semi-global solutions of ∂¯b with lp (1 ≤ p ≤ ∞) bounds on strongly pseudoconvex real hypersurfaces in cn (n ≥ 3)

Publicacions Matem`atiques, Vol 43 (1999), 535–570. SEMI-GLOBAL SOLUTIONS OF ¯ ∂b WITH Lp(1 ≤p≤∞)BOUNDS ON STRONGLY PSEUDOCONVEX REAL HYPERSURFACES IN Cn(n≥3) C. H. Chang and H. P. Lee Abstract Let Mbe an open subset of a compact strongly pseudoconvex hypersurface {ρ=0}defined by M=D×Cn−m∩{ρ=0}, where 1 ≤m≤n−2, D={σ(z1,... ,z m)<0}⊂Cmis strongly pseudoconvex in Cm.For¯ ∂bclosed (0,q) forms fon M, we prove the semi-global existence theorem for ¯ ∂bif 1 ≤q≤n−m−2, or if q=n−m−1 and fsatisfies an additional “moment condition”. Most importantly, the solution operator satisfies Lpestimates for 1≤p≤∞with p= 1 and ∞included. 1. Introduction and Main Results Let Mbe a connected open subset of a compact strongly pseudoconvex hypersurface {ρ=0}in Cndefined by M=D×Cn−m∩{ρ=0}, where 1≤m≤n−2, D={σ(z1,... ,z m)<0}⊂Cmis strongly pseudoconvex in Cm. We assume that ρ,σ∈C3are strictly plurisubharmonic in neighborhoods of {ρ≤0}and {σ≤0}respectively, and dρ ∧dσ =0 on ∂M.For¯ ∂bclosed (0,q) forms fon M, we study in this paper the semi-global solvability of ¯ ∂bu=fin Lpspaces, 1 ≤p≤∞. Global solution with Lpestimates for ¯ ∂bon compact strongly pseudoconvex hypersurfaces was obtained by Folland-Stein [Fol-St] for 1 ≤q<n−1, and by Henkin [He], Skoda [Sk] for 1 ≤q≤n−1. Using the explicit integral representation for the solution operator, Henkin [He] also gave the first local solution with supnorm estimate (i.e. C0estimate). Local solution with Lpestimates, 1 <p<∞, was obtained by Shaw [Sh1]. Our main results state as follows: Keywords. ¯ ∂b, CR manifold. 1991 Mathematics subject classifications: 32C16, 32F40. Partially supported by NSC grant 84-2121-M-001-011. 536 C. H. Chang, H. P. Lee Theorem 1. Let Mbe as in the above. Then there exists a linear operator Lmapping Lp (0,q)(M)to Lp (0,q−1)(M),1≤p≤∞with (1.1) ||Lf||p≤C||f||p,where Cis independent of p. Moreover, Lfsolves the equation (1.2) ¯ ∂bu=f provided that f∈Lp (0,q)(M)is ¯ ∂bclosed in distribution sense, and either (i) 1 ≤q≤n−m−2,or (ii) q=n−m−1,f∈C1(¯ M)and for any smooth ¯ ∂bclosed (n, m−1) form hdefined in a neighborhood Vhof ∂M, the condition (1.3) ∂M f∧h=0 ∀>0small holds, where M={ρ=0}∩{σ<−}and ∂M⊂Vh. (1.3) is necessary and sufficient for ¯ ∂bto have C1solution in Mat this critical degree. From this theorem we have Corollary 1. Let Mbe as in Theorem 1. The ranges of the ¯ ∂boperator from Lp (0,q−1)(M)to Lp (0,q)(M),1≤p≤∞,1≤q≤n−m−1 are closed. When 1≤q≤n−m−2, they are exactly sets of ¯ ∂b-closed forms in Lp (0,q)(M), and for q=n−m−1, the range is the Lp-closure of ¯ ∂b(C∞ (0,n−m−2)(¯ M)). Corollary 2. Let Mbe as in Theorem 1. The ¯ ∂b-closed (0,q)forms with C∞(¯ M)coefficients, 0≤q≤n−m−2, are dense in ¯ ∂b-closed forms with Lp(M)coefficients, 1≤p<∞. Let σ,ρbe defined as above, one may study the ¯ ∂boperator on the CR manifold ˜ M={ρ=0}∩{σ>0}. Arguments parallel to the proof of Theorem 1 with suitable changes (see Remark 3 in Section 2) give the following: Solutions of ¯ ∂bwith Lp(1 ≤p≤∞)bounds 537 Theorem 1’. Let ˜ Mbe as in the above. Then there exists a linear operator ˜ Lmapping Lp (0,q)(˜ M)to Lp (0,q−1)(˜ M),1≤p≤∞with (1.1’) ||˜ Lf||p≤C||f||p,where Cis independent of p. Moreover, ˜ Lfsolves (1.2) provided that f∈Lp (0,q)(˜ M)is ¯ ∂bclosed in distribution sense, and either (i) m≤q≤n−3,or (ii) q=n−2,f∈C1(¯ ˜ M)and for any (n, 0) form hdefined in a neighborhood Vhof ∂˜ Mwith holomorphic coefficient, the moment condition (1.3’) ∂˜ M f∧h=0 ∀>0small holds, where ˜ M={ρ=0}∩{σ>}and ∂˜ M⊂Vh. (1.3’) is necessary and sufficient for ¯ ∂bto have C1solution in ˜ Mat this critical degree. Results parallel to Corollaries 1 and 2 obviously hold, we leave them to readers. For the critical degree qwhich is n−m−1 for Mand n−2 for ˜ M, we have Corollary 3. Let M, and ˜ Mbe as in the above. (a) Suppose any smooth ¯ ∂b-closed (n, m−1) form hdefined in a neighborhood of ∂M can be approximated by ¯ ∂b-closed forms in C0(¯ M) on ∂M. Then (1.2) is solvable on Mwith Lpestimates at degree q=n−m−1. (b) Suppose any function holomorphic in a neighborhood of ∂˜ Mcan be approximated on ∂˜ Mby functions holomorphic in ¯ ˜ M. Then (1.2) is solvable with Lpestimates for q=n−2on ˜ M. This condition is satisfied if ∂˜ Mis Runge. Combine Theorem 1 and Theorem 1’ together, we can generalize the special example of [B] as follows: Corollary 4. Suppose Mcan also be defined in the form of ˜ M,in other words, there is a real function ˜σ∈C3(C˜m),1≤˜m≤n−2, which is strictly plurisubharmonic such that M=˜ M={˜σ>0}∩{ρ=0}. Then (1.2) is solvable with Lpestimates 1≤p≤∞for 1≤q≤ n−m−2or ˜m≤q≤n−3. 538 C. H. Chang, H. P. Lee Under the same assumption we also have Corollary 5. Let Mbe as in Corollary 4. Suppose in addition that both mand ˜mare greater than 1with ∂M ={˜σ=0}∩{σ=0}, and that {˜σ≤0}∩{σ≤0}has a neighborhood system consisting of Runge domains. Then (1.2) is solvable with Lpestimates 1≤p≤∞for q=n−2. In particular, the assertion holds if there exists a decreasing sequence {j},j→0such that {σ< j}∩{˜σ< j}are convex for all j. This is because Hartogs’ theorem gives that every function holomorphic in a neighborhood of ∂M can be extended holomorphically into a neighborhood of {˜σ≤0}∩{σ≤0}which is Runge by assumption. It turns out that ∂M is Runge. Corollary 5 now follows from Corollary 3. When m= 1, the present paper also provides a uniform proof for Lpestimates 1 ≤p≤∞of local solutions with p= 1 and p=∞ included. Moreover, the constant in (1.1) is independent of p. So our results improve [Sh1] and give a complete and uniform proof for the existence of a local solution of ¯ ∂bwith Lpestimates. The idea of estimation used here is also applied in a forthcoming paper [C-L3] to obtain Lpestimates for ¯ ∂-operators in the piecewise smooth pseudoconvex domain {ρ<0}∩D×Cn−m. The results of this paper can be used to obtain Lpestimates for the solution of ¯ ∂bon some special compact piecewise smooth strongly pseudoconvex surfaces. We think this is an interesting direction. It is also interesting to have Lpestimates for ¯ ∂bon larger classes of surfaces than those discussed here. When q=n−m−1 necessary and sufficient conditions for the solvability of ¯ ∂bon Mare given in [C-L1], while in [C-L2] we investigate the (non)solvability of ¯ ∂bat degrees q≥n−m−1onM(respectively, q≤m−1, on ˜ M) and we observe much more complicated phenomenon when m>1. The solvability of ¯ ∂b-operators in various CR-manifolds under various norms have been extensively studied in past few years, see [Ma-Mi], [Ro], [Sh3], [Sh4], [Mi-Sh1], [Mi-Sh2] and references there. We outline here the plan of the paper. In Section 2, we derive the solution operator for ¯ ∂b-closed (0,q) form fwith coefficients in C1(¯ M) (denote by f∈C1 (0,q)(¯ M)). The solution operator consists of two integral operators: one is an integral over Mdefined by Henkin’s kernel Ω(r,r∗) for ¯ ∂bon {ρ=0}; the other one is a boundary integral defined by the kernel Ω(r,r∗,s) which involves not only the Leray sections r,r∗of {ρ=0}but also the Leray section sfor the lower dimensional strongly pseudoconvex domain {σ<0}⊂Cm. Section 2 contains definitions Solutions of ¯ ∂bwith Lp(1 ≤p≤∞)bounds 539 of various kernels and their properties. It was proved in [He] that the kernel Ω(r,r∗) defines a linear operator from Lpto Lp(and even better). Thus our main task is to show the kernel Ω(r,r∗,s) defines an Lp-bounded operator, 1 ≤p≤∞. We sketch the major difficulties and how we overcome them. When Lpestimates are concerned, usually we transform by Stokes’ theorem the boundary integral of f∧Ω(r,r∗,s) to the integral of f∧¯ ∂Ω−(r,r∗,s) over M, where Ω−(r,r∗,s) is an extension of Ω(r,r∗,s)to M. By Classical results of Singular Integral Operators it suffices to prove the L1norms of ¯ ∂Ω−(r,r∗,s)(ζ,z) w.r.t. ζ,zrespectively are bounded uniformly in z,ζrespectively. As sis the Leray section of a strongly pseudoconvex domain in Cm,m<n, the L1(M)-norm of ¯ ∂Ω−(r,r∗,s)(ζ,z) grows logarithmically as zapproaches boundary. To overcome this difficulty, we first transform part of Ω(r,r∗,s) to kernels involving r,r∗,s and the Bochner-Martinelli kernel form Ω(b)ofCm. This kind of transformation was used in Sergeev-Henkin, see [Se-He] for extensive discussions. The resulting kernels then take the advantages of the following properties of Ω(b): (i) ¯ ∂Ω0(b) = 0 off the diagonal, where Ω0(b)isthe component of Ω(b) of degree 0 in d¯z, (ii) 0<a<|ζ|<b Ω0(b) = 0, (iii) apparently its order of singularity is 2m−1 without using any coordinate transformation. This is done in Section 2 under the title Reductions. Let Kdenote an arbitrary component of kernels resulting from the above transformation. To estimate the L1norm of ¯ ∂K, the most difficult part lies in the fact that there are points where ∂ρ/∂zj=0,∀j>m(they are characteristic points when m= 1). This is because j>m |∂ρ/∂zj| is essentially related to the jacobian of the coordinate transformation that will make the L1norm of ¯ ∂Kconverge. By a closer observation of the integrals to be estimated, we see that for most critical terms, their integrands in a certain sense contain factors of the jacobians of the needed coordinate transformations in their numerators. Therefore, we decompose the domains of integration Vby comparing the sizes of jacobians with, say |ζ−z|τ,0<τ<1 to be determined, for ζ∈Cn we write ζ=(ζ,ζ) with ζ∈Cmand ζ ∈Cn−m. To illustrate, e.g. we let V=V1∪V2, where V1=  ζ∈V: j>m |∂ρ/∂zj|≤|ζ−z|τ  , V2=  ζ∈V: j>m |∂ρ/∂zj|>1 2|ζ−z|τ  . 540 C. H. Chang, H. P. Lee Then in V1the order of singularity is reduced by τ, so with suitably chosen τ, the integral over V1is finite, while in V2we are able to perform coordinate transformation with controllable jacobian. On the other hand, to guarantee that coordinate transformations are of finite multiplicity, we incorporate the technique first introduced by Range-Siu [Ra-Siu]. We remark that the strong pseudoconvexity also plays an important role, see (3.15) and Section 4. These estimates are summarized as three key lemmas stated in Section 3 and proved in Section 4. Using these lemmas we prove the L1norm of ¯ ∂Kis bounded by a constant independent of p. Thus assertions of Theorem 1 hold for ¯ ∂bclosed f∈C1 (0,q)(¯ M). We then apply the classical mollification procedure of K. O. Friedrichs to complete the proof of Theorem 1. These are contained in Section 3. We would like to thank Mei-Chi Shaw for very helpful discussions. 2. Notations, the Solution Operator and Reductions Throughout the paper, the constants C,cdenote positive numbers which may vary from time to time. We adopt the convention that Cn={(x1+√−1x2,... ,x 2j−1+ √−1x2j,... ,x 2n−1+√−1x2n)}∼ ={(x1,...,x 2n)}=R2n. In this paper, we often write ζ∈Cnas (ζ,ζ) where ζ∈Cmand ζ ∈Cn−m, (respectively, x=(x,x ) where x∈R2mand x ∈R2(n−m)). Similarly, for differential forms we write df =(df,df), and ∂f =(∂f,∂f), where d·,d·denote respectively the differentials with repect to the first 2mvariables and those to the last 2(n−m) variables, likewise for ∂· and ∂·. Also we have dζ =dζ1∧···∧dζn,dζ=dζ1∧···∧dζmand dζ =dζm+1 ∧···∧dζn. We denote by |f|=( |fI|2)1 2the length of aq-form f=fIdxI. For fixed a∈Cmthe notation Γadenotes the fibre {(a,ζ)∈M}of Mover a. We denote by xˆ j:= {x1,... ,ˆxj,... ,x 2n}, the coordinate system obtained from that of R2nby deleting the coordinate function xj; and we denote by dxˆ j:= dx1∧···∧ dxj∧···∧dx2n, the corresponding 2n−1 form. The notation Bδ(z) denotes the Euclidean ball centered at z∈Cn with radius δ.“AB” means the quotient |A|/|B|is a nonvanishing function bounded from above. We recall that Mis an open subset of the real hypersurface {ρ=0}. Let vMbe the measure on Minduced by the Lebesgue measure in the Solutions of ¯ ∂bwith Lp(1 ≤p≤∞)bounds 541 Euclidean space. Let νbe the 1-form dual to the unit normal at points of {ρ=0}, explicitly, ν=|dρ|−1dρ =|dρ|−1(n 1∂ρ/∂zjdzj+∂ρ/∂ ¯zjd¯zj). Then dvMis the (2n−1)-form such that ν∧dvM=(−2i)−ndz ∧d¯z= dx1∧dx2∧···∧ dx2n=dvR2nas forms in R2n. It is well-known that any differential form defined in a neighborhood of Min R2ncan be uniquely written on Mas the sum of the component tangential to Mand that normal to M. In particular, if the differential form gis of degree 2n−1, its tangential component can be expressed in terms of dvM, i.e. g=gMdvM+dρ∧R, where Risa(2n−2)-form, and it is clear that it is gMthat matters for integrations over M. In view of the above, the function gMcan be determined by (2.2) gMdvR2n=|dρ|−1dρ ∧g. gMdvMis referred as the component of gtangential to M. In arguments followed we will use the general area and coarea formula given in [Si], see also [F]. We also need consider various maps gfrom l-dimensional submanifolds M⊂RNto Rk. When l≤kthe generalized jacobian is defined by Jg(x)={det(dgx)∗◦(dgx)}1 2, and dgx:TxM→ Rkdenotes the induced linear map. And when l>kthe generalized jacobian is defined by J∗g(x)={det(dgx)◦(dgx)∗}1 2. In particular, we note that Remark 1. If M={*=0}is a real hypersurface in RN, and gis a map from Mto RNsuch that one of the component of g, say g1is *, then Jg =|dg1∧···∧dgN|/|d*|. The Solution Operator. We introduce the following notations and exterior calculus developed by Harvey and Polking [H-P]: Let E1,... ,Eαbe a collection of n-tuples of C2functions in (ζ,z)∈ Cn×Cn, following Harvey-Polking we define (2.3) Ω(E1,... ,Eα)= E1,dζ E1,ζ−z∧···∧ Eα,dζ Eα,ζ−z ∧ λ1+···+λα=n−α¯ ∂ζ,zE1,dζ E1,ζ−zλ1 ∧···∧¯ ∂ζ,zEα,dζ Eα,ζ−zλα where x, y=xiyifor vectors x,yin Cnand dζ here is understood to be the n-vector (dζ1,... ,dζ n). Then Ω is C1away from the singular set 542 C. H. Chang, H. P. Lee A=α 1{(ζ,z),Ej,ζ−z=0}. We can rewrite Ω as Ω(E1,... ,Eα)= n−1 0Ωq(E1,... ,Eα), where Ωqis the sum of components of Ω which are of degree qin d¯z. Outside the singular set Awe have the following identity: (2.4) ¯ ∂ζ,zΩ(E1,... ,Eα)= j (−1)jΩ(E1,... , Ej,... ,Eα). To construct the kernel we need following results of [For]: For any strongly pseudoconvex domain X={*<0}⊂CN, where *∈Ck,k≥2, is strictly plurisubharmonic in a neighborhood of ¯ X, there exists >0 and H(ζ,z)∈Ck−1(X×X), X={z∈Cn,*(z)<} satisfying (2.5) H(ζ,·) is holomorphic in X, (2.6) ∃hj(ζ,z)∈Ck−1(X×X), j=1,... ,N, holomorphic in z, such that H(ζ,z)= N  1 hj(ζ,z)(ζj−zj), (2.7) ∃c>0, such that ∀z∈¯ X,ζ∈¯ X 2ReH(ζ,z)≥*(ζ)−*(z)+c|ζ−z|2, (2.8) dζH(ζ,z)|z=ζ=∂*(ζ). For the strongly pseudoconvex domain {ρ<0}⊂Cn, let r(ζ,z)= (r1,... ,rn), rj,j=1,... ,n be the n-tuple function and its components corresponding to (2.6). We use s(ζ,z), sj,j=1,... ,m to denote those for {σ<0}⊂Cm. We denote by s(ζ,z) the map (s1(ζ,z),... ,sm(ζ,z),0,... ,0). Let r∗(ζ,z)=(r∗ 1(ζ,z),... ,r∗ n(ζ,z)), where r∗ j(ζ,z)=−rj(z,ζ). Thus r,r∗and sare C2in a neighborhood of M×M. We remark that (2.7) implies there exists c>0 such that −σ(ζ)+Res,ζ−z≥−σ(ζ)/2−σ(z)/2(2.9) +c|ζ−z|2≥c|ζ−z|2,for ζ, z ∈¯ D, Rer,ζ−z≥c|ζ−z|2,for ζ, z ∈¯ M,(2.10) Rer∗,ζ−z≥c|ζ−z|2,for ζ, z ∈¯ M.(2.11) Solutions of ¯ ∂bwith Lp(1 ≤p≤∞)bounds 543 We define Ω(r,r∗), Ω(r,s), Ω(r∗,s), and Ω(r,r∗,s), according to formula (2.3). In view of the fact that sdepends only on ζ,z, we see that the exponents of ¯ ∂ζs,dζ s,ζ−zin Ω(r,s), Ω(r∗,s) and Ω(r,r∗,s) must be ≤m−1. For any (0,q) form fon Mwhose coefficients are C1up to ¯ M, following Shaw [Sh1] (see also Henkin [He]), we have for z∈M, (2.12) (−1)qf(z)=¯ ∂bM f∧Ω(r,r∗) +M ¯ ∂bf∧Ω(r,r∗)−∂M f∧Ω(r,r∗). In fact, in (2.12) Mcan be any open subset of {ρ=0}with C2boundary. Using the identity (2.4), we can rewrite the boundary integral in (2.12) as (2.13) ∂M f∧Ω(r,s)−∂M f∧Ω(r∗,s)−∂M f∧¯ ∂ζΩ(r,r∗,s) +(−1)q+1 ¯ ∂b∂M f∧Ω(r,r∗,s). Since Ω(r,s)isan(n, n−2) form in ζ,f∧Ω(r,s)isan(n, n−2+q) form and q≥1, so the integral against ∂M must be null by type consideration. As for ∂M f∧Ω(r∗,s), we note that f∧Ω(r∗,s)isan(n, n−2) form in ζ only when β=n−2−q, where βis the exponent of ¯ ∂ζs,dζ s,ζ−zin Ω(r∗,s). But as β≤m−1, ∂M f∧Ω(r∗,s) is zero if q≤n−m−2 again by type consideration. If q=n−m−1, then Ω(r∗,s)isa¯ ∂b-closed form in a neighborhood of ∂M, thus for (0,n−m−1) form fthat satisfies (1.3) ∂M f∧Ω(r∗,s) is null. We therefore arrive at the following homotopy formula for (0,q) forms fwith 1 ≤q≤n−m−2, or q=n−m−1 and fsatisfies (1.3). (2.14) (−1)qf(z)=M ¯ ∂bf∧Ωq(r,r∗)(ζ,z) +(−1)q+1 ∂M ¯ ∂bf∧Ωq(r,r∗,s)(ζ,z) +¯ ∂bM f∧Ωq−1(r,r∗)(ζ,z) +(−1)q∂M f∧Ωq−1(r,r∗,s)(ζ,z) =Sq(¯ ∂bf)+¯ ∂bSq−1f. 550 C. H. Chang, H. P. Lee Estimation of K1(ζ,z). A straightforward calculation shows that |K1(ζ,z)|is bounded by the sum of following functions: |A1dρ ∧r,dζ∧s,dζ∧¯ ∂σ ∧dζ ˆ i1,ˆ i2∧d¯ ζˆ l1,ˆ l2| (|r|+|ζ−z|2)(n−q−α)|ζ−z|2q(|σ|+|s|+|ζ−z|2)(α+1) (3.10) |A2dρ ∧r,dζ∧s,dζ∧dζ ˆ i1,ˆ i2∧d¯ ζˆ j| (|r|+|ζ−z|2)(n−q−α)|ζ−z|2q(|σ|+|s|+|ζ−z|2)α (3.11) |A3dρ ∧r,dζ∧s,dζ∧dζ ˆ i1,ˆ i2∧d¯ ζˆ j| (|r|+|ζ−z|2)(n−q−α+1)|ζ−z|2q(|σ|+|s|+|ζ−z|2)α (3.12) |A4dρ ∧r,dζ∧dζˆ i∧d¯ ζˆ j| (|r|+|ζ−z|2)(n−q−α)|ζ−z|2q(|σ|+|s|+|ζ−z|2)α (3.13) |A5dρ ∧s,dζ∧dζˆ i∧d¯ ζˆ j| (|r|+|ζ−z|2)(n−q−α)|ζ−z|2q(|σ|+|s|+|ζ−z|2)α (3.14) where 1 ≤α≤m−1, i1<i 2,dζ ˆ i1,ˆ i2=dζ1∧···∧ dζi1∧···∧ dζi2∧···∧dζn denotes the (n−2,0) form in Cnwith indice i1,i 2being deleted, likewise for dζˆ i,d¯ ζˆ l1,ˆ l2and d¯ ζˆ j, and Aj,j=1,... ,5 are functions continuous on ¯ M. Specifically, A1contains coefficients of r+r∗and ¯ ∂s,ζ−zas factors; A2contains the coefficient of ¯ ∂(r+r∗); A3contains coefficients of ¯ ∂r,ζ −zand r+r∗;A4contains those of ¯ ∂sand r+r∗, while A5 contains those of ¯ ∂rand r+r∗. Thus when |ζ−z|<δ,A1,A4,A5are O(|ζ−z|) functions, while A2=O(1) and A3=O(|ζ−z|2) by (3.9). In view of (2.9)-(2.11) and (3.9), we see that Lemma 1 implies that (3.10)-(3.14) are integrable over the subregion {|ζ−z|<d}∩Bδ(z). As for points in W=R2d∩{|ζ−z|<d}∩{|ζ−z|>δ}, (2.10) and (2.11) imply that |r|+|ζ−z|2,|ζ−z|2are bounded away from zero by constants indepedent of ζ,z. We cover Wby two parts: W1={ζ∈W:|∂ρ(ζ)|≤|ζ−z|1 2} W2={ζ∈V:|∂ρ(ζ)|>1 2|ζ−z|1 2}. For points ζwith |∂ρ(ζ)|≤|ζ−z|t,0<t≤1, the fact that Re∂ρ,ζ −z≥c|ζ−z|2implies (3.15) |ζ −z|≤c|ζ−z|t 2 for some constant cindependent of ζ,z. Solutions of ¯ ∂bwith Lp(1 ≤p≤∞)bounds 551 Thus (3.15’) |ζ−z|≤c|ζ−z|t 2. Denote by Dthe denominator of (3.10), then for ζ∈W1(3.15’) with t=1 2and (2.9)-(2.11) give D−1|ζ−z|−(2n−2α+8)(|ζ−z|2)−α(|ζ−z|2)−α. Since α≤m−1, it is easy to see that the integration of (3.10) over W1is bounded by a constant independent of z. In case {|∂ρ|>1 2|ζ−z|1 2}, we use general coarea formula applied to the projection map πfrom Cnto Cm. Elementary calculation shows that J∗π=|dρ|. Thus the integral of (3.10) over W2is bounded by Cπ(W2) (|σ|+|s|+|ζ−z|2)−(α+1)|ζ−z|−1 2,1≤α≤m−1. The standard coordinate transformation for {σ<0}⊂Cmas defined in [Ra, Chapter V, Lemma 3.4], denoted here as h zand an elementary integration give a finite bound for this integral which is independent of z. Similarly, integrals of (3.11)-(3.14) over Ware bounded by constants independent of λ,z. In summary R2d∩{|ζ−z|<d}|K1(ζ,z)|dv(ζ)<C, Cis independent of λ,z. Estimation of K2(ζ,z). A straightforward calculation shows that |K2(ζ,z)|is bounded by the sum of the following functions: |B1dρ ∧r,dζ∧dζˆ i∧d¯ ζˆ j| (|r|+|ζ−z|2)(n−q−m)|ζ−z|2q|ζ−z|2m−1 (3.16) |B2dρ ∧r,dζ∧dζˆ i∧d¯ ζˆ j| (|r|+|ζ−z|2)(n−q−m+1)|ζ−z|2q|ζ−z|2m−1 (3.17) |B3dρ ∧dζ ∧d¯ ζˆ j| (|r|+|ζ−z|2)(n−q−m)|ζ−z|2q|ζ−z|2m−1 (3.18) where i>mby type consideration, and B1,... ,B 3are functions continuous on M, with the property that when |ζ−z|<δ,B1=O(1), B2=O(|ζ−z|2) and B3=O(|ζ−z|). 552 C. H. Chang, H. P. Lee We proceed as in estimating the integral of |K1(ζ,z)|. For fixed z∈Rd, the integrability of (3.16)-(3.18) over {|ζ−z|<d}∩Bδ(z) follows from (3.7) of Lemma 3. And integrations over Ware calculated by methods similar to those for (3.10)-(3.14). Thus R2d|K2(ζ,z)|dv(ζ) is bounded by a constant independent of z. Estimation of K3(ζ,z). |K3(ζ,z)|is bounded by the sum of the following functions: |C1dρ ∧r,dζ∧s,dζ∧¯ ∂σ ∧dζ ˆ i1,ˆ i2∧d¯ ζˆ l1,ˆ l2| (|r|+|ζ−z|2)n−q−m|ζ−z|2q(|σ|+|s|+|ζ−z|2)m−α+1|ζ−z|2α−1 (3.19) |C2dρ ∧r,dζ∧s,dζ∧dζ ˆ i1,ˆ i2∧d¯ ζˆ j| (|r|+|ζ−z|2)n−q−m|ζ−z|2q(|σ|+|s|+|ζ−z|2)m−α|ζ−z|2α (3.20) |C3dρ ∧r,dζ∧dζˆ i∧d¯ ζˆ j| (|r|+|ζ−z|2)n−q−m|ζ−z|2q(|σ|+|s|+|ζ−z|2)m−α|ζ−z|2α−1 (3.21) where 1 ≤α≤m−1, and i1≤m,i2>m,i>mby type considerations. And Cl=O(1), l=1,... ,3 are functions continuous on M. For fixed z∈Rd, the integrability of (3.19)-(3.21) over {|ζ−z|< d}∩Bδ(z) follows from Lemma 2. For the integral of |K3|over W, we argue as in the proof of integrability for K1(ζ,z) and get that it is bounded by a constant independent of z. We note that all estimates are valid if we drop the subscript λ.On the other hand, it is obvious that for each fixed zand for almost all ζ, we have lim λ→0|d¯ ζJ∧Sα λ(ζ,z)|=|d¯ ζJ∧¯ ∂ζ(θα(r,r∗)∧ωα −(s))(ζ,z)| lim →0|d¯ ζJ∧Sm (b)(ζ,z)|=|d¯ ζJ∧¯ ∂ζ(θm(r,r∗)∧Ω0(b))(ζ,z)| lim →0|d¯ ζJ∧Sm (s−,b )(ζ,z)|=|d¯ ζJ∧¯ ∂ζθm(r,r∗)∧¯ ∂ζΩ0 −(s,b )(ζ,z)|. Solutions of ¯ ∂bwith Lp(1 ≤p≤∞)bounds 553 Lebesgue convergence theorem gives that the coefficient of the (n, n) form dρ ∧d¯ ζJ∧Sα λconverges in L1(M) to the coefficient of dρ ∧d¯ ζJ∧ ¯ ∂ζ(θα(r,r∗)∧ωα −(s)), likewise, the coefficient of dρ∧d¯ ζJ∧Sm (b) converges to that of dρ∧d¯ ζJ∧¯ ∂ζ(θm(r,r∗)∧Ω0(b)) in L1(M) and the coefficient of dρ∧d¯ ζJ∧Sm (s−,b ) converges in L1(M) to that of dρ∧d¯ ζJ∧¯ ∂ζθm(r,r∗)∧ ¯ ∂ζΩ0 −(s,b ). We denote by Sα,Sm(b)and Sm(s−,b )operators defined respectively by forms ¯ ∂ζ(θα(r,r∗)∧ωα −(s)),¯ ∂ζ(θm(r,r∗)∧Ω0(b)) and ¯ ∂ζθm(r,r∗)∧¯ ∂ζΩ0 −(s,b ). Estimations of T,λ(b),T α ,λ(s−,b )and T,λ. As for T,λ(b), we recall that T,λ(b)=−2mφ|ζ−z| ¯ ∂ζθm λ(r,r∗)∧b,dζ∧d¯ ζ,dζm−1 and |dρ ∧d¯ ζJ∧T,λ(b)|≤dρ ∧φ|ζ−z| d¯ ζJ∧¯ ∂ζθm(rλ,r∗ λ)∧Ω0(b). (3.7) of Lemma 3 shows that the integral of |dρ∧d¯ ζJ∧T,λ(b)|is bounded by a constant of the order τ|log |for some appropriately chosen τ<1 (τindependent of z,λ). As goes to zero the integral diminishes to zero, we conclude that (3.22) lim →0lim λ→0|dρ ∧d¯ ζJ∧T,λ(b)|=0. Similarly, since Tα ,λ(s−,b ) =−2αφ|ζ−z| ¯ ∂ζθm λ(r,r∗)∧¯ ∂ζ(ωm−α −(s)∧b,dζ∧d¯ ζ,dζα−1), we have |dρ ∧d¯ ζJ∧Tα ,λ(s−,b )|≤dρ ∧d¯ ζJ∧φ|ζ−z| ¯ ∂ζθm λ(r,r∗) ∧¯ ∂ζωm−α −(s)∧b,dζ b,ζ−z∧d¯ ζ,dζ b,ζ−zα−1. 554 C. H. Chang, H. P. Lee As the coefficient of the right hand side of the above inequality are bounded by the sum of the integrands of (3.4)-(3.6), Lemma 2 shows that the integral of |dρ ∧d¯ ζJ∧Tα ,λ(s−,b )|is bounded by a constant of order τ|log |for some appropriately chosen τ<1(τindependent of z, λ). Therefore, we have lim →0lim λ→0|dρ ∧d¯ ζJ∧Tα ,λ(s−,b )|=0. It remains to estimate d¯ ζJ∧T,λ.AsT,λ =−2mφ(|ζ−z| )θm λ(r,r∗)∧ d¯ ζ,dζm, the length of the tangential part of d¯ ζJ∧T,λ is bounded by the following −1|O(|ζ−z|)dρ ∧r,dζ∧dζˆ i∧d¯ ζˆ j| (|r|+|ζ−z|2)|(n−q−m)|ζ−z|2q|ζ−z|(2m−1) where by type consideration both i,jare >m. Then (3.8) of Lemma 3 gives that the integral of |dρ∧d¯ ζJ∧T,λ|over Mis bounded by a constant independent of λ,z(and even independent of ). This holds if we drop the subscript λin T,λ, we denote such form by T. Thus we have that the coefficient of dρ ∧d¯ ζJ∧T,λ as well as that of dρ ∧d¯ ζJ∧Tare in L1(M). Lebesgue convergence theorem implies that the coefficient of dρ ∧d¯ ζJ∧T,λ converges in L1to that of dρ ∧d¯ ζJ∧T. On the other hand, we observe that for any ¯ ∂b-closed (0,q)-form fand 0 < 1< 2<d, M∩{|ζ−z|=2} f(ζ)∧θm(r,r∗)∧Ω0(b) −M∩{|ζ−z|=1} f(ζ)∧θm(r,r∗)∧Ω0(b) ={1<|ζ−z|<2}∩M f(ζ)∧¯ ∂ζθm(r,r∗)∧Ω0(b). In particular, if f=d¯ ζJ(3.7) of Lemma 3 (or the estimate of K2(ζ,z)) asserts that the right hand side of the above formula is bounded by |1 8 2log 2−1 8 1log 1|, this and (3.22) imply that the coefficient of the (n, n) form dρ ∧d¯ ζJ∧Tforms a Cauchy sequence in L1(M), hence it converges in L1.We denote by Tthe operator which assigns to all possible monomial d¯ ζJthe limit of dρ ∧d¯ ζJ∧Tas →0. Solutions of ¯ ∂bwith Lp(1 ≤p≤∞)bounds 555 Let T=αSα+Sm(b)+Sm(s−,b )+T. We have shown that if we denote by K(ζ,z) the tangential part of d¯ ζJ∧T, then for each fixed z∈Rd, the integral of |K(ζ,z)|over R2dis bounded by a constant independent of z. In view of the definition of r∗, (2.9)-(2.11) and (3.9), etc., the whole process of estimation is valid with minor modifications if we switch roles of ζand z. Classical results of singular integrals (see e.g. [Ra, Appendix]) then implies K(ζ,z) defines an operator which maps Lp(M) to itself, 1 ≤p≤∞. We have proved the assertion of Theorem 1 for ¯ ∂b-closed (0,q)forms with coefficients in C1(¯ M). To complete the proof of Theorem 1, we proceed as in [Sh1], for 1 ≤p<∞, let f∈Lp (0,q)(M)be¯ ∂b-closed in distribution sense, we approximate fby a mollified sequence of (0,q) forms fk given by the mollification method of Friedrichs [Fr] whose coefficients are C1in a neighborhood of ¯ Mk, where Mk={ρ=0}∩{r<1 k}, and fk→f, ¯ ∂bfk→0inLp (0,q)(M), Lp (0,q+1)(M) respectively. If 1 ≤q≤n−m−2, Remark 2 and previous arguments applied to Mkimplies that there exists gk∈C1(Mk) such that ¯ ∂bgk=¯ ∂bfkand ||gk||p→0. Since both fkand gkare C1in a neighborhood of ¯ Mk+1, the above assertion implies there exists uk∈C1(Mk+1) such that ¯ ∂buk=fk−gkand ||uk||p≤C||fk−gk||p in Mk+1. We remark that from the above proof for ¯ ∂b-closed C1 (0,q)(¯ M) forms it is clear that the constant can be chosen independent of small perturbations of the domain M, so it depends neither on pnor on k. As the solution operator established via (2.14) is linear, we see that uk+l−uksolves ¯ ∂bv=fk+l−gk+l−fk+gkon Mk+l+1 and satisfies ||uk+l−uk||Lp(Mk+l+1)≤C||fk+l−fk−(gk+l−gk)||Lp(Mk+l+1)→0, when k→∞. Therefore, {uk}forms a Cauchy sequence in Lp(M), its limit u must satisfy ¯ ∂bu=f, and ||u||p≤C||f||p. For the case p=∞, let {fk}be a fixed mollified sequence of fobtained by Friedrichs’ method. Then, as f∈L∞ (0,q)(M)isinLp (0,q)(M) for all 1 ≤p<∞, we see that fk∈Lp(Mk), fk→f,¯ ∂bfk→0inLp(M) norms for all 1 ≤p<∞.We argue as in the proof for p<∞to obtain gk’s, and uk’s by the very solution operator established via (2.14). It is clear that they are independent of p. Now the same argument also gives that {uk}forms a Cauchy sequence in Lp(M) for all 1 ≤p<∞. So, for each 1 ≤p<∞there exists a(0,q−1) form up∈Lp(M) such that ¯ ∂bup=fand ||up||p≤C||f||p, where Cis independent of p,1≤p<∞. However, it is easy to check that up=up,∀p, p∈[1,∞), as Mis a bounded domain. We denote it by u. Well-known results in Function Spaces then imply that uis in L∞ (0,q−1)(M) and ||u||∞≤C||f||∞. We have completed the proof of the Lp-boundedness of the solution operator. Corollaries 1 and 2 are derived easily from Theorem 1 and its proof. 556 C. H. Chang, H. P. Lee 4. Proofs of Lemmas We note that R2dcan be written as finite union of subsets Ri,j, such that in Ri,j,|∂ρ ∂xi ∂σ ∂xj−∂ρ ∂xj ∂σ ∂xi|≥κ, where 2n2κis the lower bound of |dρ ∧dσ|on ¯ R2d. And it can also be written as finite union of Rk,l where in Rk,l,|∂ρ ∂ζk ∂σ ∂ζl−∂ρ ∂ζl ∂σ ∂ζk|≥|∂ρ ∂ζp ∂σ ∂ζq−∂ρ ∂ζq ∂σ ∂ζp|,∀p, q. Clearly, estimation over each V∩Ri,j ∩Rk,l suffices to give proofs of Lemmas 13. Fix an arbitrary Ri,j ∩Rk,l, it will be apparant from the proof that we may assume without loss of generality, say i, k =1,j, l= 2, and for simplicity denote again by Vthe intersection of it with V. We remark that if |∂ρ|>cfor some positive constant cin V, using standard coordinate transformations for strongly pseudoconvex domains and strongly pseudoconvex hypersurfaces, it is not hard to see Lemmas 1-3 hold. Difficulties arise because of the presence of points where ∂ρ= 0. Therefore, we have to divide regions of integration into subregions where one could perform “coordinate transformations” with controllable jacobians. And the use of (3.15), (3.15’) in proofs demonstrates the strong pseudoconvexity of {ρ<0}also plays an important role. Proof of Lemma 1: (A) Proof of (3.1): Let V1={ζ∈V:|∂ρ(ζ)|≤|ζ−z|1 8,|∂ρ ∧∂σ|≤|ζ−z|τ} V2={ζ∈V:|∂ρ(ζ)|≤|ζ−z|1 8,|∂ρ ∧∂σ|>1 2|ζ−z|τ} V3={ζ∈V:|∂ρ(ζ)|>1 2|ζ−z|1 8,|ζ−z|1 4≥|ζ −z|} V4={ζ∈V:|∂ρ(ζ)|>1 2|ζ−z|1 8,|ζ−z|1 4≤|ζ −z|}. Denote by Jl,l=1,... ,4 respectively integrals over Vl. Denote by Dthe denominator of (3.1), then (3.15’) with t=1 8gives (4.1) D−1|ζ−z|−(2n−2p+2)(|σ|+|ζ−z|2)−(p−1 32 ). Solutions of ¯ ∂bwith Lp(1 ≤p≤∞)bounds 557 In Ri,j =R1,2, we define a map Θ1=Θ 1 1,2:R1,2→R2nas follows: we write by y=(y0,... ,y 2n−1) the coordinate in the target space, let (4.2) y0=ρ, y1=σ, y2=x3,... ,y 2n−1=x2n, then the generalized jacobian JΘ1={det(dΘ1 x)∗◦(dΘ1 x)}1 2is given by |∂ρ ∂x1 ∂σ ∂x2−∂ρ ∂x2 ∂σ ∂x1|≥κ>0. Simple topological argument implies that (Θ1)−1(y) consists of finitely many points with its number bounded by a constant independent of y. We now apply the general area formula to the map Θ1and the integral J1and get that J1 |y1|≤C1 |˜y|≤b |y |≤C2 dy1···dy2n−1 |˜y|(2n−2p+2−2τ)(|y1|+|˜y|2)(p−1 32 )b2(τ−31 32 ), if we choose 1>τ>31 32 , where ˜y=(y2,... ,y 2m−1), y =(y2m,... ,y 2n−1), and ˜y=(˜y,y). For the integration over V2, as |∂ρ ∧∂σ|>0, we would like to define a map from Rk,l =R1,2to R2nso that y0=ρ,y1=σ,y2=s, y3=x5,... ,y 2n−2=x2n, and y2n−1=rwith |∂ρ ∂ζ1 ∂σ ∂ζ2−∂ρ ∂ζ2 ∂σ ∂ζ1|2as its generalized jacobian. Yet this map may fail to have finite multiplicity over R2d, so we modify the map using the technique first introduced by Range-Siu [Ra-Siu] (see also Shaw [Sh1]), namely, let Pρ,Pσ,Ps,Pr, be respectively the second order Taylor polynomials of ρ,σ,s, and r expanded at z. They have the following properties: (4.3) ρ(ζ)=Pρ(ζ,z)+o(|ζ−z|2), where ois uniform in z, (4.4) dζρ=dζPρ(ζ,z)+o(|ζ−z|), where ois uniform in z, (4.5) σ(ζ)=Pσ(ζ,z)+o(|ζ−z|2), where ois uniform in z, (4.6) d ζσ=d ζPσ(ζ,z)+o(|ζ−z|), where ois uniform in z, (4.7) s=Ims,ζ−z=Ps(ζ,z)+o(|ζ−z|2), where ois uniform in z, (4.8) d ζIms,ζ−z=d ζPs(ζ,z)+o(|ζ−z|), where ois uniform in z, (4.9) r=Imr,ζ−z=Pr(ζ,z)+o(|ζ−z|2), where ois uniform in z, (4.10) dζImr,ζ−z=dζPr(ζ,z)+o(|ζ−z|), where ois uniform in z. 558 C. H. Chang, H. P. Lee We may shrink δso that for |ζ−z|<δand |ζ−z|<δthere exists a constants c>0 independent of zsuch that (4.11) |s|+|ζ−z|2≥c(|Ps(ζ,z)|+|ζ−z|2), (4.12) |σ|+|s|+|ζ−z|2≥c(|Pσ(ζ,z)|+|Ps(ζ,z)|+|ζ−z|2), (4.13) |r|+|ζ−z|2≥c(|Pr(ζ,z)|+|ζ−z|2). These follow directly from Taylor expansions for C2functions and (2.9)-(2.11). For each fixed z∈Rd, we now define the map Θ2:R1,2→R2nby y0=Pρ(ζ,z), y1=Pσ(ζ,z), y2=Ps(ζ,z), yk=xk+2,3≤k≤2n−2, y2n−1=Pr(ζ,z). In view of (4.4), (4.6), (4.8), (4.10), we have (4.14) JΘ2=|dρ ∧dr ∧dσ ∧ds ∧2n 5dxk|+o(|ζ−z|)|∂ρ ∧∂σ| +o(|ζ−z|)|∂ρ ∧∂σ|+O(|ζ−z|2) =|∂ρ/∂ζ1∂σ/∂ζ2−∂ρ/∂ζ2∂σ/∂ζ1|2+o(|ζ−z|)|∂ρ ∧∂σ| +o(|ζ−z|)|∂ρ ∧∂σ|+O(|ζ−z|2). In particular, if |∂ρ ∧∂σ|>|ζ−z|τ, we then have |∂ρ/∂ζ1∂σ/∂ζ2− ∂ρ/∂ζ2∂σ/∂ζ1|>1 2n2|ζ−z|τand (4.14) implies that JΘ2>c|ζ−z|2τ for some constant cindependent of ζ,z. Moreover, we have the following estimates: c≤|∂ρ/∂ζ1∂σ/∂ζ2−∂ρ/∂ζ2∂σ/∂ζ1|2/JΘ2≤C(4.15) |∂ρ ∧∂σ|/JΘ2≤C|ζ−z|−τ.(4.16) Clearly, (4.15) is equivalent to (4.15’) c≤|∂ρ ∧∂σ|2/JΘ2≤C. From (4.12), (4.13) and (3.15’) we have that (4.17) D−1(|Pr|+|ζ−z|2)−(n−q−p+1 2)|ζ−z|−2q−3 4 (|Pσ|+|Ps|+|ζ−z|2)−(p−3 128 ). Solutions of ¯ ∂bwith Lp(1 ≤p≤∞)bounds 559 We now apply the general area formula to the map Θ2and J2, then (4.15’) and (4.17) give J2 |yj|≤C1,j=1,2,2n−1 |˜y|≤b |y |≤C2 dy1···dy2n−1 (|y2n−1|+|˜y|2)(n−q−p+1 2)|˜y|2q+3 4(|y1|+|y2|+|˜y|2)(p−3 128 ) where ˜y=(y3,... ,y 2m−2), y =(y2m−1,... ,y 2n−2), and ˜y=(˜y,y), if m>2; and when m=2,˜yis void. Direct computation gives the integral is bounded by Cb1 4. In case {|∂ρ|>1 2|ζ−z|1 8}, we use general coarea formula applied to πthe projection map from Cnto Cm. Note that J∗π=|dρ|. And for almost all ζ∈π(¯ M) the set V3∩Γζis contained in a 2n−2m−1 submanifold by the Sard-type theorem [Si, p. 56]. For such ζ, we use the coordinate transformation h ζdefined as follows: In the subset of Mwhere ∂ρ= 0, for j>m, let Vj={ζ∈M, ∂ρ=0,|∂ρ/∂ζj|≥|∂ρ/∂ζl|∀l>m}. Fix a Vj, say j=n, the map h ζ from Vn∩Γζ∩Bδ(z)toR2n−2mis defined by t 0=Pρ(ζ,z), t 1=x2m+1, t 2n−2m−2=x2n−2,t 2n−2m−1=Pr(ζ,z). Then previous argument for JΘ2also applies here, and we have Jh ζ≃|∂ρ|−1(|∂ρ/∂ζn|2+O(|ζ− z|)|∂ρ|+O(|ζ−z|2)). In particular, if |∂ρ|>|ζ−z|τ, reasoning as before, Jh ζ≥c|ζ−z|τand c≤|∂ρ|/Jh ζ≤C. Moreover, the map has the property that the inverse image |h ζ −1(t)|is a finite set with uniformly bounded cardinal number independent of ζand t. The condition |ζ−z|1 4≥|ζ −z|implies that Jh ζ|ζ−z|1 8.As for the image π(V)inCm, we use the coordinate transformation h zfor {σ<0}⊂Cmas defined in [Ra, Chapter V, Lemma 3.4] which has its jacobian bounded from below. Thus, J3 |yj|≤C1,j=1,2 |˜y|≤b       |y2n−1|≤C1 |y |≤C2 dy2m+1 ···dy2n−1 (|y2n−1|+|˜y|2)(n−q−p+1 2)|˜y|2q       dy1···dy2m (|y1|+|y2|+|˜y|2)p|˜y|1 4 where ˜y=(y3,... ,y 2m), y =(y2m+1,... ,y 2n−2), and ˜y=(˜y,y). Again direct computation shows that it is bounded by Cb3 4if p>2, and by Cb3 4|log b|if p=2. 566 C. H. Chang, H. P. Lee While in V3, as |ζ−z|κτ ≤|ζ −z|, let ϑ>1 be a constant such that ϑκτ < 1, we have D−1(|r|+|ζ−z|2)−(n−q−m)|ζ−z|−(2q−ϑ) (|σ|+|ζ−z|2)−1|ζ−z|−(2m−2+ϑκτ). Using coordinate transformation ˜ hfor π(V3) and arguing as in the proof of (3.1) for J4, give J3 |yj|≤C1,j=1, |˜y|≤b      Γζ dvΓζ |˜y|2n−2m−ϑ     dy1···dy2m (|y1|+|˜y|2)|˜y|2m−2+ϑκτ b1−ϑκτ |log b| where ˜y=(y2,... ,y 2m), and ˜y=(˜y,x 2m+1,... ,x 2n). The estimation of (3.6) follows the same idea as that of (3.5), and it is simpler, we skip them. Proof of Lemma 3: (A) Proof of (3.7): Let V1={ζ∈V:|∂ρ(ζ)|≤|ζ−z|1 8} V2=ζ∈V:|∂ρ(ζ)|>1 2|ζ−z|1 8,|ζ−z|1 4≥|ζ −z| V3=ζ∈V:|∂ρ(ζ)|>1 2|ζ−z|1 8,|ζ−z|1 4≤|ζ −z|. Denote by Jl,l=1,2,3 respectively integrals over Vl. Since in V1,|∂ρ|≤|ζ−z|1 8is small, we must have |∂ρ|large, say >1 2, and V1could be written as finite union of submanifolds each with xˆ j,j≤2mas local coordinate system, such that coordinate maps defined by y0=ρ,y1=x1,... ,y j=xj+1,... ,y 2m=x2m+1,... ,y 2n−1=x2n, 1≤j≤2mhave their generalized jacobian bounded from below by 1 4n. Therefore, it is obvious we have J1 a≤|y|≤b |y |≤C2 dy1···dy2n−1 |y|2(n−m)|y|(2m−1−1 8)b1 8|log b|−a1 8|log a|, where y=(y1,... ,y 2m−1), y =(y2m,... ,y 2n−1), and y=(y,y). Solutions of ¯ ∂bwith Lp(1 ≤p≤∞)bounds 567 In V2, we use general coarea formula applied to the projection map π, and we use the coordinate transformation h ζfor the fibre Γζ.Asinthe proof for Lemma 2, we have Jh ζ|ζ−z|1 8in V2. We get J2 a≤|y|≤b            |y2n−1|≤C1 |y |≤C2 dy2m+1 ···dy2n−1 (|y2n−1|+|˜y|2)(n−q−m)|˜y|2q            dy1···dy2m |y|(2m−1+ 1 8) b7 8|log b|−a7 8|log a|if a>0, b7 8|log b|if a=0, where y=(y1,... ,y 2m), y =(y2m+1,... ,y 2n−2), and ˜y=(y,y). In V3, again, we use general coarea formula applied to π. The condition |ζ−z|1 4≤|ζ −z|gives that |ζ−z|−1≤|ζ−z|−1 4, thus we have J3 a≤|x|≤b    Γζ dvΓζ |x|2n−2m−2  dx1···dx2m |x|(2m−1+ 1 2)b1 2−a1 2 where x=(x1,... ,x 2m), and we have used the fact that |x|−(2n−2m−2) is integrable over Γζfor almost all ζwith finite bound, since the volume of Γζis bounded by constant independent of ζ. Thus (3.7) holds if we let e=1 8and Clarge. (B) Proof of (3.8): Let V1={ζ∈V:|∂ρ(ζ)|≤|ζ−z|1 2} V2=ζ∈V:|ζ−z|1 4≥|∂ρ(ζ)|>1 2|ζ−z|1 2 V3=ζ∈V:|∂ρ(ζ)|>1 2|ζ−z|1 4. Denote by Jl,l=1,2,3 respectively integrals over Vl. 568 C. H. Chang, H. P. Lee For V1, we argue exactly as in the proof of the corresponding case of (3.7) and get J1 |y|≤b |y |≤C2 dy1···dy2n−1 |y|(2n−2m−1)|y|(2m−2) b where y=(y1,... ,y 2m−1), y =(y2m,... ,y 2n−1), and y=(y,y). To calculate the integral over V2, as in previous proofs, we use general coarea formula applied to π. The condition |∂ρ(ζ)|≤|ζ−z|1 4implies J2 a≤|x|≤b    Γζ dvΓζ |x|2n−2m−11 4  dx1···dx2m |x|(2m−1) b where x=(x1,... ,x 2m), and we have used the fact that |x|−(2n−2m−11 4) is integrable over Γζwith uniform bound for almost all ζ. In V3, we use general coarea formula applied to π, and the coordinate transformation h ζfor the fibre Γζ. The condition |∂ρ(ζ)|>1 2|ζ−z|1 4 implies c≤|∂ρ(ζ)|/Jh ζ≤Cfor some constants c,Cindependent of ζand z. Moreover, |h ζ −1(y)|is bounded by a non-negative integer uniformly in ζand z. 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Institute of Mathematics Academia Sinica Taipei, Taiwan REPUBLIC OF CHINA e-mail: [email protected] e-mail: [email protected]w Primera versi´o rebuda el 2 de setembre de 1998, darrera versi´o rebuda el 26 de maig de 1999