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PRECISE SUBELLIPTIC ESTIMATES FOR A CLASS OF SPECIAL DOMAINS 9 TRAN VU KHANH AND GIUSEPPE ZAMPIERI 0 0 2 Abstract. For the ∂¯-Neumann problem on a regular coordinate n domain Ω ⊂ Cn+1, we prove ǫ-subelliptic estimates for an index a ǫ which is in some cases better than ǫ = 1 (m being the multi- J 2m plicity) as it was previously provedby Catlin and Cho in [4]. This 7 alsosuppliesamuchsimplifiedproofoftheexistingliterature. Our approachis founded onthe method by Catlin in [3] whichconsists ] V in constructing a family of weights {φδ} whose Leviformis bigger C than δ−2ǫ on the δ-strip around ∂Ω. . MSC: 32F10, 32F20, 32N15, 32T25 h t a m [ 1. Introduction 2 v Regular coordinate domains have ǫ-subelliptic estimates; we discuss 0 6 here about the optimal ǫ. These domains are defined by 5 2 N 2. (1.1) 2Rezn+1 + |fj(z)|2 < 0, for (z,zn+1) ∈ Cn ×C, 1 j=1 8 X 0 where the f ’s are holomorphic functions in a neighborhood of 0 which j : v satisfy f = f(z ,...,z ) and f (0,...,0,z ) 6= 0. We denote by m the j 1 j j j j Xi smallest index such that ∂mjf 6= 0 and define m := Π m . The zj j j j=1,...,n r a D’Angelo’s type D is the order of contact of ∂Ω with any complex 1- dimensional complex variety. For these domains it is readily seen that |f |2 > |z|2m which implies D ≤ 2m. Moreover, according to Catlin j j ∼ [1],wemusthaveǫ ≤ 1 and,conversely, itisaconjecturebyD’Angelo’s P D [7] that ǫ ≥ 1 . (Indeed, the conjecture is formulated for more general 2m special domains and in this case the integer m is the multiplicity.) We present a simplified proofof a recent result by Catlin andCho [4] which gives positive answer to the conjecture for coordinate domains. More important, we find an intermediate number 1 ≤ γ ≤ 1 obtained 2m 2 D by combining vanishing orders of the f ’s in different directions and j prove subelliptic estimates forǫ coinciding withthis new number γ.For 2 instance, consider the domain defined by (1.1) for the choice f = zm1 1 1 1 2 T.V. KHANHAND G. ZAMPIERI and then (1.2) f (z) = zmj +zlj , l ≤ m , j = 2,...,n. j j j−1 j j−1 According to Theorem 2.3, we set γ = 1 , define inductively γ = 1 m1 j min li γ and write γ for γ . (In particular, when l = m and m ≥ i≤j−1mj i n j j−1 j m for any j, then γ = min 1 .) Then we have ǫ-subelliptic estimates j−1 j i≤j mi for ǫ = γ. 2 Note that we always have γ ≥ 1 with equality holding only if l = 1 2 2m j for any j. On the other hand we have γ 1 (1.3) ≥ , 2 D which shows that our index of subellipticity is optimal in this example. To prove (1.3), we have just to notice that the curve C → Cn+1, τ 7→ (τγγ1,...,τγnγ−1,τ,0), has order of contact equal to 2. On the other hand, the presence of ǫ- γ subelliptic estimates for ǫ = γ assured by our theorem, yields equality 2 in (1.3). 2. Precise subellipticity index for a class of special domains Let z = (z ,...,z ) be coordinates in Cn and (z,z ) coordinates 1 n n+1 in Cn ×C. We deal with regular coordinate domains Ω ⊂ Cn+1, that is, domains defined by (1.1) for f holomorphic which satisfy f = j j f (z ,...,z ) and ∂mjf 6= 0 for some m . For these domains we consider j 1 j zj j j the ∂¯-Neumann problem and, in particular, the ǫ-subelliptic estimates. These are of the type (2.1) |||u|||2 < ||∂¯u||2+||∂¯∗u||2+||u||2, ǫ 0 0 0 ∼ for any C∞(Ω¯)-form u of degree k ≥ 1 in the domain D of ∂¯∗. Here c ∂¯∗ |||·||| isthetangentialSobolevnormofindex ǫ.This issaidanestimate ǫ withǫ-fractionalgainofderivative. It isclassical (see [5])that it implies the local hypoellipticity of the ∂¯-Neumann problem. The canonical solution u of ∂¯u = f, that is, the solution orthogonal to ker∂¯, is C∞ up to ∂Ω precisely at those points of ∂Ω where f is C∞. In particular, the Bergman projection preserves C∞ smoothness. The following theorem has been recently obtained by [4]; we give here a much simplified proof. PRECISE SUBELLIPTIC ESTIMATES FOR A CLASS OF SPECIAL DOMAINS3 Theorem 2.1. Let Ω ⊂ Cn+1 be a regular coordinate domain defined by (1.1) with the f ’s satisfying f = f (z ,...,z ). Write j j j 1 j (2.2) f = zmj +O(z ,...,z ,zmj+1); j j 1 j−1 j then ǫ-subelliptic estimates hold for ǫ = 1 . 2m1·...·mn Proof. According to Catlin [3] it suffices to find a family of bounded weights {ϕδ} for δ ց 0 whose Levi form satisfies ϕδ u u¯ ≥ ij ij i j δ−m1·.1..·mn|u|2 over the strip of Ω about the boundary SPδ = {z ∈ Ω : −r(z) < δ}. Once the functions ϕδ have been found, we have to de- form them to new functions ϕ˜δ, bounded and plurisubharmonic not only in S but in the whole Ω, satisfying the same Levi conditions δ as the ϕδ on S ; for this we refer to Lemma 2.2 after the end of the δ 2 proof of Theorem 2.1. To define the functions ϕδ, we put γ = 1, 0 γ = 1 , choose α ≥ 1 and put α = αm ...m . Note that j m1·...·mj j j+1 n (α −1)−α (m −1) = α −1. We choose a cut off function χ which j−1 j j j satisfies suppχ ⊂ (0,2), χ ≡ 1 on (0,1), ( and also take a small constant c to be specified later. We also rewrite the inequality (1.1) which defines Ω as r < 0 and define (2.3) −r +δ n mj−1 1 δ(mj−h)γj ϕδ = −log( )+ log |∂hf |2 + δ |log ∗| zj j |logδ|(mj−h)αj j=1 h=1 (cid:18) (cid:19) XX n |z |2 |z |2 +δγj j j +c χ( )log , δγj δγj j=1 (cid:18) (cid:19) X where ∗ = δγj(mj−h) . Notice here that log∗ ≃ logδ. The weights ϕδ |logδ|(mj−h)αj that we have defined are bounded in the strip S . We use the notations δ A = δ−1|∂f ·u|2 Bh = δ−(mj−h)γj|∂∂ f ·u|2. j j j zj j We also denote by cC the Levi form of the third term of (2.3) applied j to u; note that C > δ−γj|u |2 for |z | ≤ δγj, but, otherwise, C can j j j j ∼ take negative values; however, |C | < δ−γj|u |2. We also set j j ∼ D = A + Bh +cC . j j j j hX≤kj 4 T.V. KHANHAND G. ZAMPIERI We first prove an auxiliary statement: the assumption (2.4) Di ≥ δ−γio|logδ|αio−1|uio|2 + δ−γi|ui|2, Xi≤io i≤Xio−1 implies, for any j ≥ i +1 o (2.5) D ≥ δ−γj|logδ|αj−1 |u |2. i i i≤j i≤j X X In fact, by iteration, it suffices to prove the statement for i = j − 1. o For that, we first notice that δ−γi ≥ δ−γio|logδ|k for any k and for any i ≤ i −1. It follows o (2.6) A + D > δ−γj−1|log|αj−1−1 |∂ f |2|u |2 − |u |2 + D j i zj j j i i ∼ ! i≤j−1 i≤j−1 i≤j−1 X X X > δ−γj−1|log|αj−1−1|∂ f |2|u |2. zj j j ∼ Assume atthis point |∂ f | ≥ δ(mj−1)γj ;then (2.6) canbe continued zj j |logδ|(mj−1)αj by ≥ δ−γj−1+(mj−1)γj|logδ|(αj−1−1)−(mj−1)αj ≥ δ−γj|logδ|αj−1|u |2. j Notice that this controls cC when this gets negative; this happens in j all cases which follow; hence we avoid to recall it at each step. If not, we can assume |∂ f | ≤ δ(mj−1)γj and then use B1. We have zj j |logδ|(mj−1)αj j (2.7) |logδ|(mj−1)αj B1 + D > δ−γj(mj−1) |∂2 f |2|u |2 − |u |2 + D j i ∼ |log∗| zj j j i ! i i≤j−1 i≤j−1 i≤j−1 X X X > δ−γj−1+γj|logδ|(mj−1)αj−1|∂2 f |2|u |2. ∼ zj j j If |∂2 f | ≥ δ(mj−2)γj , then (2.7) can be continued by zj j |logδ|(mj−2)αj ≥ δ−γj−1+γj+γj−1−2γj|logδ|αj−1|u |2. j If |∂2 f | ≤ δ(mj−2)γj , we pass to B2. In this way we jump from zj j |logδ|(mj−2)αj j ∂h−1f to ∂hf until we reach Bmj−1. At this stage we have |∂mj−1f | ≤ zj j zj j j zj j δγj which yields readily |logδ|αj−1 Bmj−1 ≥ δ−γj|logδ|αj−1|u |2. j j PRECISE SUBELLIPTIC ESTIMATES FOR A CLASS OF SPECIAL DOMAINS5 This proves that (2.4) for i = j − 1 implies (2.5). By iteration the o conclusion is true for any i ≤ j−1. We now prove that, for any value o of z j (2.8) D > δ−γj|u |2, i j ∼ i≤j X whereas, when |z | ≥ δγj j either δ−γj|logδ|αj−1|u |2 j (2.9) D > i ∼ (or δ−γj−1|zj|2(mj−1)|uj|2. i≤j X We proceed by induction over j. The first step j = 1 is easy. In fact C > δ−γ1|u |2 for |z | ≤ δγ1 and 1 1 1 ∼ A > max(δ−γ1,δ−1|z |2(m1−1))|u |2 for |z | ≥ δγ1. 1 1 1 1 ∼ In particular, when cC takes negative values, these are controlled by j A for suitably small c. This proves (2.8) and (2.9) for j = 1. Suppose 1 that (2.8) and (2.9) are true up to step j − 1 and prove them for j. If |z | ≤ δγj, then C > δ−γj|u |2. Otherwise, assume |z | ≥ δγj. We j j j j ∼ show now that, under this assumption, we must have (2.4) for some i ≤ j − 1 unless the second alternative in (2.9) holds. In fact, let o |z | > |z |mj−1; then |z | > δγj(mj−1) ≥ δγj−1−γj and therefore j−1 j j−1 ∼ ∼ δ−γj−2|z |2(mj−1−1) ≥ δ−γj−2+γj(mj−1−1)(mj−1) j−1 = δ−γj−2+γj−2−γj−1−γjmj−1+γj ≥ δ−γj−1|logδ|k for any choice of k. On the other hand, for any choice of i ≤ j −2, we have δ−γi ≥ δ−γj−1|logδ|k. This implies (2.4) for i = j−1 which implies in turn (2.5). Otherwise, o we assume |z | < |z |mj−1. Now, we point our attention to z . If j−1 j j−2 ∼ |z | > |z |mj−1, then |z | ≥ δγj(mj−1) ≥ δγj−1−γj and hence j−2 j j−2 ∼ δ−γj−3|z |2(mj−2−1) ≥ δ−γj−3+(mj−2−1)(γj−1−γj) j−2 = δ−γj−3+mj−2γj−1 ≥ δ−γj−2|logδ|k. 6 T.V. KHANHAND G. ZAMPIERI On the other hand, we have for any i ≤ j − 3, δ−γi ≥ δ−γj−2|logδ|k. This implies (2.10) D ≥ δ−γj−2|logδ|αj−2−1 |u |2. i i i≤j−2 i≤j−2 X X The same argument which shows that (2.4) implies (2.5) also serves in proving that (2.10) implies (2.4); in turn, (2.4) implies (2.5). We can therefore assume |z | < |z |mj−1. j−2 j ∼ We repeat the argument that we developed for i = j−1 and i = j−2 for any other index i ≤ j −1. This can be explained by the fact that the second of (2.9) implies the first at each step i ≤ j − 1 (with the cases i = j −1 and i = j −2 having already been proved). In fact, if |z | ≥ |z |mj−1 and hence |z | ≥ δγj−1−γj, then i j i δ−γi−1|z |2(mi−1) ≥ δ−γi−1+(γj−1−γj)(mi−1) i ≥ δ−γi−1+miγj−1 ≥ δ−γi|logδ|k. This yields (2.4) and thus also (2.5) unless (2.11) |z | ≤ |z |mj−1 for any i ≤ j −1. i j On the other hand, when (2.11) holds, then 1 |∂ f | > |z |2(mj−1) − |z |2 > |z |2(mj−1). zj j ∼ j 2 i ∼ j i≤j−1 X It follows A + D ≥ δ−1|∂f ·u|2 + D j i j i i≤j−1 i≤j−1 X X (2.12) > δ−γj−1(|z |2(mj−1)|u |2 − |u |2)+ D j j i i ∼ i≤j−1 i≤j−1 X X > δ−γj−1|z |2(mj−1)|u |2. j j ∼ Note that this is in any case > δ−γj|u |2 and also that it controls cC˜ , j j ∼ for suitable c when C˜ gets negative. So, inthis case we have the second j alternative in (2.9). This concludes the proof of the theorem. (cid:3) It remains to prove the technical lemma which shows how to modify the functions ϕδ to ϕ˜δ so that they are plurisubharmonic on the whole of Ω. PRECISE SUBELLIPTIC ESTIMATES FOR A CLASS OF SPECIAL DOMAINS7 Lemma 2.2. There are ϕ˜δ, plurisubharmonic and bounded on Ω and such that ϕ˜δ = e2δr +e−2ϕδ on S2δ, (0 on Ω\Sδ. Proof. We take θ : R → R+, t 7→ θ(t), convex increasing, that is, satisfying θ˙ ≥ 0, θ¨≥ 0 and such that 0 for t ≤ 2e−2, θ = t for t ≥ e−1, ( set ψδ := e2δr +e−2ϕδ and define ϕ˜δ := θ◦ψδ. Remember that ϕδ take values in [0,1] and notice that S ⊂ {z : ψδ ≥ e−1} and Ω\S ⊂ {z : δ δ 2 ψδ ≤ 2e−2}; thus ϕ˜δ = ψδ on S and ϕ˜δ = 0 on Ω\S . δ δ 2 (cid:3) We now prove subelliptic estimates for a class of domains with a better index ǫ ≥ 1 ; in particular, for the domains defined by 2m1·...·mn (1.1), this index coincides with the optimal value ǫ = 1. To achieve our D goal, we have to specify the vanishing order of f in different directions j z for i ≤ j. i Theorem 2.3. Let Ω ⊂ Cn+1 be a regular coordinate domain defined by (1.1) with f = f (z ,...,z ) and ∂mjf 6= 0. We denote by li, i < j, j j 1 j zj j j the vanishing order of f in z , that is, we write j i (2.13) f = zmj +O (zlj1,...zljj−1,zmj) for li ≤ m . j j j 1 j−1 j j i Assume that each O contains no power of z of degree ≤ m −1 and j j j define γ = 1 and, inductively, 1 m1 li j γ = min γ j i i≤j−1mj and also write γ for γ . Then we have ǫ-subelliptic estimates for any n ǫ = γ. 2 Proof. Let k be the highest power ≤ m −1 of z in O and let r < 0 j j j j be the inequality which defines Ω. We define −r +δ n |z |2 |z |2 (2.14) ϕδ = −log( )+c χ( j )log j +1 δ δγj δγj j=1 (cid:18) (cid:19) X and prove that for any u ∈ Cn, ϕδ satisfy ϕδ u u¯ ≥ δ−γ|u|2 over ij ij i j n the strip S . We denote by c ϕ the secoPnd term in the right hand δ j j=1 P 8 T.V. KHANHAND G. ZAMPIERI side of (2.14) and define A = δ−1|∂f ·u|2, C = ∂∂¯ϕ (u,u¯), D = A +cC . j j j j j j j We wish to prove that (2.15) D > δ−siγi|z |2(si−1)|u |2 for any s ≤ m . i i i i i ∼ i≤j i≤j X X It is easy to prove the first step, that is, D > δ−sγ1|z |2(s−1)|u |2 for any s ≤ m . 1 1 1 1 ∼ Suppose we have already proved (2.15) for any i ≤ j −1. We have to prove that (2.16) D ≥ δ−sγj|z |2(s−1)|u |2 for any s ≤ m . i j j j i≤j X We recall here that li ≤ m and notice that γ ≤ 1 ; in particular, j i i mi liγ ≤ 1. We have j i A + D > δ−1|∂f ·u|2 + δ−siγi|z |2(si−1)|u |2. j i j i i ∼ i≤j−1 i≤j−1 X X We fix our choice of the s ’s as s = li which are smaller than m ; thus i i j i liγ ≤ 1. It follows j i Aj + Di ≥ δ−ljiγi |∂fj ·u|2 +|zi|2(lji−1)|ui|2 . i≤Xj−1 i≤Xj−1 h i On the other hand |∂fj ·u|2 > |zj|2(mj−1)|uj|2 − |zi|2(lji−1)|ui|2, ∼ i≤j−1 X and therefore, since liγ ≥ m γ for any i ≤ j −1, we conclude j i j j (2.17) A + D > δ−mjγj|z |2(mj−1)|u |2. j i j j ∼ i≤j−1 X Thisproves (2.15)forthechoices = m .Weprovenow (2.15)fors = 1; j for this we have to call into play C . We have j 1 (2.18) C +δ−mjγj|z |2(mj−1) > (δ−γj +δ−mjγj|z |2(mj−1))|u |2. j j j j 2 ∼ In fact, if |z |2 ≤ δγj, then C ≥ δ−γj|u |2. If, instead, |z | ≥ j j j j δγj, and thus C gets negative, we have on our side the fact that j δ−mjγj|z |2(mj−1)|u |2 ≥ δ−γj|u |2 and therefore it controls cC for j j j j suitably small c. From (2.17) we conclude that A + cC + D is j j i i≤j−1 P PRECISE SUBELLIPTIC ESTIMATES FOR A CLASS OF SPECIAL DOMAINS9 bigger than the right side of (2.18), which yields (2.15) for s = 1 and s = m . The estimate (2.15) for general s with 1 ≤ s ≤ m is just a j j combination of the two opposite cases s = 1 and s = m . j (cid:3) Remark 2.4. The theorem applies in particular to the class of examples described by (1.2). We have a final statement which collects in a unified frame the con- clusions of Theorems 2.1 and 2.3. Theorem 2.5. Let Ω ⊂ Cn+1 be a regular coordinate domain defined by (1.1) for f = f (z ,...,z ) which satisfy (2.13). Define γ = 1 and, j j 1 j 1 m1 inductively min 1 γ if O contains some power of z in degree ≤ m −1, i≤j−1mj i j j j γ = j min lji γ otherwise i≤j−1mj i and also write γ for γ . Then we have ǫ-subelliptic estiates for ǫ = γ.  n 2 Proof. We use in this situation the family of weights −r +δ 1 δ(mj−h)γj ϕδ = −log( )++ log |∂hf |2 + δ |log∗| zj j |logδ|(mj−h)αj {j:Xkj6=0}hX≤kj (cid:18) (cid:19) n |z |2 |z |2 j j +c χ log +1 . δγj δγj j=1 (cid:18) (cid:19) (cid:18) (cid:19) X The proof of he thorem is a combination of those of Theorem 2.1 and 2.3. (cid:3) Example 2.6. Let us consider in C4 the domain defined by 2Rez +|z6|2 +|z4 −z z |2 +|z4 −z3 +z |2 < 0. 4 1 2 1 2 3 2 1 Here γ = 1, γ = 1 and γ = 3 ; we have ǫ-subelliptic estimates for 1 6 2 6·4 3 6·4·4 ǫ = γ3. 2 References [1] D.Catlin—Necessaryconditionsforsubellipticityofthe∂¯-Neumannprob- lem, Ann. of Math. 117 n.2 (1983), 147-171 [2] D. Catlin—Boundaryinvariantsofpseudoconvexdomains,Ann. of Math. 120 (1984), 529-586 [3] D. Catlin—Subelliptic estimates for the ∂¯-Neumann problem on pseudoconvex domains, Ann. of Math. 126 (1987), 131-191 10 T.V. KHANHAND G. ZAMPIERI [4] D. Catlin and J.S. Cho—Sharp estimates for the ∂¯-Neumann problem on regular coordinate domains, arXiv:0811.0830v1, (2008) [5] G.B. Folland and J.J. Kohn—The Neumann problem for the Cauchy- Riemann complex, Ann. Math. Studies, Princeton Univ. Press, Princeton N.J. 75 (1972) [6] J.D’Angelo—Realhypersurfaces,orderofcontact,andapplications,Ann. of Math. 115 (1982), 615–637 [7] J. D’Angelo—Severalcomplex variables and the geometry of real hyper- surfaces, CRC Press (1992) [8] J.J. Kohn—Subellipticity of the ∂¯-Neumann problem on pseudoconvex domains: sufficient conditions, Acta Math. 142 (1979),79–122156 (2002), 213–248 [9] T.V. Khanh and G. Zampieri—Subellipticity of the ∂¯-Neumann prob- lemonaweaklyq-pseudoconvex/concavedomain,arXiv:0804.3112v(2008) Dipartimento di Matematica, Universita` di Padova, via Belzoni 7, 35131 Padova, Italy E-mail address: [email protected], [email protected]