OF SEPARATELY ANALYTIC FUNCTIONS

SINGULAR SETS

OF SEPARATELY ANALYTIC FUNCTIONS

by

Zbigniew B locki

Abstract. In this paper we complete the characterization of singular sets of separately analytic functions.

In the case of functions of two variables it was earlier done by J. Saint Raymond and J. Siciak.

-1.Introduction. If Ω is an open subset of R ⁿ

×· · ·×R ⁿ

, then we say that a function f : Ω −→ C is p−separately analytic (1 ≤ p < s), if for every x ⁰ = ¡

x ⁰ ₁ , . . . , x ⁰ _s ¢

∈ Ω and for every sequence 1 ≤ i 1 < · · · < i p ≤ s the function

¡ x i

, . . . , x i

¢ −→ f ¡

x ⁰ ₁ , . . . , x i

, . . . , x i

, . . . , x ⁰ _s ¢ is analytic in a neighbourhood of

³

x ⁰ _i

, . . . , x ⁰ _i

´

. For a p−separately analyitc function f in Ω let

A (f ) := {x ∈ Ω : f is analytic in a neighbourhood of x}

denote its set of analycity, and S (f ) := Ω \ A (f ) - its singular set.

If X and Y are any sets, S ⊂ X ×Y and ¡

x ⁰ , y ⁰ ¢

∈ X ×Y , then we denote S ¡ x ⁰ , • ¢

© :=

y ∈ Y : ¡ x ⁰ , y ¢

∈ S ª , S ¡

•, y ⁰ ¢ := ©

x ∈ X : ¡ x, y ⁰ ¢

∈ S ª .

The following theorems characterize singular sets of separately analytic functions:

Theorem A. If f is p−separately analytic in Ω, then for every sequence 1 ≤ j 1 <

· · · < j _q ≤ s, where q := s − p, the projection of S (f ) on R ⁿ

× · · · × R ⁿ

is pluripolar (in C ⁿ

× · · · × C ⁿ

).

Theorem B. Let S be a closed subset of Ω such that for every sequence 1 ≤ j 1 <

· · · < j q ≤ s, where q := s − p, the projection of S on R ⁿ

× · · · × R ⁿ

is pluripolar. Then there exists p−separately anlytic function f in Ω such that S = S (f ).

Theorem C. Let f be a p−separately analytic in Ω. If 1 ≤ k < s, then for quasi almost all x ∈ R ⁿ

× · · · × R ⁿ

(that is for x ∈ R ⁿ

× · · · × R ⁿ

\ P , where P is pluripolar) S (f (x, •)) = S (f ) (x, •).

Theorems A and B in case s = 2, p = n 1 = n 2 = 1 were proved by Saint Raymond [2].

This result was generelized by Siciak [5], who proved theorem A for p ≥ s/2 and theorem B. The aim of this paper is to give a proof of theorem C and, as a trivial consequence, we get theorem A.

0.Preliminaries. We need the following two theorems:

Siciak’s theorem ([3]; see also [4], theorem 9.7). Let for j = 1, . . . , s D _j = D ¹ _j ×

· · · × D ⁿ _j

, D ^t _j - open sets in C, symmetric with respect to x _t -axis (t = 1, . . . , n _j ), K _j = K _j ¹ ×· · ·×K _j ⁿ

, K _j ^t - closed intervals in D _j ^t ∩R. Let f be a separately holomorphic function in

X :=

[ s j=1

K 1 × · · · × D j × · · · × K s

(that is for every (x 1 , . . . , x s ) ∈ K 1 × · · · × K s and for every j = 1, . . . , s the function f (x 1 , . . . , x j−1 , •, x j+1 , . . . , x s ) is holomorphic in D j ). Then f can be extended to a holo- morphic function in a neighbourhood of X. ¹

Bedford-Taylor theorem on negligible sets [1]. If {u _j } _j∈J is a family of plurisubharmonic functions locally bounded from above then the set

½

z ∈ D : u (z) := sup

j∈J

u _j (z) < u ^∗ (z)

¾

is pluripolar (u ^∗ denotes the upper regularization of u).

1.Proofs.

Theorem C ⇒ theorem A: We may assume that (j ₁ , . . . , j _q ) = (1, . . . , q). Then it is enough to take k = q and see that for x ∈ R ⁿ

× · · · × R ⁿ

S (f (x, •)) = ∅.

Proof of theorem C: We can write

R ⁿ

× · · · ×R ⁿ

= (R ⁿ

× · · · ×R ⁿ

)× · · · ×(R ⁿ

× · · · ×R ⁿ

)

×(R ⁿ

× · · · ×R ⁿ

)× · · · ×(R ⁿ

× · · · ×R ⁿ

) , where a = [k/p], b = [(s − k) /p]. Then f is separately analytic (that is 1−separately analytic) with respect to such variables. Therefore it is enough to prove theorem C for p = 1. Let {X ν × Y ν } _ν∈N be a countable family of closed intervals in (R ⁿ

× · · · × R ⁿ

) × (R ⁿ

× · · · × R ⁿ

) such that S _∞

ν=1 X ν × Y ν = Ω. It is clear that the set

© x ∈ R ⁿ

× · · · × R ⁿ

: S (f (x, •)) ⊆ _/ S (f ) (x, •) ª

is contained in _∞ [

ν=1

© x ∈ X _ν : S (f (x, •)) ∩ Y _ν ⊆ _/ S (f ) (x, •) ∩ Y _ν ª .

Hence we may assume that f is separately analytic in a closed interval I 1 × · · · × I s ⊂ R ⁿ

× · · · × R ⁿ

(that is analytic in some open neighbourhood of this interval).

To prove theorem C we have to show that the set Z _f,k := ©

x ∈ I ₁ × · · · × I _k : S (f (x, •)) ⊆ _/ S (f ) (x, •) ª is pluripolar.

For (x, y) ∈ (I 1 × · · · × I k ) × (I k+1 × · · · × I s ) such that y ∈ A (f (x, •)) define

Q f,k (x, y) := sup

|α|≥1

¯ ¯

¯ ¯ 1 α!

∂ ^|α| f

∂y ^α (x, y)

¯ ¯

¯ ¯

1/|α|

1In fact we use the Siciak’s theorem under additional assumption that f is bounded. In this case the proof of

the theorem is much simpler - it can be deduced from theorem 2a in [3].

(of course Q f,k (x, y) < +∞ and f (x, •) is holomorphic in the polydisc P (y, 1/Q f,k (x, y))).

For y ∈ I k+1 × · · · × I s let

F f,k (y) := {x ∈ A (f ) (•, y) : Q f,k (•, y) is not upper semicontinuous at x} . Theorem C is proved by induction with respect to k. First assume that k = 1.

1 ⁰ The projection of S (f ) on I 2 × · · · × I s is nowhere dense in R ⁿ

× · · · × R ⁿ

, that is there exists U - open, dense subset of I 2 × · · · × I s such that I 1 × U ⊂ A (f ). In particular A (f ) is dense in I 1 × · · · × I s .

Induction with respect to s. The same proof applies to the case s = 2 and to the step s − 1 ⇒ s. We have

I 1 = [a 1 , b 1 ] × · · · × [a n

, b n

] . Define for m ∈ N

I ₁ ^m :=

½

z ∈ C ⁿ

: max

1≤t≤s dist (z t , [a t , b t ]) < 1/m

¾ ,

E m :=

(

y 1 ∈ I 2 × · · · × I s : f (•, y 1 ) is holomorphic in I ₁ ^m , sup

z∈I

|f (z, y 1 ) | ≤ m )

. We have E _m ⊂ E _m+1 , S _∞

m=1 E _m = I ₂ × · · · × I _s . First we want to show that the set U 1 := S _∞

m=1 intE m is dense in I 2 × · · · × I s . Let Y ⁰ be a closed interval in I 2 × · · · × I s , and H - a family of closed intervals which form a countable base of topology in Y ⁰ . For x 1 ∈ I 1 the set A (f (x 1 , •)) is dense: this is trivial if s = 2 and follows from the inductive assumption if s ≥ 3. Therefore, if for H ∈ H we denote

A H := {x 1 ∈ I 1 : f (x 1 , •) is analytic in H} , it follows that S

H∈H A H = I 1 . We claim that there exists H 0 ∈ H such that the set A H

is determinig for functions holomorphic in a complex neighbourhood of I 1 . Indeed, suppose it is not so. Then all the sets A H (H ∈ H) are nowhere dense in I 1

and by the Baire theorem we get a contradiction. Hence, by the Montel’s lemma, the sets E m ∩ H 0 (m ∈ N) are closed, and, again by the Baire theorem, U 1 ∩ H 0 6= ∅.

Therefore U 1 is open, dense in I 2 × · · · × I s . Analogously to I ₁ ^m and U 1 we define sets I _j ^m and U j (j = 2, . . . , s, m ∈ N). Let us take a closed interval K 2 × · · · × K s ⊂ U 1 . Since U j are dense we can find closed intervals e K 1 ⊂ I 1 , e K j ⊂ K j (j = 2, . . . , s) and m ∈ N such that for j = 1, . . . s

K e 1 × · · · × e K j−1 × e K j+1 × · · · × e K s ⊂ U j

and is f separately holomorphic and bounded by m in the set [ s

j=1

K e 1 × · · · × I _j ^m × · · · × e K s .

Hence, by the Siciak’s theorem, I 1 × e K 2 × · · · × e K s ⊂ A (f ).

2 ⁰ For y 1 ∈ U the set F f,1 (y 1 ) is pluripolar.

Since I 1 × {y 1 } ⊂ A (f ) we see that there exist D - complex neighbourhood of I 1

and B - complex neighbourhood of y 1 such that f is holomorphic in D × B. By the Bedford-Taylor theorem

N :=

(

z ∈ D : ϕ (z) := sup

|α|≥1

¯ ¯

¯ ¯ 1 α!

∂ ^|α| f

∂y ₁ ^α (z, y 1 )

¯ ¯

¯ ¯

1/|α|

< ϕ ^∗ (z) )

is pluripolar, and of course F f,1 ⊂ N .

3 ⁰ If V is a countable and dense subset of U then Z f,1 ⊂ S

y

∈V F f,1 (y 1 ).

Take x ⁰ ₁ ∈ Z f,1 . We can find y ₁ ⁰ ∈ I 2 × · · · × I s such that ¡

x ⁰ ₁ , y ₁ ⁰ ¢

∈ S (f ), but y ₁ ⁰ ∈ A ¡

f ¡

x ⁰ ₁ , • ¢¢

. Hence f ¡ x ⁰ ₁ , • ¢

is holomorphic in the polydisc P ¡ y ₁ ⁰ , 1 ±

Q f,1

¡ x ⁰ ₁ , y ⁰ ₁ ¢ ¢

⊂ C ^N , where N := n ₂ +· · ·+n _s . Let λ be such that 0 < λ ≤ 1/4 and (1 − λ) ^−1−N < 2 and let r := min ©

1, 1 ± Q f,1

¡ x ⁰ ₁ , y ₁ ⁰ ¢ ª

. For y 1 ∈ ϑ := P ¡

y ₁ ⁰ , λr ¢

⊂ C ^N we have

f ¡

x ⁰ ₁ , y ₁ ¢

= X

α

1 α!

∂ ^|α| f

∂y ^α

¡ x ⁰ ₁ , y ⁰ ₁ ¢ ¡

y ₁ − y ₁ ⁰ ¢ _α .

We deduce that

¯ ¯

¯ ¯

¯ 1 β!

∂ ^|β| f

∂y ₁ ^β

¡ x ⁰ ₁ , y 1

¢ ¯

¯ ¯

¯ ¯ ≤ Q f,1

¡ x ⁰ ₁ , y ⁰ ₁ ¢ _|β| X

α

(α + β)!

α! β! λ ^|α|

= Q f,1

¡ x ⁰ ₁ , y ₁ ⁰ ¢ _|β|

(1 − λ) ^−|β|−N , hence

Q f,1

¡ x ⁰ ₁ , y 1

¢ ≤ (1 − λ) ^−1−N Q f,1

¡ x ⁰ ₁ , y ₁ ⁰ ¢

< 2 /r .

By 1 ⁰ there exists e y ₁ ∈ ϑ ∩ V . It is enough to show that x ⁰ ₁ ∈ F _f,1 (e y ₁ ). Assume it is not so, that is Q f,1 (•, e y) is upper semicontinuous at x ⁰ ₁ . Therefore there exists a closed interval K - neighbourhood of x ⁰ ₁ in I ₁ such that for x ₁ ∈ K

Q _f,1 (x ₁ , e y) < 2 /r .

The function f (x 1 , •) is holomorphic in a neighbourhood of e y 1 (because e y 1 ∈ U , hence (x 1 , e y 1 ) ∈ A (f )) and so is holomorphic in the polydisc P (e y 1 , 1/Q f,1 (x 1 , e y 1 )). We have

P (e y ₁ , 1/Q _f,1 (x ₁ , e y ₁ )) ⊃ P (e y ₁ , r /2 ) ⊃ ϑ,

hence for x 1 ∈ K f (x 1 , •) is holomorphic in ϑ. Moreover for y 1 ∈ ϑ we have

|f (x 1 , y 1 )| ≤ X

α

Q f,1 (x 1 , y 1 ) ^|α| (λr) ^|α| ≤ X

α

µ 1 2

¶ _|α|

= 2 ^N .

Let U 1 and I ₁ ^m be as in the proof of 1 ⁰ . Take a closed interval H ⊂ ϑ ∩ U 1 . We can find m such that f is separately holomorphic (as a function of two variables: x 1 ∈ I 1

and y 1 ∈ I 2 × · · · × I s ) and bounded by m in K × ϑ ∪ I ₁ ^m × H. By the Siciak’s theorem

¡ x ⁰ ₁ , y ₁ ⁰ ¢

∈ A (f ) - contradiction.

By 2 ⁰ and 3 ⁰ we deduce that Z f,1 is pluripolar. Thus we have proved the first inductive step: we have shown that theorem C is true for k = 1 and any s ≥ 2. Now let k ≥ 2 and assume that theorem C is true for k − 1 and any s ≥ k.

4 ⁰ The set

W := {y ∈ I _k+1 × · · · × I _s : S (f (•, y)) = S (f ) (•, y)}

is dense in I k+1 × · · · × I s .

As we have just shown theorem C is true for k = 1. Using this k times for any k > 1 we see that for quasi almost all x s ∈ I s , . . . , for quasi almost all x k+1 ∈ I k+1

we have

S (f (•, x k+1 , . . . , x s )) = S (f ) (•, x k+1 , . . . , x s ) . In particular W is dense.

5 ⁰ For y ∈ W the set F f,k (y) is pluripolar.

If L ⊂⊂ A (f ) (•, y), then in the same way as in the proof of 2 ⁰ we show that F f,k (y) ∩ L is pluripolar.

6 ⁰ If W ⁰ is a countable and dense subset of W , then the set R := Z f,k \ [

y∈W

¡ S (f (•, y)) ∪ F f,k (y) ¢

is pluripolar.

Take any x ⁰ ∈ R. By the definition of Z _f,k we can find y ⁰ ∈ I _k+1 × · · · × I _s such that ¡

x ⁰ , y ⁰ ¢

∈ S (f ), but y ⁰ ∈ A ¡ f ¡

x ⁰ , • ¢¢

. Denote g := f ¡

x ⁰ ₁ , . . . , x ⁰ _k−1 , • ¢

. First we want to show that ¡

x ⁰ _k , y ⁰ ¢

∈ A (g). Assume ¡

x ⁰ _k , y ⁰ ¢

∈ S (g). We have y ⁰ ∈ A ¡

g ¡

x ⁰ _k , • ¢¢

, therefore x ⁰ _k ∈ Z g,1 . By 3 ⁰ we can find y ∈ W ⁰ such that x ⁰ _k ∈ F g,1 (y), that is Q _g,1 (•, y) is not upper semicontinuous at x ⁰ _k . By the definition of R and W we have

x ⁰ ∈ A (f (•, y)) \ F _f,k (y) = A (f ) (•, y) \ F _f,k (y) , whence Q f,k (•, y) is upper semicontinuous at x ⁰ _k . In particular Q f,k

¡ x ⁰ ₁ , . . . , x ⁰ _k−1 , •, y ¢

= Q g,1 (•, y) is upper semicotinuous at x ⁰ - contradiction. Thus we have ¡

x ⁰ _k , y ⁰ ¢

∈ A (g), hence

¡ x ⁰ _k , y ⁰ ¢

∈ S (f ) ¡

x ⁰ ₁ , . . . , x ⁰ _k−1 , • ¢

\ S ¡ f ¡

x ⁰ ₁ , . . . , x ⁰ _k−1 , • ¢¢

, and so ¡

x ⁰ ₁ , . . . , x ⁰ _k−1 ¢

∈ Z f,k−1 . We have shown that the projection of R on I 1 ×

· · · × I _k−1 is contained in Z _f,k−1 which is, by the inductive assumption, pluripolar. In

particular R is pluripolar.

By the inductive assumption theorem C is true for any separately analytic function of k variables, hence for such functions theorem A is true as well. In particular for y ∈ I k+1 ×· · ·×I s the set S (f (•, y)) is pluripolar. Therefore, by 4 ⁰ , 5 ⁰ and 6 ⁰ , Z f,k is pluripolar.

The proof of theorem C is complete.

Acknowledgements. I would like to thank professor Siciak for calling my attention to the problem, for help in solving it and precious discussions on this material.

References

[1] E. Bedford, B. A. Taylor, A new capacity for plurisubharmonic functions, Acta Math. 149 (1982), 1-40.

[2] J. Saint Raymond, Fonctions s´epar´ement analytiques, Ann. Inst. Fourier 40 (1990), 79-101.

[3] J. Siciak, Analyticity and separate analyticity of functions defined on lower dimen- sional subsets of C ⁿ , Zeszyty Nauk. UJ, Prace Mat. 13 (1969), 53-70.

[4] J. Siciak, Separately analytic functions and envelopes of holomorphy of some lower dimensional subsets of C ⁿ , Ann. Pol. Math. 22 (1969), 145-171.

[5] J. Siciak, Singular sets of separately analytic functions, Coll. Math. 60/61(1990), 281-290.

Jagiellonian University Institute of Mathematics Reymonta 4

30-059 Krak´ow, Poland