Some gap power series in multidimensional setting

We study extensions of classical theorems on gap power series of a complex variable to the multidimensional case. 1. Power series with Ostrowski gaps. Let

be a power series in C N , i.e. a series of homogeneous polynomials Q j of N complex variables of degree j.
The set D given by the formula D := {a ∈ C N ; the sequence (1.1) is convergent in a neighborhood of a} is called a domain of convergence of (1.1).
Key words and phrases.Plurisubharmonic functions, negligible sets in C N , power series, lacunary power series, multiple power series.
If ψ * is finite, then it is plurisubharmonic and absolutely homogeneous (i.e.ψ * (λz) = |λ|ψ * (z), λ ∈ C, z ∈ C N ).Therefore, the domain of convergence D is either empty, or it is a balanced (i.e.λz ∈ D for all λ ∈ C with |λ| ≤ 1 and z ∈ D) domain of holomorphy.Every balanced domain of holomorphy is a domain of convergence of a series (1.1).
For every balanced domain D in C N there is a unique nonnegative function h (so-called Minkowski functional of D) such that h(λz) = |λ|h(z) for all λ ∈ C and z ∈ C N , and D = {z ∈ C N ; h(z) < 1}.In particular, if D is a domain of convergence of (1.1), then h(z) ≡ ψ * (z).
It is known that a balanced domain in C N is a domain of holomorphy if and only if its Minkowski functional h is an absolutely homogeneous plurisubharmonic function.
(1.5) lim sup for all compact sets K ⊂ D, where Definition 1.1.We say that a function f holomorphic in a neighborhood of a point and Moreover, the maximal domain of existence G = G f of f is identical with the maximal domain of existence of f − f o .Definition 1.2.We say that a function f holomorphic in a neighborhood of a point z o possesses Ostrowski's gaps relative to a sequence of positive integers {n k }, if {n k } is increasing and there exists a sequence of positive real numbers {q k } such that q k → 0 as k → ∞ and lim j→∞,j∈I If the conditions of Definition 1.2 are satisfied, consider two cases.If m := lim inf k→∞ q k n k is finite, then the function f is entire, so that f has Ostrowski's gaps (m k , n k ] according to Definition 1 for any sequence m k , n k satisfying 1 o .
If lim inf k→∞ q k n k = ∞, then f possesses Ostrowski's gaps ( q k l n k l , n k l ] for a suitable chosen increasing subsequence k l of positive integers. We say that a compact subset K of C N is polynomially convex if K is identical with its polynomially convex hull K := {a ∈ C N ; |P (a)| ≤ P K for every polynomial P of N complex variables}.We say that an open set Ω in C N is polynomially convex, if for every compact subset K of Ω the polynomially convex hull K of K is contained in Ω.
The following theorem is known (see [7]).It is a multidimensional version of the classical Ostrowski's Theorem (see Theorem 3.1.1 in [1]).
Theorem 1.If a holomorphic function f possesses Ostrowski's gaps (m k , n k ] at a point z o ∈ C N , then the maximal domain of existence G = G f of f is one-sheeted and polynomially convex.Moreover, for every compact subset K of G we have where is the nth partial sum of the Taylor series development of f around z o . where The following result gives an N -dimensional version of W. Luh's Theorem 1 in [4].In particular, it says that if a function f holomorphic in a domain G in C N possesses Ostrowski's gaps at some point z o ∈ G, then f possesses the same property at every other point a of the maximal domain of existence of f .
By 2 o and Theorem 1 we get the following: Proof of Theorem 2. 1 o .Without loss of generality we may assume that Given a fixed point a ∈ G f , we have (Observe that s n k is a polynomial of degree at most m k ), and 0 < r < min(dist(a, ∂G f ), dist(z o , ∂G f )).By Theorem 1 there exist M > 1 and 0 < θ < 1 such that Therefore, by Cauchy inequalities, Let {k l } be an increasing sequence of natural numbers such that By (1.8) we get The choice of the sequence {k l } does not depend on a ∈ G f .Therefore, f possesses Ostrowski's gaps m k l , at every point a of G f (according to Definition 1.1).The proof of the case 1 o is ended.
2 o .Observe that for z − a ≤ 1 2 r we have By (1.7) and (1.9) we get It follows that the sequence of plurisubharmonic functions {u k } is locally uniformly upper bounded in C N , and Therefore, the plurisubhamonic function u * = const.
2. E. Fabry's Theorem.Now we shall present a multidimensional version of E. Fabry's Theorem (Theorem 2.2.1 in [1]).Let f be a function of N complex variables holomorphic in a neighborhood of 0 with a gap Taylor series development Due to Fabry we know that Theorem 3 is true for N = 1.It is also well known (by Bedford-Taylor Theorem on negligible sets) that the set E := {z ∈ C N ; ψ(z) < ψ * (z)} is pluripolar.Therefore, in particular, the set E is of 2N -dimensional Lebesgue measure zero.Suppose Theorem 3 is not true for some N > 1.Then there is a function Therefore, there is a sequence We know that the 2N -dimensional Lebesgue measure v 2N (E) = 0. Therefore, by the sub-mean-value property, for every k ≥ 1 there is a point It is clear that the sequence {z k } satisfies the following properties: The function f (λb) (resp., g(λb)) is holomorphic in ∆ (resp., in G R ), and f (λb) = g(λb) for λ ∈ G r .Therefore, f (λb) = g(λb) on ∆ ∩ G R .It follows that g(λb) is an analytic continuation of f (λb) across λ = 1, contrary to the Fabry Theorem for N = 1.We have got a showing that Theorem 3 is true.
Remark.The present proof of Theorem 3 -with no assumption on the continuity of the function ψ * -is a joint result of the author and Professor Azimbay Sadullaev.

Fatou-Hurwitz-Polya
Theorem.First we shall state Fatou-Hurwitz -Polya Theorem for a series of homogeneous polynomials of N complex variables.
Theorem 4. Let f be a function holomorphic in a neighborhood of 0 ∈ C N .Let be its Taylor series development around 0. Then there exists a sequence = { j } with j ∈ {−1, 1} (resp., j ∈ {0, 1}) such that the function For N = 1 this theorem (with j ∈ {−1, 1}) is due to Fatou-Hurwitz-Polya (Theorem 4.2.8 in [1]).Now, we shall present an N -dimensional version of the Fatou-Hurwitz-Polya theorem for N -tuple power series We shall see that Theorems 4 and 5 are direct consequences of the following Lemma 3.2.
Let X := {0, 1} N (resp.{−1, 1} N ) be the space of all sequences x = (x 1 , x 2 , . . . ) where x j = 0, or x j = 1 (resp.x j = −1, or x j = 1) for j = 1, 2, . . . .Endow X in the topology determined by the metric One can easily check that X is a complete metric space, and therefore, it has Baire property.Moreover, in the topology a sequence {x(n)} of elements of X converges to an element x ∈ X if and only if for every Remark 3.1.Let {f k } be a sequence of holomorphic functions in an open subset Ω of C n .Then the following three conditions are equivalent: (1) the series (2) the series ∞ 1 f k converges locally normally in Ω, i.e. for every point a of Ω there exists a neighborhood U of a such that the series ∞ Suppose now (1) is true, and let E(a, r) for all z ∈ E(a, r 2 ) and k ≥ 1.By Lebesgue monotonous convergence theorem the series ∞ 1 µ k is convergent, and so is the series We shall see that our extensions of the classical Fatou-Hurwitz-Polya Theorem (Theorem 4.2.8 in [1]) are a direct consequence of the following Lemma 3.2 (slight modification of Lemma 5, p. 97 in [5]).Then, either the series ∞ 1 f k is normally convergent on a neighborhood of a, or there exists a subset R of X of the first category such that for every x ∈ X \ R the function f x (z) := k x k f k (z), z ∈ B, has a singular point at a (in other words, f x cannot be analytically continued to any neighborhood of a).
Proof.Given a natural number m, let R m denote the set of all x ∈ X such that there exists a holomorphic function fx on E m (where E m is the polydisk E m := E a, 1 m with center a and radius 1 m ) such that | fx (z)| ≤ m on the polydisk, and fx (z It is clear that the set R := ∞ 1 R m ≡ {x ∈ X ; f x has an analytic continuation across a}.
The lemma will be proved if we show that the following two claims are true.

Claim 1.
The set R m is closed in the space X .Claim 2. If the interior of R m is not empty, then the series ∞ 1 f k is normally convergent on a neighborhood of a.
Indeed, if the series f x := ∞ 1 x k f k converges normally on no neighborhood U of a, then for every m ≥ 1 the set R m is closed and has empty interior.Hence, the set R := ∞ 1 R m ≡ {x ∈ X ; f x has an analytic continuation fx across a} is of the first category, and for every x ∈ X \ R the function f has a singular point at a, i.e. f x has no analytic continuation across a.We say that a function fx holomorphic on a polydisk E with center a is an analytic continuation of f x across a, if fx (z) = f x (z) on B ∩ E.
Proof of Claim 1.Let {x(j)} be a sequence of elements of R m convergent to x ∈ X .Let {h j } ≡ { fx(j) } be a sequence of holomorphic functions on E m such that |h j (z)| ≤ m on E m and h j (z) = f x(j) (z) on the intersection B ∩ E m for j ≥ 1 .Observe that for every k o there exists j o such that and for all j > j o .It follows that the sequence {h j } is convergent at each point of B ∩ E m .By Vitali's theorem the sequence {h j } is locally uniformly convergent on E m to a holomorphic function h bounded by m and identical with f x on E m ∩ B, which shows that x ∈ R m .

Proof of Claim 2.
If R m has a nonempty interior, then there exist
Then Then there exists a subset R of X of the first category such that for every point x ∈ X \ R the holomorphic function f x (z) := ∞ k=1 x k f k (z), z ∈ D, cannot be continued analytically across any boundary point of D.

Theorem 3 .
a} is a domain of convergence of (2.1).If lim k→∞ k m k = 0, then the domain of convergence D of the series (2.1) is identical with the maximal domain of existence G f of f .Proof.Without loss of generality we may assume that D = C N .
r).Let b o be a fixed point of ∂D such that b o − z o = r.Since the ball B(z o , r) is non-thin at the point b o ,we have lim sup z→bo,z∈B(zo,r) has no analytic continuation across any boundary point of the domain of convergence D := {ψ * (z) < 1} of series (3.0), where ψ(z) := lim sup j→∞ j |Q j (z)|.

Lemma 3 . 2 .
Let X denote any of the two metric spaces {0, 1} N or {−1, 1} N .Let {f k } be a sequence of holomorphic functions in an open neighborhood Ω of the closure of a ball B = B(w, r) such that the series ∞ 1 |f k (z)| converges at every point z ∈ B. Let a be a boundary point of B.

2 k∈ACorollary 3 . 3 .
|u k (z)| ≤ k∈A (x k (1) − x 2 (2)) f k (z) = | fx(1) (z) − fx k (2) (z)| ≤ 2m.Hence, by the analogous argument as in the proof of the case 1, we get∞ k=1 |f k (z)| ≤ M + 4m, z ∈ E m ,which ends the proof of the case 2. Let {f k } be a sequence of holomorphic functions on an open set Ω ⊂ C N .Let D denote the set of all points a in Ω such that the series ∞ 1 f k is absolutely convergent at every point of a neighborhood of a. Assume that the sum ϕ(z) := ∞ 1 |f k (z)| is locally bounded in D, and D ⊂ Ω.Let X be any of the two metric spaces {0, 1} N or {−1, 1} N .