The $\Box_b$-heat equation on finite type CR manifolds with comparable Levi form
UDC 517.9 The main purpose of this paper is to study the initial-value problems for the heat equations associated with the operator $\Box_b$ on compact CR manifolds of finite type. The critical component of our analysis is the condition called $D^{\epsilon}(q)$ and introduced by K. D. Koenig [A...
Saved in:
| Date: | 2019 |
|---|---|
| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2019
|
| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/1500 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
| Download file: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507293738598400 |
|---|---|
| author | Ly, Kim Ha Лі, Кім Ха |
| author_facet | Ly, Kim Ha Лі, Кім Ха |
| author_sort | Ly, Kim Ha |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T08:57:29Z |
| description | UDC 517.9
The main purpose of this paper is to study the initial-value problems for the heat
equations associated with the operator $\Box_b$ on compact CR manifolds of finite type.
The critical component of our analysis is the condition called $D^{\epsilon}(q)$ and
introduced by K. D. Koenig [Amer. J. Math. -- 2002. -- {\bf 124}. -- P. 129--197].
Actually, it states that the $\min\{q, n-1-q\}$th smallest eigenvalue of the Levi
form is comparable with the largest eigenvalue of the Levi form. |
| first_indexed | 2026-03-24T02:07:01Z |
| format | Article |
| fulltext |
UDC 517.9
Ly Kim Ha (Univ. Sci., Vietnam Nat. Univ., Ho Chi Minh City)
THE 2\bfitb -HEAT EQUATION ON FINITE TYPE CR MANIFOLDS
WITH COMPARABLE LEVI FORM*
РIВНЯННЯ 2\bfitb -ТЕПЛОПРОВIДНОСТI НА CR МНОГОВИДАХ
СКIНЧЕННОГО ТИПУ ЗI СПIВВИМIРНИМИ ФОРМАМИ ЛЕВI
The main purpose of this paper is to study the initial-value problems for the heat equations associated with the operator
2b on compact CR manifolds of finite type. The critical component of our analysis is the condition called D\epsilon (q) and
introduced by K. D. Koenig [Amer. J. Math. – 2002. – 124. – P. 129 – 197]. Actually, it states that the \mathrm{m}\mathrm{i}\mathrm{n}\{ q, n - 1 - q\} th
smallest eigenvalue of the Levi form is comparable with the largest eigenvalue of the Levi form.
Основною метою цiєї роботи є вивчення початкових задач для рiвнянь теплопровiдностi, асоцiйованих з оператором
2b, на компактних CR многовидах скiнченного типу. Критичним компонентом нашого аналiзу є так звана умова
D\epsilon (q), що була запропонована К. Д. Кьонiгом [Amer. J. Math. – 2002. – 124. – P. 129 – 197]. Фактично вона встановлює,
що \mathrm{m}\mathrm{i}\mathrm{n}\{ q, n - 1 - q\} -найменше власне значення форми Левi є спiввимiрним iз найбiльшим власним значенням
форми Левi.
1. Introduction. In Riemannian geometry, the Laplace – Beltrami operator defined on a Riemannian
manifold M is \Delta = d\ast d. In order to study the relation between geometry and analysis on M, a
well-known approach is to use the heat equation associated to the Laplace – Beltrami operator. Let u
be defined on (0,\infty )\times M. We say that u solves the heat equation on M when
\partial u
\partial s
+\Delta u = 0 on (0,\infty )\times M.
Moreover, we are also interested in the initial-value problem for the heat equation. That is, finding a
function u(s, x) solving the heat equation on M and satisfying
\mathrm{l}\mathrm{i}\mathrm{m}
s\rightarrow 0+
u(s, \cdot ) = f
with convergence in an appropriate norm on M. It is well-known that there is a unique fundamental
solution H(s, x, y) of the initial-value problem so that
u(s, x) =
\int
M
H(s, x, y)f(y)dV (y),
where dV (\cdot ) is the volume form on M. The kernel H(s, x, y) is a smooth function only for s > 0.
For s = 0, it agrees with the delta distribution of the diagonal, and it is obviously not smooth. The
smoothness is a consequence of the ellipticity of the Laplace – Beltrami operator.
The study of the heat equation and the heat kernel for operators of the Laplace-type has numer-
ous applications, including heat-kernel proofs of the Atiyah – Singer index theorem and its various
generalizations.
In the present work, we will consider one analogue of the heat equation in Cauchy – Riemann
(CR) geometry. That is an equation associated to the \Box b-heat operator. Here, the operator \Box b
* This research was supported by the Vietnam National Foundation for Science and Technology Development (grant
№ 101.02-2017.06).
c\bigcirc LY KIM HA, 2019
1082 ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 8
THE 2b -HEAT EQUATION ON FINITE TYPE CR MANIFOLDS WITH COMPARABLE LEVI FORM 1083
is the Laplacian associated to the \=\partial b-complex on a given CR manifold. Unfortunately, none of
these are elliptic on CR manifolds. Hence, the classical analysis approach in Riemannian geometry
does not allow us to deal with the \Box b-heat equation. For non degenerate CR manifolds, the study
of the \Box b-heat equation culminates with Beals – Greiner – Stanton [1]. They actually built a class
of pseudodifferential operators with a full symbolic calculus that allows us to construct explicit
parametrices for the \Box b-heat equation on (p, q)-forms under the condition Y (q). In the strictly
pseudoconvex case, this was used to derive a full short-time asymptotic for the heat kernel in terms
of local pseudo-Hermitian invariants (i.e., universal polynomial in the covariant derivatives of the
curvature and torsion tensors of the Tanaka – Webster connection). These results are the complete
analogues of the results of the heat equation associated with Laplace type operators in the Riemannian
setting.
The condition Y (q) cannot hold on weakly pseudoconvex CR manifolds that are not strongly
pseudoconvex. For finite type CR manifolds there were various attempts to give estimates in terms
of the associated Carnot – Carathéodory metric for the fundamental solution of the \Box b-equation. In
particular, Nagel – Stein [8] established such estimates for the \Box b-heat equation on finite type domains
in \BbbC 2.
The main purpose of this paper is to give an attempt to extend Nagel – Stein’s results to finite type,
compact CR manifolds of real dimension \geq 5 by using the D\epsilon (q)-condition introduced by K. Koenig
[6]. Using this condition, K. Koenig [6] established that the Kohn – Laplacian has an inverse that
belongs to a class of operators called nonisotropic smoothing (NIS) operators. This implies that its
Schwartz kernel (i.e., the Green function of \Box b) satisfies suitable metric distance estimates. The
present work is also motivated to the fourth level in Fefferman’s hierarchy [4], deriving estimates
directly from the singularities of the integral kernels.
The main result of this paper is as follows:
Theorem 1.1. Let M be a pseudoconvex, finite type, compact CR manifold for which the range
of \=\partial b is closed in L2 and which satisfies the D\epsilon (q0) condition. Then, for every s > 0, the heat
solution operators e - s2b is a NIS operator of order zero on (0, q)-forms, with q0 \leq q \leq n - 1 - q0,
and associated estimates are uniform in s > 0.
The paper is organized as follows. We will recall the definition of the operator \Box b and its
properties, see [10] for all notions. Section 3 includes a short review on the class of NIS operators.
The last section contains the full proof of Theorem 1.1, which is divided into two main steps:
Theorems 4.3 and 4.4.
2. The hypoellipticity of the 2\bfitb -heat operator. Throughout this paper M is a compact oriented
CR manifold of dimension (n - 1) with n \geq 3. The existence of a CR structure means there is a rank
(n - 1) complex subbundle T 1,0(M) of the complexified tangent bundle \BbbC T (M) = T (M) \otimes \BbbR \BbbC
such that
(1) T 1,0(M) \cap T 0,1(M) = \{ 0\} , where T 0,1(M) = T 1,0(M),
and
(2) if Z and W are smooth sections of T 1,0(M), then [Z,W ] is also a smooth section of T 1,0(M).
We always assume that the manifold M is equipped with a Hermitian metric on \BbbC T (M) so
that T 1,0(M) is orthogonal to T 0,1(M). Denote by \eta (M) the orthogonal complement of T 1,0(M)\oplus
\oplus T 0,1(M). Let T \ast 1,0(M) and T \ast 0,1(M) be the dual bundles of T 1,0(M) and T 0,1(M), respectively.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 8
1084 LY KIM HA
For 0 \leq p, q \leq n - 1, the vector bundle \Lambda p,q(M) is defined as
\Lambda p,q(M) = \Lambda pT \ast 1,0(M)\oplus \Lambda qT \ast 0,1(M).
The tangential CR complex \=\partial b : C\infty (\Lambda p,q(M)) \rightarrow \scrC \infty (\Lambda p,q+1(M)) is defined by
\=\partial b := \pi p,q+1 \circ d,
where \pi p,q+1 is the orthogonal projection of \Lambda p+q+1(M) onto \Lambda p,q+1(M) and d is the exterior
differentiation. Let \=\partial \ast b be the formal adjoint of \=\partial b in L2
(p,q)(M), where L2
(p,q)(M) is the closure of
C\infty (M,\Lambda p,q(M)) with respect to the appropriate pre-Hermitian inner product. The operator \Box b is
the Laplacian associated to the \=\partial b-complex
\Box b = \=\partial b \=\partial
\ast
b +
\=\partial \ast b
\=\partial b.
The associated initial-value problem is to find a function u on (0,\infty )\times M such that
\frakH [u](s, x) :=
\biggl[
\partial
\partial s
+\Box b
\biggr]
u(s, x) = 0 for s > 0 and x \in M,
\mathrm{l}\mathrm{i}\mathrm{m}
s\rightarrow 0+
u(s, \cdot ) = \phi (\cdot ) in L2
(0,q)(M).
Let U be a suitable small open subset of M. We pick an orthogonal basis \{ \omega 1, , . . . , \omega n - 1,
\omega 1, \omega 2, . . . , \omega n - 1, \omega 0\} of T \ast (U) such that \{ \omega 1, . . . , \omega n - 1\} is a frame of \Lambda 1,0(U), and \omega 0 is a
real annihilator of T 0,1. Next, let \{ L1, . . . , Ln - 1, L1, . . . , Ln - 1, T\} be the (local) basis dual to
\{ \omega 1, . . . , \omega n - 1, \omega 1, \omega 2, . . . , \omega n - 1, \omega 0\} .
Definition 2.1. The Levi matrix associated with the Levi form is a Hermitian matrix\bigl(
ckj
\bigr)
k,j=1,...,n - 1
presented by
[Lk, Lj ] = ickjT, \mathrm{m}\mathrm{o}\mathrm{d} (L1, . . . , Ln - 1, L1, . . . , Ln - 1).
Now, we will consider what is called the comparable Levi form condition on the CR manifolds.
For 1 \leq q \leq n - 1, let \sigma q denote any of the
\biggl(
n - 1
q
\biggr)
sums of q eigenvalues \lambda j of the Levi matrix
(ckj ) and \tau =
\sum n - 1
j=1
\lambda j be the trace.
Definition 2.2 [6]. We say that the D\epsilon (q) condition holds on M if there exists \epsilon > 0 such that
\epsilon \tau \leq \sigma q \leq (1 - \epsilon )\tau on M, for all possible \sigma q.
Moreover, for 1 \leq q0 \leq n - 2, if the D\epsilon (q0) condition holds in U, so does the D\epsilon (q) condition also
holds for all \mathrm{m}\mathrm{i}\mathrm{n}(q0, n - 1 - q0) \leq q \leq \mathrm{m}\mathrm{a}\mathrm{x}(q0, n - 1 - q0). For this reason, we will always assume
that 1 \leq q0 \leq
n - 1
2
, that means the range of D\epsilon (q0) is for q \in [q0, n - 1 - q0].
Theorem 2.1. Assume U \subset M (suitable small open subset) is of finite commutator type and
satisfies the D\epsilon (q0) condition defined as above, for 1 \leq q0 \leq (n - 1)/2. Then the heat operator \frakH
acting on (0, q) forms is hypoelliptic in (0,\infty )\times U with q0 \leq q \leq n - 1 - q0.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 8
THE 2b -HEAT EQUATION ON FINITE TYPE CR MANIFOLDS WITH COMPARABLE LEVI FORM 1085
Here and in what follows, \lesssim and >
\sim
denote inequality up to a positive constant. Moreover, we
will use \approx for the combination of \lesssim and >
\sim
.
Proof. It suffices to prove the theorem on (0, q0)-forms.
Since the finite type and D\epsilon (q0) conditions, in the distributional sense, we have
\| \frakH [u]\| 2L2
(0,q0)
((0,\infty )\times M) = \| \partial su\| 2L2
(0,q0)
((0,\infty )\times M) + \| \Box bu\| 2L2
(0,q0)
((0,\infty )\times M)+
+\langle \partial su,\Box bu\rangle L2
(0,q0)
((0,\infty )\times M) + \langle \Box bu, \partial su\rangle L2
(0,q0)
((0,\infty )\times M).
But \Box b is self-adjoint and \partial \ast s = - \partial s, then
\| \frakH [u]\| 2L2
(0,q0)
((0,\infty )\times M) = \| \partial su\| 2L2
(0,q0)
((0,\infty )\times M) + \| \Box bu\| 2L2
(0,q0)
((0,\infty )\times M).
Let u = u(s, x), for s > 0 and x \in U. The condition of finite commutator type for L1, . . . , Ln - 1,
\=L1, . . . , \=Ln - 1 on U also implies the finite commutator type for L1, . . . , Ln - 1, \=L1, . . . , \=Ln - 1, \partial s on
(0,\infty )\times U. Since the D\epsilon (q) condition holds, we have obtained the well-known subelliptic estimate
for the heat operator and the maximal estimate for \Box b [6]. Then these imply that
\| u\| 2H\epsilon 0 \lesssim \| \frakH [u]\| 2L2
(0,q0)
((0,\infty )\times M) + \| u\| 2L2
(0,q0)
((0,\infty )\times M).
Let \zeta , \zeta 1 be smooth real-valued cutoff functions supported in U, with \zeta \prec \zeta 1 (i.e., \zeta = 1 on supp\zeta 1).
For any \delta \in \BbbR and N > 0, by the same method to prove Theorem 8.2.9 in [3], the following estimate
holds:
\| \zeta u\| H\delta +\epsilon 0 \leq C\delta ,N (\| \zeta 1\frakH [u]\| H\delta + \| \zeta 1u\| H - N ).
Therefore, \frakH is hypoelliptic on all (0, q0)-forms defined on (0,\infty )\times U.
Theorem 2.1 is proved.
3. Spaces of homogeneous type. In this section, we also assume that the holomorphic vectors
fields L1, . . . , Ln - 1, \=L1, . . . , \=Ln - 1 defined on U \subset M satisfy the condition of finite commutator
type and the condition D\epsilon (q). The real vector fields X1, . . . , X2n - 2 are defined by Xj = \mathrm{R}\mathrm{e}Lj ,
Xn+j - 1 = \mathrm{I}\mathrm{m}Lj , j = 1, . . . , n - 1. For each finite sequence i1, . . . , ik of integers with 1 \leq ij \leq
\leq 2n - 2, setting I = (i1, . . . , ik) and the length | I| = k. We can write the commutator
[Xik , [Xik - 1
, . . . , [Xi2 , Xi1 ], . . .]] = \lambda i1...ikT, \mathrm{m}\mathrm{o}\mathrm{d} (X1, . . . , X2n - 2),
where \lambda i1...ik \in C\infty (U).
Definition 3.1. For x \in U and r > 0, the size functions are defined by
\Lambda l(x) =
\left( \sum
2\leq | I| \leq l
| \lambda i1...ik(x)|
2
\right) 1
2
, l \geq 2,
and
\Lambda (x, r) =
m\sum
l=2
\Lambda l(x)r
l.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 8
1086 LY KIM HA
Definition 3.2. For each x, y \in U, the natural nonisotropic distance \rho M (x, y) corresponding
to the vector fields X1, . . . , X2n - 2 is defined by
\rho M (x, y) = \mathrm{i}\mathrm{n}\mathrm{f}
\biggl\{
\delta > 0 : there exists a continuous, piecewise smooth map
\phi : [0, 1] \rightarrow U such that \phi (0) = x, \phi (1) = y,
and \phi \prime (t) =
\sum 2n - 2
j=1
\alpha j(t)Xj almost everywhere,
with | \alpha j(t)| < \delta , for j = 1, . . . , 2n - 2
\biggr\}
.
The nonisotropic ball centered at x \in U, with radius r > 0 is given by
BM (x, r) = \{ y \in U : \rho M (x, y) < r\} .
For any x, y \in U, we also define V (x, y) = | BM (x, \rho M (x, y))| .
Let \BbbB 0 denote the unit ball (defined by the Euclidean metric) in \BbbR 2n - 1. For x \in U and r > 0,
we set
\Phi x,r(u) = exp(ru1X1 + . . .+ ru2n - 2X2n - 2 + \Lambda (x, r)u2n - 1T )(x),
where u = (u1, . . . , u2n - 1) \in \BbbB 0. There is R0 > 0 depending on the manifold M so that for
all 0 < r < R0, the map \Phi x,r is a diffeomorphism of the unit ball \BbbB 0 to its image. Hereafter,
0 < r < R0 when we have calculations on the exponential map \Phi x,r. Now, let
\widetilde BM (x, r) = \Phi x,r(\BbbB 0),
that is
\widetilde BM (x, r) =
\Bigl\{
y \in U : y = exp(a1X1 + . . .+ a2n - 2X2n - 2 + aT )(x),
where | aj | < r for j = 1, . . . , 2n - 2, and | a| < \Lambda (x, r)
\Bigr\}
.
We have the following facts about the size function \Lambda and the above families of nonisotropic balls,
which were proved in [9].
Theorem 3.1. Assume that U \subset M of finite commutator type, there exists R0 > 0 such that:
(1) There are positive constants C1, C2 so that for all x \in U and 0 < r < R0,
BM (x,C1r) \subset \widetilde BM (x, r) \subset BM (x,C2r).
(2) There are two constants C3, C4 > 0 such that for all x, y \in U
C3 \leq
VM (x, y)
(\rho M (x, y))2n - 2\Lambda (x, \rho M (x, y))
\leq C4.
(3) Let Jx,r(u) denote the Jacobian matrix of \Phi x,r(u). Then | \mathrm{d}\mathrm{e}\mathrm{t}(Jx,r(u))| \approx r2n - 2\Lambda (x, r)
uniformly in x and 0 < r < R0.
(4)
\bigm| \bigm| \bigm| \bigm| \partial \alpha \partial u\alpha \mathrm{d}\mathrm{e}\mathrm{t}(Jx,r(u))
\bigm| \bigm| \bigm| \bigm| \lesssim r2n - 2\Lambda (x, r) uniformly in x and 0 < r < R0, for each multiindex \alpha .
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 8
THE 2b -HEAT EQUATION ON FINITE TYPE CR MANIFOLDS WITH COMPARABLE LEVI FORM 1087
For any function f \in C1(\BbbB 0), the scaled pullbacks to \BbbB 0 of the vector fields Xj are defined by
( \widehat Xjf)(u) = ( \widehat Xjf)x,r(u) = r(Xj
\v f)(\Phi x,r(u)), j = 1, . . . , 2n - 2,
where \v f(y) = f \circ \Phi - 1
x,r(y) for y \in \widetilde BM (x, r). Therefore, \widehat X1, . . . , \widehat X2n - 2 may be written (in the
u-coordinates) as linear combinations of the vector fields
\partial
\partial u1
, . . . ,
\partial
\partial u2n - 1
. Also, we define the
scaled pullback to \BbbB 0 of the function \phi on \widetilde BM (x, r) by\widehat \phi (u) = \phi (\Phi x,r(u))
for u \in \BbbB 0.
The following facts are also from [2, 9].
Theorem 3.2. (1) The coefficients of the \widehat Xj (expressed in u-coordinates), together with their
derivatives, are bounded above uniformly in x and r.
(2) The vector fields \widehat X1, . . . , \widehat X2n - 2 are of finite commutator type on \BbbB 0, and\bigm| \bigm| \bigm| \mathrm{d}\mathrm{e}\mathrm{t}( \widehat X1, . . . , \widehat X2n - 2, Z)
\bigm| \bigm| \bigm| > C
for a commutator Z (of the \widehat Xj ) of length \leq m such that \widehat X1, . . . , \widehat X2n - 2, Z span the tangent space
(C > 0 is independent of x and r).
In particular, we can write
\partial
\partial uj
=
2n - 2\sum
l=1
bjl \widehat Xl + bj,2n - 1Z,
so that bj,l and its derivatives are bounded above uniformly in x and r, for j, l = 1, . . . , 2n - 1.
Now, let us define the CR structure on \BbbB 0 determined by the following vector fields:\widehat Lj = \widehat Xj + i \widehat Xn+j - 1,
\widehat \=Lj = \widehat Xj - i \widehat Xn+j - 1,
and the basis of (0, 1)-forms dual to \widehat \=L1, . . . ,
\widehat \=Ln - 1 by \widehat \=\omega 1, . . . , \widehat \=\omega n - 1.
We consider the equation on \widetilde B(x, r)
\=Lj\phi = f.
From the definition, \widehat \=Lj
\widehat \phi = r(\=Lj\phi )(\Phi x,r(u)) = rf(\Phi x,r(u)) = r \widehat f = r\widehat \=Lj\phi .
So, \widehat \=Lj\phi = r - 1\widehat \=Lj
\widehat \phi is the scaled pullback of the equation \=Lj\phi = f. Now, \=L\ast
j = - Lj + aj , for some
aj \in C\infty (U), we also define \widehat \=L\ast
j = - \widehat Lj + r\widehat aj .
Similarly, we obtain \widehat \=L\ast
j\phi = r - 1\widehat \=L\ast
j
\widehat \phi . We also define the pullbacks \widehat \=\partial b and \widehat \=\partial \ast b of the operators \=\partial b and
\=\partial \ast b by
\widehat (\=\partial b\phi ) = r - 1 \widehat \=\partial b\widehat \phi ,
and
\widehat (\=\partial \ast b\phi ) = r - 1\widehat \=\partial \ast b \widehat \phi ,
respectively.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 8
1088 LY KIM HA
Lemma 3.1 [6]. Let (\=\partial b)B and (\=\partial \ast b )
B be the operators defined in the terms of the CR structure
on the unit ball \BbbB 0 which is determined by the vector fields \{ \widehat Lj ,
\widehat \=Lj\} . Then \widehat \=\partial b = (\=\partial b)
B and
r\widehat \=\partial \ast b - (\=\partial \ast b )
B is a differential operator of order zero on \BbbB 0 uniformly in x and r.
Finally, we also extend the map \Phi x,r on \BbbB 0 to the map \Phi (s,x),r on \BbbR \times \BbbB 0 by
\Phi (s,x),r(s, u) =
\bigl(
r - 2s,\Phi x,r(u)
\bigr)
with 0 < r < R0.
The scaled pullback of the heat equation on \BbbR \times M to \BbbR \times \BbbB 0 is\biggl( \biggl(
\partial
\partial s
+\Box b
\biggr)
\phi (s, x)
\biggr) \widehat
= r - 2 \partial
\partial s
\widehat \phi (s, u) + r - 2\widehat \Box b
\widehat \phi (s, u),
where \widehat \Box b =
\widehat \=\partial b\widehat \=\partial \ast b +\widehat \=\partial \ast b \widehat \=\partial b is defined on \BbbB 0.
Next, under the condition of finite commutator type on M, we also define the parabolic non-
isotropic metric on \BbbR \times M. Recall that the family of dilation \delta \lambda on \BbbR \times M is
\delta \lambda (s, x) = (\lambda 2s, \lambda x),
for s \in \BbbR , x \in M and the parameter \lambda > 0.
Definition 3.3. Denote by Y the vector field
\partial
\partial s
on \BbbR , then the family of the vector fields
\{ Y,X1, . . . , X2n - 2\} also satisfies the condition of finite commutator type. For every p = (s, x),
q = (t, y) \in \BbbR \times U, the following function is finite:
\rho \BbbR \times M (p, q) = \rho M (x, y) +
\sqrt{}
| s - t| .
This distance associates to the corresponding balls B\BbbR \times M ((s, x), r) on \BbbR \times M. Note that\bigm| \bigm| B\BbbR \times M ((s, x), | t - s| )
\bigm| \bigm| \approx (t - s)2| BM (x, | t - s| )| .
Let \widetilde \BbbB 0 denote the unit ball in \BbbR 2n. For each (u, u0) \in \widetilde \BbbB 0 the exponential mapping on \BbbR \times M is
\widetilde \Phi (s,x),r(u, u0) = exp(r2u0Y + ru1X1 + . . .+ ru2n - 2X2n - 2 + \Lambda (x, r)u2n - 1T )(s, x)
and \widetilde B\BbbR \times M ((s, x), r) = \widetilde \Phi (s,x),r(\widetilde \BbbB 0).
Now, we briefly recall the definition of the class of NIS operators on the CR manifold M. For more
discussions, see [6 – 8].
Let \scrD \prime (M) be the space of distributions defined on M and \BbbI k be the set of multiindexes
(\alpha 1, . . . , \alpha 2n - 2) such that
\sum 2n - 2
j=1 \alpha j = k for k = 0, 1, . . . .
Definition 3.4. An operator \scrT : C\infty
0 (M) \rightarrow \scrD \prime (M) is called a NIS operator of order \kappa \geq 0
if the following conditions hold:
(1) There is a function T0(x, y) \in C\infty (M \times M \setminus \Delta M ) (smooth of the diagonal) so that if
\phi , \psi \in C\infty
0 (M) have disjoint supports,\bigl\langle
\scrT [\phi ], \psi
\bigr\rangle
=
\int
M
\int
M
\phi (y)\psi (x)T0(x, y)dV (x)dV (y).
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 8
THE 2b -HEAT EQUATION ON FINITE TYPE CR MANIFOLDS WITH COMPARABLE LEVI FORM 1089
(2) For any s \geq 0, there exist parameters \alpha (s) < \infty , \beta < \infty such that if \zeta , \zeta \prime \in C\infty (M),
\zeta \prec \zeta \prime , then there is a constant Cs so that
\| \zeta \scrT [f ]\| s \leq Cs(\| \zeta \prime f\| \alpha (s) + \| f\| \beta ) (3.1)
for all f \in C\infty (M).
(3) For any \alpha \in \BbbI k, \beta \in \BbbI l, there exists a constant Ck,l so that
| X\alpha
xX
\beta
y T0(x, y)| \leq Ck,l\rho M (x, y)\kappa - k - lVM (x, y) - 1.
(4) For any ball BM (x0, r) \subset U, for each integer k \geq 0, there is a positive integer Nk and a
constant Ck so that if \phi \in C\infty
0 (BM (x0, r)) and \alpha \in \BbbI k, we have
\mathrm{s}\mathrm{u}\mathrm{p}
x\in BM (x0,r)
| X\alpha
x \scrT [\phi ](x)| \leq Ckr
\kappa - k \mathrm{s}\mathrm{u}\mathrm{p}
y\in M
\sum
| J | \leq Nk
r| J | | XJ [\phi ](y)| .
(5) The above conditions also hold for the adjoint operator T \ast with kernel T0(y, x).
We also generalize this definition to a class of operators defined on (0, q)-forms. Let \scrT be an
operator from C\infty
0,q1
(M) into C\infty
0,q2
(M) and \phi =
\sum \prime
| I| =q1
\phi I \=\omega I \in C\infty
0,q1(U), then
\scrT [\phi ](x) =
\sum
| J | =q2
\prime (\scrT [\phi ])J(x)\=\omega J ,
where
(\scrT [\phi ])J(x) =
\sum
| I| =q1
\prime \bigl\langle \scrT [\phi I(x)\=\omega I ], \=\omega J
\bigr\rangle
L2 .
Then we define \scrT IJ [g](x) = \langle \scrT [g(x)\=\omega I ], \=\omega J\rangle L2 for g \in C\infty
0 (U). We say that \scrT is a NIS operator
of order \kappa on (0, q1)-form if and only if \scrT and \scrT \ast satisfy the estimate in condition (2) of the
Definition 3.4 and each \scrT IJ is a NIS operator of order \kappa on functions.
Example 3.1 (Szegö projections). Let \scrS q and \scrS \prime
q denote the orthogonal projections in L2
(0,q)(M)
onto ker( \=\partial 0,qb ) and ker( \=\partial \ast 0,qb ), respectively, where \=\partial 0,qb and \=\partial \ast 0,qb mean \=\partial b, \=\partial
\ast
b acting on (0, q)-forms.
We can rewrite these operators by
\scrS q[\phi ](x) =
\sum
| J | =q2
\prime
\sum
| I| =q1
\prime \langle \scrS q[\phi I(x)\=\omega I ], \=\omega J\rangle L2 \=\omega J =
\sum
| J | =q2
\prime
\left( \sum
| I| =q1
\prime \scrS IJ
q [\phi I ](x)
\right) \=\omega J ,
\scrS \prime
q[\phi ](x) =
\sum
| J | =q2
\prime
\sum
| I| =q1
\prime \langle \scrS \prime
q[\phi I(x)\=\omega I ], \=\omega J\rangle L2 \=\omega J =
\sum
| J | =q2
\prime
\left( \sum
| I| =q1
\prime (\scrS IJ
q )\prime [\phi I ](x)
\right) \=\omega J
for \phi =
\sum
| I| =q1
\phi I \=\omega I . Now, by the Riesz representation theorem,
\scrS q[\phi ](x) =
\sum
| J | =q2
\prime
\left( \sum
| I| =q1
\prime
\int
M
SIJ
q (x, y)\phi I(y)dV (y)
\right) \=\omega J ,
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 8
1090 LY KIM HA
\scrS \prime
q[\phi ](x) =
\sum
| J | =q2
\prime
\left( \sum
| I| =q1
\prime
\int
M
(SIJ
q )\prime (x, y)\phi I(y)dV (y)
\right) \=\omega J ,
where SIJ
q (x, y) and (SIJ)\prime (x, y) are the respective Schwartz kernels of \scrS IJ
q [\cdot ] and (\scrS IJ
q )\prime [\cdot ].
In [6], the author showed that the operators \scrS q and \scrS \prime
q are the NIS operators of order zero with
q0 \leq q \leq n - 1 - q0.
As a consequence, we have the following proposition.
Proposition 3.1. Let \alpha be a multiindex with | \alpha | = k \geq 1. For 0 \leq j \leq [k/2] , there are NIS
operators Aj,1, . . . , Aj,2n - 2, Aj,2n - 1 smoothing of order zero such that
X\alpha (I - \scrH q) =
[k/2]\sum
j=0
\Biggl(
2n - 2\sum
l=1
(Aj,lXl) +Aj,2n - 1
\Biggr)
\Box
j
b.
Here the operator \scrH q is the orthogonal projection in L2
0,q onto its harmonic subspace. In particular,
if k = 2j, Aj,1 = Aj,2 = . . . = Aj,2n - 2 = 0.
Proof. The proof uses the fact (which is established in [6]) that the relative inverse K to \Box b is a
NIS operator of order 2, in the cases of comparable Levi form. Here the modification to Proposition
3.4.7 in [8] is that now \Box bK = I - \scrH q instead of I - \scrS q in dimension n = 2.
The scaling method also provides following Sobolev type theorem.
Theorem 3.3. Assume that the D\epsilon (q) and finite type conditions hold on M. Then there are a
constant C and an even integer Lm so that if f \in C\infty (U), then, for all x \in U and all r \leq r0,
\mathrm{s}\mathrm{u}\mathrm{p}
BM (x,r)
| f | \leq C| BM (x, r)| -
1
2
\sum
0\leq | I| \leq Lm,| I| even
r| I| \| XIf\| L2(BM (x,2r)).
Let f \in \Lambda 0,q\prime (C\infty (M)) \cap L2
0,q\prime (M) with q \leq q\prime \leq n - 1 - q. Moreover, if f \in (\mathrm{k}\mathrm{e}\mathrm{r}(\Box b))
\bot , then
\mathrm{s}\mathrm{u}\mathrm{p}
BM (x,r)
| f | \leq C| BM (x, r)| -
1
2
Lm/2\sum
j=0
r2j\| \Box j
bf\| L2
(0,q)
.
Proof. We apply the scaling method introduced above. From the property (1) in Theorem 3.1,
we have
\mathrm{s}\mathrm{u}\mathrm{p}
y\in BM (x,r)
| f(y)| \leq \mathrm{s}\mathrm{u}\mathrm{p}
y\in \widetilde BM (x,C1r)
| f(y)| \leq \mathrm{s}\mathrm{u}\mathrm{p}
u\in \BbbB 0
| f(\Phi x,C2r(u))| .
Set F (u) = f(\Phi x,C2r(u)) for u \in \BbbB 0. Let G(u) = F (u)\theta (u), where \theta \in C\infty
0 (\BbbR 2n - 1), \theta = 1 on
\BbbB 0, and \theta = 0 outside the ball \BbbB (0, 2) \subset \BbbR 2n - 1. So,
\mathrm{s}\mathrm{u}\mathrm{p}
u\in \BbbB 0
| F (u)| \leq \mathrm{s}\mathrm{u}\mathrm{p}
u\in \BbbR 2n - 1
| G(u)| \leq
\int
\BbbR 2n - 1
| \widehat G(\xi )| dV (\xi ) \leq
\leq \| (1 + | \xi | 4)1/N \widehat G(\xi )\| L2(\BbbR 2n - 1)\| (1 + | \xi | 4) - 1/N\| L2(\BbbR 2n - 1) \leq
(N can be chosen large enough to guarantee integrability)
\leq C
\sum
0\leq | I| \leq 2, | I| even
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\biggl(
\partial
\partial u
\biggr) I
F
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
L2(\BbbB (0,2))
.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 8
THE 2b -HEAT EQUATION ON FINITE TYPE CR MANIFOLDS WITH COMPARABLE LEVI FORM 1091
Now, from the statement (2) in Theorem 3.2, there is a positive integer number l depending on m
(the constant l should be the higher power to make the Hölder’s inequality work) such that
\mathrm{s}\mathrm{u}\mathrm{p}
\BbbB 0
| F | \leq C
\sum
0\leq | I| \leq 2l, | I| even
\|
\bigl( \widehat X\bigr) IF\| L2(\BbbB (0,2)).
Then, after rescaling pullback, by Theorem 3.1, we have the first statement. The analogue of the first
statement for forms is immediately obvious. In order to estimate XIf in the terms of \Box
j
bf, with
f = f - \scrH q[f ], we will apply the basic decomposition in Proposition 3.1. Since | I| is even, there
exists a NIS operator of smoothing of order zero AI such that
XI(I - \scrH q) = AI\Box
| I|
2
b .
Therefore, if f is orthogonal to the null space of \Box b, we obtain the second assertion.
Theorem 3.3 is proved.
4. Proof of the main theorem. 4.1. The heat kernel. For convenience, we recall some results
for the heat semigroup of unbounded operators e - s2b via Hilbert space theory.
Theorem 4.1. Let \phi \in L2
(0,q)(M) for q \in [q0, n - 1 - q0], then:
(1) \mathrm{l}\mathrm{i}\mathrm{m}s\rightarrow 0 \| e - s2b [\phi ] - \phi \| L2
(0,q)
(M) = 0;
(2) for s > 0, \| e - s2b [\phi ]\| L2
(0,q)
(M) \leq \| \phi \| L2
(0,q)
(M);
(3) if \phi \in \mathrm{D}\mathrm{o}\mathrm{m}(\Box b), then \| e - s2b [\phi ] - \phi \| L2
(0,q)
(M) \leq s\| \Box b[\phi ]\| L2
(0,q)
(M);
(4) for s > 0 and j nonnegative integer, \| (\Box b)
je - s2b [\phi ]\| L2
(0,q)
(M) \leq
\biggl(
j
e
\biggr) j
s - j\| \phi \| L2
(0,q)
(M);
(5) e - s2b\scrH q[\phi ] = \scrH qe
- s2b [\phi ] = \scrH q[\phi ];
(6) e - s2b [\phi ] = (I - \scrH q)e
- s2b [\phi ] +\scrH q[\phi ] = e - s2b(I - \scrH q)[\phi ] +\scrH q[\phi ];
(7) for any \phi \in L2
(0,q)(M) and any s > 0, the Hilbert space valued form e - s2b [\phi ] satisfies
[\partial s +\Box b][e
- s2b [\phi ]] = 0 for s > 0,
\mathrm{l}\mathrm{i}\mathrm{m}
s\rightarrow 0
e - s2b [\phi ] = \phi in L2
(0,q)((0,\infty )\times M);
(8) for any s \geq 0, e - s2b is a self-adjoint operator on L2
(0,q)(M).
From Proposition 4.1 in [6], the operator \scrH q does exactly equal to \scrS q + \scrS \prime
q - I. Therefore, \scrH q
is a NIS operator of order zero, with q0 \leq q \leq n - 1 - q0. This fact and Theorem 4.1 imply the
pseudolocal property (3.1) for \scrT = e - s2b uniformly in s > 0.
Lemma 4.1. Let M be a pseudoconvex finite type compact CR manifold for which the range of
\=\partial b is closed in L2 and which satisfies the D\epsilon (q0) condition. Let | \alpha | = a \geq 0 and K \subset M be a
compact set. Choose an integer N so that N\epsilon 0 > 2n - 1 + a. Then there is a constant C such that
for each s > 0, if x \in K, and for all \phi \in L2
(0,q)(M) with q0 \leq q \leq n - 1 - q0,
| X\alpha e - s2b [\phi ](x)| \leq C(1 + s - N )\| \phi \| L2
(0,q)
(M).
As a consequence of the condition of finite commutator type, for any derivative D on M,
| D\alpha e - s2b [\phi ](x)| \leq C(1 + s - N )\| \phi \| L2
(0,q)
(M).
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 8
1092 LY KIM HA
Proof. Choose \zeta \in C\infty
0 (M) with \zeta (x) = 1 for all x \in K. Then, pick cutoff functions
\zeta \prec \zeta 1 \prec . . . \prec \zeta N = \zeta \prime . By Sobolev imbedding heorem, we have
| X\alpha e - s2b [\phi ](x)| = | X\alpha \zeta (x)e - s2b [\phi ](x)| \leq C\| \zeta e - s2b [\phi ]\| H2n - 1+a .
Applying the basic subelliptic estimate, we get
\| \zeta e - s2b [\phi ]\| H2n - 1+a \leq C
\bigl[
\| \zeta 1\Box be
- s2b [\phi ]\| H2n - 1+a - \epsilon 0 + \| \zeta 1e - s2b [\phi ]\| L2
\bigr]
.
If we repeat this argument N times, by (4) in Theorem 4.1, we will obtain
\| \zeta e - s2b [\phi ]\| H2n - 1+a \leq C
N\sum
j=0
\| \zeta \prime \Box j
be
- s2b [\phi ]\| L2
(0,q)
(M) \leq C(1 + s - N )\| \phi \| L2
(0,q)
(M).
Lemma 4.1 is proved.
We also have integral kernels for X\alpha e - s2b [\phi ].
Lemma 4.2. For each x \in M and s > 0, and for each multiindex | \alpha | = a, there exist unique
functions HIJ
s,x,\alpha \in L2(M), where | I| = | J | = q, q \in [q0, n - 1 - q0], so that
X\alpha e - s2b [\phi ](x) =
\sum
| J | =q
\prime
\left( \sum
| I| =q
\prime
\int
M
HIJ
s,x,\alpha (y)\phi I(y)dV (y)
\right) \=\omega J
or in short
X\alpha e - s2b [\phi ](x) =
\int
M
Hs,x,\alpha (y)\phi (y)dV (y),
where \phi =
\sum \prime
| I| =q\phi I \=\omega I . Moreover, if K \subset M is compact and if C is the corresponding constant in
Lemma 4.1, then, if x \in K, \sum
I,J
\int
M
| HIJ
s,x,\alpha (y)| 2dy \leq C2(1 + s - N )2.
Proof. For each s > 0, x \in M, we define the mapping \phi \mapsto \rightarrow X\alpha e - s2b [\phi ](x). By Lemma 4.1,
this functional is bounded. Moreover, since
X\alpha e - s2b [\phi ](x) =
\sum
| J | =q
\prime \sum
| I| =q
\prime
\langle X\alpha e - s2b [\phi I \=\omega I ], \=\omega J\rangle L2(x)\=\omega J ,
and so by the Riesz representation theorem, there exist functions HIJ
s,x,\alpha \in L2(M) so that
X\alpha ((e - s2b)IJ [\phi I ](x)) = \langle X\alpha e - s2b [\phi I \=\omega I ], \=\omega J\rangle (x) =
\int
M
HIJ
s,x,\alpha (y)\phi I(y)dV (y).
Hence, by duality, we obtain \sum
I,J
\prime
\int
M
| HIJ
s,x,\alpha (y)| 2dy \leq C2(1 + s - N )2.
Lemma 4.2 is proved.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 8
THE 2b -HEAT EQUATION ON FINITE TYPE CR MANIFOLDS WITH COMPARABLE LEVI FORM 1093
Definition 4.1. We define the following (0, q)-forms, for q \in [q0, n - 1 - q0],
HI
y (s, x) =
\sum
| J | =q
\prime
HIJ(s, x, y)\=\omega J(x),
HJ
x (s, y) =
\sum
| I| =q
\prime
HIJ(s, x, y)\=\omega I(y),
and, for s > 0,
Hs(x, y) =
\sum
| I| =q
| J | =q
\prime
HIJ(s, x, y)\omega J(x)\otimes \omega I(y).
It turns out that e - s2b [\phi ](x) =
\sum \prime
| J | =q
\langle HJ
x (s, \cdot ), \phi \rangle L2
(0,q)
(M)\=\omega J .
For fixed s > 0, since HIJ(s, x, y) = HIJ
s,x,0, for all I, J, Lemma 4.2 says that the maps
y \mapsto \rightarrow HIJ(s, x, y) belong to L2(M) for all I, J, and
e - s2b [\phi ](x) =
\sum
| J | =q
\prime
\left( \sum
| I| =q
\prime
\int
M
HIJ(s, x, y)\phi I(y)dV (y)
\right) \=\omega J .
We denote these sums as
\int
M
H(s, x, y)\phi (y)dV (y).
Theorem 4.2. For each fixed s > 0 and x \in M, the function y \mapsto \rightarrow HIJ(s, x, y) belongs to
L2(M), so each integral above converges absolutely. Moreover, each component HIJ(s, x, y) of
H(s, x, y) satisfies
(1) for s > 0 and x, y \in M, HIJ(s, x, y) = HJI(s, y, x);
(2) [\partial s + (\Box b)x][H
I
y ](s, x) = [\partial s + (\Box b)y][H
J
x ](s, y) = 0 and, hence,
[\partial s + (\Box b)x][Hs(x, y)] = [\partial s + (\Box b)y][Hs(x, y)] = 0;
(3) for any integer j, k \geq 0,
(\Box b)
j
x(\Box b)
k
yH
IJ
s (x, y) = (\Box b)
j+k
x HI
y (s, x) = (\Box b)
j+k
y HJ
x (s, y);
(4) for each s > 0 and y \in M, for any nonnegative integer j, each function
x \mapsto \rightarrow (\Box b)
j
xH
I
y (s, x)
is orthogonal to \mathrm{k}\mathrm{e}\mathrm{r}(\Box b).
Proof. The proof of Theorem 4.2 is almost identical to the one in [8] (Theorem 5.1.2), hence it
is omitted.
These heat kernels also provide the fundamental solutions for the initial-value problem on the
whole space \BbbR \times M.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 8
1094 LY KIM HA
Definition 4.2. For \psi \in \Lambda 0,q(C\infty
0 (\BbbR \times M)), we set
\langle \BbbH x, \psi \rangle = \mathrm{l}\mathrm{i}\mathrm{m}
\epsilon \rightarrow 0+
\infty \int
\epsilon
\int
M
\widetilde H(s, x, y)\psi (s, y)dV (y) ds = \mathrm{l}\mathrm{i}\mathrm{m}
\epsilon \rightarrow 0+
\infty \int
\epsilon
\int
M
H(s, x, y)\psi (s, y)dV (y) ds,
where \widetilde H ’s components are
\widetilde HIJ(s, x, y) =
\Biggl\{
HIJ(s, x, y), if s > 0,
0, if s \leq 0.
Proposition 4.1. The limit defining \BbbH x exists. Moreover,
[\partial s + (\Box b)y][\BbbH x] = \delta 0 \otimes \delta x
in the sense of distributions (in component-wise), i.e.,\bigl\langle
\BbbH x, [ - \partial s +\Box b]\psi
\bigr\rangle
= \psi (0, x).
Proof. Setting \psi s(y) = \psi (s, y), so \psi s \in \Lambda 0,q(C\infty (M)). Choose a positive integer N so that
N\epsilon >
2n - 1
2
. Choose \zeta \prec \zeta 1 \prec . . . \prec \zeta N = \zeta \prime with \zeta (x) = 1. Then, once again, by Sobolev
imbedding theorem and the basic subelliptic estimate applied N times, we obtain\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\int
M
H(s, x, y)\psi (y)dV (y)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| = \bigm| \bigm| \zeta e - s2b [\psi s](x)
\bigm| \bigm| \leq
\leq C\| \zeta e - s2b [\psi s]\| N\epsilon \leq
\leq C[\| \zeta 1\Box b[e
- s2b [\psi s]]\| (N - 1)\epsilon + \| \zeta 1e - s2b [\psi s]\| 0] \leq
\leq . . . repeating N -times as above . . . \leq
\leq C
N\sum
j=0
\| \zeta \prime \Box j
b[e
- s2b [\psi s]]\| 0.
Moreover, since the operators \Box b and e - s2b commute,\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\int
M
H(s, x, y)\psi (y)dV (y)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq C
N\sum
j=0
\| \zeta \prime e - s2b [\Box j
b\psi s]\| 0 \leq C
N\sum
j=0
\| \Box j
b\psi s\| 0.
The right-hand side is uniformly bounded in s, and then, taking integral on [\eta 1, \eta 2], we have\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\eta 2\int
\eta 1
\int
M
H(s, x, y)\psi (y)dV (y) ds
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq C| \eta 2 - \eta 1| \mathrm{s}\mathrm{u}\mathrm{p}
s
N\sum
j=0
\| \Box b\psi s\| 0.
We see that the left-hand side goes to zero as \eta 2 \rightarrow 0, so the limit defining \BbbH x exists. Again, let
\psi \in \Lambda 0,q(C\infty
0 (\BbbR \times M), then
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 8
THE 2b -HEAT EQUATION ON FINITE TYPE CR MANIFOLDS WITH COMPARABLE LEVI FORM 1095
\bigl\langle
\BbbH x, [ - \partial s +\Box b]\psi s
\bigr\rangle
= \mathrm{l}\mathrm{i}\mathrm{m}
\epsilon \rightarrow 0
\infty \int
\epsilon
e - s2b [[ - \partial s +\Box b]\psi s] ds =
= - \mathrm{l}\mathrm{i}\mathrm{m}
\epsilon \rightarrow 0
\infty \int
\epsilon
\int
M
H(s, x, y)\partial s\psi (s, x)dV (y) ds+
+ \mathrm{l}\mathrm{i}\mathrm{m}
\epsilon \rightarrow 0
\infty \int
\epsilon
\int
M
H(s, x, y)\Box b\psi (s, y)dV (y) ds. (4.1)
Now, for the first term, we get
-
\infty \int
\epsilon
\int
M
H(s, x, y)\partial s\psi (s, x)dV (y) ds =
=
\sum
| J | =q
\prime
\left( \sum
| I| =q
\prime
\int
M
HIJ(\epsilon , x, y)\psi I(\epsilon , y)dV (y)
\right) \=\omega J+
+
\infty \int
\epsilon
\sum
| J | =q
\prime
\langle \partial sHJ
x (s, \cdot ), \psi (s, \cdot )\rangle \=\omega J ds. (4.2)
For the second term,
\infty \int
\epsilon
\int
M
H(s, x, y)\Box b\psi (s, y)dV (y) ds =
\infty \int
\epsilon
\sum
| J | =q
\prime
\langle HJ
x (s, \cdot ),\Box b\psi (s, \cdot )\rangle \=\omega J ds =
=
\infty \int
\epsilon
\sum
| J | =q
\prime
\langle (\Box b)yH
J
x (s, \cdot ), \psi (s, \cdot )\rangle \=\omega J ds. (4.3)
Hence, since [\partial s + (\Box b)y]H
J
x (s, y) = 0, (4.1), (4.2), and (4.3) imply
\langle \BbbH x, [ - \partial s +\Box b]\psi s\rangle = \psi (0, x) = \langle \delta 0 \otimes \delta x, \psi \rangle .
Proposition 4.1 is proved.
A remark that, by translation, we also have
\langle \BbbH x, [ - \partial s+t +\Box b]\psi \rangle = \psi (t, x).
4.2. Pointwise estimates for the heat kernel. We begin by recalling the scaled pullback of the
heat equation on \BbbR \times M to \BbbR \times \BbbB 0 by
((\partial s +
\widehat \Box b)
\widehat \phi (s, u)) = r2[(\partial s +\Box b)\phi (s, x)]
\wedge .
Now, using the above changing of the variable \Phi (s,x0),r, with s > 0, x0 \in M, u, v \in \BbbB 0, we define
the pullback of the heat kernel H(s, x, y):
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 8
1096 LY KIM HA
W IJ(s, u, v) =W IJ
x0,r(s, u, v) = HIJ(r2s,\Phi x0,r(u),\Phi x0,r(v))
for each | I| = | J | = q, q0 \leq q \leq n - 1 - q0, and 0 < r < R0. Hence, from previous sections, we
have
[\partial s + (\widehat \Box b)u][W
I
v ](s, u) = 0,
[\partial s + (\widehat \Box b)v][W
J
u ](s, v) = 0,
where W I
v (s, u) and W J
u (s, v) are defined by the same formulas as HI
y (s, x) and HJ
x (s, y).
In the same way, for s > 0, and \phi \in \Lambda 0,q(C\infty
0 (\BbbB 0)), we define the scaled pullback of e - s2b as
\BbbW s[\phi ](u) =
\int
\BbbB 0
W (s, u, v)\phi (v)dv =
\int
\BbbB 0
H(r2s,\Phi x0,r(u),\Phi x0,r(v))\phi (v)dv.
The key point is that the norm of the operator \BbbW s is bounded on L2
(0,q)(\BbbB 0).
Lemma 4.3. There is a constant C which is independent of x0, r and s > 0 so that
\| \BbbW s[\phi ]\| L2
(0,q)
(\BbbB 0) \leq C| B(x0, r)| - 1\| \phi \| L2
(0,q)
(\BbbB 0).
Proof. Let x \in B(x0, r), by changing of variables, we have
\BbbW s[\phi ](\Phi
- 1
x0,r(x)) =
\int
\BbbB 0
H(r2s,\Phi x0,r(\Phi
- 1
x0,r(x)),\Phi x0,r(v))\phi (v)dv =
=
\int
M
H(r2s, x, y)(\phi \circ \Phi - 1
x0,r)(y)Jx0,r(\Phi
- 1
x0,r)\Phi
- 1
x0,r(y)(y)dV (y) =
= e - r2s2b [(\phi \circ \Phi - 1
x0,r)Jx0,r\Phi
- 1
x0,r](x).
Since \| e - r2s2b [(\phi \circ \Phi - 1
x0,r)Jx0,r\Phi
- 1
x0,r]\| L2 \leq \| (\phi \circ \Phi - 1
x0,r)Jx0,r\Phi
- 1
x0,r\| L2 , it follows that\int
M
| \BbbW s[\phi ](\Phi
- 1
x0,r(x))|
2dV (x) \leq
\int
M
| \phi (\Phi - 1
x0,r(x))|
2(Jx0,r(\Phi
- 1
x0,r(x)))
2dV (x) =
=
\int
\BbbB 0
| \phi (u)| 2(Jx0,r\Phi
- 1
x0,r(\Phi x0,r(u)))
2Jx0,r\Phi x0,r(u)du \leq
\leq C| B(x0, r)| - 1
\int
\BbbB 0
| \phi (u)| 2du.
The last inequality is derived from the facts that Jx0,r\Phi
- 1
x0,r(\Phi x0,r(u)) = Jx0,r\Phi x0,r(u)
- 1, and
Jx0,R\Phi x0,R(u) \geq C - 1| B(x0, r)| for 0 < r < R0 according to Theorem 3.1.
On the other hand,\int
M
| \BbbW s[\phi ](\Phi
- 1
x0,r(x))|
2dV (x) \geq C - 1| B(x0, r)|
\int
\BbbB 0
| \BbbW s[\phi ](u)| 2du.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 8
THE 2b -HEAT EQUATION ON FINITE TYPE CR MANIFOLDS WITH COMPARABLE LEVI FORM 1097
Hence, we obtain
\| \BbbW s[\phi ]\| L2
(0,q)
(\BbbB 0) \leq C| B(x0, r)| - 1\| \phi \| L2
(0,q)
(\BbbB 0).
Lemma 4.3 is proved.
Next, we will obtain local estimates for the functions HIJ ’s, | I| = | J | = q, q \in [q0, n - 1 - q0].
Theorem 4.3. Let j, k, l be nonnegative integers. For every positive integer N, there is a
constant CN = CN,j,k,l so that if | \alpha | = k, | \beta | = l,
| \partial jsX\alpha
xX
\beta
yH
IJ(s, x, y)| \leq
\left\{ CN\rho (x, y)
- 2j - k - l| B(x, \rho (x, y))| - 1
\biggl(
s
\rho (x, y)2
\biggr) N
, if s \leq \rho (x, y)2,
CNs
- j - k/2 - l/2| B(x,
\surd
s)| - 1, if s \geq \rho (x, y)2,
for all (s, x, y) with \rho \BbbR \times M ((s, x), (0, y)) = | s| 1/2 + \rho (x, y) \leq 1.
The proof is based the scaling method which was introduced M. Christ [2], and then deve-
loped in higher dimensions by K. Koenig [6]. We need the following subelliptic estimate for the
scaled pullback of \Box b operator on \BbbB 0 which is a consequence of the subelliptic estimate of \Box b and
Theorem 3.1.
Proposition 4.2. Fix \zeta , \zeta \prime \in C\infty
0 (\BbbB 0) with \zeta \prec \zeta \prime . For smooth (0, q)-forms, q \in [q0, n - 1 - q0],
\phi =
\sum \prime
| K| =q\phi K
\=\widehat \omega K on \BbbB 0 and \delta \geq 0, we have
\| \zeta \phi \| 2\delta +\epsilon \leq C\delta
\Bigl(
\| \zeta \prime \widehat \=\partial b\phi \| 2\delta + \| \zeta \prime \widehat \=\partial \ast b\phi \| 2\delta + \| \zeta \prime \phi \| 20
\Bigr)
,
where C\delta is a positive constant independent of x and 0 < r < R0.
As a consequence, for q \in [q0, n - 1 - q0], the heat operator \partial s+\widehat \Box b also satisfies the subelliptic
estimate
\| \zeta \phi \| 2\delta +\epsilon \leq C\delta
\Bigl(
\| \zeta \prime [\partial s +\widehat \Box b]\phi \|
2
\delta + \| \zeta \prime \phi \| 20
\Bigr)
,
for all smooth (0, q)-forms \phi on \BbbR \times \BbbB 0.
Proof of Theorem 4.3. We will prove the theorem with N = 0 first. By compactness, if
R0 \leq | s| 1/2+\rho (x, y) \leq 1, the estimates are trivial. Hence, it suffices to show that the estimates hold
when | s| 1/2 + \rho (x, y) \leq R0. Now, let fix (s0, x0) \in \BbbR \times M, and let (s, x) \in \BbbR \times M be another
point so that \rho \BbbR \times M ((s0, x0), (s, x)) = r \leq R0. There exists a unique point (t0, v0) \in ( - 1, 1) \times \BbbB 0
such that (s, x) = (s0 + r2t0,\Phi x0,r(v0)). Let \tau > 0 such that | t0| 1/2 + | v0| \geq \tau .
For (t1, u), (t2, v) \in ( - 1, 1)\times \BbbB 0, we put
W\#((t1, u), (t2, v)) = H(r2(t2 - t1),\Phi x0,r(u),\Phi x0,r(v)),
in the sense that
\bigl(
W\#
\bigr) IJ
((t1, u), (t2, v)) = HIJ(r2(t2 - t1),\Phi x0,r(u),\Phi x0,r(v)) for all | I| =
= | J | = q. Then
[ - \partial t1 + (\widehat \Box b)u][(W
\#)Iv] = 0,
[\partial t2 + (\widehat \Box b)v][(W
\#)Ju ] = 0,
and
[\partial jsX
\alpha
xX
\beta
yH](r2(t2 - t1),\Phi x0,r(u),\Phi x0,r(v)) = r - 2j - k - l[\partial jt2
\widehat X\alpha
u
\widehat X\beta
vW
\#]((t1, u), (t2, v)).
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 8
1098 LY KIM HA
Now, for \phi \in C\infty
0 (( - 1, 1)\times \BbbB 0), set
\scrT \#[\phi ](t1, u) =
\int \int
\BbbR \times \BbbB 0
W\#((t1, u), (t2, v))\phi (t2, v)dvdt2
in the sense as above, i.e.,
\Bigl(
\scrT \#[\phi ](t1, u)
\Bigr)
J
=
\left( \sum
| I| =q
\prime
\int \int
\BbbR \times \BbbB 0
(W\#)IJ((t1, u), (t2, v))\phi I(t2, v)dvdt2
\right)
J
.
Then
\scrT \#[\phi ](t1, u) =
\sum
| J | =q
\prime \Bigl(
\scrT \#[\phi ](t1, u)
\Bigr)
J
\widehat \=\omega J .
The nonisotropic balls
B1 =
\biggl\{
(t1, u) : | t1| 1/2 + | u| < 1
3
\tau
\biggr\}
,
B2 =
\biggl\{
(t2, v) : | t2 - t0| 1/2 + | v - v0| <
1
3
\tau
\biggr\}
are disjoint.
Choose cutoff functions \zeta \prec \zeta \prime \prec \zeta \prime \prime \in C\infty
0 (B2) with \zeta (t0, v0) = 1, and \eta \prec \eta \prime \in C\infty
0 (B1)
with \eta (0, 0) = 1. Then, by the Sobolev inequality and the basic subelliptic estimate for the operator
\partial t2 +\widehat \Box b, we have\bigm| \bigm| \bigm| [\partial jt2 \widehat X\alpha
u
\widehat X\beta
v (W
\#)J ]((0, 0), (t0, v0))
\bigm| \bigm| \bigm| = \bigm| \bigm| \bigm| \zeta (t0, s0)[\partial jt2 \widehat X\alpha
u
\widehat X\beta
v (W
\#)J ]((0, 0), (t0, v0))
\bigm| \bigm| \bigm| \leq
\leq C
\bigm\| \bigm\| \bigm\| \zeta \prime (W\#)J((0, 0), (\cdot , \cdot ))
\bigm\| \bigm\| \bigm\|
2n+j+k+l
\leq
\leq C
\biggl[ \bigm\| \bigm\| \bigm\| \zeta \prime \prime [\partial t2 + (\widehat \Box b)v](W
\#)J((0, 0), (\cdot , \cdot ))
\bigm\| \bigm\| \bigm\|
2n+j+k+l - \epsilon
+
+
\bigm\| \bigm\| \bigm\| \zeta \prime \prime (W\#)J((0, 0), (\cdot , \cdot ))
\bigm\| \bigm\| \bigm\|
0
\biggr]
\leq
\leq C
\bigm\| \bigm\| \bigm\| \zeta \prime \prime (W\#)J((0, 0), (\cdot , \cdot ))
\bigm\| \bigm\| \bigm\|
0
\leq
\leq C \mathrm{s}\mathrm{u}\mathrm{p}
\phi \in C\infty (B2)
\| \phi \| =1
\bigm| \bigm| \bigm| \scrT \#[\zeta \prime \phi ](0, 0)
\bigm| \bigm| \bigm| ,
where the last estimate follows from the fact that [\partial t2 +
\widehat \Box b]W
\#((0, 0), (t, s)) = 0 on B1 containing
the support of \zeta \prime .
Now, to estimate the term with the supremum sign, again, we use the basic subelliptic for
- \partial t1 + (\widehat \Box b)u,
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 8
THE 2b -HEAT EQUATION ON FINITE TYPE CR MANIFOLDS WITH COMPARABLE LEVI FORM 1099
\mathrm{s}\mathrm{u}\mathrm{p}
\phi \in C\infty (B2)
\| \phi \| =1
\bigm| \bigm| \bigm| \scrT \#[\zeta \prime \phi ](0, 0)
\bigm| \bigm| \bigm| = \mathrm{s}\mathrm{u}\mathrm{p}
\phi \in C\infty (B2)
\| \phi \| =1
\bigm| \bigm| \bigm| \eta (0, 0)\scrT \#[\zeta \prime \phi ](0, 0)
\bigm| \bigm| \bigm| \leq
\leq C \mathrm{s}\mathrm{u}\mathrm{p}
\phi \in C\infty (B2)
\| \phi \| =1
\bigm\| \bigm\| \bigm\| \eta \scrT \#[\zeta \prime \phi ]
\bigm\| \bigm\| \bigm\|
2n
\leq
\leq C \mathrm{s}\mathrm{u}\mathrm{p}
\phi \in C\infty (B2)
\| \phi \| =1
\left[
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \eta \prime [ - \partial t1 + (\widehat \Box b)u]\scrT
\#[\zeta \prime \phi ]\underbrace{} \underbrace{}
=0 on B1 containing supp(\eta \prime )
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
2n - \epsilon
+
\bigm\| \bigm\| \bigm\| \eta \prime \scrT \#[\zeta \prime \phi ]
\bigm\| \bigm\| \bigm\|
0
\right] \leq
\leq C \mathrm{s}\mathrm{u}\mathrm{p}
\phi \in C\infty (B2)
\| \phi \| =1
\bigm\| \bigm\| \bigm\| \scrT \#[\zeta \prime \phi ]
\bigm\| \bigm\| \bigm\|
0
\leq
\leq C\| \scrT \#\| .
Therefore, we have shown that\bigm| \bigm| \bigm| \partial jsX\alpha
xX
\beta
yH(r2t0, x0, x)
\bigm| \bigm| \bigm| \leq Cr - 2j - k - l\| \scrT \#\| .
The last step is to estimate the norm \| \scrT \#\| . Let \phi , \psi be (0, q)-forms whose coefficients are
C\infty
0 (( - 1, 1) \times \BbbB 0), and let \phi s(v) = \phi (s, v), \psi t(u) = \psi (t, u). Then, in the sense as above, we
get \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\int \int
\BbbR \times \BbbB 0
\scrT \#[\phi ](t, u)\psi (t, u) du dt
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| =
=
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\sum
| J | =q
\prime
\int \int
\BbbR \times \BbbB 0
\left( \sum
| I| =q
\prime
\int \int
\BbbR \times \BbbB 0
(W\#)IJ((t, u), (s, v))\phi I(s, v)dv ds
\right) \psi J(t, u) du dt
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| =
=
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\sum
| J | =q
\prime \sum
| I| =q
\prime
\int \int
\BbbR \times \BbbB 0
\int \int
\BbbR \times \BbbB 0
(W\#)IJ((t, u), (s, v))\phi I(s, v)\psi J(t, u) ds dt du dv
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| =
=
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\sum
| J | =q
\prime \sum
| I| =q
\prime
\int \int \int \int
HIJ(r2(s - t),\Phi x0,x(u),\Phi x0,r(v))\phi I(s, v)\psi J(r, u) ds dt du dv
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| =
=
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\sum
| J | =q
\prime \sum
| I| =q
\prime
\int \int \int \int
HIJ(r2s,\Phi x0,x(u),\Phi x0,r(v))\phi I(s+ t, v)\psi J(t, u) ds dt du dv
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq C
\int \int
\BbbR 2
\| \BbbW s[\phi s+t]\| L2
(0,q)
(\BbbB 0)\| \psi t\| L2
(0,q)
(\BbbB 0) dsdt.
Now, by Lemma 4.3, \| \BbbW s[\phi s+t]\| L2
(0,q)
\leq C| B(x0, r)| - 1\| \phi s+t\| L2
(0,q)
(\BbbB 0). Then
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 8
1100 LY KIM HA\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\int \int
\BbbR \times \BbbB 0
\scrT \#[\phi ](t, u)\psi (t, u) du dt
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq C| B(x0, r)| - 1\| \phi \| \BbbR \times \BbbB 0\| \psi \| \BbbR \times \BbbB 0 .
Hence, we obtain\bigm| \bigm| \bigm| \partial jsX\alpha
xX
\beta
yH(s, x, y)
\bigm| \bigm| \bigm| \leq C(\rho \BbbR \times M (s, x), (0, y)) - 2j - k - l| B(x, \rho \BbbR \times M (s, x), (0, y))| - 1. (4.4)
This implies the statement of Theorem 4.3 when N = 0. To deal with the case N > 0, we must use
the facts that s \mapsto \rightarrow HIJ(s, x, y) is a smooth function when x \not = y, and vanishes to infinite order as
s\rightarrow 0 by Proposition 4.1. Hence, applying Taylor’s formula and integrating by parts, we get
| HIJ(s, x, y)| \leq 1
(N - 1)!
s\int
0
| \partial Nt H(t, x, y)| (s - t)N - 1dt \leq
\leq C0
1
(N - 1)!
\rho (x, y) - 2N | B(x, \rho (x, y)| - 1
s\int
0
(s - t)N - 1dt \leq
\leq C0
1
N !
\biggl(
s
\rho (x, y)2
\biggr) N
| B(x, \rho (x, y)| - 1,
when s \leq \rho (x, y), and replace \rho (x, y) by s1/2 to obtain the expected estimate. This argument also
provides the same results when s \geq \rho (x, y). Finally, applying (4.4), estimates for other derivatives
of HIJ(s, x, y) are handled in the same way.
Theorem 4.3 is proved.
Next, the action of the heat operator on bump functions is provided.
Theorem 4.4. Fix s > 0, for each multiindex \alpha , there is an integer N\alpha and a constant C\alpha so
that if \phi \in \Lambda 0,q(C\infty
0 (B(x, r))), then
| X\alpha
x e
- s2b [\phi ](x)| \leq C\alpha r
- | \alpha | \mathrm{s}\mathrm{u}\mathrm{p}
y\in M
\sum
| \beta | \leq N\alpha
r| \beta | | X\beta \phi (y)| . (4.5)
Proof. By Sobolev type Theorem 3.3 and the argument before, for \phi \in (ker\Box b)
\bot and 0 < r <
< R0, we have
r| \alpha | | X\alpha e - s2b [\phi ](x)| \leq
\leq C| BM (x, r)| - 1/2
\sum
0\leq | \beta | \leq Lm,| \beta | even
r| \beta | +| \alpha | \| X\alpha +\beta e - s2b [\phi ]\| L2
(0,q)
\leq
\leq C| BM (x, r)| - 1/2
Lm\sum
l=0, l even
rl+| \alpha |
| \alpha | +l
2\sum
j=0
\| e - s2b [(\Box b)
j\phi \| L2
(0,q)
\leq
\leq C| BM (x, r)| - 1/2
Lm\sum
l=0, l even
rl+| \alpha |
l+| \alpha | \sum
| \beta | =0
\| X\beta \phi \| L2
(0,q)
,
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 8
THE 2b -HEAT EQUATION ON FINITE TYPE CR MANIFOLDS WITH COMPARABLE LEVI FORM 1101
which yields the desired estimate in this case.
If r \geq R0, applying Theorem 3.1 for r = R0, we obtain
| X\alpha e - s2b [\phi ](x)| \leq Cr - | \alpha | | BM (x,R0)| - 1/2
Lm\sum
l=0, l even
R
l+| \alpha |
0
l+| \alpha | \sum
| \beta | =0
\| X\beta \phi \| L2
(0,q)
(BM (x,2r)) \leq
\leq C \prime r - | \alpha |
Lm+| \alpha | \sum
| \beta | =0
r| \beta | \mathrm{s}\mathrm{u}\mathrm{p}
B(x,2r)
| X\beta \varphi | .
For the last inequality, note that r \leq cR0 since M is compact. This yields (4.5) for \phi \in (ker\Box b)
\bot .
The general case follows from Theorem 4.1 and the fact that \scrH q is NIS of order zero.
Theorem 4.4 is proved.
References
1. Beals R., Greiner P. C., Stanton N. K. The heat equation on a CR manifold // J. Different. Geom. – 1984. – 20. –
P. 343 – 387.
2. Christ M. Regularity properties of the \=\partial b equation on weakly pseudoconvex CR manifolds of dimension 3 // J. Amer.
Math. Soc. – 1988. – 1. – P. 587 – 646.
3. Chen S. C., Shaw M. C. Partial differential equations in several complex variables // Stud. Adv. Math. – 2001. – 21.
4. Fefferman C. On Kohn’s microlocalization of \=\partial problem // Modern Methods in Complex Analysis: Princeton Conf.
in Honor of Gunning and Kohn / Eds T. Bloom, D. W. Catlin, J. P. D’Angelo, Y. T. Siu. – Princeton Univ. Press,
1995. – P. 119 – 134.
5. Hörmander L. Hypoelliptic second order differential equations // Acta. Math. – 1967. – 119. – P. 147 – 171.
6. Koenig K. D. On maximal Sobolev and Hölder estimates for tangential Cauchy – Riemann operator and boundary
Laplacian // Amer. J. Math. – 2002. – 124. – P. 129 – 197.
7. Nagel A. Analysis and geometry on Carnot – Caratheodory spaces // http://www.math.wisCedu/ nagel/2005Book.pdf.
8. Nagel A., Stein E. M. The 2b -heat equation on pseudoconvex manifolds of finite type in \BbbC 2 // Math. Z. – 2001. –
238. – S. 37 – 88.
9. Nagel A., Stein E. M., Wainger S. Balls and metrics defined by vector fields I: Basic properties // Acta Math. – 1985. –
155. – P. 103 – 147.
10. Zampieri G. Complex analysis and CR geometry // Univ. Lect. Ser. – Providence: Amer. Math. Soc., 2008. – 43. –
200 p.
Received 09.08.16,
after revision — 30.07.17
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 8
|
| id | umjimathkievua-article-1500 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:07:01Z |
| publishDate | 2019 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/7d/503c53f1595c07703ec25094c7e1c67d.pdf |
| spelling | umjimathkievua-article-15002019-12-05T08:57:29Z The $\Box_b$-heat equation on finite type CR manifolds with comparable Levi form Рiвняння $\Box_b$ -теплопровiдностi на CR многовидах скiнченного типу зi спiввимiрними формами Левi Ly, Kim Ha Лі, Кім Ха UDC 517.9 The main purpose of this paper is to study the initial-value problems for the heat equations associated with the operator $\Box_b$ on compact CR manifolds of finite type. The critical component of our analysis is the condition called $D^{\epsilon}(q)$ and introduced by K. D. Koenig [Amer. J. Math. -- 2002. -- {\bf 124}. -- P. 129--197]. Actually, it states that the $\min\{q, n-1-q\}$th smallest eigenvalue of the Levi form is comparable with the largest eigenvalue of the Levi form. УДК 517.9 Основною метою цiєї роботи є вивчення початкових задач для рiвнянь теплопровiдностi, асоцiйованих з оператором $\Box_b$, на компактних CR многовидах скiнченного типу. Критичним компонентом нашого аналiзу є так звана умова $D^{\epsilon}(q)$, що була запропонована К. Д. Кьонiгом [Amer. J. Math. -- 2002. -- {\bf 124}. -- P.~129--197]. Фактично вона встановлює, що $\min\{q, n-1-q\}$ -найменше власне значення форми Левi є спiввимiрним iз найбiльшим власним значенням форми Левi. Institute of Mathematics, NAS of Ukraine 2019-08-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1500 Ukrains’kyi Matematychnyi Zhurnal; Vol. 71 No. 8 (2019); 1082-1101 Український математичний журнал; Том 71 № 8 (2019); 1082-1101 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1500/484 Copyright (c) 2019 Ly Kim Ha |
| spellingShingle | Ly, Kim Ha Лі, Кім Ха The $\Box_b$-heat equation on finite type CR manifolds with comparable Levi form |
| title | The $\Box_b$-heat equation on finite type CR manifolds with comparable Levi form |
| title_alt | Рiвняння $\Box_b$ -теплопровiдностi на CR многовидах
скiнченного типу зi спiввимiрними формами Левi |
| title_full | The $\Box_b$-heat equation on finite type CR manifolds with comparable Levi form |
| title_fullStr | The $\Box_b$-heat equation on finite type CR manifolds with comparable Levi form |
| title_full_unstemmed | The $\Box_b$-heat equation on finite type CR manifolds with comparable Levi form |
| title_short | The $\Box_b$-heat equation on finite type CR manifolds with comparable Levi form |
| title_sort | $\box_b$-heat equation on finite type cr manifolds with comparable levi form |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1500 |
| work_keys_str_mv | AT lykimha theboxbheatequationonfinitetypecrmanifoldswithcomparableleviform AT líkímha theboxbheatequationonfinitetypecrmanifoldswithcomparableleviform AT lykimha rivnânnâboxbteploprovidnostinacrmnogovidahskinčennogotipuzispivvimirnimiformamilevi AT líkímha rivnânnâboxbteploprovidnostinacrmnogovidahskinčennogotipuzispivvimirnimiformamilevi AT lykimha boxbheatequationonfinitetypecrmanifoldswithcomparableleviform AT líkímha boxbheatequationonfinitetypecrmanifoldswithcomparableleviform |