Sharper estimates of Ohsawa--Takegoshi L 2 -extension theorem in higher dimensional case
aa r X i v : . [ m a t h . C V ] F e b SHARPER ESTIMATES OF OHSAWA–TAKEGOSHI L -EXTENSIONTHEOREM IN HIGHER DIMENSIONAL CASE SHOTA KIKUCHI
Abstract.
In [6], Hosono obtained sharper estimates of the Ohsawa–Takegoshi L -extentiontheorem by allowing the constant depending on the weight function for a domain in C . In thisarticle, we show the higher dimensional case of sharper estimates of the Ohsawa–Takegoshi L -extention theorem. To prove the higher dimensional case of them, we establish an ana-logue of Berndtsson–Lempert type L -extension theorem by using the pluricomplex Greenfunctions with poles along subvarieties. As a special case, we consider the sharper estimatesin terms of the Azukawa pseudometric and show that the higher dimensional case of sharperestimate provides the L -minimum extension for radial case. Introduction
The
Ohsawa–Takegoshi L -extension theorem [9] states the following: Let Ω ⊂ C n be abounded pseudoconvex domain, V a closed complex submanifold of Ω and ϕ a plurisubhar-monic function on Ω. Suppose that a holomorphic function f ∈ O ( V ) is given. Then thereexists a holomorphic function F ∈ O (Ω) satisfying the L -estimate R Ω | F | e − ϕ ≤ C R V | f | e − ϕ ,where C is the positive constant that is independent of the weight ϕ and a given f ∈ O ( V ).In [3], [4] and [5], the optimal L -extension theorem was proved. This means that wecan determine the positive constant C in the best possible way. In particular, Berndtsson–Lempert [3] proved the optimal L -estimate under the assumption that there exists a negativeplurisubharmonic function G on Ω such that(1) log d V ( z ) − B ( z ) ≤ G ( z ) ≤ log d V ( z ) + A ( z ) , where d V ( z ) is the distance between z ∈ Ω and V , A ( z ) and B ( z ) are continuous functionson Ω. The function B ( z ) appears in the L -estimate as follows: A negative plurisubhar-monic function G satisfying (1) is called a Green-type function on Ω with poles along V [6].Characterizing pairs (Ω , V ) of a domain Ω ⊂ C n and a closed subvariety V ⊂ Ω admitting aGreen-type function with poles along V in terms of their geometry is an open problem.Hosono proposed in [6] an idea of getting an L -estimate sharper than the one of Berndtsson–Lempert type L -extension theorem by allowing constants depending on weight functions.In other words, we can determine the positive constant C ′ depending on weight functionssharper than the optimal constant C . In this article, we call this result Hosono’s sharperestimate . Specifically, the idea used in the proof of Hosono’s sharper estimate is as follows:
Mathematics Subject Classification.
Key words and phrases.
Ohsawa–Takegoshi L extension theorem, pluricomplex Green functions . Let Ω ⊂ C be a bounded domain containing 0 and ϕ a subharmonic function with ϕ (0) = 0.At first, we construct the pseudoconvex domain ˜Ω = { ( z, w ) ∈ C : z ∈ Ω , | w | < e − ϕ ( z ) } in C . For the pseudoconvex domain ˜Ω and a closed complex submanifold ˜ V = { z = 0 } ofit, we construct the Green-type function ˜ G using solutions of the Dirichlet problem for com-plex Monge–Amp`ere equation. By using Berndtsson–Lempert type L -extension theorem for˜Ω, ˜ V and ˜ G , we can get an L -estimate sharper than the one of Berndtsson–Lempert type L -extension theorem. As an application of Hosono’s sharper estimate, in the case where Ωis the unit disc {| z | < } in C and ϕ is a radial subharmonic function, Hosono was able todetermine the L -minimum extension of the function 1 on the subvariety { } .In this article, for a bounded pseudoconvex domain in C n and a closed complex submanifoldof it with some conditions, we generalize Hosono’s sharper estimate. In general, for a closedsubmanifold, it is difficult to construct the Green-type function because we do not knowwhether the logarithmic distance from a closed complex submanifold is plurisubharmonic.To generalize this result, we establish an analogue of Berndtsson–Lempert type L -extensiontheorem by using the theory of the pluricomplex Green function with poles along subvarieties[10]. Our first main theorem is as follows. Theorem 1.1 (Theorem 2.4) . Let Ω be a bounded pseudoconvex domain in C n , V a closedcomplex submanifold of Ω with codimension k such that V has bounded global generators ψ = ( ψ , . . . , ψ k ) and there exists a positive constant C such that C ≤ | J ψ | near V where J ψ is a Jacobian of ψ for suitable coordinates. We state about J ψ in Section 2. Let ϕ be aplurisubharmonic function on Ω and G Ω ,V the pluricomplex Green function on Ω with polesalong V . Assume that there exist some continuous function B on Ω such that log | ψ ( z ) | − B ( z ) ≤ G Ω ,V ( z ) . Then for any holomorphic function f on V with Z V | f | e − ϕ +2 kB < ∞ , there exists a holo-morphic function F on Ω such that F | V = f and Z Ω | F | e − ϕ ≤ Cσ k Z V | f | e − ϕ +2 kB , where σ k is the volume of the unit ball in C k . In Section 2, we give a geometric interpretation for the assumptions of Theorem 1.1.Under the conditions in Theorem 1.1, we consider a pseudoconvex domain ˜Ω in C n + k defined by ˜Ω = { ( z, w ) ∈ C n + k : z ∈ Ω , | w | < e − ϕ ( z ) k } and a closed complex submanifold ˜ V of ˜Ω such that˜ V = { ˜ ψ = · · · = ˜ ψ k = 0 } , where ˜ ψ i ( z, w ) := ψ i ( z ) are holomorphic functions on ˜Ω. Let ˜ G be the pluricomplex Greenfunction on ˜Ω with poles along ˜ V or a subsolution of it such that there exists a continuous IGHER DIMENSIONAL CASE OF SHARPER ESTIMATE 3 function ˜ B ( z, w ) on ˜Ω such thatlog | ˜ ψ ( z, w ) | − ˜ B ( z, w ) ≤ ˜ G ( z, w ) . Then, by using Theorem 1.1, we can obtain the following higher dimensional case of Hosono’ssharper estimate.
Theorem 1.2 (Theorem 3.1) . Under the above setting, the following statements hold. (1)
For any holomorphic function f on V with Z V | f | e − ϕ +2 kB < ∞ , there exists a holo-morphic function F on Ω such that F | V = f and Z Ω | F | e − ϕ ≤ C Z ˜ V | ˜ f | e k ˜ B , where C is a positive constant determined from Theorem 1.1 and the holomorphic function ˜ f on ˜ V is defined by ˜ f ( z, w ) := f ( z ) . (2) Suppose that ˜Ω is a strictly pseudoconvex domain and − B ( z ) is a plurisubharmonicfunction. Then one can make the estimate in (1) strictly sharper than the one in Theorem1.1, i.e., there exist functions ˜ G and ˜ B satisfying the above conditions such that the estimate Z ˜ V | ˜ f | e k ˜ B < σ k Z V | f | e − ϕ +2 kB holds. As an application of Theorem 1.2, we can show that for a unit ball Ω = B n in C n , a closedcomplex submanifold V = { z = · · · = z k = 0 } = { z ′ = 0 } and a radial plurisubharmonicfunction ϕ with respect to V , i.e., ϕ ( z ) = ϕ ( | z ′ | ), one can obtain the L -minimum extensionof holomorphic functions f on V .In addition, we consider the sharper estimates in terms of the Azukawa pseudometric.Specifically, we aim at the comparison with the result which was obtained in [7]. To considerit, we prove the following result. Theorem 1.3 (Theorem 4.3) . Let Ω be a bounded pseudoconvex domain in C n , V be aclosed complex submanifold defined by V = { z = · · · = z k = 0 } and ϕ be a plurisubharmonicfunction on Ω . Let G Ω ,V be the pluricomplex Green function on Ω with poles along V . Weassume that there exists the limit A Ω ,V,w ( X ) := lim λ → ( G Ω ,V ( λX, w ) − log | λ | ) , where (0 , . . . , , w ) ∈ V and = X ∈ C k . We define I Ω ,V,w := { X ∈ C n | A Ω ,V,w ( X ) < } .Then for any holomorphic function f on V with R V vol( I Ω ,V,w ) | f | e − ϕ < ∞ , there exists aholomorphic function F on Ω such that F | V = f and Z Ω | F | e − ϕ ≤ Z V vol( I Ω ,V,w ) | f | e − ϕ . S. KIKUCHI
Using the above result, we study a sharper estimate in a specific example.This article is organized as follows. In Section 2, we define the pluricomplex Green functionwith poles along subvarieties and introduce some properties of it. We also prove Theorem 1.1following the argument of the proof of Berndtsson–Lempert type L -extension theorem [8].In Section 3, we prove Theorem 1.2 by using Theorem 1.1. In Section 4, as a special case,we consider the sharper estimates in terms of the Azukawa pseudometric. In particular, weprove Theorem 1.3 and study a specific example. In addition, as an application of Theorem1.2, in the case where Ω = B n is the unit ball in C n , V = { z = · · · = z k = 0 } is a closedsubmanifold and ϕ is a radial plurisubharmonic function with respect to V , we prove thatone can obtain the L -minimum extension of holomorphic functions f on V .2. Establishment an analogue of Berndtsson–Lempert type L -extensiontheorem Let Ω be a domain in C n and V a anlytic subvariety of Ω. In other words, for any z ∈ V ,there exist a neighborhood U of z and holomorphic functions ψ , . . . , ψ k on U such that V ∩ U = { ψ = · · · = ψ k = 0 } . Let O Ω be the sheaf of germs of locally defined holomorphicfunctions on Ω and I V be the coherent ideal sheaf of V in O Ω . Definition 2.1 ([10]) . Let Ω be a domain in C n and V a anlytic subvariety of Ω. Theclass F V consists of all negative plurisubharmonic functions u on Ω such that for any z ∈ Ω,there exist local generators ψ , . . . , ψ k of I V near z and a constant C depending on u andgenerators such that u ≤ log | ψ | + C near z where we denote ψ = ( ψ , . . . , ψ k ). Definition 2.2 ([10]) . The pluricomplex Green function G Ω ,V with poles along V is the upperenvelope of all functions in F V , i.e., for any z ∈ Ω, G Ω ,V ( z ) := sup { u ( z ) | u ∈ F V } . When F V = ∅ , we define G Ω ,V = −∞ . The pluricomplex Green function G Ω ,V have thefollowing important property. Theorem 2.3 ([10]) . If V is closed, then G Ω ,V ∈ F V . In other words, if V is closed, G Ω ,V is plurisubharmonic without upper semi-continuousregularization.Here, we assume that V has bounded global generators ψ . This means that there existbounded holomorphic functions ψ , . . . , ψ k on Ω such that V = { ψ = · · · = ψ k = 0 } . Then, by boundedness of generators, we can take a positive constant M such that | ψ | M < | ψ | M ∈ F V . Since F V = ∅ , there exists the pluricomplex Green function G Ω ,V withpoles along V . IGHER DIMENSIONAL CASE OF SHARPER ESTIMATE 5
And, since V is submanifold, for any z ◦ ∈ V , we can take the coordinate z = ( z , . . . , z n )near z ◦ such that J ψ := det ∂ ( ψ , . . . , ψ k ) ∂ ( z , . . . , z k ) = 0 . In particular, in the case where V has bounded global generator, for any z ∈ Ω, we canfind a re-ordering of linear coordinates z ′ := ( z , . . . , z k ) and z ′′ := ( z k +1 , . . . , z n ) such that J ψ = det ∂ ( ψ ,...,ψ k ) ∂ ( z ,...,z k ) = 0.In this setting, we establish an analogue of the Berndtsson–Lempert type L -extensiontheorem. Theorem 2.4.
Let Ω be a bounded pseudoconvex domain in C n , V a closed complex subman-ifold of Ω with codimension k such that V has bounded global generators ψ = ( ψ , . . . , ψ k ) andthere exists a positive constant C such that C ≤ | J ψ | near V . Let ϕ be a plurisubharmonicfunction on Ω and G Ω ,V the pluricomplex Green function on Ω with poles along V . Assumethat there exists some continuous function B on Ω such that (2) log | ψ ( z ) | − B ( z ) ≤ G Ω ,V ( z ) . Then for any holomorphic function f on V with Z V | f | e − ϕ +2 kB < ∞ , there exists a holo-morphic function F on Ω such that F | V = f and Z Ω | F | e − ϕ ≤ Cσ k Z V | f | e − ϕ +2 kB , where σ k is the volume of the unit ball in C k . Remark 2.5.
Theorem 2.4 is obtained even if we use an another function in F V satisfying(2) in substitution for G Ω ,V . In general, if we use an another function in F V satisfying (2),the L -estimate obtained from the conclusion becomes rough. Proof.
To prove Theorem 2.4, we follow the argument in the proof of [8, Theorem 2.1].Let A (Ω , ϕ ) = A (Ω) be a Hilbert space of holomorphic functions F on Ω with R Ω | F | e − ϕ < ∞ . We may assume that ϕ is continuous.Let F ◦ ∈ A (Ω , ϕ ) be an arbitrary L -extension of f to Ω. Here, we consider a sequencesof domains Ω j such that Ω j approximates Ω from inside. From now on, we will discuss onΩ j . For simplicity, we omit the subscript j . S. KIKUCHI
Let F be the L -minimum extension of f . The L -norm || F || A (Ω) of F is equal to the L -norm k F ◦ k A (Ω) /A (Ω) ∩I V of F ◦ , where we denote I V := { F ∈ O (Ω) : F | V = 0 } . Actually, k F k A (Ω) = sup = e ∈ A (Ω) | ( e, F ) A (Ω) |k e k A (Ω) = sup = e ∈I ⊥ V | ( e, F ) A (Ω) |k e k A (Ω) = sup = ξ ∈ A (Ω) ∗ ∩ Ann I V | h ξ, F i |k ξ k A (Ω) ∗ = sup = ξ ∈ A (Ω) ∗ ∩ Ann I V | h ξ, F ◦ i |k ξ k A (Ω) ∗ = || F ◦ || A (Ω) /A (Ω) ∩I V , where we denote Ann I V := { ξ ∈ A (Ω) ∗ : ξ | A (Ω) ∩I V = 0 } . Then we deal with the linearform ξ in the following way. For a fixed smooth function g on V with compact support, wedefine a linear functional ξ g on A (Ω) by h ξ g , h i := σ k Z V h ¯ ge − ϕ +2 kB , h ∈ A (Ω , ϕ ) . The set of such functionals ξ g is a dense subspace of ( A (Ω) /A (Ω) ∩ I V ) ∗ . Therefore, the L -norm k F ◦ k A (Ω) /A (Ω) ∩I V can be written assup g | h ξ g , F ◦ i ||| ξ g || A (Ω) ∗ . For p > t ∈ C with Re t ≤
0, we define ϕ t,p ( z ) := ϕ ( z ) + p max (cid:26) G ( z ) − Re t , (cid:27) . For any fixed p , by the convexity theorem in [2, Theorem 1.1], t log || ξ g || A (Ω ,ϕ t,p ) ∗ is subharmonic. In particular, ϕ t,p is convex in Re t . Therefore, we can assume that t ∈ R ≤ .The following lemma describes the asymptotic behavior of || ξ g || A (Ω ,ϕ t,p ) ∗ when t → −∞ . Wewill write || ξ g || A (Ω ,ϕ t,p ) ∗ = || ξ g || t,p for simplicity. Lemma 2.6 ([3, Lemma 3.2]) . For fixed p > , it follows that || ξ g || t,p e kt = O (1) . when t → −∞ . In particular, || ξ g || t,p e kt is increasing in t . Let F t,p be the L -minimum extension of f in A (Ω , ϕ t,p ). By Lemma 2.6, e − kt || F t,p || A (Ω ,ϕ t,p ) = sup g h ξ g , F ◦ i e kt || ξ g || t,p IGHER DIMENSIONAL CASE OF SHARPER ESTIMATE 7 is decreasing in t . Therefore, it follows that || F || A (Ω ,ϕ ) ≤ e − kt || F t,p || A (Ω ,ϕ t,p ) ≤ e − kt || F ◦ || A (Ω ,ϕ t,p ) . For fixed t <
0, the right hand side of the last inequalities converges to e − kt Z { G< t } | F ◦ | e − ϕ ! when p → ∞ .The subscript j will be specified from here. We prepare the following lemma. Lemma 2.7.
Let χ ≥ be a continuous function on ¯Ω and integrable on V . Then, it followsthat lim sup t →−∞ e − kt Z Ω j ∩ Ω t χ ≤ Cσ k Z Ω j ∩ V χe kB , where we denote Ω t = (cid:26) G Ω ,V < t (cid:27) . Proof.
By using the assumption on G Ω ,V , we can denoteΩ t = {| ψ | < e t + B } . Since the submanifold Ω j ∩ V is compact in C n , there exists a finite open covering { U i } Ni =1 such that there exists a change of numbering of linear coordinates depending on i so that wehave J ψ = det ∂ ( ψ , . . . , ψ k ) ∂ ( z , . . . , z k ) = 0 on each U i .Let t < V , there exists U i suchthat the given point in U i , we have e − kt Z Ω j ∩ Ω t χ = e − kt Z Ω j ∩{| ψ | By the continuity of χ , we can calculate the right-hand side of (3) as follow: Ce − kt Z Ω j ∩ V dz ′′ Z {| ψ | In this section, we prove the higher dimensional case of Hosono’s sharper estimate by usingTheorem 2.4. Our setting is as follows.Let Ω be a bounded pseudoconvex domain in C n and V a closed complex submanifoldof Ω with codimension k such that V has bounded global generators ψ = ( ψ , . . . , ψ k ).Suppose that there exists a positive constant C such that 1 C ≤ | J ψ | near V . Let ϕ be aplurisubharmonic function on Ω and G Ω ,V the pluricomplex Green function on Ω with polesalong V . Assume that there exists some continuous function B on Ω such thatlog | ψ ( z ) | − B ( z ) ≤ G Ω ,V ( z ) . Let ˜Ω be a pseudoconvex domain in C n + k defined by˜Ω = { ( z, w ) ∈ C n + k : z ∈ Ω , | w | < e − ϕ ( z ) k } and ˜ V a closed complex submanifold of ˜Ω such that˜ V = { ˜ ψ = · · · = ˜ ψ k = 0 } , IGHER DIMENSIONAL CASE OF SHARPER ESTIMATE 9 where ˜ ψ i ( z, w ) := ψ i ( z ) are holomorphic functions on ˜Ω. Let ˜ G be a pluricomplex Green func-tion on ˜Ω with poles along ˜ V or an another function in F V such that there exists continuousfunction ˜ B ( z, w ) on ˜Ω such thatlog | ˜ ψ ( z, w ) | − ˜ B ( z, w ) ≤ ˜ G ( z, w ) . Then the following theorem holds. Theorem 3.1. Under the above setting, the following statements hold. (1) For any holomorphic function f on V with Z V | f | e − ϕ +2 kB < ∞ , there exists a holo-morphic function F on Ω such that F | V = f and Z Ω | F | e − ϕ ≤ C Z ˜ V | ˜ f | e k ˜ B , where C is a positive constant determined from Theorem 2.4 and the holomorphic function ˜ f on ˜ V is defined by ˜ f ( z, w ) := f ( z ) . (2) Suppose that ˜Ω is a strictly pseudoconvex domain and − B ( z ) is a plurisubharmonicfunction. Then one can make the estimate in (1) strictly sharper than one in Theorem 2.4,i.e., there exist functions ˜ G and ˜ B satisfying the above conditions such that the estimate Z ˜ V | ˜ f | e k ˜ B < σ k Z V | f | e − ϕ +2 kB holds. Proof. (1) By applying Theorem 2.4 with the trivial metric e − ˜ ϕ ≡ f ( z, w ) := f ( z ) on ˜ V , we get a holomorphic function ˜ F on ˜Ω satisfying the propertiesthat ˜ F | ˜ V = ˜ f and Z ˜Ω | ˜ F | ≤ Cσ k Z ˜ V | ˜ f | e k ˜ B = Cσ k Z ˜ V | f | e k ˜ B . We consider a holomorphic function F ( z ) := ˜ F ( z, 0) on Ω. For any z ∈ V , we have F ( z ) = ˜ F ( z, 0) = ˜ f ( z, 0) = f ( z ) , i.e., F | V = f holds. For fixed z ∈ Ω, by the mean value inequality, we have | F ( z ) | = | ˜ F ( z, | ≤ σ k e − ϕ ( z ) Z | w | G < u , we can choose˜ B = − ˜ u .Then it is sufficient to prove the following inequality. Z ˜ V | f | e k ˜ B < σ k Z V | f | e − ϕ +2 kB . First, we calculate each side of the above inequality separately: Z ˜ V | f | e k ˜ B = Z V | f ( z ) | (cid:18)Z | w | Remark 3.2. When B ( z ) is a continuous function on Ω with log | ψ ( z ) | − B ( z ) ≤ G Ω ,V ( z )on Ω, we can obtain a continuous function B ′ ( z ) on Ω such that − B ′ ( z ) is plurisubharmonicand log | ψ ( z ) | − B ′ ( z ) ≤ G Ω ,V ( z ) on Ω. In fact, by [1, Theorem D], there exists − B ′ ∈ P SH ( ˜Ω) ∩ C ( ¯˜Ω) such that ( dd c ( − B ′ )) n + k = 0 on ˜Ω and − B ′ = − B on ∂ ˜Ω . Then, since log | ˜ ψ | − B ′ = log | ˜ ψ | − B ( z ) ≤ G Ω ,V ( z ) ≤ ∂ ˜Ω, from the maximum principle, it follows that log | ˜ ψ | − B ′ ≤ | ψ | − B ′ ( z, ≤ G Ω ,V ( z ) on Ω.But, we do not know whether we can obtain the sharper estimates after replacing thefunction B with B ′ . 4. Special cases Toward sharper estimates of the Ohsawa–Takegoshi L -extension theorem interms of the Azukawa pseudometric. The L -estimate obtained from the conclusionwill be sharper if B ( z ) is smaller, i.e., G Ω ,V ( z ) is bigger. Therefore, we need to take thepluricomplex Green function to obtain a better L -estimate. As a special case, the followingresult was obtained in [7]. Theorem 4.1 ([7]) . Let Ω be a bounded pseudoconvex domain in C n , w a point in Ω and ϕ a plurisubharmonic function on Ω . Let g Ω ,w be the pluricomplex Green function on Ω with apole at w and A Ω ,w the Azukawa pseudometric. We assume that there exists the limit A Ω ,w ( X ) = lim λ → ( g Ω ,w ( w + λX ) − log | λ | ) . Then there exist a holomorphic function f on Ω such that f ( w ) = 1 and Z Ω | f | e − ϕ ≤ vol( I Ω ,w ) e − ϕ ( w ) , where I Ω ,w is the Azukawa indicatrix defined by I Ω ,w := { X ∈ C n : A Ω ,w ( X ) < } and vol( I Ω ,w ) is the euclidean volume of I Ω ,w . Remark 4.2. In [11], it is shown that the assumptions of Theorem 4.1 holds on a boundedhyperconvex domain.When the submanifold V is { z = · · · = z k = 0 } , we can generalize Theorem 4.1 by usingthe pluricomplex Green function with poles along V . Theorem 4.3. Let Ω be a bounded pseudoconvex domain in C n , V a closed complex subman-ifold defined by V = { z = · · · = z k = 0 } and ϕ a plurisubharmonic function on Ω . Let G Ω ,V be the pluricomplex Green function on Ω with poles along V . We assume that there exists thelimit A Ω ,V,w ( X ) := lim λ → ( G Ω ,V ( λX, w ) − log | λ | ) , where (0 , . . . , , w ) ∈ V and = X ∈ C k . We define I Ω ,V,w := { X ∈ C n : A Ω ,V,w ( X ) < } .Then for any holomorphic function f on V with R V vol( I Ω ,V,w ) | f | e − ϕ < ∞ , there exists aholomorphic function F on Ω such that F | V = f and Z Ω | F | e − ϕ ≤ Z V vol( I Ω ,V,w ) | f | e − ϕ . Proof. It is sufficient to prove the following lemma. Lemma 4.4. Let χ ≥ be a continuous function on ¯Ω . Then we have lim sup t →−∞ e − kt Z Ω t χ ≤ Z V vol( I Ω ,V,z ′′ ) χ, where Ω t := (cid:26) G Ω ,V < t (cid:27) and (0 , . . . , , z ′′ ) ∈ V . Proof. For any δ > t , by continuity of χ , we have e − kt Z Ω t − δ χ ≤ e − kt Z V ( χ (0 , z ′′ ) + ǫ ) dz ′′ Z { G Ω ,V ( z ′ ,z ′′ ) < t − δ } dz ′ . Here, for any (0 , . . . , , z ′′ ) ∈ V , we consider the value of e − kt Z { G Ω ,V ( z ′ ,z ′′ ) < t − δ } dz ′ . Using thesubstitution z ′ = e t ˜ z ′ , we have(4) e − kt Z { G Ω ,V ( z ′ ,z ′′ ) < t − δ } dz ′ = Z { G Ω ,V ( e t ˜ z ′ ,z ′′ ) − log e t < − δ } d ˜ z ′ . By the assumptions of G Ω ,V , take the upper limits with respect to t of both sides of (4), thenthe right-hand side of (4) converges to something whose magnitude is at most Z { A Ω ,V,z ′′ (˜ z ′ ) ≤− δ } d ˜ z ′ . This value can be estimated as follow: Z { A Ω ,V,z ′′ (˜ z ′ ) ≤− δ } d ˜ z ′ ≤ Z { A Ω ,V,z ′′ (˜ z ′ ) < } d ˜ z ′ = vol( I Ω ,V,z ′′ ) . Therefore, by δ → 0, we can get the following inequalitylim sup t →−∞ e − kt Z Ω t χ ≤ Z V vol( I Ω ,V,z ′′ ) χ. (cid:3) IGHER DIMENSIONAL CASE OF SHARPER ESTIMATE 13 By replacing Lemma 2.7 in the proof of Theorem 2.4 with Lemma 4.4, we can proveTheorem 4.3. (cid:3) Example 4.5. Here, for n ≥ 2, we consider a unit ball Ω = B n in C n and ϕ ( z ) = − n log(1 −| z | ). In this situation, the pluricomplex Green function g B n , ( z ) with a pole at 0 is equalto log | z | and the Azukawa pseudometric A B n , ( X ) is equal to log | X | . Since the Azukawaindicatrix is I B n , = {| X | < } , therefore we have vol( I B n , ) = σ n . On the other hand, since˜Ω = {| w | + | z | < } in C n and ˜ V = {| w | < } , the pluricomplex Green function G ˜Ω , ˜ V withpoles along ˜ V is equal to log | z | p − | w | and A ˜Ω , ˜ V ,w ( X ) is equal to log | X | − log(1 − | w | ).Since I ˜Ω , ˜ V ,w = {| X | < (1 − | w | ) } , we have vol( I ˜Ω , ˜ V ,w ) = σ n (1 − | w | ) n . Then Z ˜ V vol( I ˜Ω , ˜ V ,w ) = Z | w | < σ n (1 − | w | ) n = σ n Z S n − Z (1 − r ) n r n − drdS = σ n µ n Z (1 − r ) n r n − dr = σ n µ n B ( n, n + 1)= σ n µ n n )Γ( n + 1)Γ(2 n + 1)= σ n µ n n − n !(2 n )! , where µ n is the volume of S n − , B is the Beta function and Γ is the Gamma function. Forany n ≥ 2, since µ n = 2 π n Γ( n ) = 2 π n ( n − µ n n − n !(2 n )! < . Therefore, in this situation, it follows that R ˜ V vol( I ˜Ω , ˜ V ,w ) < vol( I B n , ). From this observation,we can expect that the sharper estimates of Ohsawa–Takegoshi L -extension theorem interms of the Azukawa pseudometric holds.4.2. Radial case in C n . In [6], Hosono obtained the L -minimum extension of the function 1on the subvariety { } by applying Hosono’s sharper estimate to the case where Ω = {| z | < } is a unit disc in C and ϕ is a radial subharmonic function, i.e., ϕ ( z ) = ϕ ( | z | ). Similarly,in this subsection, we obtain the L -minimum extension of holomorphic functions f on V in the setting where Ω = B n is a unit ball in C n , V is a closed submanifold defined by V = { z = · · · = z k = 0 } = { z ′ = 0 } and ϕ is a radial plurisubharmonic function withrespect to V , i.e., ϕ ( z ) = ϕ ( | z ′ | ), by applying Theorem 3.1 in this setting. For z ∈ Ω, we denote z = ( z ′ , z ′′ ) where z ′ = ( z , . . . , z k ) , z ′′ = ( z k +1 , . . . , z n ). We mayassume that ϕ (0) = 0. Under the above setting, we can write ϕ ( z ) = ku (log | z ′ | ) where u isa convex increasing function on R < . Assume that u is strictly increasing and for fixed z ′′ ,lim t → log(1 −| z ′′ | ) − u ( t ) = ∞ . Define the plurisubharmonic function ψ by ψ ( w ) := − u − ( − log | w | ) . Then Proposition 4.6. For fixed z ′′ , we have (5) Z | z ′ | + | z ′′ | < e − ϕ = Z | w | < e − kψ ( w ) . Proof. Let µ k be the volume of S k − . First, we calculate the left hand side of (5). Usingthe substitution 2 log r = t , we have Z | z ′ | + | z ′′ | < e − ϕ = Z | z ′ | < −| z ′′ | e − ku (log | z | ) = µ k Z e − ku (log r ) r k − dr = µ k Z log(1 −| z ′′ | ) −∞ e − ku ( t ) e kt dt. (6)Next, we calculate the right-hand side of (5). At first, using the substitution 2 k log r = t , wehave Z | w | < e − kψ = Z | w | < e ku − ( − log | w | ) = µ k Z e ku − ( − log r ) r k − dr = µ k k Z −∞ e ku − ( − tk ) e t dt. (7)Then letting u − ( − tk ) = s , we see that the right hand side of (7) is equal to(8) µ k Z log(1 −| z ′′ | ) −∞ e ks e − ku ( s ) u ′ ( s ) ds. IGHER DIMENSIONAL CASE OF SHARPER ESTIMATE 15 And finally, using the substitution ks − ku ( s ) = q , we calculate the difference between (6)and (8) as follows: µ k Z log(1 −| z ′′ | ) −∞ e − ku ( t ) e kt dt − µ k Z log(1 −| z ′′ | ) −∞ e ks e − ku ( s ) u ′ ( s ) ds = µ k k Z log(1 −| z ′′ | ) −∞ k (1 − u ′ ( s )) e ks − ku ( s ) ds = µ k k Z −∞−∞ e q dq = 0 . (cid:3) We define ˜ G ( z, w ) := log | z | + ψ ( w ). From Proposition 4.6 and Theorem 2.4, we infer thatfor any holomorphic function f on V , there exists a holomorphic function F on B n such that F | V = f and Z B n | F | e − ϕ ≤ Z | z ′ | + | z ′′ | < | f (0 , z ′′ ) | e − ϕ . Therefore, in this case, by the above inequality and the mean value inequality, we can obtainthat the L -minimum extension with respect to ϕ is F ( z ) = f (0 , z ′′ ) in holomorphic functionson B n with F | V = f .In the general case, for any ǫ > 0, we define ku ǫ (log | z ′ | ) := ϕ ( z ) − ǫ log(1 − | z ′′ | − | z ′ | ).Then u ǫ is a strictly increasing and satisfies that for fixed z ′′ , lim t → log(1 −| z ′′ | ) − u ǫ ( t ) = ∞ .Therefore, for any ǫ > 0, we can obtain that the L -minimum extension with respect to ku ǫ (log | z | ) is F ( z ) = f (0 , z ′′ ) in holomorphic functions on B n with F | V = f . Finally, by ǫ → 0, we get the conclusion. Acknowledgment. The author would like to thank Prof. Ryoichi Kobayashi and Dr.Genki Hosono for valuable comments. References [1] E. Bedford and B. A. Taylor. The Dirichlet problem for a complex Monge–Amp`ere equation, Invent.Math., , 1-44, 1976[2] B. Berndtsson. Curvature of vector bundles associated to holomorphic fibrations, Ann. of Math. ,531-560, 2009[3] B. Berndtsson and L. Lempert. A proof of the Ohsawa–Takegoshi theorem with sharp estimates, J.Math. Soc. Japan, (4),1461-1472, 2016[4] Z. B locki. Suita conjecture and the Ohsawa-Takegoshi extension theorem, Invent. Math., , 149-158,2013[5] Q. Guan. and X. Zhou. A solution of an L extension problem with an optimal estimate and applications,Ann. of Math. (2), , 1139-1208, 2015[6] G. Hosono. On sharper estimates of Ohsawa-Takegoshi L -extension theorem, J. Math. Soc. Japan. ,909-914, 2019[7] G. Hosono. Subharmonic variation of Azukawa pseudometrics for balanced domains, arXiv:1811.07154 [8] G. Hosono. A simplified proof of optimal L -extension theorem and extensions from non-reduced sub-varieties, arXiv:1910.05782[9] T. Ohsawa. and K. Takegoshi. On the extension of L holomorphic functions, Math. Z. , 197-204,1987[10] A. Rashkovskii and R. Sigurdsson. Green functions with singularities along complex spaces, Internat. J.Math., , 333-355, 2005[11] W. Zwonek. Regularity properties of Azukawa metric, J. Math. Soc. Japan. , 899-914, 2000 Graduate School of Mathematics, Nagoya University, Furocho, Chikusa-ku, Nagoya, 464-8602, Japan Email address ::