$\bar\partial$-problem in fiber bundles for decreasing $(0,1)$-forms

In this paper we consider the $\bar\partial$-problem in fiber bundles (fibers biholomorphic to $\mathbb C^k$, $k\geq 1$), namely the equation $\bar\partial\sigma =\omega$ for $(0,1)$-forms $\omega$ which decrease along the fibers. The order of decrease is slightly more than one. The important fact is that we do not assume that $\omega$ has compact support. The main theorem says that the equation has a solution which also decreases along fibers, however, not necessarily with the order as the original form. Existence of solution of the above mentioned $\bar\partial$-problem can be applied in various situations in Complex Analysis, in particular, to the Hartogs extension phenomenon.


Introduction
Inhomogeneous Cauchy-Riemann equation orB-problem, namely the equationBσ " ω, where ω is a closed pp, qq form, is a basic problem in Complex Analysis. Moreover,B-problem has deep consequences on Algebraic Geometry, Partial Differential Equations and other areas of mathematics. The existence of solution of theB-problem has applications to integral formulas, holomorphic extension, holomorphic approximations or vanishing of cohomology groups.
Usually for an explicit solution of the equationBσ " ω, say in C n , where ω is a p0, 1q-closed form, we need that the support of ω is compact. If the support of ω is not compact, then we need additional assumptions on the domain where the equation is considered, for instance, Runge domains (see [14]). In Runge domains, we know that a solution of the equationBσ " ω exists, but since the solution is not given explicitly, we cannot control some properties of σ in terms of properties of ω.
In this paper we will consider the equationBσ " ω in fiber bundles X and in the case of p0, 1qforms ω which decrease along the fibers. Roughly speaking, a fiber bundle is a complex manifold X together with a holomorphic projection π onto another complex manifold B, called the base, and we will be assuming that the fibers π´1ppq are biholomorphic to C k (k " 1, 2, . . . is fixed). Moreover, locally a chart X s , X s Ă X, is biholomorphic with U sˆC k , where U s is a domain in C n , using the coordinates pz, wq " pz 1 , . . . , z n , w 1 , . . . , w k q. In the paper [7] a similar problem was considered but for forms ωpz, wq which have compact support along the fibers. In various situations, the compactness of supp ω is a too restrictive assumption. In this paper we drop the condition of compactness of support along the fibers and replace it by a much weaker condition that the form is decreasing along the fibers. Roughly speaking (more precisely it is defined in Section 2), in each chart X s , the form ωpz, wq " a 1 pz, wq dz 1`. . .`a n pz, wq dz n`b1 pz, wq dw 1`. . .`b k pz, wq dw k is decreasing along the fibers if the coefficients of the form are of class C 1 and there exist ε ą 0 and a constant C ą 0 such that the coefficients satisfy the conditions lim |w|Ñ8 a α pz, wq " 0, |b γ pz, wq| ď C 1`}w} 1`ε , |Db γ pz, wq| ď where D is any partial derivative of first order with respect to real variables x β " Re z β or y β " Im z β or u δ " Re w δ or v δ " Im w δ , β " 1, . . . , n, δ " 1, . . . , k. Here, }w} 1`ε is a short notation for |w 1 | 1`ε`. . .`|w k | 1`ε , but |w δ | is the standard module of a complex number, |w| " a |w 1 | 2`. . .`|w k | 2 . Of course, it is possible to choose other, almost equivalent conditions to (1), however, it seems that (1) is the most convenient and "symmetric" with respect to variables w 1 , . . . , w k .
The main result can be stated (more precise version is in Theorem 2, beginning of Section 3): Theorem 1 (Main Theorem). Let X be a fiber bundle and let ωpz, wq be a p0, 1q-form that vanishes along the fibers. Then there exists a globally defined C 1 -function Bpz, wq on X such that lim |w|Ñ8 Bpz, wq " 0 (in each coordinate chart) andBB " ω.
The theorems like Theorem 1 or from [7] are very useful in the Hartogs-type extension (see [13], [12]) of holomorphic or Cauchy-Riemann functions in a wide class of complex manifolds, see [15], [16], [18], [2], [3], [4], [5], [6], [9], [10]. Applications are given in [7] and we do not repeat them here. The advantage of the Main Theorem over the results in [7] is that it will allow us to apply the existence of extension to functions or forms on complex manifolds which are not necessarily holomorphic outside some set. It is enough that the functions or forms vanish at a point of the manifold or vanish on a complex submanifold, and after removing the point or the submanifold, we get a fiber-type bundle, like in [6] or toric manifolds [8]. As an illustration of this, we provide an example of such situation in the last Section 4.

Definitions and notation
Fiber bundles. Let X be a complex connected manifold of complex dimension n`k, where n, k " 1, 2, . . .. Assume that X has a locally finite, open cover tX s u sPS , where X s is biholomorphically equivalent to U sˆC k , where U s is a domain in C n . The coordinates on X s are of the form pz, wq " pz s , w s q " pz s1 , . . . , z sn , w s1 , . . . , w sk q " pz 1 , . . . , z n , w 1 , . . . , w k q and they give a biholomorphism from X s onto U sˆC k . The real and imaginary parts of the coordinates are The transition functions are of the form pz s , w s q " pf st pz t q, g st pz t , w t qq , s, t P S, the function f st depends on z t only, and moreover the second mapping g st gives a biholomorphism We also use the following notation, Purposely we do not put the index δ at w 1 , hopefully there will be no confusion.
p0, 1q-forms. In this paper we will deal only with functions and p0, 1q-forms. The functions and the forms will be defined on the fiber bundle X or in the local coordinate domain X s . A p0, 1q-form defined on X s can be written ω s pz s , w s q " ωpz, wq " Always we will be assuming that the form is of class C 1 , i.e., the coefficients a α pz, wq and b γ pz, wq are C 1 in the domain when the form is considered. We have a natural split of the form into two parts, the so-called a-part (or z-part) and b-part (or w-part): apz, wq " n ÿ α"1 a α pz, wq dz α , bpz, wq " The form ω isB-closed ifBω " 0, which is equivalent to the compatibility conditions Decreasing forms. Let ω be a p0, 1q-form defined on a fiber bundle X. We say that the form is decreasing along the fibers if in each coordinate domain X s « U sˆC k , for any compact set K Ă U s , the coefficients satisfy the conditions: There exist ε " εps, Kq ą 0 and a constant C " Cps, K, εq such that lim |w|Ñ8 a α pz, wq " 0 uniformly with respect to z P K, α " 1, . . . , n; uniformly with respect to z P K, γ " 1, . . . , k; |Db γ pz, wq| ď C 1`|w 1 | 1`ε`. . .`|w k | 1`ε uniformly with respect to z P K, γ " 1, . . . , k; where D is any partial derivative Here the uniform on K limit means, e.g. for the first limit above, The number ε ą 0 in (5) can be arbitrarily small. When ε is chosen smaller, then the conditions in (5) become weaker.
We note that if a closed p0, 1q-form ωpz, wq is decreasing along the fibers, then the a-part of the form (as in (3)) is uniquely determined by the b-part. Namely we have: suppose that bpz, wq " 0, then from the compatibility conditions (4) we obtain Ba α Bw γ pz, wq " 0 α " 1, . . . , n, γ " 1, . . . , k, which gives that the coefficients a α pz, wq are holomorphic with respect to w. Since these coefficients are decreasing to zero with respect to w as |w| Ñ 8, we have that a α pz, wq " 0.

Solution of theB-problem and proof of the theorem
The main goal of this section is to prove the following Theorem 2 Let X be a fiber bundle and let ω " ωpz, wq be a p0, 1q-form that vanishes along the fibers, as it is defined in (5). Then there exists a globally defined C 1 -function Bpz, wq on X such that 1.BB " ω.
2. In each coordinate map X s , as defined in Section 2 (other notation also from this section), the solution satisfies the estimate |Bpz, wq| ď const when z is running over a compact set and the constant depends on this compact set.
3. The function Bpz, wq is bounded in each coordinate chart, uniformly when z is running over a compact set.

Definition of the function B δ pz, wq
Our goal is to solve the equationBσ " ω in the fiber bundle X. We have to be careful with the method (as in [7] of solving the problem since the functions b δ pz, wq, δ " 1, . . . , k, from (2) do not have compact support with respect to w. We define If ζ " ξ`iη, then dζ^dζ " pdξ`idηq^pdξ´idηq " dξ^dξ´idξ^dη`idη^dξ`dη^dη "´2idξ^dη.
We note that the integrals (6) or (7) make sense because the function b δ is decreasing with respect to the variables w 1 , . . . , w δ´1 , ζ, w δ`1 , . . . , w k . Namely we have: Suppose that w δ is running over a compact set K δ Ă C. Then from the assumption on the decreasing condition, there exists ε ą 0 and a constant C such thaťˇˇˇb Since w δ is running over a compact set K δ Ă C, possibly increasing the constant C to r C, the last quotient can be estimated by Clearly the integral of the function on the right above exists and is finite. Namely we havěˇˇˇż

Estimates of the function B δ pz, wq along the fibers
We investigate the behavior of the function B δ pz, wq along the fibers. By assumption, the function b δ pz, wq and its first order derivatives satisfy the conditions (5). We will estimate the function B δ pz, wq defined in (6) or (7). We use the notation as in the beginning of Section 2.

First estimate of B δ pz, wq
In this subsection we will prove the following Lemma 1 With all the notation as above, we have the estimate where the constant C is the same as in the assumption (5) of the decreasing condition of coefficients of the p0, 1q-form bpz, wq.

Remark 1
We note that the second inequality above holds between the integrals, not as implication of a corresponding inequality between the integrands. Actually the inequality between the integrands is not true.
Proof Lemma 1. We have the following estimates: In the next sequence of estimates we switch to polar coordinates, Obviously the last integral depends on }w 1 }; also it depends on |w δ | only because of the integration with respect to θ. Moreover it is clear that in the case k ě 2 if w δ is running over a compact set, then the integral is tending to zero when }w 1 } Ñ 8; if k " 1, the term }w 1 } 1`ε is absent.
Now we look at the integral (9) (after dropping the constant C): ż We ask the question: With fixed w 1 , for what w δ the above integral takes the biggest value?
If we look at this integral geometrically, the answer is: For such w δ at which |w δ`ζ | is the smallest when ζ is zero, that is, w δ " 0. Another explanation of this property is that we want the fraction 1 |ζ| to take the maximum value while at the same time |w δ`ζ | takes the smallest value.
If the geometric argument is correct, we have the inequality ż Because the geometric explanation is heuristic, now we look at the above inequality more algebraically.
Since we know that the integral (10) depends only on |w δ |, let us examine the function Our near goal is to calculate F 1 pxq, x ą 0, however, at this moment we do not know if we can enter with the derivative under the integral sign. First we calculate the derivative of the integrand (actually of the second factor) and estimate it. Since the derivative is a local notion, we assume that x is running over a compact interval I, I Ă p0, 8q. We havěˇˇˇB where C I is a constant that depends on the interval I only.
Consequently, we have an estimate of the derivative with respect to x of the integrand in F pxq: for ζ P C and any fixed w 1 . This estimate allows us to enter with derivative of F pxq under the integral sign.
To be more precise, we can apply the Lebesgue Dominated Limit Theorem, see R. Sikorski [22] or R.G. Bartle [1] or H.L. Royden -P.M. Fitzpatrick [20], and a modified mean value theorem for vector-valued functions, see e.g., W. Rudin [21] p. 113, or S.M. Nikolsky [19] p. 119. We show all these applications in detail in Subsection 3.3, because of that, we do not repeat them here -the arguments are exactly the same.
Using the same type of estimates as for the derivative F 1 pxq, we get that the function F pxq is continuous on the interval r0, 8q.
From (9) and (11), immediately we obtain B δ pz, wq is bounded for all z, w, and B δ pz, wq ÝÑ 0 as }w 1 } Ñ 8 pif it is not voidq, however, we do not know its behavior with respect to w δ . So we will take a look at this case.

Second estimate of B δ pz, wq
Actually this subsection gives a geometric explanation of the property that the function F pxq defined in (12) is decreasing. An additional property which we prove here is lim The main result in this section is the following Lemma 2 The function |B δ pz, wq| is bounded with respect to w and uniformly for z running in a compact set. Moreover, we have where the constant C is the same as in the assumption (5) of the decreasing condition of coefficients of the p0, 1q-form bpz, wq.
Proof Lemma 2. We consider again the following integral From the considerations in the previous subsection, we know that F depends on |w δ | only, i.e., we have F pw δ q " F p|w δ |q. Geometrically it should be clear that F p|w δ |q Ñ 0 as |w δ | Ñ 8, as is shown on Fig. 2.

Calculation of the derivative
In this subsection we will prove the property that we can enter with differentiation under the integral sign. Since differentiation is a local notion, without any loss of generality we can assume that w 1 , . . . , w k are running over a compact set in C k .

Lemma 3 We have the property
The same is true for derivatives BB δ Bw γ pz, wq, BB δ Bz α pz, wq, BB δ Bz α pz, wq. Moreover, all the derivatives are bounded with respect to w uniformly with z running over a compact set. Also Actually the above holds for any kind of first order derivatives.
Proof Lemma 3. The first two equations in (19) are obvious, so we have to show that we can enter with differentiation under the integral sign (the third equation). One way is to apply theorems about differentiation of integrals with parameter, for instance see S.M. Nikolsky [19], pp. 118 -146, but the theorems from the book do not exactly apply to our case, and some small work is required. Another way is to prove that the differentiation can be moved inside the integral by using the Lebesgue Dominated Convergence Theorem. We choose the latter method.
We consider B{Bw γ as the Wirtinger derivative Of course, it is enough to prove our request for the derivative B{Bu γ since the calculations are the same for other derivatives: B{Bv γ , B{Bx α , B{By α .
If γ " δ, calculations are practically the same as in the case γ ‰ δ. The only difference is that instead of taking w γ`h and w δ`ζ , we use w δ`ζ`h . So the formula (25) also holds for γ " δ.
Also the same way the proof of (25) goes if we differentiate with respect to v γ " Im w γ instead of u γ " Re w γ . Because of that, we have In exactly the same way we prove the property, of entering with differentiation under the integral sign, for all other first-order derivatives. The last part of the lemma follows immediately from Section 3.2, where we replace B δ pz, wq by DB δ pz, wq, where D is any first-order derivative.

Using the compatibility conditions
As the next step, we use the compatibility conditions (4) in the integration below: At this point we use the Bochner-Martinelli formula in the case of one variable. Here we recall this formula: Theorem (see [17], Corollary 1.1.5, p. 24).

30)
The first integral in (28) can be estimateďˇˇˇˇ1 Next, we consider the second integral in (28). We note that the integral taken over the entire plane C exists because of the following estimates:ˇˇˇˇ1 From the estimates (30) and (32) we see that the integral over the entire plane C 1 2πi Finally, combining (28) and the above estimates (30) and (32), and taking the limit when R Ñ 8, we get the formula b γ pz, wq " 1 2πi Comparing the formulas (27) and (33), we obtain BB δ Bw γ pz, wq " b γ pz, wq, γ, δ " 1, . . . , k, or equivalentlyB w B δ pz, wq " bpz, wq. We note that the right-hand side of the equation does not depend on δ. Take two functions B δ pz, wq and B δ 1 pz, wq which satisfȳ B w B δ pz, wq "B w B δ 1 pz, wq orB w rB δ pz, wq´B δ 1 pz, wqs " 0 which implies w ÝÑ B δ pz, wq´B δ 1 pz, wq is holomorphic with respect to w.

Globally defined form on the fiber bundle
In this subsection we repeat the argument given in [7], pp. 562 -563, that the functions Bpz, wq " B s pz, wq, s P S, defined in each coordinate domain X sˆC k , actually can be put together and give a global function.
In another chart, say X t » U 1ˆCk , the form ω 1 pz 1 , w 1 q coincides with ωpz, wq on the intersection pU X U 1 qˆC k and we have the corresponding functions Of course, also we have We will prove that Bpz, wq " B 1 pz 1 , w 1 q on the domain where they both are defined. To avoid a confusion, here w 1 is just another point w, i.e., w 1 " pw 1 1 , . . . , w 1 k q.
Using (36) we have

End of the proof of the theorem
We recall the main points what we proved: In each coordinate map X s " U sˆC k , we have a solution Bpz, wq " B s pz, wq that satisfies 1. Bpz, wq is bounded with respect to w and Bpz, wq ÝÑ 0 as }w 1 } Ñ 8 or |w δ | Ñ 8 for some δ, uniformly in each chart X s " U sˆC k with respect to z running over a compact set.
4 Application of the Main Theorem to bundles over CP 1 In this section we show a simple application of the main theorem in the situation that other results cannot be applied. Let CP 2 be the 2-dimensional complex projective space. We know that C 2 can be embedded into CP 2 and we can write i.e., we can compactify C 2 by adding CP 1 at infinity (see [7] or [11]).
Similarly, if we remove a point p from CP 2 , then we obtain a vector bundle E over CP 1 with onecomplex dimensional fiber (see [7]).
We will make some comments about the latter case because it is more interesting and directly the main theorem can be applied. Namely, let ω be a closed p0, 1q form defined on CP 2 that vanishes to order two or more at the point p, mentioned above. This form ω can be considered as a form defined on the vector bundle E. Of course, this form is closed and, applying the main theorem, there is a solution of the equationBσ " ω. Moreover the solution σ vanishes along the fibers of E, however, not necessarily has compact support along fibers. From the main theorem also we can deduce how fast the function ω is approaching zero if the argument goes to "infinity" along the fibers. Of course, this speed depends on the order of decreasing of the form ω.