Domains of existence for finely holomorphic functions
aa r X i v : . [ m a t h . C V ] M a r DOMAINS OF EXISTENCE FOR FINELY HOLOMORPHICFUNCTIONS
BENT FUGLEDE, ALAN GROOT, AND JAN WIEGERINCK
Abstract.
We show that fine domains in C with the property that they are Euclidean F σ and G δ , are in fact fine domains of existence for finely holomorphic functions. Moreover regular fine domains are also fine domains of existence. Next we show that fine domainssuch as C \ Q or C \ ( Q × i Q ), more specifically fine domains V with the properties thattheir complement contains a non-empty polar set E that is of the first Baire category inits Euclidean closure K and that ( K \ E ) ⊂ V , are not fine domains of existence. Introduction
It was already known to Weierstrass that every domain Ω in C is a domain of existence ,roughly speaking, it admits a holomorphic function f that cannot be extended analyticallyat any boundary point of Ω. In his thesis [Bo94] Borel showed, however, that it may bethat f can be (uniquely) extended to a strictly larger set X in an ”analytic” way, albeitthat X is no longer Euclidean open. This eventually led Borel to the introduction of hisCauchy domains and monogenic functions, cf. [Bo17]. Finely holomorphic functions onfine domains in C as introduced by the first named author, are the natural extension andsetting for Borel’s ideas, see [Fu81a], also for some historic remarks on Borel’s work. Inthis paper we will study fine domains of existence , roughly speaking, fine domains in C that admit a finely holomorphic function that cannot be extended as a finely holomorphicfunction at any fine boundary point. Definition 2.1 contains a precise definition. Herethe results are different from both the classical Weierstrass case of one variable and theclassical several variable case, where Hartogs showed that there exist domains D ( D ∗ in C n with the property that every holomorphic function on D extends to the larger domain D ∗ .Our results are as follows. In Section 2 we show that every fine domain that is a Euclidean F σ as well as a Euclidean G δ is a fine domain of existence. Euclidean domains are of thisform and therefore are fine domains of existence. We also show that regular fine domainsare fine domains of existence. Noting that on Euclidean domains holomorphic and finelyholomorphic functions are the same, our Theorem 2.9 includes Weierstrass’ theorem. InSection 3, however, we show that fine domains V with the property that their complementcontains a non-empty polar set E that is of the first Baire category in its Euclidean closure Mathematics Subject Classification.
Key words and phrases. finely holomorphic function, domain of existence.We are grateful to Jan van Mill for many enlightening discussions. K (in particular, E has no Euclidean isolated points) and ( K \ E ) ⊂ V , are not finedomains of existence.Pyrih showed in [Pyr] that the unit disc is a fine domain of existence, and that as acorollary of the proof, the same holds for simply connected Euclidean domains by theRiemann mapping theorem.The starting point of our research was the following observation. Edlund [Edl] showedthat every closed set F ⊂ C admits a continuous function f : F → C such that the graphof f is completely pluripolar in C . It is easy to see that by construction this f is finelyholomorphic on the fine interior of F . In [EEW] the main result can be phrased as follows: If a finely holomorphic function on a fine domain D admits a finely holomorphic extensionto a strictly larger fine domain D ′ , then the graph of f over D is not completely pluripolar. Hence, if D is the fine interior of a Euclidean closed set, every fine component of D is afine domain of existence. This is extended by our Theorem 2.9 because the fine interior ofany finely closed set is regular.In the next section we recall relevant results about the fine topology and fine holomorphy.1. Preliminaries on the fine topology and finely holomorphic functions
Recall that a set E ⊂ C is thin at a ∈ C if a / ∈ E or else if there exists a subharmonicfunction u on an open neighborhood of a such thatlim sup z → a,z ∈ E \{ a } u ( z ) < u ( a ) . The fine topology was introduced by H. Cartan in a letter to Brelot as the weakest topologythat makes all subharmonic functions continuous. He pointed out that E is thin at a if andonly if a is not in the fine closure of E . The fine topology has the following known features.Finite sets are the only compact sets in the fine topology, which follows easily from thefact that every polar set X ⊂ C is discrete in the fine topology, it consists of finely isolatedpoints. The fine topology is Hausdorff, completely regular, Baire, and quasi–Lindel¨of, i.e.a union of finely open sets equals the union of a countable subfamily and a polar set,cf. e.g. [ArGa, Lemma71.2] . For our purposes fine connectedness is important.Recall the following Theorem 1.1 (Fuglede) . The fine topology on C is locally connected . That is, for any a ∈ C and any fine neighborhood U of a , there exists a finely connected finely open neighborhood V of a with V ⊂ U . See [Fu72, p.92] (or see [EMW] for a proof using only elementary properties of subhar-monic functions).In fact, the fine topology is even locally polygonally arcwise connected as was shown in[Fu72], but we need something stronger.
Definition 1.2. A wedge is a polygonal path consisting of two line segments [ a, w ] and[ w, b ] of equal length.The result we need is as follows, OMAINS OF EXISTENCE FOR FINELY HOLOMORPHIC FUNCTIONS 3
Theorem 1.3 ([Fu80]) . Let U be a finely open set in C and α > . Then for every w ∈ U ,there exists a fine neighborhood V of w such that any two distinct points a, b ∈ V can beconnected by a wedge contained in U of total length less than α | a − b | . Lyons, see [Lyo, p. 16] had already proven that a and b can be connected by a polygonalpath consisting of two line segments of total length less than α | a − b | and his proof essentiallycontained Theorem 1.3. For an even stronger result see Gardiner, [Gar, Theorem A].We will also need the following elementary lemma. Let B ( x, r ) denote the open discwith radius r about x ∈ C , B ( x, r ) its closure, and let C ( x, r ) denote its boundary. Lemma 1.4.
Let V be a fine neighborhood of a ∈ C . Then there exists C > such thatfor C > C and every n ∈ N there exists t ∈ [ C − n − , C − n ] with C ( a, t ) ⊂ V .Proof. We can assume a = 0. A local basis at 0 for the fine topology consists of thesets B (0 , r, h ) = { z ∈ B (0 , r ) : h ( z ) > } , where r > h is subharmonic on B (0 , r )and h (0) = 1. See [EMW, Lemma 3.1] for a proof. Thus for some h and r we have V ⊃ W := { z ∈ B (0 , r ) : h ( z ) > / } and B (0 , r ) \ V is contained in F := B (0 , r ) \ W ,which is Euclidean open, hence an F σ , that is thin at 0, because W contains the finelyopen set B (0 , r, h ) which contains 0. Theorem 5.4.2 in [Ran] states for given r > > Z E x dx < Γ < ∞ , where E := { s : 0 < s < r | ∃ θ with se iθ ∈ F } . Let C = max { e Γ , /r } . For C > C each interval of the form [ C − n − , C − n ] is containedin (0 , r ), but can not be contained in E . Hence there exists t ∈ [ C − n − , C − n ] such that C (0 , t ) ∩ F = ∅ , that is, C (0 , t ) ⊂ W ⊂ V . (cid:3) For more information on the fine topology see [Do, Part I, Chapter XI], [ArGa, ChapterVII].There are several equivalent definitions for finely holomorphic functions on a fine domainin D ⊂ C , cf. [Fu81, Fu81a, Fu88, Lyo]. Definition 1.5.
A function f on a fine domain D is called finely differentiable at z ∈ D with fine complex derivative f ′ ( z ) if there exists f ′ ( z ) ∈ C such that for every ε > V ⊂ D of z such that (cid:12)(cid:12)(cid:12)(cid:12) f ( z ) − f ( z ) z − z − f ′ ( z ) (cid:12)(cid:12)(cid:12)(cid:12) < ε for all z ∈ V. In other words, the limit ( f ( z ) − f ( z )) / ( z − z ) exists as z − z → f is finely holomorphic on D if and only if f is finely differentiable at everypoint of D and f ′ is finely continuous on D .We will use the following characterizations of fine holomorphy. BENT FUGLEDE, ALAN GROOT, AND JAN WIEGERINCK
Theorem 1.6 ([Fu81a]) . Let f be a complex valued function on a fine domain D . Thefollowing are equivalent (1) The function f is finely holomorphic on D . (2) Every point z ∈ D admits a fine neighborhood V ⊂ D such that f is a uniformlimit of rational functions on V . (3) The functions f and z zf ( z ) are both (complex valued) finely harmonic functionson D . In the following theorem we collect properties of finely holomorphic functions that indi-cate how much this theory resembles classical function theory.
Theorem 1.7 ([Fu81]) . Let f : D → C be finely holomorphic. Then (1) The function f has fine derivatives f ( k ) of all orders k , and these are finely holo-morphic on D . (2) Every point z ∈ D admits a fine neighborhood V ⊂ D such that for every m =0 , , , . . . (1.1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ( w ) − m − X k =0 f ( k ) ( z ) k ! ( w − z ) k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:14) | w − z | m is bounded on V × V for z = w . (3) At any point z of D the Taylor expansion (1.1) uniquely determines f on D . (Ifall coefficients equal 0, then f is identically .) We now introduce finely isolated singularities.
Definition 1.8.
Let D be a fine domain, a ∈ D , and f finely holomorphic on D \ { a } . • If f extends as a finely holomorphic function to all of D , then f has a removablesingularity at a . • If f ( z ) → ∞ for z → a finely, f has a pole at a . • If f has no pole at a nor a removable singularity, then f has an essential singularityat a . Theorem 1.9.
Let D be a fine domain and a ∈ D . Let f be finely holomorphic on D \ { a } and suppose that f is bounded on a fine neighborhood V of a . Then a is a removablesingularity of f .Proof. (As indicated in [Fu88].) We apply Theorem 1.6, no. 3. Clearly f and z zf ( z )are bounded, finely harmonic functions on V \ { a } . [Fu72, Corollary 9.15] states that thesefunctions extend to be finely harmonic on all of D and again by Theorem 1.6 f is finelyholomorphic on D . (cid:3) Fine domains of existence
For a set A in C , we denote its fine interior by int f A , its fine closure by cl f A and its fineboundary by ∂ f A . OMAINS OF EXISTENCE FOR FINELY HOLOMORPHIC FUNCTIONS 5
Definition 2.1.
A fine domain U is called a fine domain of existence if there exists afinely holomorphic function f on U with the property that for every fine domain V thatintersects ∂ f U and for every fine component Ω of V ∩ U , the restriction f | Ω admits nofinely holomorphic extension to V . Definition 2.2.
Let U be a fine domain in C . If there exist compact sets K n ⊂ U suchthat U = S ∞ n =1 K n with K ⊂ int f K ⊂ K ⊂ int f K ⊂ · · · , then the sequence { K n } iscalled a fine exhaustion of U . If, moreover, the K n have the property that every boundedcomponent of C \ K n contains a point from C \ U , we call { K n } a special fine exhaustion of U .We have the following lemma, which is a consequence of the Lusin–Menchov propertyof the fine topology. Lemma 2.3 ([LMN, Corollary 13.92]) . Let U be a finely open set and let K be a compactsubset of U . Then there exists a Borel finely open set V such that K ⊂ V ⊂ V ⊂ U . Proposition 2.4.
Let U be a fine domain that is a Euclidean F σ . Then U admits a specialfine exhaustion.Proof. Let U = S ∞ n =1 F n for compact sets F ⊂ F ⊂ · · · . Fix a strictly increasing sequence( r n ) n ≥ tending to infinity such that F n ⊂ B (0 , r n ) for all n ≥
1. We put K := F . Forthe construction of K , note that F ∪ K (= F ) is a compact subset of U . By Lemma2.3, there exists a finely open set V such that F ∪ K ⊂ V ⊂ V ⊂ U . Then the set K := V ∩ B (0 , r ) is compact andint f K = int f V ∩ int f B (0 , r ) ⊃ V ∩ B (0 , r ) ⊃ F ⊃ F = K . As induction hypothesis, suppose that for some n ≥
2, we have found compact sets K , . . . , K n such that K j ⊂ int f K j +1 for all 1 ≤ j ≤ n − F j ⊂ K j for all 1 ≤ j ≤ n and K j ⊂ B (0 , r j +1 ) for all 1 ≤ j ≤ n and that we have found finely open sets V , . . . , V n suchthat F j ∪ K j − ⊂ V j ⊂ V j ⊂ U for all 2 ≤ j ≤ n .We now prove the induction step. Note that the set F n +1 ∪ K n is a compact subset of U .By the previous lemma, there exists a finely open set V n +1 such that F n +1 ∪ K n ⊂ V n +1 ⊂ V n +1 ⊂ U . The set K n +1 := V n +1 ∩ B (0 , r n +1 ) is compact, contained in U and in B (0 , r n +2 )and int f K n +1 = int f V n +1 ∩ int f B (0 , r n +1 ) ⊃ V n +1 ∩ B (0 , r n +1 ) ⊃ F n +1 ∪ K n and therefore int f K n +1 ⊃ K n and K n +1 ⊃ F n +1 . This proves the induction step.Consequently, we can find compact subsets K , K , . . . of U such that K ⊂ int f K ⊂ K ⊂ int f K ⊂ · · · and F n ⊂ K n for all n ≥
1. It follows that U = S ∞ n =1 K n , which provesthat U admits a fine exhaustion.We will next adapt this fine exhaustion so that every bounded component of C \ K n contains a point from C \ U . Observe that K n = C \ S j D nj , where for each n the D nj are the countably, possibly infinitely, many mutually disjoint open components of C \ K n .Let D n be the unbounded component. Then for any n and any finite or infinite sequence1 j j · · · the set K n ∪ S k D nj k is compact too. For every n we set K ∗ n = K n ∪ S k D nj k , BENT FUGLEDE, ALAN GROOT, AND JAN WIEGERINCK S W S W b q Figure 1.
Replacing a small circular arc.where the D nj k are those components, if any, of C \ K n that are completely contained in U .We claim that K ∗ n ⊂ int f ( K ∗ n +1 ).Indeed, if x ∈ K n then x ∈ int f K n +1 ⊂ int f ( K ∗ n +1 ). Now let x ∈ D nj k ( ⊂ U ). For theproof that x ∈ K ∗ n +1 we may suppose that x / ∈ K n +1 . Then x belongs to a (necessarilybounded) component of C \ K n +1 that is completely contained in D nj k ⊂ U , hence x ∈ K ∗ n +1 .It follows that D nj k ⊂ K ∗ n +1 and as D nj k is Euclidean open, D nj k ⊂ int f ( K ∗ n +1 ). This provesthe claim. (cid:3) We will need the following lemma. Figure 1 illustrates its content.
Lemma 2.5.
Let D be an open disc, U a finely open subset of D , and let a ∈ C ∩ U where C is an open circular arc contained in D with the property that D \ C has two components.Then there exists a sequence ( r j ) j of positive numbers decreasing to 0, such that for every r j there exists a compact set C ′ (= C ′ j ) ⊂ U , which is the union of four arcs, such that (1) D \ C ′ is connected; (2) C ′ \ B ( a, r j ) = C \ B ( a, r j ) ; (3) Every wedge ℓ = [ p, b ] ∪ [ b, q ] with | p − b | = | b − q | > r j that meets C also meets C ′ .Proof. By Theorem 1.3 there exists a fine neighborhood V ⊂ U of a such that any twodistinct points p, q ∈ V can be connected by a wedge L ⊂ U of length less than √ | p − q | .Lemma 1.4 provides us with a sequence ( r j ) j such that C ( a, r j ) ⊂ V .After scaling and rotating we may assume that C = C ( − i, ∩ D and that a = 0. Wecan also assume all r j less than 1 and so small that B (0 , r j ) ⊂ D . Now we fix r = r j and OMAINS OF EXISTENCE FOR FINELY HOLOMORPHIC FUNCTIONS 7 set C = ( C \ ( B (0 , r ) ∩ { Re z < } )) ∪ (cid:0) { re iθ : π/ θ π/ } (cid:1) .C is contained in V along with C (0 , r ), therefore ir , respectively − ir , (both on C ) canbe connected to 0(= a ) ∈ V ⊂ U by a wedge W , respectively W , of length less than √ r contained inside U . Observe that the wedges W , W are contained in B (0 , r ), and in fact W is contained in the closed square S with diagonal [0 , ir ] and similarly W is containedin the closed square S with diagonal [0 , − ir ], see Figure 1.Let ℓ = [ p, b ] ∪ [ b, q ] be any wedge with | p − b | = | b − q | > r that meets C andassume that ℓ does not meet C . Then one of its constituting segments, say [ b, q ], meets C ∩ B (0 , r ) ∩ { Re z < } and because | q − b | > r , it meets C (0 , r ) in a point re iθ with − π/ < θ < π/
6. Therefore, as r <
1, [ b, q ] ∩ ( S ∪ S ) is contained in { z : | Im z | 6) = r/ } ∩ ( S ∪ S ), as indicated in Figure 1.Now we define L j = W j ∩ { z : | Im z | r/ } ⊂ U , j = 1 , C ′ = C ∪ L ∪ L ⊂ U is the the union of three circular arcs and one arc consisting of two straight segments, allcontained in U . Clearly ℓ meets C ′ and D \ C ′ is connected. (cid:3) Proposition 2.6. Let U ( C be a nonempty fine domain that is a Euclidean F σ and let ∅ 6 = F ⊂ F ⊂ · · · be an increasing sequence of compact sets in C \ U . Then there existsequences ( K n ) n ≥ and ( L n ) n ≥ of nonempty compact sets such that (a) K ⊂ int f K ⊂ K ⊂ int f K ⊂ · · · and S ∞ n =1 K n = U , (b) d ( K n , L n ) > for all n ≥ , (c) L n ⊂ K n +1 for all n ≥ , (d) for any n ≥ , every two points w, z ∈ C \ ( K n ∪ L n ) in the same component of C \ K n also lie in the same component of C \ ( K n ∪ L n ) , (e) every bounded component of C \ ( K n ∪ L n ) contains a point from C \ U , (f) every wedge of length at least /n that meets F n and K n , also meets L n .Moreover, there exists a sequence ( f ν ) ν ≥ of rational functions with poles in C \ U such that (1) the sup-norm k f n − f n − k K n < / n for all n ≥ and (2) | f n | > n on L n for all n ≥ .The sequence ( f ν ) ν converges uniformly on any K n to a function f that is finely holomorphicon U and has the property that | f | > n − / n on L n for any n ≥ .Proof. For ( K n ) we take a special fine exhaustion of U , U = S ∞ n =1 K n , which exists inview of Proposition 2.4. We next construct a sequence ( L n ) n ≥ of compact sets such that d ( L n , K n ) > L n ⊂ K n +1 for all n ≥ n ≥ 1. The set F n ⊂ C \ U is compact and is therefore covered (uniquely) bythose finitely many components of C \ K n that meet F n . We denote these componentsby D n , D n , . . . , D nk n . Since K n and F n are compact and K n ∩ F n = ∅ , it follows that δ n := d ( K n , F n ) > 0. Consequently,(2.1) { z ∈ C : d ( z, K n ) < δ n } ∩ F n = ∅ . BENT FUGLEDE, ALAN GROOT, AND JAN WIEGERINCK Since K n ⊂ int f K n +1 , by Lemma 1.4 each z ∈ ∂K n is the center of a circle C ( z, ε z )contained in int f K n +1 of radius ε z less than δ n . We will now construct L n by construct-ing the intersections L n ∩ D nk (with some abuse of notation) in each of the components D n , D n , . . . , D nk n and taking the union of these sets.Fix a component D nk . Since ∂D nk ⊂ ∂K n is compact and is covered by the open disks B ( z, ε z ) where z ∈ ∂D nk , it follows that there are finitely many points z , . . . , z m ∈ ∂D nk such that ∂D nk ⊂ S mj =1 B ( z j , ε z j ). Note that ( ∂ S mj =1 B ( z j , ε z j )) ∩ D nk is a finite unionof closed circular arcs C l –and possibly finitely many points disjoint from these arcs– ofthe components of S mj =1 ∂B ( z j , ε z j ) \ S mk =1 B ( z k , ε z k ). We define a compact set L ∗ n,k := S Ml =1 C l ⊂ D nk . Also note that every path γ , in particular every wedge, that connects apoint z ∈ F n ∩ D nk and a point w ∈ K n must intersect one of the arcs C l in L ∗ n,k , because d ( z, K n ) > δ n > ε z in view of (2.1), and w ∈ K n ⊂ C \ D nk . The open set C \ L ∗ n,k has onlyfinitely many, say N > 2, components as each of the C l belongs to the boundary of two ofthese components. We will replace L ∗ n,k by L n ∩ D nk in the following way to achieve that C \ ( L n ∩ D nk ) is connected and the properties (a), (b) and (c) are kept.To do this, note that C belongs to the boundary of two components of C \ L ∗ n,k . Pick apoint a of C that is not an endpoint of this closed arc. Since a ∈ int f K n +1 \ K n , clearly a / ∈ F n +1 ⊃ F n , and again since a lies in the relative interior of the arc C , we see that forall r > B ( a, r ) ∩ K n = ∅ = B ( a, r ) ∩ F n and B ( a, r ) ∩ L ∗ n,k is contained inthe relative interior of C .We apply Lemma 2.5 with D = B ( a, r ), U = int f K n +1 ∩ B ( a, r ), r j < /n and obtaina compact C ′ . Then C \ ( C ′ ∪ S Ml =2 C l ) has N − C belongs to theboundary of two different components of C \ ( C ′ ∪ S Ml =2 C l ) we replace it by C ′ likewise,reducing the number of components of the complement by one again. If C belongs to theboundary of only one component, we just put C ′ = C . Proceeding in this way, we end with L n ∩ D nk := S Ml =1 C ′ l , the complement of which is connected. By setting L n := S nk =1 ( L n ∩ D nk ),we now find that any two points w, z ∈ C \ ( K n ∪ L n ) that are in the same component of C \ K n are in the same component of C \ ( K n ∪ L n ) as well, while Lemma 2.5 guaranteesthat (f) is satisfied. Also, note that L n ⊂ int f K n +1 \ K n , so that K n ∩ L n = ∅ . It followsthat d ( K n , L n ) > n ≥ 1. We have obtained sequences ( K n ) n ≥ and ( L n ) n ≥ ofcompact sets that satisfy the properties (a) – (f).Because of property (b), we can apply Runge’s theorem recursively for each n to theholomorphic function g n that equals 0 on an open set containing K n and is equal to P n − j =1 max z ∈ L n {| R j ( z ) |} + n + 1 on an open set containing L n . Thus there exists for each n a rational function R n with | R n | < / n , on K n ;(2.2) | R n | > n − X j =1 max z ∈ L n {| R j ( z ) |} + n, on L n , (2.3) OMAINS OF EXISTENCE FOR FINELY HOLOMORPHIC FUNCTIONS 9 while R n has at most one pole in a preassigned point in each of the bounded componentsof C \ ( K n ∪ L n ). Because of property (e) these poles may be taken from C \ U . Set f k = k X j =1 R j . Then f k clearly satisfies (1) and (2). It now follows from (1) and (a) that there exists afunction f on U such that f k → f uniformly on any K n . Note that f is finely holomorphicon U , because if z ∈ U , then z ∈ K n for some n ≥ K n +1 is acompact fine neighborhood of z on which f k → f uniformly. Furthermore, it follows from(2) and (c) above that for any n ≥ z ∈ L n | f n ( z ) | ≥ | R n ( z ) | − n − X j =1 | R j ( z ) | > n. Finally, | f | > n − / n follows immediately from (1) and (2). (cid:3) We shall prove in Theorem 2.9 that under suitable conditions on the fine domain, thefunction that we have just constructed admits no finely holomorphic extension outside thedomain. We need some facts about (regular) fine domains.It is known that every regular fine domain U is an F σ because C \ U is a base in thesense of Brelot, and hence a G δ . Moreover, we need the following Lemma, which is (iv) of[ArGa, Theorem 7.3.11]. Lemma 2.7. Every finely closed set A ⊂ C is the disjoint union of a (Euclidean) F σ anda polar set. Lemma 2.8. Let ∅ 6 = U ⊂ V be fine domains and let E be a polar subset of C \ U . Theneither there exists a point a ∈ ( ∂ f U ∩ V ) \ E , or V = U ∪ e for the polar set e := ( V \ U ) ∩ E ,whereby e is empty and hence U = V in case U is a regular fine domain.Proof. If ( ∂ f U ∩ V ) \ E is empty then ( ∂ f U ∩ V ) ⊂ E ∩ ( V \ U ) = e , hence the fine boundary ∂ f U ∩ ( V \ e ) of U relative to V \ e is empty. Since V \ e is finely open and finely connectedalong with V , by [Fu72, Theorem 12.2], and since U = ∅ this means that U = V \ e , thatis, V = U ∪ e as claimed. If U is regular then e is empty because e and U are disjoint and V is finely open. (cid:3) Theorem 2.9. Let U ⊂ C be a non-empty fine domain that is either (1) both Euclidean F σ and Euclidean G δ , or (2) a regular fine domain. Then U is a fine domain of existence.Proof. In both cases U is an F σ , and we can write U = S ∞ n =1 K n with K j ⊂ K j +1 compactsubsets of U . Because of Lemma 2.7, C \ U = F ∪ E , where F = S ∞ n =1 F n with F j ⊂ F j +1 compact in C \ U , is a Euclidean F σ and E is a polar set which we can assume to be emptyin Case (1). Let L j ⊂ K j +1 be compact sets and f a finely holomorphic function on U asconstructed in Proposition 2.6.Let V be a fine domain that meets ∂ f U and let Ω be a fine component of U ∩ V . Then ∂ f Ω ∩ V ⊂ ∂ f U ∩ V , because if x ∈ ∂ f Ω ∩ V ∩ U there would be a finely connected fine neighborhood of x contained in V ∩ U , contradicting that Ω is a fine component of V ∩ U .Suppose that f | Ω admits a finely holomorphic extension ˜ f defined on V . By Lemma 2.8applied to Ω in place of U , there is a point z ∈ ( ∂ f Ω ∩ V ) \ E , whereby in Case (1) E = ∅ andhence z ∈ ( ∂ f Ω ∩ V ) ⊂ ∂ f U ∩ V ⊂ F . In Case (2), if V = Ω ∪ e , then U ∪ V = U ∪ e , whichis impossible because U is a regular domain. Hence z ∈ ( ∂ f Ω ∩ V ) \ E = ( ∂ f U ∩ V ) \ E ⊂ F .Again by Proposition 2.6 there is a compact fine neighborhood V of z in V on which ˜ f is the uniform limit of rational functions with poles off V . In particular, ˜ f is continuousand bounded on V , say | ˜ f | ≤ M on V . Choose a fine domain V ⊂ V containing z andsuch that, for given α > 1, any two distinct points a, b of V can be joined by a wedge oftotal length less than α | a − b | contained in V , see Theorem 1.3.Choose z ∈ Ω \ { z } . By the above and by Lemma 1.4, there exists for every ε > < r < ε with r < | z − z | such that C ( z , r ) ⊂ V . Choose z ∈ Ω with | z − z | < r (possible since z ∈ ∂ f Ω ⊂ ∂ Ω). There is a path γ ⊂ Ω joining z and z and meeting C ( z , r ) at a point w of Ω ∩ V . Thus C ( z , r ) ∩ Ω = ∅ . Recall that also z ∈ V . Let l = [ w , b ] ∪ [ b , z ] with | w − b | = | b − z | be a wedge of total length less than α | w − z | contained in V . Since w ∈ U , we have w ∈ K n for all n large enough. Also, z ∈ F ,hence z ∈ F n for all n large enough. Therefore we can take an n ≥ w ∈ K n and z ∈ F n and such that n − / n − > M and 4 /n < | w − b | + | b − z | hold simultaneously. Then by Proposition 2.6 (f), l meets L n ∩ Ω.Let z ∈ l ∩ L n ∩ Ω. Since z ∈ l ∩ Ω ⊂ V , we have | ˜ f ( z ) | ≤ M . On the other hand,since z ∈ L n ∩ Ω, we have ˜ f ( z ) = f ( z ) and by (2) in Proposition 2.6 | ˜ f ( z ) | = | f ( z ) | ≥ n − / n − > M , which is a contradiction. We conclude that f does not admit a finelyholomorphic extension and that U is a fine domain of existence. (cid:3) In the rest of this section we extend the above theorem a little. Lemma 2.10. Let U and U be fine domains of existence, and suppose that ( ∂ f U ) ∩ ( ∂ f U ) = ∅ . Then every fine component of U ∩ U is a fine domain of existence.Proof. The assertion amounts to U ∩ U being a “finely open set of existence” (if nonvoid)in the obvious sense. Because ( ∂ f U ) ∩ ( ∂ f U ) = ∅ we have, writing U ∩ U = U , ∂ f U = ( U ∩ ∂ f U ) ∪ ( U ∩ ∂ f U ) . By hypothesis there exists for i = 1 , h i on U i such that forany fine domain V that intersects ∂ f U i and any fine component Ω i of U i ∩ V , h i | Ω i doesnot extend finely holomorphically to V .Let V be any fine domain that intersects ∂ f U and let Ω be a fine component of V ∩ U .Without loss of generality we may assume that V intersects U ∩ ∂ f U , and by shrinking V that V ⊂ U . Then Ω is a fine component of U ∩ V = U ∩ ( U ∩ V ) = U ∩ V .The function h := h | U + h | U is then finely holomorphic on U . Because h is finelyholomorphic on V ⊂ U , the function h | Ω is extendible to V if and only if h | Ω is extendibleover V , and that is not the case. Therefore U is a finely open set of existence. (cid:3) Theorem 2.11. Every fine domain U ⊂ C such that the set I of irregular fine boundarypoints for U is both an F σ and a G δ is a fine domain of existence. OMAINS OF EXISTENCE FOR FINELY HOLOMORPHIC FUNCTIONS 11 Proof. In the above lemma take U = U r (the regularization of U ) and U = C \ I . Since I is polar, U is a fine domain along with C . Since U is finely connected so is U r . In fact, U r = U ∪ E , where E denotes the polar set of finely isolated points of C \ U . If U r = V ∪ V with V , V finely open and disjoint then U = U r \ E = ( V \ E ) ∪ ( V \ E )with V \ E and V \ E finely open and disjoint. It follows that for example V \ E = ∅ andhence V = ∅ , showing that U r indeed is finely connected. By hypothesis, the complementof U and hence U itself is an F σ and a G δ . Then by Theorem 2.9 U and U are finedomains of existence. By Theorem 2.9, U ∩ U = U r \ I = U is indeed a fine domain ofexistence. (cid:3) Fine domains that are not domains of existence Proposition 3.1. Let ( a n ) be a sequence in C that converges to a . Suppose that V n is afine neighborhood of a n of the form V n = V n ( a n , h n , r ) = { z ∈ B ( a n , r ) : h n ( z ) > } , where h n is a subharmonic function on B ( a n , r ) such that h n ( a n ) = 1 / and h n < on B ( a n , r ) . Then S ∞ n =1 V n is a (possibly deleted) fine neighborhood of a .Proof. Let r < r . Then there exists n > n > n the function h n is definedon B ( a, r ). Let h = ( sup n > n { h n | B ( a,r ) : n > n } ) , on B ( a, r ) and let h ∗ denote its upper semi-continuous regularization. Then h ∗ is subhar-monic, h ∗ 1, and h ∗ ( a ) > lim sup n →∞ h ( a n ) > / 2. Let V = { z ∈ B ( a, r ) : h ∗ ( z ) > } .Then V is a fine neighborhood of a , since h ∗ is finely continuous. Because the set X = { h < h ∗ } \ { a } is polar, it is finely closed, therefore the set V = V \ X is afine neighborhood of a .We claim that V \ { a } ⊂ S n > n V n . Indeed, if z ∈ V \ { a } then h ( z ) = h ∗ ( z ) > h n ( z ) > n > n , and z ∈ V n . (cid:3) Proposition 3.2. Let E be a non-empty polar set in C and suppose that E is of the firstBaire category in its Euclidean closure K . Suppose that f is finely holomorphic on a finelyopen set V such that K \ E ⊂ V . Then there exist a Euclidean open ball B that meets K ,and a finely open fine neighborhood V of K ∩ B such that f is bounded on V \ E .Proof. Let x ∈ V \ E . Denote by U x a finely open subset of V containing x and having theproperty stated in Theorem 1.3 applied to V . By shrinking we may arrange that U x hasthe form U x = U x ( x, h x , r x ) = { z ∈ B ( x, r x ) : h x ( z ) > } , where h x is a subharmonic function on B ( x, r x ) such that h x ( x ) = 1 / h x < B ( x, r x ).Let X j be the set of x ∈ K \ E such that r x > /j , and | f | j on U x and that U x satisfies Lemma 1.4 with C = j . Then K \ E = S j X j , hence by the Baire category theorem, there exist j and an open Euclidean ball which we may assume to be the unitdisc D , such that ∅ 6 = K ∩ D ⊂ X j (the Euclidean closure of X j ).Let w ∈ K ∩ D . Then w = lim n →∞ x n for a sequence ( x n ) n in X j . As r x n > /j ,Proposition 3.1 gives us that S n U x n is a (possibly deleted) fine neighborhood of w onwhich | f | j and W w := { w } ∪ S n U x n is a fine neighborhood of w . If w ∈ ( K \ E ) ∩ D , | f | j on W w . In fact, f is finely holomorphic on V , hence | f | is finely continuous on thefinely open set V ∩ W w which contains w because V ⊃ K \ E . If | f ( w ) | > j then | f | > j on some fine neighborhood Z of w , which contradicts that Z meets W w \ { w } because C has no finely isolated points.The set V = S w ∈ K ∩ D W w is a finely open fine neighborhood of K ∩ D with the propertythat on | f | j on V \ E . (cid:3) Remark . The reader should be aware that the condition E is of the first Baire categoryin its Euclidean closure K prevents sets like E = { /n, n = 1 , , . . . } . Indeed, E can not bewritten as countable union of nowhere dense subsets in its Euclidean closure K = E ∪ { } ,because this set E is relatively open in K . Theorem 3.4. Suppose that E and K are as in Proposition 3.2, and that V is a finedomain such that V ∩ E = ∅ and K \ E ⊂ V . Then (1) E ⊂ ∂ f V and (2) V is not a finedomain of existence.Proof. For (1) observe that ∂ f V = ∂V , because V is connected and therefore not thin atany of its Euclidean boundary points. Thus it suffices to show that E ⊂ ∂V . We have E = S ∞ n =1 F n , a countable union of nowhere dense subsets of K , which is of the secondcategory in itself. Therefore, E is contained in the closure of K \ E . If not, suppose to reacha contradiction, that O were an open set in K not meeting K \ E , then O = S ∞ n =1 ( F n ∩ O ),a countable union of nowhere dense sets, contradicting that K is of the second category.As K \ E is contained in the domain V , we find that E must be contained in the Euclideanclosure of V and hence in ∂ f V = ∂V because V ∩ E = ∅ .For (2) let f be finely holomorphic on V . By Proposition 3.2 there exist an open ball B that meets K (and hence E ) and a finely open set V containing K ∩ B such that f is bounded on V ∩ V ∩ B ⊂ V \ E . Because E is polar, V ∩ V is a deleted finely openfine neighborhood of every point in E ∩ B , hence each point a of E ∩ B is a finely isolatedsingularity of f . Theorem 1.9 implies that f extends over each a ∈ E ∩ B and hence hasa finely holomorphic continuation to the finely open set V ∪ [( V ∪ E ) ∩ B ] = V ∪ ( E ∩ B ),which contains V properly because ( E ∩ B ) ⊂ ∂ f V . (cid:3) Example 3.5. Let V = C \ Q , E = Q , and K = R . 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