Hq
t(with
respect to h·,·it)
(17) |kϕ|k2
t≤CQb,t(ϕ, ϕ).
From H¨ormander [H¨or65], Theorem 1.1.2, (17) is equivalent to the closed range of ¯
∂b:L2
0,q(M)→
L2
0,q+1(M) and ¯
∂∗
b,t :L2
0,q(M)→L2
0,q−1(M) where both operators are defined with respect to
h·,·it. By H¨ormander [H¨or65], Theorem 1.1.1, this means that ¯
∂∗
b,t :L2
0,q+1(M)→L2
0,q(M) and
¯
∂b:L2
0,q−1(M)→L2
0,q(M) also have closed range. Thus, the Kohn Laplacian b,t on (0, q)-forms
also has closed range and Gq,t exists and is a continuous operator on L2
0,q(M).
17

6.2. Hodge theory and the canonical solutions operators. We now prove the existence of a
Hodge decomposition and the existence of the canonical solution operators. Unlike the standard
computations for the ¯
∂-Neumann operators and complex Green operators in the pseudoconvex case,
we only have the existence of the complex Green operator Gq,t at a fixed level qand not for all
1≤q≤n−1. (hence, we cannot commute Gq,t with either ¯
∂bor ¯
∂∗
b,t). If Hq
tis the projection of
L2
0,q(M) onto Hq
t= null( ¯
∂b)∩null( ¯
∂∗
b,t) = {ϕ∈L2
0,q(M)∩Dom( ¯
∂b)∩Dom( ¯
∂∗
b,t) : Qb,t (ϕ, ϕ) = 0},
then we know
ϕ=¯
∂b¯
∂∗
b,tGq,t ϕ+¯
∂∗
b,t ¯
∂bGq,tϕ+Hq
tϕ.
We now find the canonical solution operators. Let ϕbe a ¯
∂b-closed (0, q)-form that is orthogonal
to Hq
t. Then Hq
tϕ= 0, so
ϕ=¯
∂b¯
∂∗
b,tGq,tϕ+¯
∂∗
b,t ¯
∂bGq,tϕ.
We claim that ¯
∂∗
b,t ¯
∂bGq,tϕ= 0. Following [Nic06], we note that
0 = ¯
∂bϕ=¯
∂b¯
∂∗
b,t ¯
∂bGq,tϕ,
so
0 = h¯
∂b¯
∂∗
b,t ¯
∂bGq,tϕ, ¯
∂bGq,tϕit=|k ¯
∂∗
b,t ¯
∂bGq,tϕ|k2
t.
Thus, ¯
∂∗
b,t ¯
∂bGq,tϕ= 0 and the canonical solution operator to ¯
∂bis given by ¯
∂∗
b,tGq,t. A similar
argument shows that the canonical solution operator for ¯
∂∗
b,t is given by ¯
∂bGq,t.
In this paragraph, we will assume that all forms are perpendicular to Hq
t. For ϕ∈Dom(b,t), it
follows that
ϕ=Gq,tb,tϕ=b,tGq,t ϕ.
We will show that
(18) ¯
∂b¯
∂∗
b,tGq,t =Gq,t ¯
∂b¯
∂∗
b,t and ¯
∂∗
b,t ¯
∂bGq,t =Gq,t ¯
∂∗
b,t ¯
∂b.
Observe that
¯
∂bα= 0 =⇒α=¯
∂b¯
∂∗
b,tGq,tα=Gq,t ¯
∂b¯
∂∗
b,tα(19)
and
¯
∂∗
b,tβ= 0 =⇒β=¯
∂∗
b,t ¯
∂bGq,tβ=Gq,t ¯
∂∗
b,t ¯
∂bβ.(20)
Next, we claim that
(21) ¯
∂bϕ= 0 =⇒¯
∂bGqϕ= 0
and
(22) ¯
∂∗
b,tϕ= 0 =⇒¯
∂∗
b,tGqϕ= 0.
Indeed, we have that ϕ⊥ Hq
t, so ϕ=¯
∂b¯
∂∗
b,tGq,tϕ+¯
∂∗
b,t ¯
∂bGq,tϕ. Since Range ¯
∂∗
b,t ⊥null ¯
∂b,¯
∂bϕ= 0
implies that ¯
∂∗
b,t ¯
∂bGq,tϕ= 0. Since Range( ¯
∂b)⊥null( ¯
∂∗
b,t), ¯
∂∗
b,t ¯
∂bGq,tϕ= 0 implies ¯
∂bGq,tϕ= 0, as
desired. A similar argument shows (22). To show (18), observe that we can write ϕ=α+βwhere
¯
∂bα= 0 and ¯
∂∗
b,tβ= 0. Thus, by (19) and (22),
¯
∂b¯
∂∗
b,tGq,tϕ=¯
∂b¯
∂∗
b,tGq,t (α+β) = ¯
∂b¯
∂∗
b,tGq,tα=Gq,t ¯
∂b¯
∂∗
b,tα=Gq,t ¯
∂b¯
∂∗
b,tϕ.
A similar argument with (20) and (21) proves that ¯
∂∗
b,t ¯
∂bGq,tϕ=Gq,t ¯
∂∗
b,t ¯
∂bϕ, finishing the proof of
(18).
18

6.3. Closed range of ¯
∂b:Hs
0,q(M)→Hs
0,q+1(M)and ¯
∂∗
b,t :Hs
0,q(M)→Hs
0,q−1(M).We start with
an argument to show closed range of ¯
∂b:Hs
0,q(M)→Hs
0,q+1(M) and ¯
∂∗
b,t :Hs
0,q(M)→Hs
0,q−1(M).
Combining Proposition 4.1 and Lemma 4.8, if tis sufficiently large, then
|kΛsϕ|k2
t≤C
t|k ¯
∂bΛsϕ|k2
t+|k ¯
∂∗
b,tΛsϕ|k2
t+Ctkuk2
s−1
≤C
t|kΛs¯
∂bϕ|k2
t+|kΛs¯
∂∗
b,tϕ|k2
t+|k[¯
∂b,Λs]ϕ|k2
t+|k[¯
∂∗
b,t,Λs]ϕ|k2
t+Ctkϕk2
s−1.
As a consequence of Lemma 3.3, [ ¯
∂∗
b,t,Λs] = Ps+tPs−1where Psand Ps−1are pseudodifferential
operators of order sand s−1, respectively. Additionally, [ ¯
∂b,Λs] is a pseudodifferential operator
of order s. Consequently,
|kΛsϕ|k2
t≤C
t|kΛs¯
∂bϕ|k2
t+|kΛs¯
∂∗
b,tϕ|k2
t+|kΛsϕ|k2
t+Ctkϕk2
s−1.
Choosing tlarge enough and ϕ∈Hs
0,q(M) allows us to absorb terms to prove
kϕk2
s=kΛsϕk2
0≤Ct|kΛsϕ|k2
t≤Ct|kΛs¯
∂bϕ|k2
t+|kΛs¯
∂∗
b,tϕ|k2
t+kϕk2
s−1
≤Ctk¯
∂bϕk2
s+k¯
∂∗
b,tϕk2
s+kϕk2
s−1.
Thus, ¯
∂b:Hs
0,q(M)→Hs
0,q+1(M) and ¯
∂∗
b,t :Hs
0,q(M)→Hs
0,q−1(M) have closed range.
6.4. Continuity of the complex Green’s operator in Hs
0,q(M).We now turn to the harder
problem of showing continuity of the complex Green operator Gδ
q,t in Hs
0,q(M), s > 0. We use an
elliptic regularization argument. Let Qδ
b,t(·,·) be the quadratic form on H1
0,q(M) defined by
Qδ
b,t(u, v) = Qb,t (u, v) + δQdb(u, v)
where Qdbis the hermitian inner product associated to the de Rham exterior derivative db, i.e.,
Qdb(u, v) = hdbu, dbvit+hd∗
bu, d∗
bvit. The inner product Qdbhas form domain H1
0,q(M). Conse-
quently, Qδ
b,t gives rise to a unique, self-adjoint, elliptic operator δ
b,t with inverse Gδ
q,t.
From Proposition 4.1 and Lemma 4.8, if tis large enough, then for ϕ∈Dom( ¯
∂b)∩Dom( ¯
∂∗
b,t),
we have the estimate
(23) |kϕ|k2
t≤K
tQb,t(ϕ, ϕ) + Ctkϕk2
−1.
Now let ϕ∈Hs
0,q(Ω). Since δ
b,t is elliptic, Gδ
q,tϕ∈Hs+2
0,q (M). Then
(24) kGδ
q,tϕk2
s=kΛsGδ
q,tϕk2
0≤Ct|kΛsGδ
q,tϕ|k2
t.
We now concentrate on finding a bound for |kΛsGδ
q,tϕ|k2
tthat is independent of δ. By (23),
(25) |kΛsGδ
q,tϕ|k2
t≤K
tQb,t(ΛsGδ
q,tϕ, ΛsGδ
q,tϕ) + Ct,skGδ
q,tϕk2
s−1.
Observe that if (Λs)∗,t is the adjoint of Λsunder the inner product h·,·it, then
hΛsu, vit= (u, ΛsH−1
tv)0=hu, HtΛsH−1
tvit=hu, (Λs+ [Ht,Λs]H−1
t)vit
implies that (Λs)∗,t = Λs+ [Ht,Λs]H−1
t. Therefore, it is a standard consequence of Lemma 3.1 in
[KN65] (or Lemma 2.4.2 in [FK72]) that
Qb,t(ΛsGδ
q,tϕ, ΛsGδ
q,tϕ)≤Qδ
b,t(ΛsGδ
q,tϕ, ΛsGδ
q,tϕ)≤ |hΛsϕ, ΛsGδ
q,tϕit|+CkGδ
q,tϕk2
s+Ct,skGδ
q,tϕk2
s−1
≤ |kΛsϕ|kt|kΛsGδ
q,tϕ|kt+kGδ
q,tϕk2
s−1
≤Ktkϕk2
s+C|kΛsGδ
q,tϕ|k2
t+Ct,skGδ
q,tϕk2
s−1
(26)
19

where C > 0 does not depend on δor t.
Plugging (26) into (25), we see that
|kΛsGδ
q,tϕ|k2
t≤K
tKtkϕk2
s+C|kΛsGδ
q,tϕ|k2
t+Ct,skGδ
q,tϕk2
s−1.
If tis sufficiently large, then it follows that
(27) |kΛsGδ
q,tϕ|k2
t≤Ktkϕk2
s+Ct,skGδ
q,tϕk2
s−1
since |kΛsGδ
q,tϕ|k2
t<∞(recall that Gδ
q,tϕ∈Hs+2
0,q (M)). Plugging (27) into (24), we have the bound
(28) kGδ
q,tϕk2
s≤Ktkϕk2
s+Ct,skGδ
q,tϕk2
s−1.
We now turn to letting δ→0. Observe that Ktand Ct.s are independent of δ. We have shown that
if ϕ∈Hs
0,q(M), then {Gδ
q,tϕ: 0 < δ < 1}is bounded in Hs
0,q(M). Thus, there exists a sequence
δk→0 and ˜u∈Hs
0,q(M) so that Gδk
q,tu→˜uweakly in Hs
0,q(M). Consequently, if v∈Hs+2
0,q (M),
then
lim
k→∞ Qδk
b,t(Gδk
q,tu, v) = Qb,t (˜u, v).
However,
Qδk
b,t(Gδk
q,tu, v) = (u, v) = Qb,t(Gq,tu, v),
so Gq,tu= ˜uand (28) is satisfied with δ= 0. Thus, Gq,t is a continuous operator on Hs
0,q(M).
6.5. Continuity of the canonical solution operators in Hs
0,q(M).Continuity of ¯
∂bGq,t and
¯
∂∗
b,tGq,t will follow from the continuity of Gq,t. Unfortunately, we cannot apply Proposition 4.1 to
either ¯
∂bGq,tϕor ¯
∂∗
b,tGq,tϕbecause neither are (0, q)-forms. Instead, we estimate directly:
k¯
∂bGq,tϕk2
s+k¯
∂∗
b,tGq,tϕk2
s≤Ct(|kΛs¯
∂bGq,tϕ|k2
t+|kΛs¯
∂∗
b,tGq,t ϕ|k2
t)
=CthΛsϕ, ΛsGq,tϕit+hΛs¯
∂bGq,tϕ, [Λs,¯
∂b]Gq,tϕit+h[¯
∂∗
b,t,Λs]¯
∂bGq,tϕ, ΛsGq,tϕit
+hΛs¯
∂∗
b,tGq,tϕ, [Λs,¯
∂∗
b,t]Gq,tϕit+h[¯
∂b,Λs]¯
∂∗
b,tGq,tϕ, ΛsGq,t ϕit
≤Ct,s(kϕk2
s+kGq,tϕk2
s)≤Ct,skϕk2
s.
6.6. The Szeg¨o projection Sq,t.The Szeg¨o projection Sq,t is the projection of L2
0,q(M) onto
ker ¯
∂b. We claim that
Sq,t =I−¯
∂∗
b,t ¯
∂bGq,t =I−Gq,t ¯
∂∗
b,t ¯
∂b.
The second equality follows from (18). Observe that if ϕ∈null( ¯
∂b), then (I−Gq,t ¯
∂∗
b,t ¯
∂b)ϕ=ϕ,
as desired. If ϕ⊥null( ¯
∂b), then ϕ⊥ Hq
t, so ϕ=¯
∂∗
b,t ¯
∂bGq,tϕ+¯
∂b¯
∂∗
b,tGq,tϕ. We claim that ϕ=
¯
∂∗
b,t ¯
∂bGq,tϕ. Let u=¯
∂∗
b,t ¯
∂bGq,tϕ. Then uis the canonical solution to ¯
∂bu=¯
∂bϕ, so ¯
∂b(ϕ−u) = 0.
However, ϕ⊥null( ¯
∂b), so u=ϕ, and 0 = ϕ−u= (I−¯
∂∗
b,t ¯
∂bGq,t)ϕ, as desired.
Proposition 6.1. Let Mbe as in Theorem 1.2. If t≥Ts, then the Szeg¨o kernel Sq,t is continuous
on Hs
0,q(M).
Proof. This argument uses ideas from [BS90]. Given ϕ∈L2
0,q(M), we know that ¯
∂∗
b,t ¯
∂bGq,tϕ∈
L2
0,q(M), but we have no quantitative bound. However,
|k ¯
∂∗
b,t ¯
∂bGq,tϕ|k2
t=h¯
∂b¯
∂∗
b,t ¯
∂bGq,tϕ, ¯
∂bGq,tϕit=h¯
∂bϕ, ¯
∂bGq,tϕit≤ |kϕ|kt|k¯
∂∗
b,t ¯
∂bGq,tϕ|kt.
This proves continuity in L2
0,q(M).
Now let s > 0. It suffices to show
(29) |kΛs¯
∂∗
b,t ¯
∂bGq,tϕ|k2
t≤Cs,t|kΛsϕ|k2
t.
20

We cannot simply integrate by parts as in the L2-case because we do not know if Λs¯
∂∗
b,t ¯
∂bSq,tϕis
finite. As above, we can avoid this issue by an elliptic regularity argument. Using the operators
Gδ
q,t from §6.4, we have (if δis small enough)
|kΛs¯
∂∗
b,t ¯
∂bGδ
q,tϕ|k2
t=hΛs¯
∂b¯
∂∗
b,t ¯
∂bGδ
q,tϕ, Λs¯
∂bGδ
q,tϕit+h[¯
∂b,Λs]¯
∂∗
b,t ¯
∂bGδ
q,tϕ, Λs¯
∂bGδ
q,tϕit
+hΛs¯
∂∗
b,t ¯
∂bGδ
q,tϕ, [Λs,¯
∂∗
b,t]¯
∂bGδ
q,tϕit
≤Cs,t(|kΛsϕ|kt+|kΛs¯
∂bGδ
q,tϕ|kt)|kΛs¯
∂∗
b,t ¯
∂bGδ
q,tϕ|kt.
Using that the continuity of ¯
∂bGδ
q,t in Hs
0,q(M) is uniform in δ(for small δ), we have
(30) |kΛs¯
∂∗
b,t ¯
∂bGδ
q,tϕ|kt≤Cs,t(|kΛsϕ|kt+|kΛs¯
∂bGq,tϕ|kt)≤Cs,t|kΛsϕ|kt.
As earlier, we can take an appropriate limit as δ→0 to establish the bound in (30) with δ= 0. 
6.7. Results for levels (0, q −1) and (0, q +1).We now show continuity of the canonical solution
operators Gq,t ¯
∂∗
b,t :Hs
0,q+1(M)→Hs
0,q(M) and Gq,t ¯
∂b:Hs
0,q−1(M)→Hs
0,q(M), and the Szeg¨o
projection Sq−1,t =I−¯
∂∗
b,tGq,t ¯
∂b:Hs
0,q−1(M)→Hs
0,q−1(M). We cannot express the Szeg¨o kernel of
(0, q + 1)-forms in terms of Gq,t because the only candidate is ¯
∂bGq,t ¯
∂∗
b,t, but this object annihilates
t-harmonic forms (which ought to remain unchanged by Sq+1,t). Since Hs+1
0,q−1(M) is dense in
Hs
0,q−1(M) and Gq,t preserves Hs
0,q(M), we may assume that ϕ∈Hs+1
0,q−1(M). Then
|kΛsGq,t ¯
∂bϕ|k2
t=h¯
∂∗
b,tGq,t
|{z }
bounded in Hs
ΛsGq,t ¯
∂bϕ, Λsϕit+hΛsGq,t ¯
∂bϕ, [Λs, Gq,t ¯
∂b]ϕit
≤Cs|kΛsGq,t ¯
∂bϕ|kt|kΛsϕ|kt.
The right hand side is finite since ¯
∂bϕ∈Hs
0,q(M) by assumption. Thus, Gq,t ¯
∂b:Hs
0,q−1(M)→
Hs
0,q(M) is bounded. A similar argument shows that Gq,t ¯
∂∗
b,t :Hs
0,q+1(M)→Hs
0,q(M) is continuous.
For the Szeg¨o projection, we investigate the boundedness of
|kΛs¯
∂∗
b,tGq,t ¯
∂bϕ|k2
t=hΛs¯
∂b¯
∂∗
b,tGq,t ¯
∂bϕ, ΛsGq,t ¯
∂bϕit+hΛs¯
∂∗
b,tGq,t ¯
∂bϕ, [Λs,¯
∂∗
b,t]Gq,t ¯
∂bϕit
+h[¯
∂∗
b,t,Λs]¯
∂∗
b,tGq,t ¯
∂bϕ, ΛsGq,t ¯
∂bϕit.
Since ¯
∂bϕis ¯
∂b-closed, ¯
∂b¯
∂∗
b,tGq,t ¯
∂bϕ=¯
∂bϕ, so
hΛs¯
∂b¯
∂∗
b,tGq,t ¯
∂bϕ, ΛsGq,t ¯
∂bϕit=
hΛsϕ, Λs¯
∂∗
b,tGq,t ¯
∂bϕit+h[Λs,¯
∂b]ϕ, ΛsGq,t ¯
∂bϕit+hΛsϕ, [Λs,¯
∂∗
b,t]Gq,t ¯
∂bϕit
≤Cs(|kΛsϕ|kt|kΛs¯
∂∗
b,tGq,t ¯
∂bϕ|kt+|kΛsϕ|k2
t).
Thus, we have
|kΛs¯
∂∗
b,tGq,t ¯
∂bϕ|k2
t≤Cs,t(|kΛsϕ|kt|kΛs¯
∂∗
b,tGq,t ¯
∂bϕ|kt+|kΛsϕ|k2
t).
Using a small constant/large constant argument and absorbing terms, we have the continuity of
the Szeg¨o projection in Hs
0,q−1(M).
The continuity of the solution operator ¯
∂∗
b,tGq,t immediately gives closed range of ¯
∂bfrom
Hs
0,q−1(M) to Hs
0,q(M). Similarly, the boundedness of the operator ¯
∂bGq,t immediately gives closed
range of ¯
∂b
∗from Hs
0,q+1(M) to Hs
0,q(M).
21

6.8. Exact and global regularity for ¯
∂b.In this section, we prove that if α∈C∞
0,˜q+1 (M) satisfies
¯
∂bα= 0 and α⊥ H˜q
t, then there exists u∈C∞
0,˜q(M) so that ¯
∂bu=αwhere ˜q=qor q−1. We
follow the argument in [Nic06], Lemma 5.10. We start by showing that if kis fixed and s > k, then
Hs
0,˜q(M)∩null( ¯
∂b) is dense in Hk
0,˜q(M)∩null( ¯
∂b). Let g∈Hk
0,˜q(M)∩null( ¯
∂b). Since C∞
0,˜q(M) is
dense in Hk
0,˜q(M), there exists a sequence gj∈C∞
0,˜q(M) so that gj→gin Hk
0,˜q(M). Let t≥Tsand
set ˜gj=S˜q,tgj. By the continuity of S˜q,t in Hs
0,˜q(M), ˜gj∈Hs
0,˜q(M). Moreover, since g=S˜q,tg, it
follows that
lim
j→∞ k˜gj−gk2
k= lim
j→∞ kS˜q,t(gj−g)k2
k≤Ck,t lim
j→∞ kgj−gk2
k= 0.
Next, since α=¯
∂b¯
∂∗
b,tG˜q,t αor ¯
∂bG˜q,t ¯
∂∗
b,tαfor all sufficiently large t, by choosing an appropriate
sequence tk→ ∞, there exists uk=¯
∂∗
b,tG˜q,tkαor G˜q,tk¯
∂∗
b,tα∈Hk
0,˜q(M) so that ¯
∂buk=α. We
will construct a sequence ˜ukinductively. Let ˜u1=u1. Assume that ˜ukhas been defined so that
˜uk∈Hk
0,˜q(M), ¯
∂b˜uk=α, and k˜uk−˜uk−1kk−1≤2k−1. We will now construct ˜uk+1. Note that
¯
∂b(uk+1 −˜uk) = 0. By the density argument above, there exists vk+1 ∈Hk+1
0,˜q(M)∩null( ¯
∂b) so that
if ˜uk+1 =uk+1 +vk+1, then k˜uk+1 −˜ukkk≤2−k. Finally, set
u= ˜u1+
∞
X
k=1
(˜uk+1 −˜uk) = ˜uj+
∞
X
k=j
(˜uk+1 −˜uk), j ∈N.
The sum telescopes and it is clear that u∈Hj
0,˜q(M) for all j∈Nand ¯
∂bu=α. Thus, u∈C∞
0,˜q(M).
7. Proof of Theorem 1.1
From (3), we know that weighted L2(M) and L2(M) are equivalent spaces. Thus, from The-
orem 1.2, we know that ¯
∂b:L2
0,q−1(M)→L2
0,q(M) and ¯
∂b:L2
0,q(M)→L2
0,q+1(M) have closed
range. Again by H¨ormander, Theorem 1.1.1, this proves that ¯
∂b
∗:L2
0,q(M)→L2
0,q−1(M) and
¯
∂b
∗:L2
0,q+1(M)→L2
0,q(M) have closed range. Consequently, the Kohn Laplacian b=¯
∂b¯
∂b
∗+¯
∂b
∗¯
∂b
has closed range on L2
0,q(M) and the remainder of the theorem follows by standard arguments.
This concludes the proof of Theorem 1.1.
Remark 7.1.This is more quantitative discussion of Remark 1.3. In particular, from the proof of
Theorem 1.1, we have the closed range bound for appropriate (0, q)-forms ϕ(using (3)),
kϕk2
0≤1
ct
kϕk2
t≤C
ct
k¯
∂bϕk2
t≤CCt
ct
k¯
∂bϕk2
0.
Thus, the closed range constants for ¯
∂b,¯
∂b
∗, and bin unweighted L2(M) depend on the size of λ+
and λ−.
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