Line antiderivations over local fields and their applications

A non-Archimedean antiderivational line analog of the Cauchy-type line integration is defined and investigated over local fields. Classes of non-Archimedean holomorphic functions are defined and studied. Residues of functions are studied, Lorent series representations are described. Moreover, non-Archimedean antiderivational analogs of integral representations of functions and differential forms such as the Cauchy-Green, Martinelli-Bochner, Leray, Koppelman and Koppelman-Leray formulas are investigated. Applications to manifold and operator theories are studied.


Introduction
Line (Cauchy) integration is the cornerstone in the complex analysis and integral formulas of functions and differential forms such as the Cauchy-Green, Martinelli-Bochner, Leray, Koppelman and Koppelman-Leray formulas play very important role in it and in analysis on complex manifolds and theory of Stein and Kähler manifolds and theory of holomorphic functions (see, for example, [8,25]). In the non-Archimedean case there is not so developed Koppelman and Koppelman-Leray formulas are investigated. These studied are accomplished on domains in finite dimensional Banach spaces over local fileds and also on manifolds over local fields. All results of this paper are obtained for the first time. Finally applications of the obtained results to the theory of non-Archimedean manifolds and linear operators in non-Archimedean Banach spaces are outlined. In works of Vishik (see [11] and references therein) the theory of non-Archimedean (Krasner) analytic operators with compact spectra in C p was developed. In this article operators may have noncompact spectra in a field L such that Q p ⊂ L (may be also L ⊃ C p and L = C p ) continuing the investigation of [18].

Line antiderivation over local fields
To avoid misunderstandings we first present our specific definitions.
2.1. Notation and Remarks. Let K denotes a local field, that is, a finite algebraic extension of the field Q p of p-adic numbers with a norm extending that of Q p [27]. Denote by C p the field of complex numbers with the norm extending that of Q p [10]. If i ∈ K take α ∈ C p \ K such that there existsm ∈ N with αm ∈ K, wherem is such a minimal natural number, m =m(α), i := (−1) 1/2 . If i / ∈ K take α = i. Denote by K(α) a local field which is the extension of K with the help of α.
Suppose U is a clopen compact perfect (that is, dense in itself) subset in K and U σ := σ is its approximation of the identity: there is a sequence of maps σ l : U → U, where 0 ≤ l ∈ Z, such that (i) σ 0 is constant; (ii) σ l • σ n = σ n • σ l = σ n for each l ≥ n; (iii) there exists a constant 0 < ρ < 1 such that for each x, y ∈ U the inequality |x − y| < ρ n implies σ n (x) = σ n (y); (iv) |σ n (x) − x| < ρ n for each integer n ≥ 0. Consider spaces C n (U, L) of all n-times continuously differentiable in the sence of difference quotients functions f : U → L, where L is a field containing K with the multiplicative norm | * | L which is the extension of the multiplicative norm | * | K in K. Then there exists an antifderivation: (1) U P n : C n−1 (U, L) → C n (U, L) given by the formula: (2) U P n f (x) := ∞ l=0 n−1 j=0 f (j) (x l )(x l+1 − x l ) j+1 /(j + 1)!, where x l := σ l (x), x ∈ U, n ≥ 1 (see §80 [22]). Formula (2) shows, that if U P n is defined on C n−1 (U, K), then it is defined on C n−1 (U, Y ) for each field L which is complete relative to its norm such that K ⊂ L and a Banach space Y over L.
Since P n is the L-linear operator, then there exists the L-linear space P C n 0 (U, Y ) := P n (C n−1 (U, Y )), put P C n (U, Y ) := P C n 0 (U, Y ) ⊕ Y , where n ≥ 1, Y is a Banach space over L. For a clopen subset Ω in (K ⊕αK) m such that Ω ⊂ U m × U m consider the antiderivation Ω P n f (z) as the restriction of U m ×U m P n f (z) on Ω, (3) Ω P n f (z) := U m ×U m P n | Ω f (z) = U m ×U m P n f (z)χ Ω (z), where (4) U m ×U m P n f (z) := U P n x 1 ... U P n xm U P n y 1 ... U P n ym f (z), χ Ω (z) denotes the characteristic function of Ω, χ Ω (z) = 1 for each z ∈ Ω, χ Ω (z) = 0 for each z ∈ K 2m \Ω, z = (x, y), x, y ∈ U m ⊂ K m , x = (x 1 , ..., x m ), x 1 , ..., x m ∈ K, U P n x l means the antiderivation by the variable x l . This is correct, since each f ∈ C (0,n−1) (Ω, L) := C((0, n − 1), Ω → L) (see §I.2.4 [12]) has a C (0,n−1) -extension on U m × U m , for example, f | U m ×U m \Ω = 0. This means, that U m ×U m P n f (z) is the antiderivation defined with the help approximation of the unity on U m × U m such that U m ×U m σ = ( U σ, ..., U σ).

2.2.2.
Let v 0 , ..., v k ∈ K(α) m such that vectors v 1 − v 0 ,...,v k − v 0 are K-linearly independent, then the subset s := [v 0 , ..., v k ] := {z ∈ K(α) m : z = a 0 v 0 + ... + a k v k ; a 0 + ... + a k = 1; a 0 , ..., a k ∈ B(K, 0, 1)} is called the simplex of dimension k over K, k = dim K s. A polyhedron P is by our definition the union of a locally finite family Ψ P of simplexes. For compact P a family Ψ P can be chosen finite. An oriented k-dimensional simplex is a simplex together with a class of linear orderings of its vertices v 0 , ..., v k . Two linear orderings are equivalent if they differ on an even transposition of vertices. For a simplicial complex S let C q (S) be an Abelian group generated by simplices s q of dimension q over K and relations s q 1 +s q 2 = 0, if s q 1 and s q 2 are differently oriented simplices (see the real case in Chapter 4 [26]). Then there exists the homomorphism ∂ q : C q (S) → C q−1 (S) such that ∂ q [v 0 , ..., v q ] := q l=0 (−1) l [v 0 , ..., v l−1 , v l+1 , ..., v q ] and ∂ q [v 0 , ..., v q ] is called the oriented Kboundary of s q .
2.2.3. A clopen compact subset Ω in (K ⊕ αK) m is totally disconnected and its topological boundary is empty. Nevertheless, using the following affine construction it is possible to introduce convention about certain curves and boundaries which will serve for the antiderivation operators.
Let Ω be a locally K-convex subset in K(α) m for which there exists a sequence Ω n of polyhedra with Ω n ⊂ Ω n+1 for each n ∈ N, Ω = cl( n Ω n ), where cl(S) denotes the closure of a subset S in K(α) m . Suppose each Ω n is the union of simplices s j,n with vertices v j 0,n , ..., v j k,n , j = 1, ..., b(n) ∈ N, moreover, dim K (s j,n ∩ s j ′ ,n ) < k for each j = j ′ and each n, where k > 0 is fixed. Then define the oriented K-border ∂Ω n := j,l (−1) l [v j 0,n , ..., v j l−1,n , v j l+1,n , ..., v j k,n ]. Consider Ω n for each n such that if dim K (s j,n ∩ s j ′ ,n ) = k − 1 for some j = j ′ , then s j,n ∩s j ′ ,n = [v j 0,n , ..., v j l−1,n , v j l+1,n , ..., v j k,n ] = [v j ′ 0,n , ..., v j ′ l ′ −1,n , v j ′ l ′ +1,n , ..., v j ′ k,n ] and (l − l ′ ) is odd. For each n choose a set of vertices generating Ω n of minimal cardinality and such that the sequence {∂Ω n : n} converges relative to the distance function d(S, B) := max(sup x∈S ρ(x, B), sup b∈B ρ(b, S)), where ρ(x, B) := inf b∈B ρ(x, b) and ρ(x, b) := |x − b|. Then by our definition ξ = (q, n − 1), where spaces C ξ (K a , K b ) := C(ξ, K a → K b ) and C ξ -manifolds and uniform spaces C ξ (M, N) of all C ξ -mappings f : M → N were defined in §I.2.4 [12], 0 ≤ q ∈ Z, 0 < n ∈ Z, P C ξ+(0, 1) 0 (Ω, L b ) := P n (C ξ Lemma 3.4 [14]. Suppose that charts where Y is a Banach space over L. In particular, S C ξ+(1,0) Tensor fields over M were defined in § §3.1 and 3.5 [14]. Then the bundle of r-differential forms is the antisymmetrized bundle ψ r : , that is, φ is surjective and bijective with φ and φ −1 ∈ S C ξ+(1, 0) , where (i) Then for a k-differential C (0,n−1) -form w on φ(τ ) define (1) φ(τ ) P n w := τ P n φ * w, where φ * w is the pull back of w such that (2) φ(τ ) P n w = 0 for dim K τ = k, since w = 0 for k > dim K M. Without loss of generality take 0 ∈ U and σ 0 (0) = 0, then σ l (0) = 0 for each l ∈ N, consequently, U P n | {0} = 0. Therefore, U m P n | (U m ∩K k ×{0} m−k ) w = 0 for k < dim K Ω = m. Each such parallelepiped is the finite union of simplices satisfying conditions of §2.2.3. The orientation of ∂τ is induced by the orientations of constituting its simplices which are consistent. Consider such parallelepipeds τ j,q,l with l = 1, Since τ j,q,l P n v + τ j,q,l ′ P n v = τ j,q,l ∪τ j,q,l ′ P n v for each differential C (0,n−1) − kform v with support in U k and each l = l ′ and U k P n is the continuous operator from C (0,n−1) Using transition mappings φ i,j and considering clopen disjoint covering (vii) W j := V j \ j−1 l=1 V j of M we get (viii) M P n w = j W j P n w independent on the choise of local coordinates in M. Mention, that since |β| = 1, then B(K l , z, r) can be represented as the parallelepiped with the desribed above K-boundary ∂ c B(K l , z, r) due the ultrametric inequality. Due to (vi − viii) there is defined γ P n v for locally affine path γ, which is the P C n -manifold, that will be supposed henceforth.
Each compact manifold M has a finite dimension over K and using W j we get an embedding into K b for some b ∈ K. Let φ : Ω → M be such that φ is surjective and bijective, φ and φ −1 ∈ S C ξ+(1, 0) , which means that Then M is oriented together with Ω. Then ∂M := φ(∂Ω) is the oriented boundary. We also can consider the analytic manifold M and the analytic diffeomorphism φ. Each compact C ξ -manifold M can be supplied with the analytic manifold structure using a disjoint covering refined into At(M).
2.2.6. Theorem. Let M be a compact S C ξ or P C ξ -manifold over the local field K with dimension dim K M = k and an atlas At(M) = {(V j , φ j ) : j = 1, ..., n}, where ξ = (q, n), 1 ≤ q ∈ N, 0 ≤ n ∈ Z, then there exists a S C ξ or P C ξ -embedding of M into K nk respectively.
Proof. Let (V j , φ j ) be the chart of the atlas K) and each coordinate x l (see §2. 1 and §2.2.5). Then the mapping ψ(z) := (ψ 1 (z), ..., ψ n (z)) is the embedding into K nk , since the rank rank 2.3.1. Theorem. Let M be a compact oriented manifold over K of dimension dim K M = k > 0 with an oriented boundary ∂M and let w be a differential (k−1)-form as in §2.2.5 such that its pull back φ * w is a differential Proof. Since M is the manifold of dim K M = k > 0, then M is dense in itself and compact, hence Ω is dense in itself and compact (see Chapter 1 and Theorems 3.1.2,3.1.10 [5]) and the approximation of the identity can be applied to Ω. In view of Formulas 2.1.(1 −4) and 2.5. (1,2) on the space of C ξ differential forms operators U P n xq and U P n xs commute for each 1 ≤ q, s ≤ k.
2.3.2. Corollary. Let M be a compact oriented manifold over K of dimension dim K M = k > 0 with an oriented boundary ∂M and let w be a differential (k − 1) C (1,n−1) -form as in §2.2.5 such that its pull back φ * w = Proof. Repeating the proof of Theorem 3.1 for each term f j 1 ,...,j k−1 dz j 1 ∧ ... ∧ dz j k−1 of w and applying Formulas 3.1. (i, ii) we get the statement of this corollary.
Consider an extension of log. Denote by C p + := {z ∈ C p : |z − 1| < 1} and K(α) + := K(α) ∩ C p + . Then K(α) + is the Abelian subgroup in the additive group C p + and C p × := C p \{0} is the Abelian multiplicative group. The group C p × is divisible, that is, for each y ∈ C p × and each n ∈ N there exists x ∈ C p × such that x n = y. Let X be a proper divisible subgroup in C p × such that C p + ⊂ X. Let G be a subgroup generated by X and y ∈ C p × \ X. Suppose y n / ∈ X for each n ∈ N, then for each g ∈ G there exist unique n ∈ Z and x ∈ X such that g = y n x. Choose z ∈ C p , then put Log(g) := nz + Log(x). The second possibility is: y n ∈ X for some n ∈ N, n > 1. For each g ∈ G there exist unique n ∈ {0, 1, ..., m − 1} and x ∈ X such that g = y n x, where m := min y n ∈X;n∈N n. Since C p is divisible, there exists z ∈ C p such that z m = Log(y m ), therefore, define Log(g) := nz + Log(x). Using the Zorn's Lemma we can extend Log from C p + on C p × . In particular we can consider values of Log(i) and Log(α) using identities Log(1) = 0, i 4 = 1, α m ∈ K, α n = 1 for some minimal n ∈ N. In view of Theorem 45.9 [22] we can choose an infinite family of branches of Log indexed by Z. For the convenience put Log(0) := A.
2.4.6. Theorem. Let conditions of Theorem 2.4.2 be satisfied for each z ∈ M and let M be affine homotopic to a point, where ∂f (z,z)/∂z = 0 for each z ∈ M encompassed by ∂M. Then for each two paths γ 0 and γ 1 which are affine homotopic in M.
Proof. Using the diffeomorphism φ we can consider Ω instead of M. For each ǫ > 0 there exists a finite partition of a suitable subset Ω ǫ into finite union of parallelepipeds of diameter less, than ǫ in the proof of Theorem 2 for each such parallelepiped ξ. Therefore, there exists a sequence {γ l : l} of affine homotopy such that γ l (0, y) and γ l (β, y) are contained in the union ξ⊂Ω ∂ξ for each ǫ l = |π| l , l ∈ N. Since γ l (0, * ) P n [f dζ] = γ l (β, * ) P n [f dζ] for each l and taking l tending to the infinity we get (1) 2.4.8. Remark. The field K is locally compact, then T q is not contained in K, where T q is a group of all q n -roots b of the unity: b l = 1, l = q n , n ∈ N, q is the prime number, since dim Qp Q p T q = ∞ for Q p ⊂ K and K would be nonlocally compact whenever T q ⊂ K, which is impossible by the supposition on K. Therefore, there exists min{s ∈ N : b q ∈ K, b / ∈ K, where b = 1 is the q s+1 -root of the unity }. Hence there exists ζ ∈ K such that ζ 1/q / ∈ K. In particular, it is true for q = 2. Therefore, each local field K has a quadratic extension K(α) such that α / ∈ K. In the particular case K = Q p there exists the finite field Proof. Using the diffeomorphism φ we can consider Ω instead of M.
where f k ∈ Y , then there exist ∂f /∂z and ∂f /∂z = 0 on B. Since z ∈ Ω is arbitrary and such balls form the covering of Ω, then ∂f /∂z = 0 on Ω.
where γ and γ ǫ are affine homotopic and 0 < diam(γ ǫ ) < ǫ. In view of continuity of the operator P n there exists lim ǫ→0 γǫ P n [f dζ] = 0. Proof. Using the diffeomorphism φ we can consider Ω instead of M. Choose a marked point z 0 in M. Let η be a path joining points z 0 and z and satisfying Conditions 2.2.4. From γ P n f = 0 it follows, that η P n f does not depend on η besides points z 0 = η(0) and z = η(β), since each two points in Ω can be joined by an affine path, hence it is possible to put F (z) : 2.5.5. Corollary. Let conditions of Theorem 2.5.4 be satisfied, then f has an antiderivative F such that F ′ = f on M.
2.6.1. Lemma. Let Ω be a clopen compact subset in K m , then for Proof. Let σ be an approximation of the unity in U. In view of §2.1 it is sufficient to consider the case m = 1.
Then put in the sence of distributions: Theorem 2.2.6), Y * is the topologically dual space of all L-linear continuous functionals θ : Y → L, the valuation group Γ L of L is discrete.
2.6.3. Theorem. Let a manifold M satisfy Conditions 2.4.2 and let f satisfy 2.4.2.(2) for each z ∈ M, then the function Indeed, for each δ > 0 and for each continuous function f • φ on Ω or a continuous par- [12]), m := dim K M, there exists a finite partition of Ω q+1 into disjoint union of balls B j such that on each B j the variation var(w q ) := sup x,y∈B j |w q (x) − w q (y)| < δ, since M is compact and for each covering of M by such balls there exists a finite subcovering. Therefore, in Using the diffeomorphism φ we can consider Ω instead of M. Choose a clopen ball B := B(K ⊕ K(α), z 0 , R) ⊂ Ω containing a point z 0 ∈ Ω and its characteristic function χ := χ B . Then (f χ) 1 Lemma 2.6.1). Using the affine mapping z → (z − z 0 ) we can consider 0 instead of z 0 . Then B is the additive group. We can take R > 0 sufficiently small such that each point of B is encompassed by ∂Ω. Therefore, Formula 2.4.4.(ii) for this antiderivative, then (ii). f is (0, n)-antiderivationally holomorphic oñ Ω := {z ∈ Ω : z j is encompassed by ∂Ω j ∀j = 1, ..., m}.
(iii). f ∈ C (0,n−1) (Ω, Y ) and for each polydisc (iv) f is locally z-analytic, that is, and for every polydisc B as in (iii) and each multiorder k as in (iv) derivatives are given by (vii) the coefficients in Formula (2) are determined by the equation: The power series (2) converges uniformly in each polydics B ⊂Ω with sufficiently small b := max(R 1 , ..., R m , 1).
Proof. In view of Theorem 2.7.2 there exists a polydics B such that on it Formulas 2.7.2. (2,4) In view of Corollary 54.2 and Theorem 54.4 [22] and [1] , since the condition of the local analyticity means that the expansion coefficients a(m, f ) of the function f in the Amice polynomial basisQ m are such that lim |m|→∞ a(m, f )/P m (ũ(m)) = 0, where P m are definite polynomials (see Formulas 2.6.(i − iii) [16]).
Let Ω and f be as in 2.7.2. (i). If ζ is zero of f such that f does not coincide with 0 on each neighbourhood of ζ, then there exists , where g is analytic and φ = 0 on some neighbourhood of z.
Proof. In view of Theorem 2.7.2 there exists a neighborhood V of ζ such that f has a decomposition into converging series 2.7.2. (2). If a k = 0 for each k, then f | V = 0. Therefore, there exists a minimal k denoted by l such that To prove the theorem consider f by each variable z l . That is, consider z = z l and m = 1. Let R 3 and R 4 be such that Theorems 2.4.6 and 2.7.2.(iv,vi) ( and forming two paths γ 1 and γ 2 affine homotopic to points in W 1 and W 2 , γ 1 ⊂ W 1 , γ 2 ⊂ W 2 , such that γ 1,2 is being gone twice in one and opposite directions. This gives (3).
In view of Theorem 2.4.6 we get Formula (2).  (1) in a neighborhood of z contains an infinite family {a k = 0 : k < 0}. If z is an essentially critical point of f , r = 2, that is, K(α) = K ⊕ αK, then for each ξ ∈ AK(α) there exists a sequence {z n : n ∈ N}, lim n→∞ z n = z such that lim n→∞ f (z n ) = ξ.
The proof of these latter three theorems is analogous to the classical case (see, for example, §II.7 [25]) due to the given above Theorems 2.7.2 and 2.7.8. 2.7.14.
Theorem. Let f satisfy 2.7.2.(i) on Ω \ ν l=1 {z l } such that ∂Ω does not contain critical points z l of f and all of them are encompassed by ∂Ω, ν ∈ N. Then (i) ∂Ω P n [f (ζ)dζ] = C(α) z l ∈Ω res z l f , where res z l f is independent of n and R in 2.7.13, (ii) res z l f = a −1 , a k is as in 2.7.8. (1).
Proof. In view of Theorems 2.4.2 and 2.4.6 C(α) = C n (α) is independent of n and res z l f is independent of n and R. From 2.7.8. (2) it follows (ii).
res z l f and it is independent of R for each R > R 0 , R < ∞. In accordance with Definition 2.7.15 and Theorem 2.4.6 (iii) ∂cB R P n [f (ζ)dζ] = −res A f . Therefore, from (ii, iii) it follows (i).

Then
( , where N and P denote total numbers of zeros and poles in Ω.
Proof. Since Ω is compact, then N and P are finite. In view of Theorem 2.7.8 where z l is the pole of f and ξ l is the zero of f for each l. On the other hand,  (1).

Corollary.
Let Ω and f be as in Theorem 3.2 and f be derivationally (q, n)-holomorphic on Ω, then  (1), which are the non-Archimedean analogs of the Martinelli-Bochner and Cauchy-Green formulas respectively (see for comparison the classical complex case in [8]). 1). Impose the condition:

Definitions and Remarks.
Let Ω be a clopen compact subset in (K ⊕ αK) m , consider the differential form: ( Let M be a compact manifold over K and φ : Ω → M ֒→ (K(α)) N be a S C (q+1,n−1) -diffeomorphism (see §2.2.5). Then the diffeomorphism φ * w of the differential form w is the differential form on M. Consider these differential forms on M also and denote them by the same notation, since {φ(ζ j ) : j} are coordinates in M. Therefore, Theorems 3.2, 3.6 and Corollaries 3.3, 3.7 are true for M also due to Theorem 2.3.1 and Corollary 2.3.2, whereM := φ −1 (Ω) (see § §2.2.5 and 2.7.2). If f is a C (0,n−1) -differential form on M, then we define: (2) (B n M f )(z) := q −1 m ζ∈M P n [f (ζ) ∧w(z, ζ)] for each z ∈ M encompassed by ∂M. If f is a C (0,n−1) -differential form on M, then we define: (3) (B n ∂M f )(z) := q −1 m ζ∈∂M P n [f (ζ) ∧w(z, ζ)] for each z ∈ M encompassed by ∂M. Writew as: when deg(f ) < q + 1 by the definition of the antiderivation. Therefore, ; P n ∂M f is of bidegree (0, deg(f )). Using the notation of §3.5 define: 2.5 and 2.7.2). There exists the decomposition: (14) (0, t) in z and f is of bidegree (m, m − t − 1) in (ζ, λ). Let f be a bounded differential form on ∂M, f = f (l,s) , then Theorem. Let M be a compact manifold over K and let B n M and B n ∂M be given by §3. 8.
3.11. Corollary. Let M and f be as in Theorem 3.10 and ∂v/∂z = 0 on M. For t = 1, ..., m put for each z ∈M and satisfying Condition 3.10. (2). If∂f = 0, then u = T n t f is a solution of∂u(z) = f (z) for each z ∈M and f satisfying 3.10. (2).
(1). A (q + 1, n)-antiderivationally holomorphic vector bundle over K(α) of K(α) dimension N over M is a S C (q+1,n−1) -vector bundle over M with the characteristic fibre (K(α)) N and with (q + 1, n)-antiderivationally holomorphic atlas of local trivializations of B, that is, with a family {U j , h j } such that {U j } is a (cl)open covering of M, for each j, h j is a S C (q+1,n−1) -bundle isomorphism from B| U j onto U j ×(K(α)) N ; the corresponding transition mappings g i,j : ) N are (q + 1, n)-antiderivationally holomorphic S C (q+1,n−1) -mappings. Equipped with the atlas {B| U j , h j } the bundle B gets the structure of the S C (q+1,n−1) − (q + 1, n)-antiderivationally holomorphic manifold.
(4). A K(α)-valued differential form of degree r over M can be defined as a section of the vector bundle Λ r T * (M) K(α) , where T * (X) K(α) is the K(α) cotangent bundle of M over scalars b ∈ K(α) (see [14]). A differential form of degree r with values in a S C (q+1,n−1) − (q + 1, n)-antiderivationally holomorphic bundle (or a B-valued differential form) over M is a section of the bundle Λ r (T * (M) K(α) ) ⊗ K(α) B.
If {U j : j ∈ J} is a (cl)open covering of M such that B is S C (q+1,n−1) − (q + 1, n)-antiderivationally holomorphically trivial over each U j and {g i,j : i, j ∈ J} is the corresponding system of transition functions, then a differential form with values in M can be identified with a system {f j } of N-tuplets of differential forms on U j such that where B is S C (q+1,n−1) − (q + 1, n)-antiderivationally holomorphically trivial, the corresponding N-tuple of differential forms on U consists of (0, t)-forms, S C (q+1,n−1) − (0, t)-form, etc. Each (s, t)-form with values in a S C (q+1,n−1) (q + 1, n)-antiderivationally holomorphic vector bundle can be identified with some (0, t)-forms with values in some other n-antiderivationally holomorphic vector bundle.
3.16. Notes and Definitions. The local field K is the disjoint union of balls B(K, z j , R) for a given 0 < R < ∞, where z j ∈ K for each j ∈ N. Therefore, the antiderivation operators B j P n on B j := B(K, z j , R) induce the antiderivation operator K P n on K such that L is a field complete relative to its uniformity. Then C (q,n−1) (K l , Y ) and P C (q,n) (K l , Y ) are supplied with the inductive limit topologies induced by the embeddings Y is a Banach space over L such that K ⊂ L (see also [16,19]).
Therefore, in the standard way we get the definition of a locally compact manifold M over K of class P C (q,n) or S C (q+1,n− 1) , that is, transition mappings of charts φ i,j ∈ P C (q,n) Using charts and P C (q,n) (K l , K m ) or S C (q+1,n−1) (K l , K m ) we get the uniform space P C (q,n) (M, N) or S C (q+1,n−1) (M, N) of all mappings g : M → N of class P C (q,n) or S C (q+1,n−1) respectively, where M is the P C (q,n)manifold or S C (q+1,n−1) -manifold on K l and N is the P C (q,n) -manifold or S C (q+1,n−1) -manifold on K m correspondingly, that is, ψ i • g • φ −1 j is of class P C (q,n) or S C (q+1,n−1) for each i and j such that its domain is nonempty, where At(M) = {(V j , φ j ) : j}, At(N) = {(W j , ψ j ) : j}. The uniformity in P C (q,n) (K l , K m ) or S C (q+1,n−1) (K l , K m ) induces the uniformity in P C (q,n) (M, N) or S C (q+1,n−1) (M, N) respectively (see Remark 2.4 [16]).
Proof. At first prove, that compositions of diffeomorphisms preserve classes P C (q,n) (M, M) and S C (q+1,n−1) (M, M) respectively. For this consider two diffeomorphisms ψ, φ ∈ Dif P (q,n) (U m ) or Dif S (q+1,n−1) (U m ) simultaneously. A diffeomorphism φ is called the simplest diffeomorphism, if it has the coordinate form: x j = φ j (y 1 , ..., y m ) = y j for each j = 1, ..., k − 1, k + 1, ..., m, In C 0 (U m , K m ) there exists the polynomial Amice base {Q n (x) : n ∈ N o m } and it is also the base in C (q,n) (U m , K m ), where N o := {j : 0 ≤ j ∈ Z} (see [1,16]). The linear ordering △ in K induces the linear ordering △ in K m and hence in U m : x△y if and only if x 1 = y 1 ,...,x j−1 = y j−1 , x j △y j , where 1 ≤ j ≤ m, y j ∈ K, y = (y 1 , ..., y m ) (see §2.2.1). Take in particular, U = B(K, 0, 1). Then (β, ..., β) is the largest element in U m . Let Z K := {z ∈ K : z = t l=0 z l π l , 0 ≤ t ∈ Z, z l ∈ {0, θ 1 , ..., θ p n −1 }}, then Z K is dense in B(K, 0, 1) and Z K is countable. There are decompositions (i) ψ l (y) = n∈No m a(n, ψ l )Q n (y) and (ii) φ k (y) = n∈No m a(n, φ k )Q n (y), where a(n, ψ l ) and a(n, φ k ) ∈ K. In view of the conditions imposed on ψ l and φ k and continuity of the K-linear operators U P n x j : (iii) φ k (y) = { n∈No m a(n, ∂φ k (y)/∂y j )( U P n y jQ n (y)| y j y j,0 )} +φ k (y 1 , ..., y j−1 , y j,0 , y j+1 , ..., y m ) for each j = 1, ..., m and analogously for ψ l , where y j,0 and y j ∈ U. To show (φ k (y 1 , ..., y k−1 , ψ l (y), y k+1 , ..., y m ) − y k ) ∈ S C (q+1,n−1) (U m , K) it is sufficient to find h j : U m → K such that (iv) U P n y j h j | y j y j,0 = −h j,0 + φ k (y 1 , ..., y k−1 , ψ l (y), y k+1 , ..., y m ) − y k for each j = 1, ..., m, where h j,0 ∈ K. From (iii) and continuity of the K-linear operator U P n y j it follows, that to resolve (iv) it is sufficient to find a solution of the problem: (v) U P n y j h| y j y j,0 = ( U P n y j y t 1 | y j y j,0 )...( U P n y j y t l | y j y j,0 ) for each l ∈ N and each t k = (t k 1 , ..., t k m ) ∈ N o m , k = 1, ..., l, y t = y t 1 1 ...y tm m . On the other hand, hence Equation (v) can be simplified in the considered class of P,y j C (q+1,n−1) 0 (U m , K)functions acting on both sides of (v) by (∂/∂y j ). For each z ∈ Z K there exists a solution z h(y) of (v) for each y ∈ U m such that y j △z, since the set {u ∈ Z K : u△z} is finite. In view of (vi) and §2.1 this family { z h(y) : z ∈ Z K } can be chosen consistent, that is, z h(y) = η h(y) for each y such that y j △ min(z, η). Therefore, there exists (vii) h = lim z→β z h such that (v) is satisfied for each y ∈ U m . In particular, id ∈ S C (q+1,n−1) (M, M).
For the class P C (q,n) (U m , K) it is sufficient to find solution of the problem (viii) ( U m P n h)(y) = ( U m P n y t 1 )...( U m P n y t l ) for each l ∈ N and each t k ∈ N o m , k = 1, ..., l, |t| := t 1 + ... + t m ≥ 1. In view of (vi) and §2.1 and U m P n = U P n y 1 ... U P n ym there exists a consistent family z h satisfying (viii) for each z ∈ Z K m and each y△z such that z h(y) = η h(y) for each y△ min(z, η), where η ∈ Z K m , since the set {u ∈ Z K m : u△z} is finite, (∂/∂y 1 )...(∂/∂y m ) U m P n | (C (q,n−1) (U m ,K)) = I and the acting by (∂/∂y 1 )...(∂/∂y m ) on both sides of Equation (viii) simplifies it in the class of P C (q,n) 0 (U m , K)-functions. Then (ix) h = lim z→(β,...,β) z h is the solution of (viii). Therefore, (φ • ψ(y) − y) and (φ • ψ −1 (y) − y) belong to S C (q+1,n− 1) or P C (q,n) correspondingly. The proof above also shows, that if a bijective surjective ψ is in P C (q,n) respectively, by solving the equation of the type v(id(y) + g(y)) = −g(y) relative to the function v for known g := ψ − id. Hence using charts (Ṽ j ,φ j ) of At(M) such thatφ j (Ṽ j ) = B ⊂ U m + z j with suitable z j ∈ K m for each j and At(M) is the refinement of At(M) and B satisfies Lemma 2.6.1 (or applying the above proof to B instead of U m ), we get that (φ l • φ • ψ k •φ −1 (y) − y) belongs to P C (q,n) or S C (q+1,n−1) respectively on its domain for each l and j, where k = 1 or k = −1. Together with Lemma 2.6.1 it provides If M is compact, then P C (q,n) (M, Y ) is normable for a Banach space Y over L, K ⊂ L (see analogously Lemma 3.4 [14]). Let V = B (C (q,n−1) .., m}. In view of K-convexity of V the set W is absolutely K-convex (disked) and W is absorbing in S C (q+1,n−1) (M, Y ), since P n x j are continuous K-linear and V is absorbing in C (q,n−1) (M, Y ). Then W is bounded in the weak topology in S C (q+1,n−1) (M, Y ). Therefore, the Minkowski functional on S C (q+1,n−1) (M, Y ) generated by W induces a norm in S C (q+1,n−1) (M, Y ) (see Exer. 6.204 [19]). Each space P,x j C (q+1,n−1) (M, Y ) is complete (see analogously with Lemma 3.4 [14]), since Y is complete.
Consider the K-linear space Ψ j := P,x j C (q+1,n−1) (M, Y )∩ S C (q+1,n−1) (M, Y ) and topologies τ P,j on P,x j C (q+1,n−1) (M, Y ) and τ S on S C (q+1,n−1) (M, Y ) induced by norms in these spaces, then τ S | Ψ j ⊂ τ P,j for each j due to continuity of P n x j (for M supplied with coordinates x j due to P C (q,n) or S C (q+1,n−1)diffeomorphism with Ω as in §2.2.5) and definition of τ S , since ker(P n x j ) = {0} and due to the open mapping Theorem (14.4.1) [19] there exists the continuous K-linear operator (P n x j ) −1 : is complete relative to the above norm.
j , dom(f l,j ) =: U l,j , * is taken of the space P C (q,n) (U l,j , K m ) or S C (q+1,n−1) (U l,j , K m ). In view of the ultrametric inequality f l,j is the isometry, since f l,j − id l,j = sup n |a(n, f l,j − id l,j )| Q n , where * is the norm in P C (q,n) (U l,j , K m ) or in S C (q+1,n−1) (U l,j , K m ) respectively induced by the norm in C (q,n−1) (U l,j , K m ) and the Minkowski functional as above. Then g k,l • f l,j − id k,j ≤ max( g k,l • f l,j − f l,j , f l,j − id l,j ). Using partial difference quotients and P n and expansion coefficients in the Amice base we get, that max l,j f −1 l,j − id l,j ≤ C max l,j f l,j − id l,j , C = const > 0 is independent of f (see the proof of Theorem 2.6 [16]), consequently, Dif P (q,n) (M) and Dif S (q+1,n−1) (M) are topological groups. For noncompact M having At(M) with compact charts and using the strict inductive limit topology we can take an entourage of the diagonal in P C (q,n) (M, M) 2 or in S C (q+1,n−1) (M, M) 2 of the form {f : f l,j − id l,j ≤ |π| for each l, j ∈ λ}, where λ is a finite subset in N. In view of Theorem A.4 [15] there exists the inverse mapping f −1 l,j , which is the local diffeomorphism, when dom(f l,j ) = ∅. Finally, statements (4,5) follow from the proofs of Theorems 2.4 and 3.6 [16] modified for the considered here classes of smoothness.
3.18. Remark and Definition. Let M and N be two locally compact C (q,n) -manifolds over K and f ∈ C 1 (M, N), dim K M =: m M , dim K N =: m N . Denote by E := E(f ) := {z ∈ M : rang(d z f ) < m N } and this set is called the set of critical values of f . The nonnegative Haar measure ν on K m N as the additive group induces the measure µ on N with the help of charts, since At(N) has a disjoint refinement, where ν is normalized by the condition ν(B(K m N , 0, 1)) = 1.
Proof. Using the charts of atlases it is sufficient to prove the theorem for f : U → K m N , where U is an open subset in K m N . For m M = 0 and m N = 0 the statement is evident, therefore, consider m M ≥ 1 and m N ≥ 1.
To finish the proof use the following two lemmas.
Proof. Consider n ≥ 2, since for n = 1 there is only one partial derivative and from y ∈ E it follows y ∈ E 1 . Let y ∈ E \ E 1 , then there exists a nonzero partial derivative, for example, ∂f 1 (x)/∂x 1 at the point x = y. There exists a mapping h : U → K m N such that h(x) := (f 1 (x), x 2 , ..., x m N ) for which rang(dh(y)) = m N . In view of Theorem A.4 [15] the mapping h is the diffeomorphism of some open V = V (y) ⊂ U onto a neighborhood W ∋ z := h(y). The set E ′ of critical points for g : Proof. Take a covering of E k by a countable number of balls of radius δ > 0, δ ≤ δ 0 , where δ 0 > 0 is sufficiently small. Take one of these balls B. From the definition of E k and the Taylor formula (see Theorem 29.4 [22] and Theorem A.5 [15]) it follows, that f ( Divide B into a disjoint union of q m M balls of radius δ/q, q = p −n . Let B 1 be a ball of this partition such that B 1 ∋ x. Then each y ∈ B 1 has the form y = x + h, where |h| ≤ δ/q. Then Therefore, Lemmas 3.19.1, 2 finish the proof of Theorem 3.19. 3.21. Theorem. Let M be a compact S C (q+1,n− 1) or P C (q,n) -manifold over a local field K, dim K M = m < ∞. Then there exists a S C (q+1,n−1) or P C (q,n) -embedding τ : M ֒→ K 2m+1 and a S C (q+1,n− 1) or P C (q,n) -immersion θ : M → K 2m correspondingly. Each continuous mapping f : M → K 2m+1 or f : M → K 2m can be approximated by τ or θ relative to the norm * C 0 .
Proof. Let M ֒→ K N be the S C (q+1,n− 1) or P C (q,n) -embedding of Theorem 2.2.6. Consider the bundle of all K straight lines in K N . They compose the projective space KP N −1 . Fix the standard orthonormal (in the non-Archimedean sence) base {e 1 , ..., e N } in K N and projections on K-linear subspaces relative to this base P L (x) := e j ∈L x j e j for the K-linear span x j e j , x j ∈ K for each j. In this base consider the function (x, y) := N j=1 x j y j . Let l ∈ KP N −1 , take a K-hyperplane denoted by K ., q N } in K N (see also [21]). This provides the projection π l : K N → K N −1 l relative to the orthonormal base {q 1 , ..., q N }. The operator π l is K-linear, hence π l ∈ S C (q+1,n−1) , since P n is the K-linear operator, U P n x j λe j | b a = λ(b − a)e j for each λ ∈ K and a, b ∈ U, j = 1, ..., N.
To construct an immersion it is sufficient, that each projection π l : T x M → K N −1 l has ker[d(π l (x))] = {0} for each x ∈ M. The set of all x ∈ M for which ker[d(π l (x))] = {0} is called the set of forbidden directions of the first kind. Forbidden are those and only those directions l ∈ KP N −1 for which there exists x ∈ M such that l ′ ⊂ T x M, where l ′ = [l] + z, z ∈ K N . The set of all forbidden directions of the first kind forms the C (q,n−1) -manifold Q of dimension (2m − 1) with points (x, l), x ∈ M, l ∈ KP N −1 , [l] ∈ T x M, where C (q,n) ⊂ C (q+1,n−1) for each n ≥ 1, q ≥ 0. Take g : Q → KP N −1 given by g(x, l) := l. Then g is of class C (q,n−1) . In view of Theorem 3.19 is not contained in KP N −1 and there exists l 0 / ∈ g(Q), consequently, there exists π l 0 : M → K N −1 l 0 . Since S C (q+1,n−1) or P C (q,n) respectively is dense in C (q,n−1) , then there exists a mapping κ such that κ ∈ S C (q+1,n−1) or κ ∈ P C (q,n) is sufficiently close to π l 0 relative to * C 1 correspondingly such that κ • θ is the immersion, since M is compact. In view of Theorem 3.17 the composition κ • θ is of class S C (q+1,n−1) or P C (q,n) correspondingly. This procedure can be prolonged, when 2m < N − k, where k is the number of the step of projection. Hence M can be immersed in K 2m .
Consider now the forbidden directions of the second type: l ∈ KP N −1 , for which there exist x = y ∈ M simultaneously belonging to l after suitable parrallel translation [l] → [l] + z, z ∈ K N . The set of the forbidden directions of the second type forms the manifold S := M 2 \ ∆, where ∆ := {(x, x) : x ∈ M}. Consider ψ : S → KP N −1 , where ψ(x, y) is the straight K-line with the direction vector [x, y] in the orthonormal base. Then µ(ψ(S)) = 0 in KP N −1 , if 2m + 1 < N. Then the closure cl(ψ(P )) coinsides with ψ(P ) ∪ g(Q) in KP N −1 . Hence there exists l 0 / ∈ cl(ψ(P )). Then consider . Since S C (q+1,n− 1) or P C (q,n) correspondingly is dense in C (q,n−1) , then there exists a mapping κ such that κ ∈ S C (q+1,n−1) or κ ∈ P C (q,n) is sufficiently close to π l 0 relative to * C 1 such that κ • τ is the embedding, since M is compact. In view of Theorem 3.17 the composition κ • τ is of class S C (q+1,n− 1) or P C (q,n) correspondingly. This procedure can be prolonged, when 2m + 1 < N − k, where k is the number of the step of projection. Hence M can be embedded into K 2m+1 .
3.21.2. Remark. Theorems 3.19 and 3.21 are non-Archimedean analogs of the Sard's and Witney's theorems. In Theorem 3.21 classes of smoothness globally on M are important. Theorem 3.21 justifies the considered class of manifolds M in the theorems above about antiderivational representations of functions.
3.41. Note. In sections 3.28-3.40 it can be taken the generalization instead of Ω for a manifold M which is S C (q+1,n−1) -diffeomorphic with Ω.