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2nd Reading November 14, 2014 15:14 WSPC/S0129-167X 133-IJM 1450106 International Journal of Mathematics Vol 25, No 11 (2014) 1450106 (15 pages) c World Scientific Publishing Company DOI: 10.1142/S0129167X14501067 ¯ Lp estimates for the ∂-equation on a class of infinite type domains Int J Math 2014.25 Downloaded from www.worldscientific.com by CORNELL UNIVERSITY on 12/17/14 For personal use only Ly Kim Ha Department of Mathematics and Computer Science University of Science, Vietnam National University 227 Nguyen Van Cu Q.5 Ho Chi Minh City, Vietnam lkha@hcmus.edu.vn Tran Vu Khanh Tan Tao University, Tan Tao Avenue Tan Duc e-City, Long An Province, Vietnam Department of Mathematics National University of Singapore Blk S17, 10 Lower Kent Ridge Road Singapore 119076, Singapore khanh.tran@ttu.edu.vn mattvk@nus.edu.sg Andrew Raich Department of Mathematical Sciences SCEN 327, University of Arkansas Fayetteville, AR 72701, USA araich@uark.edu Received October 2013 Accepted 21 October 2014 Published 18 November 2014 We prove Lp estimates, ≤ p ≤ ∞, for solutions to the Cauchy–Riemann equations ¯ = φ on a class of infinite type domains in C2 The domains under consideration are a ∂u ¯ class of convex ellipsoids, and we show that if φ is a ∂-closed (0, 1)-form with coefficients ¯ = φ, then u p ≤ C φ p where the in Lp and u is the Henkin kernel solution to ∂u constant C is independent of φ In particular, we prove L1 estimates and obtain Lp estimates by interpolation ¯ Henkin solution; Henkin operator; Lp estimates for ∂; ¯ infinite type Keywords: ∂; domains Mathematics Subject Classification 2010: 32W05, 32F32, 32T25, 32T99 1450106-1 2nd Reading November 14, 2014 15:14 WSPC/S0129-167X 133-IJM 1450106 L K Ha, T V Khanh & A Raich Introduction A fundamental question in several complex variables is to establish Lp estimates for solutions of the Cauchy–Riemann equation Int J Math 2014.25 Downloaded from www.worldscientific.com by CORNELL UNIVERSITY on 12/17/14 For personal use only ¯ =φ ∂u on domains Ω ⊂ Cn In this paper, we provide the first examples of infinite type domains for which Lp bounds hold, ≤ p ≤ ∞ The domains under consideration ¯ (0, 1)-form are a class of convex ellipsoids in C2 , and we show that if φ is a ∂-closed ¯ = φ, then u p ≤ C φ p where the constant C and u is the Henkin solution to ∂u is independent of φ Specifically, we prove L1 estimates and use the Riesz–Thorin Interpolation Theorem to obtain Lp estimates by interpolating with the L∞ estimates established by Khanh [12] The L∞ estimates of Khanh generalized and were inspired by the L∞ estimates of Fornæss et al [9] We investigate domains of the following form: Ω ⊂ C2 is a smooth, bounded domain with a global defining function ρ such that: for any p ∈ bΩ, there exist a coordinates zp = Tp (z) with the origin at p where Tp is a linear transformation, and functions Fp , and rp such that Ωp = Tp (Ω) = {zp = (zp,1 , zp,2 ) ∈ C2 : ρ(Tp−1 (zp )) = Fp (|zp,1 |2 ) + rp (zp ) < 0} (1.1) or Ωp = Tp (Ω) = {zp = (zp,1 , zp,2 ) ∈ C2 : ρ(z) = Fp (x2p,1 ) + rp (zp ) < 0} where zp,j = xp,j + iyp,j , xp,j , yp,j ∈ R, j = 1, 2, and i = the functions Fp : R → R and rp : C2 → R satisfy: (1) (2) (3) (4) (1.2) √ −1 We also assume that Fp (0) = 0; F (t) Fp (t), Fp (t), Fp (t), and ( pt ) are non-negative on (0, dp ); ∂rp rp (0) = and ∂zp,2 = on bΩ with |z1,p | ≤ δ; rp is convex, where dp is the square of the diameter of Ωp and δ is a small number We will also require that each Fp satisfies a certain log integrability property This class of domains includes two well-known examples If Ω is of finite type 2m, then Fp (t) = tm at the points of type 2m On the other hand, if Fp (t) = exp(−1/tα ), then Ω is of infinite type at p, and this is our main case of interest We call our domains Ω ellipsoids because they are generalizations of real and complex ellipsoids in C2 Classically, a complex ellipsoid in Cn is a domain of the form {z = (z1 , , zn ) ∈ Cn : nj=1 |zj |2mj < 1}, and a real ellipsoid is a domain of n 2nj + y 2mj ) < 1} where the form {z = (x1 + iy1 , , xn + iyn ) ∈ Cn : j=1 (x mj , nj ∈ N, ≤ j ≤ n Our hypotheses include the following two classes of infinite 1450106-2 2nd Reading November 14, 2014 15:14 WSPC/S0129-167X 133-IJM 1450106 ¯ Lp estimates for the ∂-equation on a class of infinite type domains type domains: exp − Ω = z = (z1 , z2 ) ∈ C2 : |zj |αj j=1 and exp − Ω = z = (x1 + iy1 , x2 + iy2 ) ∈ C2 : α |x j| j j=1 ≤ e−1 ; + exp − |yj |βj ≤ e−1 , where αj , βj ∈ (0, 1) Moreover, our setting also includes domain Int J Math 2014.25 Downloaded from www.worldscientific.com by CORNELL UNIVERSITY on 12/17/14 For personal use only Ω= z = (x1 + iy1 , x2 + iy2 ) ∈ C2 : exp − |x1 |α + χ(y1 ) + |z2 |2 ≤ where χ is a convex function and χ(y1 ) = when |y1 | < δ and < α < This is a tube domain of infinite type at This domain provides an interesting example for studying necessary and sufficient conditions for local regularity of ∂¯ on domains of infinite type In fact, the canonical solution of ∂¯ associated to this tube domain has superlogarithmic estimate and is locally regular if and only if α < (see [1, 6, 14]) ¯ dating back There is a long history of proving Lp estimates for the ∂-equation, to the work of Kerzman [11] and Øvrelid [16] In [15], Krantz proved essentially optimal Lipschitz and Lp estimates on strongly pseudoconvex domains In the case that Ω is a real ellipsoid, Diederich et al obtained sharp Hă older estimates [7] while Chen et al established optimal Lp estimates for complex ellipsoids [4] See also Range [17] and Bruna and del Castillo [2] Both real and complex ellipsoids are domains of finite type, and the analysis in the referenced works depends in an essential fashion on the type In C2 , Chang et al [3] proved Lp estimates for the ¯ ∂-Neumann operator on weakly pseudoconvex domains of finite type See [4, 9] and the references within for a more complete history More recently, there has been work on supnorm estimates for the Cauchy– Riemann equations on infinite type domains in C2 Fornæss et al provided the first examples in [9] and Khanh found that the estimates hold when domains are of the type (1.1) or (1.2) [12] In particular, Khanh proved the following Theorem 1.1 ([12, Theorem 1.2]) If there exists δ > so that (1) Ω is defined by (1.1) and (2) Ω is defined by (1.2) and δ δ |log Fp (t2 )| dt < ∞ for all p ∈ bΩ, or |log (t) log Fp (t2 )| dt < ∞ for all p ∈ bΩ, ¯ then for any bounded, ∂-closed (0, 1)-form φ on Ω, the Henkin solution u on Ω ¯ satisfies ∂u = φ and u L∞(Ω) ≤C φ where C > is independent of φ 1450106-3 L∞ (Ω) , 2nd Reading November 14, 2014 15:14 WSPC/S0129-167X 133-IJM 1450106 L K Ha, T V Khanh & A Raich In this paper, we will prove the Lp -version of Theorem 1.1 Our technique yields Lp -estimates both in the finite and infinite type cases Theorem 1.2 If there exists δ > and either of the following conditions hold : (1) Ω is defined by (1.1) and (2) Ω is defined by (1.2) and δ δ |log Fp (t2 )| dt < ∞ for all p ∈ bΩ, |log(t) log Fp (t2 )| dt < ∞ for all p ∈ bΩ, ¯ then for any ∂-closed (0, 1)-form φ in Lp (Ω) with ≤ p ≤ ∞, the Henkin kernel ¯ = φ and solution u on Ω satisfies ∂u u Lp (Ω) ≤C φ Lp (Ω) , Int J Math 2014.25 Downloaded from www.worldscientific.com by CORNELL UNIVERSITY on 12/17/14 For personal use only where C > is independent of φ The following examples show that the Lp estimates in Theorem 1.2 are sharp in the case of infinite type case Example 1.1 For < α < 1, let Ω be defined by Ω = {(z1 , z2 ) ∈ C2 : e 1− |z 1|α + |z2 |2 < 1} Then for any φ ∈ Lp (Ω) with ≤ p ≤ ∞, there is a solution u of the equation ¯ = φ such that u ∈ Lp (Ω) Moreover, if p = ∞, there is no solution u ∈ Lq (Ω) ∂u with q > p The organization of the paper is as follows: we recall the construction of the Henkin solution via the Henkin kernel in Sec We prove Theorem 1.2 in Sec and discuss Example 1.1 in Sec Henkin Solution In this section, we recall the construction of the Henkin kernel and Henkin solution ¯ For complete details, see [10, 18], or for a more modern treatment, see [5] to ∂ Our construction follows the lines of Khanh [12] and Range [Chap V, Sec 1, 18] to build a Leray map that focuses on the local behavior of the kernel Definition 2.1 A C2 -valued C function G(ζ, z) = (g1 (ζ, z), g2 (ζ, z)) is called a Leray map for Ω if g1 (ζ, z)(ζ1 − z1 ) + g2 (ζ, z)(ζ2 − z2 ) = for every (ζ, z) ∈ bΩ × Ω ¯ so that A support function Φ(ζ, z) for Ω is a smooth function defined near bΩ × Ω Φ admits a decomposition Φj (ζ, z)(ζj − zj ), Φ(ζ, z) = j=1 ¯ holomorphic in z, and vanish only on the where Φj (ζ, z) are smooth near bΩ × Ω, diagonal {ζ = z} 1450106-4 2nd Reading November 14, 2014 15:14 WSPC/S0129-167X 133-IJM 1450106 ¯ Lp estimates for the ∂-equation on a class of infinite type domains ∂ρ ∂ρ For a convex domain, it is well known that G(ζ, z) = ∂ρ ∂ζ = ( ∂ζ1 , ∂ζ2 ) is a Leray map [5, Lemma 11.2.6], and Φ defined by the Leray map ∂ρ(ζ) , ∂ζj Φj (ζ, z) = j = 1, 2, is a support function for Ω ¯ ¯ The solution zj be a bounded, C , ∂-closed (0, 1)-form on Ω Let φ = j=1 φj d¯ ¯ ¯ u of the ∂-equation, ∂u = φ, provided by the Henkin kernel is given by u = T φ(z) = Hφ(z) + Kφ(z) (2.1) Int J Math 2014.25 Downloaded from www.worldscientific.com by CORNELL UNIVERSITY on 12/17/14 For personal use only where Hφ(z) = 2π Kφ(z) = 4π ∂ρ(ζ) ¯ ∂ζ1 (ζ2 ζ∈bΩ Ω ∂ρ(ζ) ¯ ∂ζ2 (ζ1 Φ(ζ, z)|ζ − z|2 − z¯2 ) − − z¯1 ) φ(ζ) ∧ ω(ζ), (2.2) φ1 (ζ)(ζ¯1 − z¯1 ) − φ2 (ζ)(ζ¯2 − z¯2 ) ¯ ω(ζ) ∧ ω(ζ), |ζ − z|4 where ω(ζ) = dζ1 ∧ dζ2 See, for example, [9, 7] To understand the Lp -norm of u, it suffices to investigate the Lp mapping properties of integral operators H and K As a consequence of the Riesz–Thorin Interpolation Theorem and Theorem 1.1, proving that T is a bounded, linear operator on L1 (Ω) suffices to establish that T is a bounded linear operator on Lp (Ω), ≤ p ≤ ∞ The L1 -estimate of |Kφ(z)| is standard and does not require interpolation Indeed, since |ζ − z|−3 ∈ L1 (Ω) in both ζ and z (separately), Lp boundedness of K, ≤ p ≤ ∞, follows from [8, Theorem 6.18] The expression for H given in (2.2) uses the fact that φ is defined on bΩ However, after an integration by parts (done below), H has an expression that is an integral over Ω Once we show that H is bounded on L10,1 (Ω), then the density ¯ ∩ ker ∂¯ in L1 (Ω) ∩ ker ∂¯ finishes the argument (the density argument (Ω) of C0,1 0,1 follows by using the argument of [5, Lemma 4.3.2], keeping in mind that a convex domain is star-shaped and we can ignore the technicalities associated with ∂¯∗ ) Alternatively, Krantz [15, p.257] presents an argument which also discusses how to pass from estimates on smooth forms on the boundary to merely Lp forms in Ω For the boundedness of H, we first begin the analysis of Hφ(z) by using Stokes’ ¯ Theorem Using the assumption that φ is ∂-closed, we observe Hφ(z) = 2π Φ1 (ζ, z)(ζ¯2 − z¯2 ) − Φ2 (ζ, z)(ζ¯1 − z¯1 ) ψ(ρ(ζ)) ∧ φ(ζ) ∧ ω(ζ), ∂¯ζ (Φ(ζ, z) − ρ(ζ))(|ζ − z|2 + ρ(ζ)ρ(z)) Ω where ψ ∈ C ∞ (R) is a cutoff function so that ≤ ψ ≤ and ψ(t) = for t ≥ −δ/8, for t ≤ −δ/4 1450106-5 2nd Reading November 14, 2014 15:14 WSPC/S0129-167X 133-IJM 1450106 L K Ha, T V Khanh & A Raich We abuse notation slightly and let H(ζ, z) be the integral kernel of H As a consequence of Tonelli’s Theorem, it suffices to prove that |H(ζ, z)φ(ζ)|dV (ζ, z) φ L1 (Ω) < ∞ (2.3) (ζ,z)∈Ω×Ω Since H(ζ, z) = when ρ(ζ) ≤ −δ/4, it suffices to prove |H(ζ, z)φ(ζ)| dV (ζ, z) (ζ,z)∈{ζ∈Ω:ρ(ζ)>−δ/4}×Ω φ L1 (Ω) Int J Math 2014.25 Downloaded from www.worldscientific.com by CORNELL UNIVERSITY on 12/17/14 For personal use only Direct calculation shows that Φ1 (ζ, z)(ζ¯2 − z¯2 ) − Φ2 (ζ, z)(ζ¯1 − z¯1 ) ψ(ρ(ζ)) (Φ(ζ, z) − ρ(ζ))(|ζ − z|2 + ρ(ζ)ρ(z)) |H(ζ, z)| ≤ ∂¯ζ |Φ(ζ, z) − ρ(ζ)|2 (|ζ − z|2 + ρ(ζ)ρ(z))1/2 + , |Φ(ζ, z) − ρ(ζ)|(|ζ − z|2 + ρ(ζ)ρ(z)) (2.4) recognizing that |ζ − z| |ζ − z| ≤ (|ζ − z|2 + ρ(ζ)ρ(z)) |ζ − z|(|ζ − z|2 + ρ(ζ)ρ(z))1/2 = (|ζ − z|2 + ρ(ζ)ρ(z))1/2 Since ρ is smooth, ρ is Lipschitz, so (ρ(ζ) − ρ(z))2 |ζ − z|2 Therefore, ρ(ζ)2 |ζ − z|2 + ρ(ζ)ρ(z), hence |ζ − z| + |ρ(ζ)| (|ζ − z|2 + ρ(ζ)ρ(z))1/2 Thus |Φ(ζ, z) − ρ(ζ)| |ζ − z| + |ρ(ζ)| (|ζ − z|2 + ρ(ζ)ρ(z))1/2 (2.5) Combining (2.4) and (2.5), we obtain |Φ(ζ, z) − ρ(ζ)|2 (|ζ − z|2 + ρ(ζ)ρ(z))1/2 |H(ζ, z)| ≤ |Φ(ζ, z) − ρ(ζ)|2 |ζ − z| (2.6) Since bΩ is compact, there exists δ > and points p1 , , pN ∈ bΩ so that bΩ is covered by{B(pj , δ)}N j=1 After changing coordinates to set pj to with the 1450106-6 2nd Reading November 14, 2014 15:14 WSPC/S0129-167X 133-IJM 1450106 ¯ Lp estimates for the ∂-equation on a class of infinite type domains transformation Tpj in Sec 1, we may assume the goal is to prove (ζ,z)∈(Ωpj ∩B(0,δ))×Ωpj (·)) φ(Tp−1 j |H(Tp−1 (ζpj ), Tp−1 (zpj ))φ(Tp−1 (ζpj ))| dV (ζpj , zpj ) j j j L1 (Ωpj ) ≈ φ L1 (Ω) , (2.7) where Ωpj = {ρpj (zpj ) := ρ(Tp−1 (zpj )) = Ppj (zpj ,1 ) + rpj (zpj ) < 0} j Int J Math 2014.25 Downloaded from www.worldscientific.com by CORNELL UNIVERSITY on 12/17/14 For personal use only and Ppj (zpj ,1 ) = Fpj (|zpj ,1 |2 ) or P (zpj ,1 ) = F (x2pj ,1 ) as in Sec Since Φ(Tp−1 (ζpj ), Tp−1 (zpj )) = Φpj (ζpj , zpj ), j j where, Φpj is the support function of Ωpj , we obtain |H(Tp−1 (ζpj ), Tp−1 (zpj ))| j j |Φpj (ζpj , zpj ) − ρpj (ζpj )|2 |ζpj − zpj | (2.8) Here and in what follows, we omit the subscript pj and still use φ(·) for φ(Tp−1 (·)) j Proof of Theorem 1.2 We will investigate the complex and real ellipsoid cases separately to show (2.7) First, however, we recall the following facts for the class of real functions F in the first part with the additional assumption that F (0) = See, for example, [9, Lemma 4] Lemma 3.1 Let F be a C convex function on [0, d] Then F (p) − F (q) − F (q)(p − q) ≥ for any p, q ∈ [0, d] If, in addition, F (0) = and F (3.1) is nondecreasing, then F (p) − F (q) − F (q)(p − q) ≥ F (p − q), (3.2) for any ≤ q ≤ p ≤ d Proof The proof of (3.1) is simple and is omitted here For (3.2), let s := p−q ≥ and g(s) := F (s+ q)− F (q)− sF (q)− F (s) Hence, g (s) = F (s+ q)− F (q)− F (s) and g (s) = F (s + q) − F (s) Using the assumption F (t) is nondecreasing, we have g (s) ≥ 0, thus g (s) is nondecreasing This implies g (s) ≥ g (0) = (since F (0) = 0) and consequently that g(s) is increasing We thus obtain g(s) ≥ g(0) = (since F (0) = 0) This completes the proof of (3.2) 1450106-7 2nd Reading November 14, 2014 15:14 WSPC/S0129-167X 133-IJM 1450106 L K Ha, T V Khanh & A Raich 3.1 Complex ellipsoid case In this subsection, Ω is defined by (1.1) Since the argument of F is |ζ1 |2 , the chain rule shows that ∂ζ∂ F (|ζ1 |2 ) = ζ¯1 F (|ζ1 |2 ) Similarly to Khanh [12, (4.1)], the convexity of r shows that Re{Φ(ζ, z)} − ρ(ζ) ≥ −ρ(z) + F (|z1 |2 ) − F (|ζ1 |2 ) − 2F (|ζ1 |2 )Re{ζ¯1 (z1 − ζ1 )} = −ρ(z) + F (|ζ1 |2 )|z1 − ζ1 |2 + (F (|z1 |2 ) − F (|ζ1 |2 ) − F (|ζ1 |2 )(|z1 |2 − |ζ1 |2 )) (3.3) Int J Math 2014.25 Downloaded from www.worldscientific.com by CORNELL UNIVERSITY on 12/17/14 For personal use only The analysis splits into two cases: (i) F (0) = and (ii) F (0) = In the first case, the hypotheses on F guarantee the existence of a δ > such that F (|ζ1 |2 ) > for any |ζ1 | < δ Hence, Re{Φ(ζ, z)} − ρ(ζ) −ρ(z) + |z1 − ζ1 |2 and |H(ζ, z)| ≤ (|ρ(z)|2 + |ImΦ(ζ, z)|2 + |ζ1 − z1 |4 )|ζ1 − z1 | The estimate in this case is the estimate for the case of a strongly pseudoconvex domain, and the result is classical and well-known Thus, we may assume that F (0) = Lemma 3.2 Let F be defined as in Sec with the additional assumption F (0) = Then, for any |z1 |, |ζ1 | ≤ d (|ρ(z) + iImΦ(ζ, z)|2 + F (|z1 − ζ1 |2 ))|z1 − ζ1 | if |ζ1 | ≥ |z1 − ζ1 |, |H(ζ, z)| if |ζ1 | ≤ |z1 − ζ1 | (|ρ(z) + iImΦ(ζ, z)|2 + F ( 12 |z1 |2 ))|z1 | (3.4) Proof Applying Lemma 3.1 to (3.3), we obtain F (|ζ1 |2 )|z1 − ζ1 |2 Re{Φ(ζ, z)} − ρ(ζ) ≥ −ρ(z) + F (|z1 |2 − |ζ1 |2 ) if < |z1 |, |ζ1 | < d, if |ζ1 | ≤ |z1 | ≤ d (3.5) We now compare the relative sizes of |ζ1 | and |z1 − ζ1 | Case |ζ1 | ≥ |z1 − ζ1 | Combining the first inequality from (3.5) with (ii) from p 2, we obtain Re{Φ(ζ, z)} − ρ(ζ) ≥ −ρ(z) + F (|z1 − ζ1 |2 ) The first line of (3.4) follows by this inequality and (2.6) 1450106-8 2nd Reading November 14, 2014 15:14 WSPC/S0129-167X 133-IJM 1450106 ¯ Lp estimates for the ∂-equation on a class of infinite type domains Case |ζ1 | ≤ |z1 − ζ1 | In this case, the estimate depends on the relative sizes of |ζ1 | and √12 |z1 | If |ζ1 | ≥ √12 |z1 |, then the argument from Case proves that Re{Φ(ζ, z)} − ρ(ζ) ≥ −ρ(z) + F (|z1 − ζ1 |2 )|z1 − ζ1 |2 ≥ −ρ(z) + F |z1 |2 , and we obtain the second estimate in (3.4) Otherwise, |ζ1 | ≤ √12 |z1 |, and this implies both |z1 | ≥ |ζ1 | and |z1 − ζ1 | ≥ (1 − √12 )|z1 | By the second case of (3.5), we observe that |z1 |2 Re{Φ(ζ, z)} − ρ(ζ) ≥ −ρ(z) + F (|z1 |2 − |ζ1 |2 ) ≥ −ρ(z) + F Int J Math 2014.25 Downloaded from www.worldscientific.com by CORNELL UNIVERSITY on 12/17/14 For personal use only and (|Re{Φ(ζ, z)} − ρ(ζ)|2 + |ImΦ(ζ, z)|2 )|ζ1 − z1 | |ρ(z) + iImΦ(ζ, z)|2 + F |z1 |2 |z1 | (3.6) This completes the proof Proof of Theorem 1.2(1) From (2.7), the only remaining estimate is the integral over (ζ, z) ∈ (Ω∩B(0, δ))×Ω We now concentrate on the integral near By Lemma 3.2, we have |H(ζ, z)φ(ζ)| dV (ζ, z) (ζ,z)∈(Ω∩B(0,δ))×Ω = (ζ,z)∈(Ω∩B(0,δ))×Ω and |ζ1 |≥|z1 −ζ1 | ··· + (ζ,z)∈(Ω∩B(0,δ))×Ω,|ζ1 |≤|z1 −ζ1 | and |z1 |≤2δ + (ζ,z)∈(Ω∩B(0,δ))×Ω,|ζ1 |≤|z1 −ζ1 | and |z1 |≥2δ ··· ··· (I) + (II) + (III), (3.7) where (ζ,z)∈(B(0,2δ)∩Ω)2 |φ(ζ)|dV (ζ, z) , (|ρ(z) + iImΦ(ζ, z)|2 + F (|z1 − ζ1 |2 ))|z1 − ζ1 | (ζ,z)∈(B(0,2δ)∩Ω)2 |φ(ζ)|dV (ζ, z) , (|ρ(z) + iImΦ(ζ, z)|2 + F ( 12 |z1 |2 ))|z1 | (I) := (II) := (III) := (ζ,z)∈(B(0,δ)∩Ω)×Ω and |z1 |≥2δ |φ(ζ)|dV (ζ, z) (|ρ(z) + iImΦ(ζ, z)|2 + F ( 12 |z1 |2 ))|z1 | (3.8) 1450106-9 2nd Reading November 14, 2014 15:14 WSPC/S0129-167X 133-IJM 1450106 Int J Math 2014.25 Downloaded from www.worldscientific.com by CORNELL UNIVERSITY on 12/17/14 For personal use only L K Ha, T V Khanh & A Raich It is easy to see that (III) (F (2δ )δ)−1 φ L1 (Ω) For the integral (I), we make the change of variables (α, w) = (α1 , α2 , w1 , w2 ) = (ζ1 , ζ2 , z1 − ζ1 , ρ(z) + iImΦ(ζ, z)) The Jacobian of this transformation is the matrix 0 0 0 0 0 0 J = −1 0 −1 0 ∂ρ(z) 0 0 ∂(Rez1 ) ∂ImΦ(ζ,z) ∂(Reζ1 ) ∂ImΦ(ζ,z) ∂(Imζ1 ) ∂ImΦ(ζ,z) ∂(Reζ2 ) ∂ImΦ(ζ,z) ∂(Imζ2 ) ∂ImΦ(ζ,z) ∂(Rez1 ) 0 0 0 0 0 0 0 0 ∂ρ(z) ∂(Imz1 ) ∂ρ(z) ∂(Rez2 ) ∂ρ(z) ∂(Imz2 ) ∂ImΦ(ζ,z) ∂(Imz1 ) ∂ImΦ(ζ,z) ∂(Rez2 ) ∂ImΦ(ζ,z) ∂(Imz2 ) To justify this coordinate change, we write zj = xj + iyj and compute ∂ Im(Φ(ζ, z)) ∂ρ(z) ∂ Im(Φ(ζ, z)) ∂ρ(z) − ∂y2 ∂x2 ∂x2 ∂y2 By a possible rotation and dilation of Ω, we can assume that ∇ρ(0) = (0, 0, 0, −1) A direct calculation then establishes that if δ is chosen sufficiently small (so that ∂ρ(z) ∂y2 dominates the other partials of ρ and |ζ − z| ≤ 4δ is small), then det(J) = Since Φ is smooth, we can assume that there exists δ > that depends on Ω, δ, and ρ so that |φ(α)| dV (α, w) (I) + F (|w |2 ))|w | (|w | 1 (α,w)∈(Ω∩B(0,δ ))×B(0,δ ) det(J) = δ φ 0 δ φ δ L1 (Ω) L1 (Ω) r1 r2 dr2 dr1 (r22 + F (r12 ))r1 log F (r12 ) dr1 < ∞ That the integral is finite follows by the hypotheses on φ and F 1450106-10 2nd Reading November 14, 2014 15:14 WSPC/S0129-167X 133-IJM 1450106 ¯ Lp estimates for the ∂-equation on a class of infinite type domains Repeating this argument with the change of variables (α, w) = (α1 , α2 , w1 , w2 ) = (ζ1 , ζ2 , z1 , ρ(z)+iImΦ(ζ, z)) for the integral (II), we can obtain the same conclusion Therefore, the estimate in the complex case is complete 3.2 Real ellipsoid case In this subsection, Ω is defined by (1.2) An argument analogous to that for (3.3) yields Re{Φ(ζ, z)} − ρ(ζ) ≥ −ρ(z) + F (ξ12 )(x1 − ξ1 )2 Int J Math 2014.25 Downloaded from www.worldscientific.com by CORNELL UNIVERSITY on 12/17/14 For personal use only + (F (x21 ) − F (ξ12 ) − F (ξ12 )(x21 − ξ12 )), where z1 = x1 + iy1 , ζ1 = ξ1 + iη1 This inequality, in turn, produces a bound of the form Re{Φ(ζ, z)} − ρ(ζ) ≥ −ρ(z) + non − negative This bound and the estimates in (2.6) allow us to reduce to the case of bounding the L1 -norm of |H(ζ, z)φ(z)| near bΩ when |z − ζ| is small Moreover, following the argument in the complex case produces the following lemma Lemma 3.3 Let F be defined as in Sec with the extra assumption that F (0) = Then, for any |x1 |, |ξ1 | ≤ δ if |ξ1 | ≥ |x1 − ξ1 |, (|ρ(z) + iImΦ(ζ, z)|2 + F ((x1 − ξ1 )2 )) (|x1 − ξ1 | + |y1 − η1 |) |H(ζ, z)| if |ξ1 | ≤ |x1 − ξ1 | (|ρ(z) + iImΦ(ζ, z)|2 + F ( 12 x21 )) ( √1 |x | + |y − η |) 1 (3.9) Proof of Theorem 1.2(2) Using Lemma 3.3, we have (ζ,z)∈(Ω∩B(0,δ))2 |H(ζ, z)φ(ζ)| dV (ζ, z) (I) + (II) + (III) (3.10) where (I) := (ζ,z)∈(Ω∩B(0,2δ))2 (II) := (ζ,z)∈(Ω∩B(0,2δ))2 (III) (F (2δ )δ)−1 φ |φ(ζ)| dV (ζ, z) ; (|ρ(z) + iImΦ(ζ, z)|2 + F ((x1 − ξ1 )2 )) (|x1 − ξ1 | + |y1 − η1 |) |φ(ζ)| dV (ζ, z) ; (|ρ(z) + iImΦ(ζ, z)|2 + F ( 12 x21 ))( √12 |x1 | + |y1 − η1 |) L1 (Ω) (3.11) 1450106-11 2nd Reading November 14, 2014 15:14 WSPC/S0129-167X 133-IJM 1450106 L K Ha, T V Khanh & A Raich We make the change of variables (α, w) = (α1 , α2 , w1 , w2 ) = (ζ1 , ζ2 , z1 − ξ1 , ρ(z) + iImΦ(ζ, z)) for (I) and (α, w) = (α1 , α2 , w1 , w2 ) = (ζ1 , ζ2 , √12 x1 − i(y1 − η1 ), ρ(z) + iImΦ(ζ, z)) for (II) Similarly to the argument above, we can check that det(J) = As above, we can therefore assume the existence of δ > depending on Ω, δ, and ρ so that (I) + (II) (α,w)∈(Ω∩B(0,δ ))×B(0,δ ) |φ(α)| dV (α, w) (|w2 |2 + F ((Rew1 )2 )(|Rew1 | + |Imw1 |) δ φ Int J Math 2014.25 Downloaded from www.worldscientific.com by CORNELL UNIVERSITY on 12/17/14 For personal use only δ δ φ δ L1 (Ω) δ L1 (Ω) φ δ L1 (Ω) r2 dr2 d(Imw1 ) d(Rew1 ) (r22 + F ((Rew1 )2 ))(|Rew1 | + |Imw1 |) log(F ((Rew1 )2 ) d(Imw1 ) d(Rew1 ) |Rew1 | + |Imw1 | log(|Rew1 |) log(F ((Rew1 )2 ) d(Rew1 ) < ∞ That the integral is finite follows by the hypotheses on φ and F This completes the proof of Theorem 1.2 Examples In this section, we present an example to show that our estimates are optimal in φ Lp (Ω) cannot hold if ≤ p < q ≤ ∞ the sense that the inequality u Lq (Ω) Specifically, let < α < 1, fix ≤ p < q ≤ ∞, and set Ω = {(z1 , z2 ) ∈ C2 : e 1− |z 1|α + |z2 |2 < 1} (4.1) ¯ We will show that there is a ∂-closed (0, 1)-form φ ∈ Lp0,1 (Ω) for which there does q ¯ = φ in Ω Indeed, let not exist a function u ∈ L (Ω) so that ∂u φ(z) = (1 − log(1 − z2 ))k d¯ z1 (1 − z2 )2/q and v(z) = (1 − log(1 − z2 ))k z¯1 (1 − z2 )2/q k (4.2) )) where k := q+2 + ∈ N The function (1−log(1−z is holomorphic on Ω with qα (1−z2 )1/q ¯ the principle branch of the logarithm < arg(1 − z2 ) < 2π The form φ is a ∂¯ closed (0, 1)-form on Ω and the function v is a solution to the equation ∂v = φ Moreover, we observe that v is L2 -orthogonal to all holomorphic functions on Ω (by Mean Value Theorem) By direct calculation (Lemma 4.1), we obtain φ ∈ Lp0,1 (Ω), v ∈ Lp (Ω), and v ∈ Lq (Ω) Let P be the Bergman projection on Ω, i.e the L2 orthogonal projection onto all holomorphic functions on Ω Recently, Khanh and Raich [13] have proven that P is a bounded operator on the Sobolev spaces Lqs (), s 0, and Hă older spaces (Ω), α > Therefore if u ∈ Lq (Ω) is a solution to 1450106-12 2nd Reading November 14, 2014 15:14 WSPC/S0129-167X 133-IJM 1450106 ¯ Lp estimates for the ∂-equation on a class of infinite type domains ¯ = φ, then v = u − P (u) is in Lq (Ω) This is impossible Therefore, there is no ∂u solution u ∈ Lq (Ω) Lemma 4.1 Let φ and v be defined in (4.2) Then φ ∈ Lp0,1 (Ω), v ∈ Lp (Ω), and v ∈ Lq (Ω) Proof We now show that φ ∈ Lp0,1 (Ω) We have |φ(z)|p dV (v) = Ω Ω Int J Math 2014.25 Downloaded from www.worldscientific.com by CORNELL UNIVERSITY on 12/17/14 For personal use only ≤ |1 − log |1 − z2 | + i arg(1 − z2 )|kp dV (z) |1 − z2 |2p/q |z2 |