(Cauchy-Szego integral/Heisenberg group/holomorphic functions). P. C. GREINER*, J. J. KOHNt, AND E. M. STEINt. * Department of Mathematics, University of ...
Proc. Nat. Acad. Sci. USA Vol. 72, No. 9, pp. 3287-3289, September 1975 Mathematics
Necessary and sufficient conditions for solvability of the Lewy equation (Cauchy-Szego integral/Heisenberg group/holomorphic functions)
P. C. GREINER*, J. J. KOHNt, AND E. M. STEINt * Department of Mathematics, University of Toronto, Toronto, Canada M5S lAl; and t Department of Mathematics, Princeton University, Jersey 08540
Princeton, New
Contributed by E. M. Stein, July 14,1975
We find the necessary and sufficient condiABSTRACT tions for the local solvability of Lewy's equation, (a/az + iz a/at) u = £ If R3 is realized as the boundary of the generalized "upper-halfpace" in C2, then the conditions are, near a point P C B3, the analytic continuability of the CauchySzego integral of f past P. In case the sufficient condition is satisfied, solutions are found that satisfy optimal regularity properties. Various generalizations are also given.
1. Main result The Lewy equation is given by [1] L~u)= ( T + i2 at~u = f where z = x + iy, and t are coordinates in R3, with a/az = ',% (a/ax - i a/ay). Lewy (in ref. 1) proved that this equation does not in general have a local solution. To state our conditions for solvability we first recall some facts about the Cauchy-Szego integral. We realize R3 = $(z,t)j as a submanifold of C2 = f(zI,Z2)j, according to the mapping [2] (z,t) -b' (z,t + iZI2) = (Z1,z2) This identifies R3 with the boundary b5D of the domain D given by ° =
I(Zlz2)IIm Z2
>
1z112i;
[3]
then b jD =
I(Zl,z2)IIm Z2 = IZ1 12
[3'] is holomorphically equivalent with the unit ball in C2. Each f E L2 (R3) (or, e.g., each distribution with compact support) gives rise to a holomorphic function, C(f), in D via the Cauchy-Szego integral
C(f)(Z1,Z2)= where S(z,w)
=
f
S(zaw)f(w)dow
1/72 [i(W2 -Z2)
-
2iiZI]-2, (Wi,W2)
[4] =
(zt + ilzI2); and duaw is the measure on b'D arising from Lebesgue measure on R3. We let Cb(f) denote the restriction of C(f) to bf (identified with R3). These boundary values exist in L2 norm if f C L2 (R3), (see §6 in ref. 2), and as a distribution, if f is a distribution with compact support. Note that if f arises as the restriction to b) of a holomorphic function f onD (which is suitably bounded), then C(f) = f. Observe also that if P C R3, then C(f) is analytically continuable past P if and only if Cb(f) is real-analytic near P. Moreover, this property depends only on the behavior of f in a neighborhood of P. 3287
THEOREM 1. Given If, then the Eq. [1] has a solution in a neighborhood of a point P C R3 if and only if the Cauchy-Szego integral Cb(f) is real-analytic in a neighborhood of P. Micro-local results for the solvability of [1], which seem to be related to the above, are given in ref. 8. We shall also extend the above result in two different directions. (i) We generalize the necessary condition to domains whose Cauchy-Szego kernel satisfies a certain analyticity condition. (ii) As far as the sufficiency condition is concerned, we find a modified "fundamental solution" for the Db Laplacian (on functions) on the Heisenberg group; when f satisfies the sufficient condition, then one can show the existence of a solution which has optimal regularity properties in terms of the function spaces studies in Folland and Stein (ref. 3). 2. The necessary condition Here we shall consider smooth domains D C Cn+ 1, n 2 1. In this section we shall for simplicity restrict ourselves to domains that are bounded, although the results extend to unbounded domains such as [3]. We recall that the CauchySzego kernel S of 0D is the reproducing kernel for the space H2( ), which consists of those holomorphic functions in which (in a suitable sense) have restrictions to bD lying in L2(bP). The Cauchy-Szeg6 integral is then given by
C(f)
S(zw)f (w)dow, fE L2(bD)
= f
[5]
D5
For further details see ref. 4, §7. Now suppose that Cauchy-Szegd kernel satisfies the following: Analyticity condition. If W is an open subset of bD, then there exists an open set D' C Cn+1, with VY C D -W such that: (i) the function S has a continuous extension to V1Y X W; (ii) for each wo C W, the function z - S(z,wo) is holomorphic in S. Assuming the above condition, we will give sufficient conditions for the holomorphic extendibility of C(f) across a boundary point P C bD. (An analogous result for the Bergman kernel appears in ref. 5.) Consider an operator A defined in a (complex) neighborhood V of P by a A =2aj-, with ajE CN(V) [6]
and such that the complex vector field A when restricted to V q bo is tangential to bMD. A then has a restriction to functions on V n b., which we denote by Ab. Denote by Ab* the formal adjoint of Ab with respect to the inner product on
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Mathematics: Greiner et al.
Proc. Nat. Acad. Sci. USA 72 (1975)
that L defined by [1] is given by L = Ab* L2(b2D. Observe -
with A = -a/ -2izl O/0z2 on C2. THEOREM 2. Suppose the above analyticity condition is satisfied, and assume that there exists an f C L2(bD)), an A of the form [6], and a u 6 L2(V n ba) such that Ab*U = f
case one can compute the resulting 0b-Laplacian on "qforms" (which we write Ob(q)) quite explicitly in terms of the vector fields Zj. In fact (see ref. 3), n
Z j zj. Egb ()=E. ill
[7]
Then C(f) has a holomorphic extension past P. There are two consequences of this theorem that we mention. (i) The Eq. [7] is not solvable whenever C(f) is not analytically continuable across P; in particular, if f is the boundary value of a (suitably bounded) holomorphic function in M which does not extend across P. (i) Suppose bO3 is real-analytic. Iff E L2(bI)) is real analytic in a neighborhood of P. then C(f) is holomorphically extendible across P. Outline of Proof of Theorem 2. In case f = 0 in a neighborhood of P. C(f) is holomorphically extendible across P because of the analyticity condition on the Cauchy-Szeg6 kernel which we assumed. Now choose p E C-(bV which is zero outside V, and so that p = 1 in a neighborhood of P. Then C[f - Ab*(pu)] = C(f), since Ab*(pu) is orthogonal to boundary values of holomorphic functions. Hence C(f) has a holomorphic extension across P since by [7] f - Ab*(pu) = 0 in a neighborhood of P. Consequence (i) is now immediate, and (ii) follows by virtue of the Cauchy-Kowalewski theorem.
Sufficient Conditions. For the purposes of dealing with the sufficiency we turn to the "half-space" C Cn+ 1 given by
One knows that when 0 < q < n, the operator b(q) is hypoelliptic and solvable, but this is not the case when q = 0, or q = n. We shall nevertheless determine the necessary and sufficient conditions for the local solvability of o30). THEOREM 3. Let Q be an open subset of Hn. Then O00°)(u) = f is solvable in Q if and only if Cb(f) is real-analytic in Q. COROLLARY. If the above condition for f is satisfied, and if f belongs to one of the spaces (see ref. 3 for definitions) Skp ((Moc.), ra(21oc.), L-(Uloc), or C@(Q), then we can find a u which belongs to Sk+ 2P (01oc), ra+2((Moc), r2(120c), or C-(Q), respectively. Observe that when n = 1, the above gives the solution of our original Eq. [1]. In fact because of [11], when u0 satisfies Ob3°)(u°) = f, then u = -Z(u0) satisfies [1]. Notice that if uO is chosen to satisfy the conclusions of the corollary, then the solution u of [1] belongs to the spaces Sk+ 1P (Qioc.), rP+i(Qloc.), rl(Qloc.), and C-(Q), respectively. The proof of Theorem 3 depends on the following lemma which gives a relative fundamental solution for Ob(°0. Let be the homogeneous kernel given by =
°
=
I(z1,... z.+)l Im Z.+,
>
zl112...
+ IznIdI,
CO log
(I2
-
i
=
f(zj,..z.+i)IIm (z,t)
Izj11%. R
X
=
X
(Iz12
where
z,,,1
Cn Cn +, given by (zly .1.. zn') The mapping
)
(Zi,
+
IZn I2.
it)-n
[8]
with its boundary b D ba)
[Iin
c0
=
2n2
(n)
[12]
Define the operator K, by K(f) = f*b
[8']
LEMMA
z'n+,) zn), Z'n+1 = t + i
(zl',
identifies Cn X R with bM. Cn X R has the structure of a group, the Heisenberg group Hn; in particular, the left-invariant vector fields Z1 = a/azj + ij a/at, j = 1, n, on the Heisenberg group are restrictions to bD of the holomorphic vector fields a/azi + 2itj a/al+,, j = 1, ...n, and the latter are tangential at bZ. Any f E L2(Hn), (or a distribution of compact support) leads to a Cauchy-Szeg6 integral C(f), defined in D by
Db * K
=
K *Eb 0)
=
I
-
Cb
[13]
'7=i ziI2,
.
C(f)(z =
S(zw)f(w)-der
.
.
[9]
with S(z,w) = C[i(in+1+-Zn + 1) - 2 zenl ZcWk]-n-1 where 2"n1r(n+ 1)/w"+', f is defined in bM via the identification of the latter with Hn, and daw corresponds to the Lebesgue volume element of Hn. The restriction of C(f) to bDf (taken on the Heisenberg group) is given by
c =
Cb (f)
=
Im f *St
[10]
e-O
when acting on distributions of compact support. In addition to the lemma one should observe that Cb2 = Cb, Cb* = Cb, Ob(O)Cb = CbOb(°) = 0. Thus, Cb is an orthogonal projection which commutes with 01(0). The meaning of [13] is that K inverts 0b(0) on the subspace orthogonal to the boundary values of holomorphic functions. We have found two proofs of this lemma. The first uses the calculations for the Euclidean Fourier transform of the fundamental solution for 01b(q) leading to lemma (4.1) in Greiner and Stein (ref. 7). (Incidentally, the identity [13] allows us to simplify and extend the results of ref. 7.) The second proof makes use of the ideas in §6, ref. 3, and is obtain= Cab with reable by differentiating the identity cJa(spa) spect to a.
We will conclude by outlining the argument proving the solvability of O,10)(u) = f, when f satisfies the conditions of Theorem 3. We may assume that f is a distribution with compact support. Write f = fi + f2, where fi = Cb(f), f2 = (I Cb) (f). Then 0(0)(ui) = fl, is solvable, with realanalytic in Q, by the Cauchy-Kowalewski theorem. However - Cb)f2 = - Cb)2f = - Cb)f = f2, SO Db30)(U2) = f2 is solvable, (by [13]), if one takes u2 = K(f2). The regularity properties claimed in the corollary then follow from the results of ref. 3. -
'
where S, = c(s + Iz12 it)-n-', and the convolution in [10] is with respect to the Heisenberg group. When f L2, the limit [10] exists in L2 norm (for further details see ref. 2). One can define the 5b complex on b-'D (see ref. 6 for the theory for a general class of boundaries); in this particular
(I
u
(I
(I
Mathematics: Greiner et al. 1. Lewy, H. (1957) "An example of a smooth linear partial differential equation without solution," Ann. Math. 66, 155-158. 2. Koranyi, A. & Vagi, S. (1971) "Singular integrals in homogeneous spaces and some problems of classical analysis," Ann. Scuola Norm. Sup. Pisa 25,575-648. 3. Folland, G. B. & Stein, E. M. (1974) "Estimates for the 0b complex and analysis on the Heisenberg group," Com. Pure Appl. Math. 27,429-522. 4. Stein, E. M. (1972) "Boundary behavior of holomorphic functions of several complex variables," Mathematical Notes (Princeton University Press, Princeton, N.J.). 5. Kohn, J. J. (1975) "Holomorphic extensions of orthogonal pro-
Proc. Nat. Acad. Sci. USA 72 (1975)
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jections into holomorphic functions," Proceedings of the AMS, Vol. 52, October 1975, in press. 6. Folland, G. B. & Kohn, J. J. (1972) "The Neumann problem for the Cauchy-Riemann complex," Ann. of Math. Studies #75 (Princeton University Press, Princeton, N.J.). 7. Greiner, P. C. & Stein, E. M. (1974) "A parametrix for the a-Neumann problem," Proceedings of the Rencontre sur Plusiers Variables Complexes et le Probleme de Neumann, Montreal 1974, to appear. 8. Sato, M., Kawai, T. & Kashiwara, M. (1973) "Micro-functions and pseudo-differential equations," Lecture Notes #287 (Springer, New York).
