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Annals of Mathematics The ¯ ∂b-complex on decoupled boundaries in Cn By Alexander Nagel and Elias M. Stein* Annals of Mathematics, 164 (2006), 649–713 The ∂¯b-complex on decoupled boundaries in Cn By Alexander Nagel and Elias M. Stein* Contents 1. Introduction 2. Definitions and statement of results 3. Geometry and analysis on Mj and on M1 × · · · × Mn 4. Relative fundamental solutions for 2b on M1 × · · · × Mn 5. Transference from M1 × · · · × Mn to M and Lp regularity of K 6. Pseudo-metrics on M 7. Differential inequalities for the relative fundamental solution K 8. Hölder regularity for K 9. Examples References 1. Introduction The purpose of this paper is to prove optimal estimates for solutions of the Kohn-Laplacian for certain classes of model domains in several complex variables. This will be achieved by applying a type of singular integral operator whose novel features (related to product theory and flag kernels) differ essentially from the more standard Calderón-Zygmund operators that have been used in these problems hitherto. (q) 1.1. Background. We consider the Kohn-Laplacian on q-forms, 2b = 2b = ∂¯b ∂¯b∗ + ∂¯b∗ ∂¯b , defined on the boundary M = ∂Ω of a smooth bounded pseudo-convex domain Ω ⊂ Cn . Our objective is the study of the (relative) inverse operator K and the corresponding Szegö projection S (when it exists), which satisfy 2b K = K 2b = I−S. By definition S is the orthogonal projection on the L2 null-space of 2b . *Research supported in part by grants from the National Science Foundation. 650 ALEXANDER NAGEL AND ELIAS M. STEIN In formulating the questions of regularity pertaining to the above, it is useful to recall Fefferman’s hierarchy [Fef95] of the levels of understanding of the problem, which we rephrase as follows: (1) Proof of C ∞ regularity. (2) Derivation of optimal Lp , Hölder, and Sobolev-space estimates of solutions. (3) Analysis of singularities of the distribution kernels of the operators K and S and derivation of the estimates in (2) from a corresponding theory of singular integrals. Now as far as the C ∞ regularity is concerned, this has been resolved in the general situation where an appropriate “finite-type” condition holds (at least ¯ for the closely connected ∂-Neumann problem) by the work of Kohn [Koh72], [Koh79], Catlin [Cat83], [Cat87], and D’Angelo [D’A82]. However, the more refined results of (2) and (3) have been obtained only in a more restrictive setting. This was carried out in a series of developments beginning with the work of Folland and Stein ([FS74]) in the strongly pseudo-convex case, and in later works of, among others, Christ ([Chr91b], [Chr88]), Fefferman and Kohn ([FK88]), Kohn ([Koh85]), McNeal ([McN89]), Nagel, Rosay, Stein, and Wainger ([NRSW89]), and Rothschild and Stein ([RS76]). This culminated in the work of Koenig ([Koe02]) on finite type domains whose Levi-form has comparable eigenvalues. At the base of these results is a version of the Calderón-Zygmund theory for the following class of singular integrals: One considers operators T of
the form T (f )(x) = T (x, y) f (y)dy whose kernels T (x, y) are distributions that are smooth away from the diagonal, that satisfy the characteristic size estimates |T (x, y)|
d(x, y)a V (x, y)−1 , and that satisfy corresponding differential inequalities and cancellation properties. Here d(x, y) is the control metric determined by the vector fields which are the real and imaginary parts of the tangential Cauchy-Riemann operators, and V (x, y) denotes the volume of the ball centered at x of radius d(x, y). It can be shown that the relative fundamental solution K and the Szegö projection S are of this type, with a = 2 for K, and a = 0 for S. As a result, one obtains for these operators maximal sub-elliptic estimates in Lp , etc. Unfortunately, while highly satisfactory, the above framework with a natural metric controlling all estimates cannot carry over in general. In fact, in more general circumstances there seem to arise a number of inequivalent metrics that control different aspects of the problem. This appears to be connected with earlier observations of Derridj [Der78] and Rothschild [Rot80] that maximal sub-ellipticity can hold only if the eigenvalues of the Levi-form are ∂ b ON DECOUPLED BOUNDARIES 651 comparable. It is the purpose of this paper to make progress in the resolution of problems (2) and (3) in an illustrative model case - that of decoupled domains. 1.2. A special case. To get a better grasp of these problems and the results we obtain, we take a closer look at the special case of a decoupled domain where Ω = {z ∈ C3 : m[z3 ] > |z1 |n + |z2 |m }, with n, m even integers. Then M = ∂Ω can be identified with {(z, t) ∈ C2 × R, z = (z1 , z2 )}, and ∂ ∂ ∂ n ∂ m − i |z1 |n−2 z1 , − i |z2 |m−2 z2 Z̄1 = Z̄2 = ∂ z̄1 2 ∂t ∂ z̄2 2 ∂t form a basis for the tangential Cauchy-Riemann vector fields. The eigenvalues λ1 , λ2 of the Levi-form at a point (z1 , z2 , t) are essentially |z1 |n−2 and |z2 |m−2 , and are not comparable. With Z̄j = 12 (Xj + iYj ), we can consider dΣ , the control metric defined by X1 , Y1 , X2 , Y2 . However, the above domain is also convex, so that there is another natural metric, which reflects the “flatness” of the boundary in different complex directions, the “Szegö metric” dS ; (see McNeal [McN94b], [McN94a], and Bruna, Nagel and Wainger [BNW88] for a real analogue). In our special case, if n ≤ m, when we measure the distance of the point p = (z1 , z2 , t) from the origin 0 we have: dΣ (0, p) ≈ |z1 | + |z2 | + |t|1/m ; dS (0, p) ≈ |z1 |m + |z2 |n + |t|. Note that dS (0, p)1/m ≈ |z1 | + |z2 |n/m + |t|1/m , and this is not equivalent to dΣ (0.p) if n = m. Thus these metrics, or powers of these metrics are in general not equivalent.  Now dΣ controls the inverse of the sub-Laplacian L = − 12 2i=1 (Zi Z̄i + Z̄i Zi ), while dS controls the Szegö kernel (the orthogonal projection on the  null-space of the operator − 2i=1 Zi Z̄i ), and some mixture of dΣ and dS arises in the fundamental solution of the operator 2b = −(Z1 Z̄1 + Z̄2 Z2 ) = 21b + 22b , which is essentially the Kohn-Laplacian acting on 1-forms. With this we can state a part of our main result obtained below, formulated in this special case, as follows: Theorem. There is an operator K so that, when it is applied to smooth functions with compact support, there is the identity K 2b = 2b K = I. Moreover (a) The four operators Z1 Z̄1 K = 1b K, Z̄2 Z2 K = 2b K, Z̄1 Z̄1 K, and Z2 Z2 K are bounded on Lp (M ) for 1 < p < ∞. (b) Let B1 , B2 be bounded functions on M , and suppose there are constants C1 , C2 so that λ1 (z1 ) B1 (z1 , z2 , t) ≤ C1 λ2 (z2 ); λ2 (z2 ) B2 (z1 , z2 , t) ≤ C2 λ2 (z1 ). 652 ALEXANDER NAGEL AND ELIAS M. STEIN 1 2 Then the two operators B1 Z̄1 Z1 K = B1 b K and B2 Z2 Z̄2 K = B2 b K are bounded on Lp (M ) for 1 < p < ∞. Here λ1 (z) = |z|m−2 and λ2 (z) = |z|n−2 are the eigenvalues of the Levi form. (c) Let B1 , B2 be bounded functions on M , and suppose there are constants C1 , C2 so that B1 (z1 , z2 , t) ≤ C1 λ2 (z2 ); B2 (z1 , z2 , t) ≤ C2 λ2 (z1 ). Then the two operators B1 Z1 Z1 K and B2 Z̄2 Z̄2 K are bounded on Lp (M ) for 1 < p < ∞. (d) K maps L∞ (M ) to the isotropic Hölder space Λα (M ), where   2 2 . α = min , n m The conclusion (b) is part of the optimal substitute for maximal subellipticity that holds in this case. 1.3. Methods used. To describe the methods used we continue with the case considered above. We begin by considering separately the component domains    M1 = (z1 , w1 ) ∈ C2  [w1 ] = |z1 |n  {(z1 , t1 ) ∈ C × R}, and    M2 = (z2 , w2 ) ∈ C2  [w2 ] = |z2 |m  {(z2 , t2 ) ∈ C × R}.  the Cartesian product M1 ×M2 and we let π be the projection We denote by M  to M given by π : (z1 , t1 ) × (z2 , t2 ) → (z1 , z2 , t1 + t2 ). of M The idea is to deduce the results about regularity of 2b on M from cor. Moreover, passing to the product allows one to responding results on M , consider various combinations of the separate metrics on each factor of M which in effect account for the different metrics on M . Our analysis proceeds as follows. (1) Analysis on each Mj : Here the key point is the use of the nonhypoelliptic “heat” semi-group e−s2j on Mj where 21 = Z1 Z̄1 , 22 = Z̄2 Z2 . (The needed estimates for this semi-group were obtained in [NS01a].) For later purposes one observes that if ∞ Kj = (e−s2j − Sj )ds, 0 where Sj is the orthogonal projection on the null-space of 2j , then Kj 2j = 2j Kj = I − Sj . 653 ∂ b ON DECOUPLED BOUNDARIES  = M1 × M2 : In finding a (relative) inverse for (2) Results on the product M  21 + 22 on M one considers ∞ K = (e−s(21 +22 ) − S1 ⊗ S2 )ds 0 and also a substitute version ∞ N = (e−s21 − S1 ) ⊗ (e−s22 − S2 )ds. 0 Now N is more tractable than K since any second order derivative in Zj and Z̄j of N turns out to be a product-type singular integral on M1 × M2 . For such singular integrals an Lp theory has been worked out in [NS04]. However, K is the desired relative inverse, since (21 + 22 )K = K(21 + 22 ) = I − S1 ⊗ S2 ; its properties can ultimately be deduced from those of N because of the identity K = N + K1 ⊗ S2 + S1 ⊗ K2 .  = M1 × M2 are translation(3) Descent to M : The operators above on M invariant in the t1 and t2 variables. Each appropriate operator T of this  −→ M to an operator kind can be transferred by the projection π: M T # on M , via the identity T # (f ) = J(T (f ◦ π)) ∞ where J(F )(z1 , w1 , z2 , w2 , t) = then applied to K to obtain K = F (z1 , w1 , t − s, z2 , w2 , s) ds. This is −∞ (K)# , the inverse of Z1 Z̄1 + Z̄2 Z2 on M . There is however a fundamental issue that arises at this point. Operators like K and N are not pseudo-local, because as product-like operators their kernels have singularities on the products of the diagonals of the Mi , and . As a result the projections of such operators not just on the diagonal of M on M are thus in general again not pseudo-local. Why then is the operator K pseudo-local? Connected with this is the question of obtaining the appropriate differential inequalities satisfied by the kernel of K away from diagonal. The resolution of these problems is connected with the key idea of “borrowing”, which allows one to pass from smoothness inherent in the t1 (and z1 ) variable to the t2 (and z2 ) variable, and vice-versa. This technique is used in several places below where it takes a number of different forms. A particularly transparent example is the identity ∂ S1 ⊗ K 2 ∂t1 # = S1 ⊗ ∂ K2 ∂t2 # which is used in obtaining conclusion (b) of the theorem above. 654 ALEXANDER NAGEL AND ELIAS M. STEIN 1.4. Previous work. Besides the results mentioned earlier which deal with the situation of comparable eigenvalues of the Levi-form, several other situations have been previously studied. The case of a decoupled domain in C3 with exactly one degenerate eigenvalue was dealt with in the paper of Machedon [Mac88] , where he also finds certain estimates for the fundamental solution which involve several metrics. In addition, Fefferman, Kohn, and Machedon [FKM90] have obtained results on Hölder regularity for 2b on boundaries of diagonalizable domains (which is a larger class of domains than we consider). In contrast, here we obtain sharp Lp and Hölder estimates, and relevant differential inequalities for the solving operators and Szegö projections. The general idea of “lifting” to a product (or “simpler” situation) is old, having already appeared in different forms in the study of the sub-Laplacian [RS76], and in [Mac88]. More recently it was used in [MRS95] to study certain operators on the Heisenberg group, and for 2b on quadratic CR manifolds of higher-codimension in [NRS01]. The operators arising in [MRS95], related to ¯ the boundary operator of the ∂-Neumann problem for the ball, which occurred in [PS86], already implicitly display the feature of the conflicting metrics which we have discussed above. There the kernels of the relevant operators arise as products of components that are homogeneous in different senses: the isotropic homogeneity reflecting the Euclidean metric, and the automorphic homogeneity of the Heisenberg group, reflecting the control metric. 1.5. Organization of the paper. Section 2 contains a review of background material and statements of the main results of the paper. The needed aspects of the geometry and analysis of each of the factors Mi and on their Cartesian product are set down in Section 3. Section 4 studies the various versions of the . This leads to Lp results on M via relative fundamental solutions of 2b on M transference, as is shown in Section 5. Section 6 deals with the various metrics on M and the resulting differential inequalities of the kernels are obtained in Section 7. In Section 8, we prove the Hölder regularity of the solutions, and in Section 9 we give examples to show that our regularity results are optimal. 2. Definitions and statement of results 2.1. Definitions. A domain Ω ⊂ Cn+1 and its boundary M are said to be decoupled if there are sub-harmonic, nonharmonic polynomials Pj such that n     Pj (zj ) ; Ω = (z1 , . . . , zn , zn+1 ) ∈ Cn+1  m[zn+1 ] > (2.1.1) j=1 n    n+1  Pj (zj ) . M = (z1 , . . . , zn , zn+1 ) ∈ C  m[zn+1 ] = j=1 655 ∂ b ON DECOUPLED BOUNDARIES We call the integer mj = 2 + degree( Pj ) the “degree” of Pj . (The actual degree of Pj may be larger, but the addition of a harmonic polynomial to Pj does not affect our analysis, and can be eliminated by a change    of variables.)  n We identify M with C ×R so that the point z1 , . . . , zn , t+i ∈M j Pj (zj ) corresponds to the point (z1 , . . . , zn , t) ∈ Cn ×R. M has real dimension 2n+1. When integrating on M , we take the measure to be Lebesgue measure on Cn × R. In addition to the boundary of a decoupled domain as in (2.1.1), we also consider Cartesian products of boundaries of domains in C2 . For 1 ≤ j ≤ n, let    Ωj = (zj , wj ) ∈ C2  m[wj ] > Pj (zj ) ;   (2.1.2)  Mj = (zj , wj ) ∈ C2  m[wj ] = Pj (zj ) .   As before, we identify Mj with C × R so that the point zj , t + iPj (zj ) corresponds to the point (zj , t). When integrating on Mj we use Lebesgue measure on C × R. The Cartesian product of these boundaries is  = M1 × · · · × Mn ⊂ C2n . M (2.1.3)  is the Shilov boundary of the product domain Ω1 × · · · × Ωn . It has Then M  with Cn × Rn real dimension 3n and real codimension n. We can identify M  corresponds so that the point p = z1 , t1 + iP1 (z1 ), . . . , zn , tn + iPn (zn ) ∈ M n n to the point (z1 , . . . , zn , t1 , . . . , tn ) = (z, t) ∈ C × R . When integrating on , we take the measure to be Lebesgue measure on Cn × Rn . M Let π : C2n → Cn+1 be the linear holomorphic mapping π(z1 , . . . , zn , w1 , . . . , wn ) = (z1 , . . . , zn , w1 + · · · + wn ).  to M . In terms of the coordinates given by This induces a mapping from M Cn × Rn and Cn × R, we have (2.1.4) π(z1 , . . . , zn , t1 , . . . , tn ) = (z1 , . . . , zn , t1 + · · · + tn ).  to M . The mapping π allows us to transfer functions from M ∞ n n # ∞ n C0 (C × R ), we define a function ϕ ∈ C0 (C × R) by setting  # ϕ (z, t) = (2.1.5) Rn−1 ϕ z, r1 , . . . , rn−1 , t − ≡ n−1  If ϕ ∈  rj dr1 · · · drn−1 j=1 ϕ(z, r) dr̃ r∈Σ(t)  where Σ(t) = {(r1 , . . . , rn ) ∈ Rn  r1 +· · ·+rn = t} and dr̃ is (n−1)-dimensional Lebesgue measure on Σ(t). 656 ALEXANDER NAGEL AND ELIAS M. STEIN . Let M be the 2.2. The ∂¯b -complex and the b operator on M and M boundary of a decoupled domain as in (2.1.1). Using coordinates (z1 , . . . , zn , t) ∈ Cn × R, bases for the Cauchy-Riemann operators of type (1, 0) and (0, 1) are given by the operators {Zj , 1 ≤ j ≤ n} and by {Z̄j , 1 ≤ j ≤ n} where ∂Pj ∂ ∂ +i (zj ) = Xj − iXn+j , ∂zj ∂zj ∂t ∂Pj ∂ ∂ Z̄j = −i (zj ) = Xj + iXn+j , ∂ z̄j ∂ z̄j ∂t Zj = (2.2.6) where {X1 , . . . , X2n } are real vector fields. Remark. For future reference, note that the operators Zj , Z̄j , and their sums and products commute with translations in the variable t. The same will also be true of the inverses or relative inverses we construct for such operators. Hence the corresponding distribution kernels K (z, t), (w, s) will be of the form K(z, w, t − s). We recall the formalism of the ∂¯b -complex on M . If f is a function, then ∂¯b [f ] = n  Z̄j [f ] dz̄j . j=1 Let ϑq denote the set of strictly increasing q-tuples of integers between 1 and n. Let J = {j1 , . . . , jq } ∈ ϑq , and let dz̄J denote the (0, q)-form dz̄j1 ∧ · · · ∧ dz̄jq . Then {dz̄J }J∈ϑq is a basis for the space of (0, q) forms and    ∂¯b [f ] ∧ dz̄J . f dz̄J = ∂¯b J∈ϑq J∈ϑq One checks that ∂¯b2 = 0. Let ∂¯b∗ denote the formal adjoint of ∂¯b so that ∂¯b∗ maps (0, q + 1)-forms to (0, q)-forms. Thus for compactly supported (0, q) and (0, q + 1) forms ϕ and ψ,     ∂¯b [ϕ], ψ q+1 = ϕ, ∂¯b∗ [ψ] q ,   where · , · q is the L2 -inner product on (0, q)-forms defined so that the forms {dz̄J }J∈ϑq are orthonormal. The Kohn-Laplacian (2.2.7) b = ∂¯b ∂¯b∗ + ∂¯b∗ ∂¯b is a second order system of partial differential operators which maps (0, q)forms to (0, q)-forms. For the decoupled boundary M , b acts as follows. For 1 ≤ j ≤ n, let (2.2.8) (+) = −Z̄j Zj ; (−) = −Zj Z̄j . j j ∂ b ON DECOUPLED BOUNDARIES For J ∈ ϑq and 1 ≤ k ≤ n set J(k) =  (+) if k ∈ J, (−) if k ∈ / J. 657 The operator b acts diagonally, and is given by     (2.2.9) ϕJ dz̄J  = J (ϕJ ) dz̄J b  J∈ϑq J∈ϑq where J = (2.2.10) n  J(k) k . k=1 ¯ Thus, the n study of the ∂b complex on M on (0, q)-forms is reduced to the study of the q operators J for J ∈ ϑq . . InWe can also consider the ∂¯b -complex on the product submanifold M stead of the vector fields (2.2.6), we set ∂Pj ∂ ∂ +i (zj ) ; ∂zj ∂zj ∂tj ∂Pj ∂ ∂ Z̄j = −i (zj ) . ∂ z̄j ∂ z̄j ∂tj Zj =  is defined in the exactly the same way as on M . If we The ∂¯b complex on M ¯ ¯∗ ¯∗ ¯ then define operators ± j as before, the operator b = ∂b ∂b + ∂b ∂b has exactly the same form as in equations (2.2.9) and (2.2.10).  → M given in equation (2.1.4) induces a mapping The mapping π : M , and hence induces a mapping dπ from from functions on M to functions on M  to tangent vectors on M . In particular, if Tj = ∂ on M  tangent vectors on M ∂tj ∂ and T = ∂t on M , then dπ(Tj ) = T for 1 ≤ j ≤ n. This justifies our use of the same notation, i.e. Zj and Z̄j for vectors fields and b for the Kohn Laplacian,  and M . The adjoint mapping dπ ∗ which carries differential forms on both M  commutes with the mappings ∂¯b . on M to differential forms on M We have also considered a mapping ϕ → ϕ# in (2.1.5) which carries  to functions on M . The following is then clear. functions on M ) so that ϕ# ∈ C ∞ (M ). Then T [ϕ# ] Proposition 2.2.1. Let ϕ ∈ C0∞ (M 0 = (Tj [ϕ])# and so in particular (Tj [ϕ])# = (Tk [ϕ])# for any 1 ≤ j, k ≤ n. 2.3. Outline of the argument. We now expand the discussion in Section 1.3 and describe the main ideas involved in the construction of relative fundamental solutions for the operators {J }. Let Wj denote either Zj or Z̄j where Wj is then a first order differential operator on M which depends