Math. Ann. 290, 3-18 (1991)
Am
9 Spdnger-Vedag1991
-problem on weakly q-convex domains Lop-Hing Ho Department of Mathematics, The Wichita State University, Wichita, KS 67208, USA Received April 23, 1990
Introduction Pseudoconvex domains has been widely accepted as the standard domain which we can analyze the •problem. The basic reason is that pseudoconvex domains are exactly the domains that the G-problem is solvable for all (p, q) forms with q > 1. However, with a more careful analysis of HSrmander's [8] theorem, we see that if we slacken our restriction on the domains we can still solve the ~--problem for (p, q) forms. There were also results [4, 5, 14, and others] on ~--problem that had been proved in the strictly q-convex domains. These are the domains where the Leviform has at least q positive eigenvalues at every point on the boundary. We want to study the case that the q eigenvalues are allowed to be zero. In allowing the eigenvalues to be zero we need more assumptions on the Levi-form. We define a class of domains that any q sums of the eigenvalue of the Levi-form is nonnegative. We will show that in this class of domains for any r > q we can solve the G-problem for (p, r) forms. We then also show that we can analyze problems related to ~-such as subellipticity of the J-Neumann problem and global regularity of the ~--problem. It appears that the known related results are equally good in the domains we consider here. This paper is arranged as follows. In Sect. 0 we give some notations and terminologies. In Sect. 1 we define the notion of q-subharmonicity. We prove several equivalences of q-subharmonicity which are analogs of the results in R 2. In Sect. 2 we define weakly q-convex domains. Again weakly q-convex domains are related to q-subharmonic functions in the same way as pseudoconvex domains are related to plurisubharmonic functions. In Sect. 3 we adapt Hrrmander's 19] proof to show that we can solve the G-problem on the weakly q-convex domains and then we give an application. In Sect. 4 we show that we can study the problem of existence of subelliptic estimates on the weakly q-convex domains. Finally, in Sect. 5 we draw an analog of Shaw's theorem [15] on global regularity of the ~--problem in the annulus between a weakly q-convex and a weakly ( n - q - 1 ) convex domain.
4
L.-H. Ho
0 Terminology and notation
Let 12be a smooth domain in tE" and 0 a C ~~defining function of f2 so that a < 0 in t2 and [00] = 1 on the boundary. In this paper LI, L2 ..... L, always means a C ~~special boundary coordinate chart in a small neighborhood of some point Xo ~ bO, i.e. L t e T 1'~ on Uc~f] with L i tangential for 1 < i < n - 1 and (L~,Lj)=6ij. The dual basis of (1, 0) forms are (ol ..... co. with co, = dr. Then cij = (Li ^/,j, 0~-e),
i, j = 1, 2, ..., n - 1,
is the Levi-form associated to Q. Also if 2 is a smooth function on t2 then we write 2r = (L~ A T~j, ~ 0 - ~ ) ,
i, j = 1, 2 ..... n.
For simplicity we will assume that all the forms are (0, q) forms and we just say q forms. It is clear that the results can be carried over to (p, q) forms for any 0 < p < n. Let u be a q form, we denote
llult -- sJ
j=l
ll ,u, IJ2 + rlull 2
Here ' means that the summation is over increasing indices. We use L,2(f2) to denote the r forms with coefficients in L2(t2), L,e(t2,loc) to denote the r forms with coefficients that are locally integrable in f2, L,2(g2,q~) to denote the r forms with coefficients in L2(O) with respect to the weight function e -~. We denote ( f , g)t,)= j f~e-~'dV t~
and Ilfll~)= ( f , f ) ( , ) . (ttfl[21 denotes the Sobolev norm of order - 1 when the subscript is without parenthesis.) In Sect. 5 q~will always be of the form ~b= t2 where t is a real number and 2 is a smooth function. In that case ( f , g)(t) denotes ( f , g)~ta)~-* is the Hilbert space adjoint of ~-of the spaces in question. Again if the norms of the spaces are defined with weight functions e -t~ we will write ~* to emphasize the dependence on t. We denote Q(f,f) = (ll0-f II2 + [10-*fI12+ [Ifll z) and t
2
*
2
Q (f,f)=(ll~fll,)+ I1~ fll(,)+ I1f I1~,)). z~C~,)(t2) denotes the space of all r forms on 12 that are smooth up to the boundary, ~(,)(U) denotes the elements in d(r)(f~) that are compactly supported in U n O and in Dom(~*), D~,)(f~)denotes smooth compactly supported r forms in t2, and C~(t2) denotes the r forms that are smooth in s9 (not necessarily up to the boundary). We use IIIftll~ to denote the tangential Sobolev norm of order e, i.e. Illfll[~= ~ ,~!_ ~[A~f(T, O)12dzdo, where
A~f""("c,e) = (1 +
IzlZ)'f(,, 6).
~-problem on weakly q-convex domains
5
Here z = (zt, ..., ~ , - t ) is the tangential direction. If f = ij~=,fjdz-J is a r form then Illflll2= E' III/AI2d
1 q-subharmonic functions
Let ~bbe a real C 2 function defined on a domain I2___IEn. For each q > 1 we define a n! square matrix O~a~(x)of order associated to ~b. In fact, the entries of the
q!(n-q)!
matrix #~)(x) are #,j(x) where I and J are increasing q tuples of integers between 1 and n. Now define t i~10ziOii d2t~ 9 ,j(x)=
* J ~2~b 0
if
l=J
if
I=(iK>,
J=<jK>,
and
i4:j,
otherwise,
where
denotes the increasing indices by reordering the set {i}wK, and ~ir i is the sign of the permutation taking iK to I, which equals to 0 if (iK> 4=I. Definition 1.1. We say that t# is q-subharmonic in a set U _ IEn if the associated matrix #~)(x) is positive semidefinite at every point x ~ U and that tp is strictly q-subharmonic if #t~ is positive definite at every point x ~ U. Lemma 1.2. The condition that dp is q-subharmonic is invariant under a unitary
change of coordinates. Proof. We may apply a calculation of Kohn [11, Proposition 4.46] to prove this. Since we need a more precise version of this kind of result we will prove this out in full details for the convenience of the reader. This computation will be used later. Assume that #r in the z coordinates. We will omit x0 from the calculations from now on. By our definition of ~to, it is not hard to see that it is equivalent to ~',' ~ ~bi~f/KfTK>O for all q form K
f= ~'fjd#
i,j
(1.1)
J
where ~bO - dz~as and fix = diKf, where I is a q tuple with increasing indices. Let (wl, w=,..., w.) be a unitary change of coordinates of (zl, z2 .... , z.). Then there is a unitary matrix (so) such that
zi= ~ s~w~,
i= 1, 2..... n.
h=l
Hence
n
wi= ~,
h=l
ShiZh,
i = 1, 2 .....
n.
We want to show that ~q~(Xo) > 0 in the w coordinates, i.e.
ZtK i, jE~
~2,k
___,,
UiKUJKg O for all q form
u = ~'~ ujd#.
6
L.-H. Ho Note that for any ui we can write
uj,...: =
Z
h2 . . . . . h, r = 1
where
A,...~,~-..~,
(1.2)
n
fit l...hq ~ . ~ Uit...iqShtil "'" Shqiq " tl,...,fq=l
We use the notation that us = 0 if J has two indices that arc equal. Thus
= E t E E K i,j k,l
=
Ski~UiK~JK k
Z'
Z e~,
K=(Itl ..... kq-l) k,|
Z
\ f , hl ..... hq=l
A~,,.,,:~,s,,:,2~, ".. s~
\j, mz,...,mq = 1
-
'
,:., ....
(q-l)!
.
.=2 ..... m,
1 -
x,k,(
(q-a)! k,l
i,t \ g
<..) "'"
x
~,h,..... h,
/
~0 by (1A). Note that we have used in line 4 and line 7 the identity kl <...
t \ h 2 . . . . . h, l
1 and 6 u in line 5 is the Kronecker delta. Lemraa 1.3. @(~ is positive semidefinite if and only if for any unitary change of coordinates {wt ..... w~} we have at x o 0~$
02r
_o.
(1.3)
Ow~O~,---~+ " " + ~wd#---~ -
Proof. Assume # (q)(Xo)is positive semidefinite. Then by Lemma 1.2 ~(~)(Xo)is also positive semidefinite in the coordinates {wt ..... w.}. Thus (1.3) holds since the lefthand side is a diagonal element of the matrix.
~-problem on weakly q-convex domains
7
Suppose ~tq)(Xo) is not positive semidefinite. By a unitary change of coordinates we may assume that the matrix is diagonal. Thus one of the diagonal element is negative. Hence (1.3) is false for some coordinates {wl .... , w,}. Theorem 1.4. Let u be a real C 2 function in a domain f2 in C". Then the following statements are equivalent to q-subharmonicity of u in f2. O2u 02u (i) ~ w l O ~ + "'" -~ dwq~ff~ >0 at any point Xo~f2 for any unitary change of coordinates {wl ..... w,). (ii) Let {wl ..... wn} be a unitary change of the original coordinates, then for any polydisc D(wl, r) x ... x/)(w~, r) in f2, we have 1
2~ !
2~ ! u(wl + rei~ .... w~ + re ~~ w~+ 1, ., w,)d01 .
.
.
.
dOq.
(iii) Let D be a q-dimensional polydisc in f2, with orthonormal edges and same radii in each side of the polydisc. I f f is an analytic polynomial in D such that u < R e f on bD, then u < R e f in D. (iv) Let K ~_I2 be a compact subset on a q-dimensional complex plane in t2. Then for every continuous function h on K which is harmonic in the interior of K and such that h > u on the boundary of K we have h ~ u on K. Proof. F r o m Lemma 1.3 we know that q-subharmonicity is equivalent to (i). (i) =:- (iv) First, consider the case u , ~ +... + Uw.,~>O for any unitary change of coordinates. If (iv) is false, then there exists a q-dimensional complex plane P in f2, a compact set K on P and a function h which is continuous on K and harmonic in the interior of K so that h > u on bK but u > h at some point inside K. Assume that the plane P lies on % + 1 . . . . = w, =0. Let xo e intK be a maximum for the function u - h . Then Uw~,l + ... +uwn,n < 0 at Xo. Contradiction. Now consider the general case that uwl,~+...+uw,v~. > 0 . Take 1 u.---u + nE IzglLThus u, satisfies (iv). So u, which is the decreasing limit of un, also satisfies (iv). (iv) => (iii) This is obvious since the real part of an analytic polynomial is harmonic. (iii) =~ (ii) Suppose ~b(0~.... ,0q) is a trigonometric polanomial so that u(w 1 + tel~
Wq + re ~~ Wq+ t ..... W,) _--<~(01, ..., 0q).
Assume that ~b(01.... , Og)= Y~a~1...k.r~.k~O~+... +t~o~) Then
f(~a, ..., ~) = ao + 2
llC~ o
ak~ ... kq
\ K =(kl ..... kq,O ..... O)
has real part which is an upper bound for u(w t +rei~ ..., w~ + re g~ wq + 1,..., w,). Thus u g R e f in D. Hence 1
w) _ao
=
2~
2~
s ... s
tz~y o
o
r
1
9.. dO~
8
L.-H. Ho
For every e > 0 we can find a trigonometric polynomial ~b~ with u < ~b~< u + e. Taking e ~ 0 , we got U(W)~_ ~
1
2,~ 2z ! .,. ! U(W1 +re ~~. . . . ,w~+rei~
, .... ,w,)dO,
...dOq.
(ii) =~ (i) Let wj = x i + iy~, 1 < j < n. We will prove that
I u ( v ~ , + ... +v~,~)dXl...
dye_>-0
for every v E C~(12) and v__>0. Then (i) follows applying integration by parts. Let r < d(supp v, ~c). Since 2~ 2n (2rc)~u(w)< y ... ~ u(wt +re ~~.... , w~ + re `~ w~+ 1..... w~)d01 ... dOq, 0
0
multiplying by v and integrate with respect to dxt ... dyq gives lu(w)
... [. v ( w l - r e i~176247
.... ,w~)d01 ... dOq-(2~)qv(w)
0
•
...dy~>O.
Taylor expand the above expression in v, divide by (2~)qr2 and let r ~ 0 we get Iu(~,,~,~ + ... + v , , , , ) d x ~
... dye__>0.
The proof is completed. Remark. In our language here a C ~ function is 1-subharmonic exactly when it is plurisubharmonic.
2 Weakly q-convex domains
Definition 2.1. Let 12 be a smooth domain in ~n, and Q a defining function of 12, then we say that 12 is weakly q-convex if at every point x0 ~ b12 we have
~, K
~ l,J
~20 i
u~r~ =
>
0
for every q form
u = ~,' u:d~:
j
such that
J
~ ~-:-u,r=O 1=1 ozt
for all
IKl=q-1.
(2.1)
[,emma 2.2. Let Q be a smooth domain in ff~". Then f2 is weakly q-convex if and only if for any xo ~ b12, any q eigenvalues ~,, .... ,2io of the Levi-form at Xo with distinct subscripts satisfies q
Y. ,~ij > O .
j=l
Proof. We will first show that the condition (2.1) is invariant under unitary change of coordinates. Assume 02Q __>
E'K~.jE~
for every q form u such that
u~Kuj~= 0
0q uiK = 0 ~ ~z~ i=l
for all K .
~--problem on weakly q-convex domains
9
Let {wl,..., w,} be a new set of coordinates under unitary transformation. We want to show that
Z'Z
00 E ~_--z-fix= 0 i = 1 ow~
for every q form f such that
for all K.
Assume that f satisfies ~
~af~ r = 0 for all K. Define the q form u as in (1.2). It aw~ L dQ is not difficult to check that u satisfies E ~ z U~K= 0 for all K. Now from the i=1
i=1
calculation in Lemma 1.2 we see that i
Since ~ ~ uir = 0 for all K, the above expression is non-negative. i = 1 uz~ By a unitary change of coordinates we may assume that 0zl .....
(00
=(0 ..... 0, l) and that
~ \~
i= 1..... n- 1
is diagonal. Thus (2.1) is equivalent
to any distinct q sums of az~d~ ~2# .>0,. . i=1, . .
n--1. This exactly means that
q
~, ,;t,~> 0 for any distinct set {i,.... , iq} of {1..... n - 1}. j=l
Corollary 2.3. If a smooth domain t2 in ~" is weakly q-convex, then 0 is weakly q + 1 convex.
Proof. This follows from the simple fact that
Note. A smooth pseudoconvex is exactly what we call a weakly 1-convex domain here. Thus by Corollary 2.3 a smooth pseudoconvex domain is weakly q-convex for q = 1 , 2 , . . . , n - 1 . Theorem 2.4. Let t2~_ffY be a bounded domain with Coo boundary and 0 be a C ~ defining function of t2. Then t2 is weakly q-convex if and only if there exists a constant C > 0 such that the function - l o g ( - 0) + Clzl 2 is q-subharmonic in I2. Proof. Assume that the function - l o g ( - 0) + Clzl 2 is q-subharmonic in t2. Then by (1.1) it implies that K
0 dzjd~ + 02 Ozj d~k + Cfjk Ujr.~kkX~ O.
for all q form u. Hence
E' r ~k ~ O ~ k
- Cot~jk) UjK~kK> 0
whenever
~ ~ 0z--~. 00 uix = 0 ~=
for all K.
l0
L.-H. H o
Thus by taking limit to the boundary we have
y: ~ ~-:-~:.,~.k,_-0 ozjVz kd2Q - - >
,honever
K j,k
_~" ~QZju,,--0
j=~
rot all K
for any xo on the boundary. Now assume that 0 is weakly q-convex, i.e. (2.1) is satisfied. Let u be a fixed q-form. We denote lul = (~' lu,12~ ~j2. We can write u = v(z)+ w(z)where v and w are /
q forms with coefficients differenfiable in z, o satisfies the condition that
Iw(z)l_-< const ~' I 2"~176 ujK In K Ij=! ~Zj
vjK(z) = 0 for all K at any point z ~ O and j=t
"
fact, let
/ v~,...h,(z)= ,=,,u~,l f
\
, = ' ,"~,1
I
It is not difficult to check that v satisfies the condition j=~ 1 ~t3Q vjK(z)= 0 and that iw(z)l_
_ ~'
c~2e
+2R~ ' ~ ~ o j ~ ) ~ ( : )
+ Y.',~J.~E~
)
wj,~z)~,~).
(2.2)
By assumption y:
d20
K ~ ~
OZj47Zk
> VjX(Z)V,x(Z) = 0
when z ~ bO.
Thus, c920 E ' ~ ~--:'-~--Vj~Z)Vk~z)-~ -CllQ(z)l lul2
E J,k vzjvzk
when z is in a small neighborhood of bO.
I~176
The rest of the terms in (2.2) is bounded below by - C 2 ~' _
K j=l ~Zj
Thus, O0 I I"l. ~=,N.,K
y, z ~_-~-__ 020 " , K - -. >~ = - < l O ( z ) l l u l ~ - C ~ ' I "
x j,k v z f i ' z t
UjK ]U].
iT-problem on weakly q-convex domains
11
Hence d 2 ( - l o g ( - ~))
=-
1Z,~2. ~ Ozo Ujr~Ukr __,t
OK
j, k 0 Z ,
k
1 ~,1~_ ~O Ujr [2
j=I~Zj
Q-2K
~Q
+ ~ Z' IJ~ ~u,~[ >-C lul2 = 3 for z in a neighborhood of the boundary. Thus it is evident that when C is large enough - log ( - 0) + C lzJ2 is q-subharmonic in f2. Remark. In fact, from the proof of Theorem 2.3 we see that if ~ is weakly q-convex
we can even find a strictly q-subharmonic exhaustion function in ft. Also, we can assume that fi is only of C 3 boundary. Lemmn 2.5 (Basic estimate). Let f2 be a weakly q-convex domain in r n and Xo ~ bf2. Then there exists a neighborhood U of Xo and a constant C > 0 such that
Ilull,~_
i,j
/
<' = C Q(u, u)
(2.3)
for all u ~ ~(a~(U). Proof. The second inequality follows from a standard expansion of Q(u, u). We
obtain the first inequality because the boundary integral is non-negative by the assumption that [2 is weakly q-convex. Definition 2.6. Let f2 be a smooth domain in ~n, we say that [2 is weakly q-concave if 0 c is weakly q-convex. It follows from Lemma 2.2 that O is weakly q-concave if and only if for any q eigenvalues 2~1, ..., ~% of the Levi-form at x0 e bf2 with distinct subscripts we have q
Z 2o~0" j=l Lemma 2.7. Let f2 be a weakly ( n - q - 1)-concave domain in IEn and xo e bf2. Then there exists a neighborhood U of Xo and a constant C > 0 such that
Y' IIL,ujII2+ Z' Z IIL,uall2+ J E B,ju,f~adS
for all u ~ ( r
(n-l)!
J
i
(2.4)
bfl l , J
Here (BH) is a positive semi-definite matrix of order
q!(n-q-1)!" Proof. By a standard expansion of Q(u, u) we get Q(u,u)=Z' ~ IIL,uj[I 2+ ~ Z' J
i=l
bfJ K
+ small error term.
Z couixftpcdS i,j
(2.5)
12
L.-H. Ho
We apply the following integration by parts formula to the terms [IL~u:l[2 for l < i < n - 1 in (2.5). IlL+u[12= IiZiul[z - ~ culul2dS + small error term. bO
Then we get Z'ilE.uJl[ 2+ Z' Z ][L,uj[[2+ I (Z' Z c,ju,ru~x- Z' Zc,[ua[2~ dS $
J
i
b~\K
i,j
J
i
j
<-5CO(u, u). Define
+ ,:.,. B/+j
+ilKF.jlKCij
0
I=(iK), otherwise.
if
J=(jK)
and
i+j,
We get (2.4) immediately if we can show that (B~:) is positive semidefinite. Assume that (c~j)is diagonal, then (B:s) is also diagonal and the diagonal elements are the negative value of (n - q - 1) sums of the eigenvalues of the Levi-form, hence (BH) is positive semidefinite. In general, we may diagonalize (cu) under a unitary transformation, and using similar calculations as in the proof of Lemma 2.2 we see that the positive semidefiniteness of B is invariant under such transformations. Hence the proof is completed. 3 ~-problem in weakly q-convex domains
In this section we want to prove the analog of the result in pseudoconvex domains for weakly q-convex domains. We need to use the results in Sect. 2 to carry out the procedure of H6rmander [9]. In fact, the key observation is that to get a good bound for [13-ffl2+ll3-*fll~ we do not need the full condition that the domain is pseudoconvex, but that a weakly q-convexity is enough. The weight function in H6rmander's proof will be replaced by a q-subharmonic function. Theorem 3.1. Let 1"2be a bounded weakly q-convex domain in C n. Then for all r > q the equation c~u= f has a solution u E L2,_t(I2, loc), with f e L2(I2, loc) and 3-f = O.
Proof. We follow H6rmander 1-9]. Since most of the techniques in the proof there can be carded over to our ease, we will just write down the main ingredients of Hfrmander's proof and point out what is new here. The key points are as follows: 1. If we can show that there are C ~ functions ~b1, ~b2, and ~b3 such that l[f l[~§ < C(]I~-*fII~+,)+ [I~'f [l~+~))
(3.1)
for all f E Dom(~nDom(3-*), then for every f e L~(O, tp2) such that 3-f= 0 we can find ueL2,_l(~2,dpO so that 3-u=f. Here 3"* refers to the adjoint of the 3- from L,2-1(O, ~t) to L,2(O,tp2) (Lemma 4.1.t of [9]). 2. If rt, is a sequence of functions in C~(~2) exhausting f2 and " dP/,~2
e-"-.
+e-+,
~--problem on weakly q-convex domains
13
for j = 1, 2 and v = 1, 2, 3,.... Then Dco(f2) is dense in Dom(~r~Dom(~Y*) for the graph norm f ~ l t f [ I , 2 + [[~-f[1~ + I]J*fl[r 9 Hence we need to prove (3.1) only for D~r)(f2). 3. Chooseo2eC
o~
n [[63r/v2
'p
suchthat ~ll~--~j < e a n d c h o o s e C x = r 1 6 2 1 6 2
and ~b3= r Then the condition (3.2) will be satisfied for any r ~ Coo(t2). What is left to be shown from above is that we can find some function r e C~176 so that we have (i) f ~ L2(f2, r (ii) The estimate (3.1) is satisfied for all feDt,~g2). Let feDtr~(I2). Then expanding II~fll~,~) and I[0*fllff,~) we have "
S~r'
--
6~2r
r
E'
E fjKfkK ~--wz-~_e- dA +
j, k = 1
tTZjUZk
~ OfJ2
~
J
j=l
t~gj e - ' d 2
=< II~-f II~,~)+ 2 II~-*f II~,~ +251flZldo21%-#d2 9 Thus we can get (3.1) if we can find r so that E'
K j , k = l ~ 9~z j
WjKW--ff_-->2(10o212+ e ~) Y~'Iwtl 2 I
k
and also that f e L2(f2, r Since ~ is a weakly q-convex domain by Theorem 2.4 we can find a strictly q-subharmonic function p on t2. Thus by taking r where Z is a suitable C Oo convex increasing function we can make ~'
~
K j,k=l
~2r
>
2
~ ~_ WjK~-~-kK_--2(18o21+e~~ O2jGZ k
2 I
and also that f e L2(I2, r o2). This completes our proof. The following corollaries exactly follows HSrmander. Corollary 4.2. Let f2 be a bounded weakly q-convex in ~E". Then for all r >__q, the equation ~u = f has a solution u e C~_ 1(0), with f e C~(t2) and J f = O. Corollary 4.3. Let f2 be a bounded weakly q-convex domain in rE" and r any q-subharmonic function in f2. I f r > q, then for every g e L2(f~, r with gg = 0 there is a solution u ~ 1_,2,_1(f2,loc) of the equation gu= g such that lul2e-*(1 + Izl2)- 2d)~=< ~ Igl2e-~d2.
4 Subelliptic estimates
Let t2 be a domain in Ir". We say that a subelliptic estimate of order e holds at Xo e I'~ for q forms for the ~--Neumann problem if there is a neighborhood U of Xo, C > 0, 0 < e < 1 such that Illu1112-< CQ(u, u) for all q forms u e ~q~(U). Most of the work on subelliptic estimates has been done on pseudoconvex domains [1-3, 11-]. There are some results on domains that are
14
L.-H. H e
not pseudoeonvex [6, 7], but the conditions imposed are not as neat as those described in the case of pseudoconvex domains. An obstacle in non-pseudoconvex domains is that it is difficult to describe the domain that can possibly give subelliptic estimates. We now follow Kohn's [11, 12] program to define a sequence of ideals of subelliptie multipliers on weakly q-convex domains. For q > 1, we define a square
(n-l)!
matrix (Cxs) of order q ! (n - q - 1) ! associated to the Levi matrix (c~j).We consider I and J as increasing q tuples of integers between I and n - 1 . Now define
~ cii
if
l =J ,
C Ax)= eixtjxc x x 0 if l = ( i K ) , 0 otherwise,
J=(jK),
and
i~j,
Similarly, for every C ~ function f in a neighborhood U of xo, we define a
(n-l)!
q! ( n - q - 1 ) ! square matrix (Au(f)) as follows:
A,Af)=
{
Y~ IL~(f)l~
if
t=J,
e::JxL,(f)Lj(f)
if
I= (iK),
0
otherwise.
J= (jK),
and
i4:j,
Consider the functions defined in a neighborhood of x 0 as germs at x0. Let a~v0-= {g e C~(xo): Igl'_~ Ifl in a neighborhood of Xo for some f in the ideal I}. Define inductively an increasing sequence of germs of functions at x0 by
I1(Xo)=
,
where the bracket ( ) denotes the ideal generated by the germs inside the bracket and Mm(xo) = {detM: MIj = Al:(fl) +... + AI~(fN) + kCIj where f~ e I,,(Xo), N is a positive integer and k is any number}. A smooth function f defined in a neighborhood of Xo is said to be a subelliptic multiplier if there exists 0 < e < 1 and C > 0 such that Illfu 1112< CQ(u, u) for all u e ~ ( U ) . Theorem 4.1. Let 12 be a smooth weakly q-convex domain in C~and x o ~ hi2. Then if 1 ~ Im(xo) for some m, then there is a subelliptic estimate for (p, q) forms at Xo. Moreover, the ideals Im are independent of the choice of Li, L2, ..., L,.
Proof. We follow K o h n [11 ] and He [6], except that we use weakly q-convexity to establish the basic estimate (2,3). We need to prove that IIIru Iit2 ___
( CHu~,Alu:) <=CQ(u, u).
(4.2)
~-.problem on weakly q-convex domains
15
If f satisfies
IIIfu ILI~5- cQ(u,u) then ~, < CQ(u, u).
(4.3)
l,d
If
Z (AIjUl, A2eu1> -
(4.4)
I,J
then [lldet(Au)u IIIff ~ CQ(u, u). The proofs of (4.1), (4.2), and (4.3) follow from Kohn [11], see also Ho [6] for the proof of (4.3). (4.4) follows from that det(Au) ~' lull2_--_C E h u u ~ j . K
l,J
[We only need to prove the case that (A~I) is diagonal in view of Proposition 1.2 and which is easy.] Hence (4.1), (4.2), and (4.4) shows that Ix(x0) is a set of subeUiptic multipliers and (4.2), (4.3), and (4.4) shows that Ira+ t(xo) is also a set of subelliptic multipliers. This concludes that if 1 ~ I~(xo) for some m, then there is a subelliptic estimate at Xo. To see that the ideals are independent of choice of coordinates, we note that the determinant of (Au(f)) and (Ca) remains unchanged under a unitary change of coordinates. The following theorem is a remark to a theorem of Post [13], who proves the same statement on pseudoconvex domains. Theorem 4.2. Let f2 be a smooth weakly q-convex domain in ff~" and x o ~ bf2. Then a subelliptic estimate holds for q forms at x o implies that a subelliptic estimate also holds for q + 1 forms at Xo. Proof. We observe that in [13] the pseudoconvexity assumption is only used for IIull~ _-
(4.5)
as in (2.6) there. Thus with our assumption of weakly q-convexity (4.5) is guaranteed by the basic estimate (2.3). The rest of the proof remains the same. Remark. Although it appears that the weakly q-convex domains we defined is a right object to study subelliptic estimates, they are not the largest class of domains that we can study such estimates. As we see from [6], we can still consider a more general class of non-pseudoconvex domains. In fact, if there is a holomorphic tangential vector field L such that ~.=
(4.6)
is non-negative and for some polynomial p we have p(L,/)2 # 0, then we have a subelliptic estimate for ( n - 1 ) forms. In our current paper we only consider the domains that satisfy c11+ ... + c , - 1 ,-l>=0
(4.7)
16
L.-H. H o
for any point on the boundary. However, the condition (4.7) can easily be verified. Moreover, the ideals of subelliptic multipliers we defined here is independent of coordinates chosen while the ideals arising from the condition (4.6) as considered in [6] is very much dependent on the choice of coordinates.
5 Global regularity for 3 on the annulus between weakly convex domains We show in this section that we can also prove global regularity theorems (up to boundary) for the annular region between some weakly convex domains. Of course, the theorem holds on a single weakly q-convex domain. Most of the ideas in here follow Kohn [113] and Shaw [15].
Theorem 5.1. Let 121 and 02 be two C ~ bounded domains in ~ with ~z c=12t and satisfying that 121 is weakly q-convex and f22 is weakly ( n - q - 1)-convex. Let 2 be a smooth function on f~ so that 2 = Izl 2 in a neighborhood of b121 and 2 = - Izl 2 in a neighborhood of b122. Let g2= I2 t - Q~. Then for n > 3 and I < q < n - 1, if ~ d(q)(f2), J~ = 0 and (~, tp) = 0 for every tp ~ ~(q)(f2) and ~*tp = O, then there exists a ue~cq_n(D) such that Ju=~. Proof. We follow Shaw 1-151. The main step is to prove the a priori estimate tJlfll~o
for fe~(,~(12).
(5.1)
If this is true, then everything in Shaw I-15] goes through without change. By a partition of unity argument, we need to prove (5.1) for f supported in a small neighborhood of the boundary since Qt is elliptic in the interior. Let f be supported in a small neighborhood of b121. Let 0 be a defining function of g21, and (c~) be the Levi matrix. Then by HGrmander's estimate we have
b~ x
.t.~
o
Z
i
l
l
a~ tllt,
e)
Thus we have (5.1) since the first term on the left-hand side of (5.2) is non-negative by assumption. Next we assume that f is supported in a small neighborhood U of b12v Now let ~?be a defining function of g2 near b122. Note that the sign of Qis negative in 12. We use the same integration by parts trick as in Shaw 115]. Integrating by parts we have J
j
/
+(1 - e)(T1 + T2)- C,Hfl[~o, where c~}= e 'a ~
(e- ,a), and
Tz= ~: X
~ c,af~+c~ - f a d s - X' ~<. Ja czzlf~]ze-t"ds
K
J
j,k
bO~ L J
where the meaning of Bz.~ is same as in the proof of Lemma 2.7.
(5.3)
~-problem on weakly q-convex domains
17
We n o w want to show that T 1 and T2 are non-negative quantities except possibly plus some small error terms. N o w 02 is weakly ( n - q - 1 ) - c o n v e x , hence Oc is weakly q-concave. Thus (BIj) is positive semi-definite and hence Tz is nonnegative. F o r T1, we note that ( 2 j k - 6sk Z )~tl~ is positive definite when n > 3. \
l
]i,j=l
..... n - - 1
If j = n or k = n we have fjK = 0 or fkK = 0 on the boundary. These terms are thus easy to handle. In fact, these terms can be a b s o r b e d in the first two terms of the right-hand side of (5.3) due to the fact that for n~J we have
iiS,illZC(iiE,(S,)ii ,)+xy.
+ C, llSll _,.
(5.4)
So combining all these we have T1 > t Y. l l f f i ~ ) + s m a l l error terms.
(5.5)
neJ
Thus combining (5.3), (5.4), and (5.5) we have
E' IIL,(TJ)IIg)+ E'E J
J
j
II~fjIl(~)+ YJ llfJIll + t EJ IlfjIl~0 n~J
ni!J
< CQt(f, f) + Ctllf II2_,.
(5.6)
Apply the inequality
IIf~ll~,)<(t-lllfjIl~ + C, IIf~ll ~-1) to those J containing n and put into (5.6) we have
tllflt~) <=CQt(f,f)+C, Ilftl2_~
for
f~)(U)
and the t h e o r e m is proved.
References 1. Catlin, D.: Necessary conditions for subellipticity of the J-Neumann problem on pseudoconvex domains. Ann. Math. (2) 117, 147-171 (1983) 2. Catlin, D.: Subelliptic estimates for the ~--Neumann problem on pseudoconvex domains. Ann. Math. (2) 126, 131-191 (1987) 3. D'Angelo, J.: Real hypersurfaces, orders of contact, and applications. Ann. Math. (2) 115, 615437 (1982) 4. Fischer, W., Lieb, I.: Lokale Kerne und beschriinkte L6sungen fiir den 5-Operator auf q-konvexen Gebieten. Math. Ann. 208, 249-265 (1974) 5. Henkin, G.M., Leiterer, J.: Andreotti-Grauert theory by integral formulas. Basel: Birkhfiuser 1988 6. Ho, L: Subellipticity of the ~--Neumann problem on nonpseudoconvex domains. Trans. Am. Math. Soc. 291, 43-73 (1985) 7. Ho, L:Subellipticityestimatefor theS-Neumannproblemforn-I forms. Tram.Am. Math. Sot., to appear 8. H6rmander, L.: L 2 estimates and existence theorems for the ~--operator. Acta Math. 113, 89-152 (1965) 9. H6rmander, L.: An introduction to complex analysis in several variables. Princeton: Van Nostrand 1966 I0. Kohn, J.J.: Global regularity for ~-on weakly pseudoeonvex manifolds. Trans. Am. Math. Soc. 81, 273-292 (1973)
18
L.-H. Ho
11. Kohn, J.J.: Subelhptieity of the ff-Neumann problem on pseudoconvex domains: sufficient conditions. Acta Math. 142, 79-122 (1979) 12. Kohn, personal communication 13. Post, S.: Finite type and subeUiptic estimates for the ~'-Neumann problem. Ph.D. thesis, Princeton University, Princeton, N.J., 1983 14. Schmalz, G.: Solution of the ~--equation with uniform estimates on strictly q-convex domains with nonsmooth boundary. Math. Z. 202, 409-430 (1989) 15. Shaw, M.: Global solvability and regularity for ~ on an annulus between two weakly pseudoconvex domains. Trans. Am. Math. Soc. 291, 255-267 (1985)
