RAPID CONVERGENCE OF RATIONAL FUNCTIONS AND PLURIPOLAR HULLS

ABSTRACT. Let f be a holomorphic function on a domain D ⊂ C n . Using the method of rapidly convergence, we study pluripolar hull in C n+1 of Γf(D) := {(z, f(z) : z ∈ D}. The first main result of this work (Theorem 3.1) states that if D is an analytic polyhedron then there exists a holomorphic function f such that Γf(D) is complete pluripolar in C n+1 . Proposition 4.2 gives a description of (Γf(D))∗ Cn+1 in case f can be rapidly approximated in capacity by a sequence of rational functions. In the course of our research, we obtain in Theorem 4.1 an improvement of an earlier result of Bloom with a simpler proof

RAPID CONVERGENCE OF RATIONAL FUNCTIONS AND PLURIPOLAR HULLS NGUYEN QUANG DIEU AND PHUNG VAN MANH A BSTRACT. Let f be a holomorphic function on a domain D ⊂ Cn . Using the method of rapidly convergence, we study pluripolar hull in Cn+1 of Γ f (D) := {(z, f (z) : z ∈ D}. The first main result of this work (Theorem 3.1) states that if D is an analytic polyhedron then there exists a holomorphic function f such that Γ f (D) is complete pluripolar in Cn+1 . Proposition 4.2 gives a description of (Γ f (D))∗Cn+1 in case f can be rapidly approximated in capacity by a sequence of rational functions. In the course of our research, we obtain in Theorem 4.1 an improvement of an earlier result of Bloom with a simpler proof. I. Introduction One of traditional problems in complex analysis is the question of holomorphic propagation e.g., find the maximal holomorphic object containing a given one. For example, let f be a holomorphic function defined on a domain D in Cn we search for its holomorphic continuation on a larger domain. A natural counterpart of holomorphic propagation in pluripotential theory is the theory of pluripolar hull. We now recall briefly some elements of pluripotential theory leading to the concept of pluripolar hull. An upper semicontinuous function u, u ̸≡ −∞, defined on a domain D ⊂ Cn is said to be plurisubharmonic if the restriction of u on every complex line is subharmonic function as a function of one variable. A subset E of D is said to be pluripolar if locally E is included in the −∞ locus of plurisubharmonic functions. According to a well-known result of Josefson, every pluripolar set E is contained in the singular locus of some global plurisubharmonic function u on Cn . Given a pluripolar subset E of D, following Poletsky and Levenberg in [LP] we define the pluripolar hull of E relative to D as follows ED∗ := {z ∈ D : ∀u ∈ PSH(D), u|E ≡ −∞ ⇒ u(z) = −∞}. Here PSH(D) denotes the cone of plurisubharmonic functions on D. It is clear that ED∗ is pluripolar. It is also obvious that if E is complete pluripolar in D i.e., E coincides with the −∞ locus of an element u ∈ PSH(D) then ED∗ = E. Conversely, Zeriahi proved in [Ze] that if ED∗ = E, E is Fσ and Gδ and if D is pseudoconvex then E must be complete pluripolar in D. For a simple explicit example of pluripolar hull we can take E = {0} × {|z| = 1} ⊂ C2 and D = C2 . Then it is easy to check, using the fact that the circle |z| = 1 is non polar in C, that the pluripolar hull ED∗ = {0} × C. Nevertheless, given a pluripolar set E ⊂ D, it is quite hard in general to determine ED∗ . A typical object in the study of pluripolar hull is the graph Γ f (D) of a holomorphic function f over a domain D ⊂ Cn . In the case where D = ∆, the unit disk in C, Levenberg, Martin and Poletsky in [LMP], continuing preceding work of Sadullaev in [Sad], showed that if f (z) = ∑k≥0 an(k) zn(k) is a gap series with radius of convergence equal to 1 and with gaps satisfying limk→∞ n(k)/n(k + 1) = 0, then Γ f (D) is complete pluripolar in C2 . A notable property of gap series emerging from the work [Sad] and [LMP] is the fact that the partial sums fm (z) = ∑0≤k≤m an(k) zn(k) converges rapidly uniformly to f on compact sets 1/n(m) of ∆ i.e., limm→∞ ∥ f − fm ∥K = 0 for every compact subset K in ∆. Later, using the rapidly 2000 Mathematics Subject Classification. Primary 41A05, 41A63, 46A32. Key words and phrases. Pluripolar hull, Rapid convergence, complete pluripolar. 1 convergence method, Poletsky and Wiegerinck constructed in Theorem 3.6 of [PW], a Cantor compact set K ⊂ C and a holomorphic function f on D := C \ K such that (Γ f (D))∗C2 is two sheeted over D. The function f is essentially constructed as a square root of a holomorphic function g which can be rapidly approximated by a sequence gm of rational functions whose zeros and poles are related to the endpoints of the intervals removed from the construction of the Cantor set K. Motivated by the previous work of Poletsky and Wiegerinck we consider the problem of describing (Γ f (D))∗Cn+1 with D is a domain in Cn and f is holomorphic on D such that f is the limit of a sequence {rm }m≥1 of rational functions which converged rapidly to f on D, see the next section for the definition of these types of convergence. Here, we always assume that the numerator and denominator of rm are polynomials of degree at most m. Our first main result is Theorem 3.1 which is a generalization of the mentioned above theorem of Levenberg, Martin and Poletsky. More precisely, for any connected analytic polyhedron D ⊂ Cn we construct a holormorphic function g on D such that Γg (D) is complete pluripolar in Cn+1 . For the proof of this result, we first deal with the special case where D = ∆n is the unit polydisk in Cn . Using gap series, we construct in this special case a holomorphic function h on ∆n continuous up to n the boundary such that Γh (∆ ) is complete pluripolar in Cn+1 . Next we let g be the composition of h with the analytic mapping that defines D as an analytic polyhedron. Then by a theorem of Coltoiu on the equivalence between closed locally complete pluripolar sets and closed globally complete pluripolar sets we conclude that Γg (D) is complete pluripolar in Cn+1 . Finally, we invoke a version of Theorem 4.6 in [EW] to conclude that the pluripolar hull of Γg (D) disjoints from the cylinder ∂ D × C. So by Zeriahi’s theorem we get complete pluripolarity of Γg (D) in Cn+1 . The next section deals with the situation when f is the limit of a sequence rm of rational functions which converged rapidly to f on D. We show in Proposition 4.2 that there exists an ”exceptional” pluripolar set E ⊂ Cn which can be somehow determined by the poles of rm such that (Γ f (D))∗Cn+1 has exactly one sheet over D \ E. Moreover, if the sequence {rm }m≥1 converges as a point z0 ∈ E which may lie outside D then (Γ f (D))∗Cn+1 contains at most one point over z0 . As an immediate step, we prove in Theorem 4.1 a result which is of independent interest, saying roughly that pointwise rapidly convergence on a non-pluripolar set implies rapidly convergence in capacity on the whole domain D. We must stress that this result has been essentially proved by Bloom (see Theorem 2.1 in [Bl]). Our point is to offer a simpler 1 (D) together with the Chern-Levineproof using compactness of PSH(D) in the topology Lloc Nirenberg inequality instead of the more delicate comparison result of Alexander and Taylor between relative capacity and global capacity as done in [Bl]. A natural question arises is whether the exceptional set E appears in Proposition 4.2 can really occur. It seems to be a hard problem and we are able to solve this in a very particular case when D is a planar domain and the set E is presumably at most countable. We close this article by providing an explicit example of a planar domain D, a holomorphic function f on D and a sequence of rational functions {rm }m≥1 such that {rm }m≥1 converges to f rapidly uniformly on compact sets of D and {rm }m≥1 goes rapidly pointwise on ∂ D except for a countable set to a bounded function. The interesting question of determining precisely (Γ f (D))∗C2 is unfortunately left open by our method. Acknowledgements. This work has been done during a visit of the first named author at the Vietnam Institute for Advanced Mathematics in the winter of 2012. He wishes to thank this institution for financial support and the warm hospitality that he receives. We also would like to thank Professor Pascal Thomas for a useful suggestion in the construction of Proposition 4.7. II. Preliminaries The following sort of compactness in PSH(D) will be useful in dealing with convergence of holomorphic functions. Lemma 2.1. Let {um }m≥1 be a sequence of plurisubharmonic functions defined on a domain D in Cn . If the sequence is uniformly bounded from above on compact subsets of D then either {um }m≥1 goes to −∞ uniformly on compacta or there exists a subsequence {um j } j≥1 converges to a plurisubharmonic function u almost everywhere on D. In the latter case lim supum j = u j→∞ outside a pluripolar subset of D. In particular, in this case, the set {z ∈ D : limm→∞ um (z) = −∞} is pluripolar. Proof. It follows from Theorem 3.2.12 in [H¨o] that either {um }m≥1 goes to −∞ uniformly on compacta or there exists a subsequence {um j } j≥1 converges to a plurisubharmonic function u 1 (D). If the convergence in L1 occurs, then we can find a subsequence of {u } in Lloc m j j≥1 that loc converges almost everywhere on D to u. The first assertion follows. For the second one, we note that u = (lim supum j )∗ everywhere on D. Using a result of Bedford-Taylor which asserts j→∞ that the set (lim supum j )∗ ̸= lim supum j is pluripolar, we conclude the lemma. j→∞ j→∞ Lemma 2.1’. Let {um }m≥1 be a sequence of pluriharmonic functions defined on a domain D in Cn . If the sequence is uniformly bounded from above on compact subsets of D then either {um }m≥1 goes to −∞ uniformly on compacta or there exists a subsequence {um j } j≥1 converges locally uniformly to a pluriharmonic function u. Proof. Assume that {um }m≥1 does not go to −∞ uniformly on compacta, then by Lemma 2.1, 1 . Using Corollary there exists a subsequence {um j } j≥1 tends to a subharmonic function u in Lloc 3.1.4 in [H¨o] we see that the convergence must be uniform on compact subsets in D. For a Borel subset E in a domain D ⊂ Cn , following Bedford and Taylor (see [Kl] p.120) we let cap(E, D) be the relative capacity of a Borel subset E in D which is defined as ∫ cap(E, D) = sup{ E (dd c u)n : u ∈ PSH(D), −1 < u < 0}. It is well known that relative capacity enjoys some important properties such as sub-additivity and monotone under increasing sequences. Moreover, pluripolar subsets of D are exactly subsets with vanishing relative capacity. We recall the following types of convergence for of measurable functions which are essentially taken from [Bl]. Definition 2.2. Let { fm }m≥1 , f be Borel measurable functions defined on a domain D ⊂ Cn . We say that the sequence { fm }m≥1 (i) converges rapidly pointwise to f on a Borel subset X ⊂ D if | f − fm |1/m goes pointwise to 0 on X; (ii) converges in capacity to f on X if for every ε > 0 we have lim cap(Xm,ε , D) = 0, m→∞ where Xm,ε := {x ∈ X : | fm (x) − f (x)| > ε }; (iii) converges rapidly in capacity to f on X if | f − fm |1/m converges in capacity to 0 on X; (iv) converges rapidly in capacity to f on D if the property (iii) holds true for every compact subset X of D. As in classical measure theory, we have the following relation between convergence in capacity and pointwise convergence. Proposition 2.3. Let { fm }m≥1 , f be Borel measurable functions defined on a domain D ⊂ Cn . If { fm }m≥1 converges in capacity (resp. rapidly in capacity) to f on a Borel subset X of D, then there exists a subsequence { fm j } j≥1 and a pluripolar subset E ⊂ X such that { fm j } j≥1 converges pointwise (resp. rapidly pointwise) to f on X \ E. Proof. Suppose that { fm }m≥1 is a sequence that converges in capacity to f on X. For every ε > 0, the hypothesis gives lim cap(Xm,ε , D) = 0, m→∞ where Xm,ε := {x ∈ X : | fm (x) − f (x)| > ε }. Hence, we can find a strictly increasing sequence {mk } such that 1 cap(Xm, 1 , D) < k , ∀m ≥ mk . 2 2k ∞ E . By the subadditive property of the relative Let us set E j := ∪∞ X and X := ∩ 1 j=1 j k= j mk , capacity we have 2k cap(E, D) ≤ cap(E j , D) ≤ ∞ ∞ k= j k= j 1 1 ∑ cap(Kmk , 21k , D) < ∑ 2k = 2 j−1 , ∀ j ≥ 1. It follows that cap(E, D) = 0 and hence E is a pluripolar set. Now, for z ∈ X \ E, it is easy to check using the definition of E j that limk→∞ fmk (z) = f (z). We are done. Finally, if { fm }m≥1 converges rapidly in capacity to f on X then by setting fm′ := | fm − f |1/m and f ′ := 0 we reduce to the previous case. We should remark that there exists pointwise convergence sequence that contains no subsequence that converges in capacity. Indeed, let {Am }m≥1 be a sequence of pairwise disjoint subset of the unit disk ∆ ⊂ C such that infm≥1 cap (Am , ∆) > 0. Then the sequence {χAm }m≥1 provides the desired example. Next, we recall some major tools that will be used in the study of pluripolar hulls. According to Levenberg and Poletsky in [LP], the negative pluripolar hull ED− of a pluripolar set E ⊂ D is defined as follows ED− = ∩{z ∈ D : u(z) = −∞, u ∈ PSH(D), u < 0, u|E = −∞}. If D is bounded and hyperconvex i.e., D admits a negative continuous exhaustion function, then by Theorem 2.4 in [LP] for every sequence {D j } of relatively compact domains in D such that D j ↑ D we have ED− = ∪ j≥1 (E ∩ D j )− Dj. Now for every subset E of D we define the pluriharmonic measure of E relative to D as follows ω (z, E, D) = − sup{u(z) : u ∈ PSH(D), u ≤ 0 on D and lim sup u(ξ ) ≤ −1 for w ∈ E}, D∋ξ →w z ∈ D. The following connection between the pluriharmonic measure and negative pluripolar hull is again due to Levenberg and Poletsky (see [LP]), ED− = {z ∈ D : ω (z, E, D) > 0}. Using these tools together with some technical facts from classical potential theory, Edigarian and Wiegerinck in [EW] completely solve the problem of describing the pluripolar hull of {(z, f (z)) : z ∈ D \ E} in D × C where E is a closed polar subset of a domain D in C and f is holomorphic on D \ E. However, up to now the case where E is non-polar seems to be untractable by these methods. At the end of this article, we give an example of a holomorphic function f defined on a domain D ⊂ C where D = C \ A, A is a countable set with A \ A is non-polar such that (Γ f (D))∗C2 has at most one sheet over every point of C except for a polar subset of A. Even in this concrete case, we are unable to compute exactly (Γ f (D))∗C2 . III. Graphs of holomorphic functions over analytic polyhedron The main result of this section is the following generalization of the mentioned above theorem due to Levenberg, Martin and Poletsky in [LMP]. It is an open and interesting problem to see if our theorem is still true for every pseudoconvex domain D in Cn . Theorem 3.1. Let Ω be a domain in Cn and D be a connected analytic polyhedron relatively compact in Ω that is defined by holomorphic functions on Ω. Then there exists a holomorphic function g on D such that Γg (D) is complete pluripolar in Cn+1 . We first prove the following special case of Theorem 3.1. This is also a high dimension analogue of an earlier result due to Levenberg-Martin-Poletsky (see Proposition 2.15 in [LMP]). Lemma 3.2. Let ∆ be the unit disk in C. Then for every n ≥ 1, there exists a holomorphic n function h on the polydisks ∆n , continuous up to the boundary such that Γh (∆ ) is complete pluripolar in Cn+1 . Proof. We rely heavily on the techniques given in Section 2 of [LMP] in which the case n = 1 has been proved. For the clarity of the exposition we consider first the case where n = 2, the necessary modifications for general n will be given at the end. Let z = (z1 , z2 ) be coordinates in C2 . According to [LMP] p.526 we can find two sequences {am } and { jm } satisfying the following conditions: jm 1) limm→∞ jm+1 = 0; 2) c|am | ≥ ∑k≥m+1 |ak |, ∀m ≥ 1, where c > 0 is independent of m; 3) ∑m≥1 j1m log |am | = −∞; 4) limm→∞ |am |1/ jm = 1. From (2), (4) we infer that the series f (ξ ) := ∑m≥1 am ξ jm has radius of convergence equal to 1 and defines a holomorphic function on ∆, continuous on ∆. By the argument in [LMP] p. 519 we also have lim |pk (ξ )| = ∞, ∀|ξ | > 1, k→∞ where pk (ξ ) := ∑1≤m≤k am ξ jm . We set h(z1 , z2 ) := 2 f (z1 ) + f (z2 ), (z1 , z2 ) ∈ ∆2 . 2 It is clear that h is continuous on ∆ and holomorphic on ∆2 . We will construct a plurisubharmonic function u on C3 such that u = −∞ exactly on Γh (∆2 ). To this end, we first note that pk converges uniformly on ∆ to f . So by Bernstein-Walsh inequality, the sequence j1k log |pk | is uniformly upper bounded on compact sets of C. It follows that the functions uk (z, w) := 1 jk+1 log |w − 2pk (z1 ) − pk (z2 )| are plurisubharmonic on C3 and uniformly bounded from above on compacta. On the other hand, from (1) we deduce that ∑k≥1 j1k < ∞, so the series u(z, w) := ∑ max{uk (z, w), −1} k≥1 defines a global plurisubharmonic function on C3 . Now we show that u(z, h(z)) = −∞ if z = (z1 , z2 ) ∈ ∆2 . Fix such a point z. For k ≥ 1 and ξ ∈ ∆, using (2) we have | f (ξ ) − pk (ξ )| = ∑ amξ jm ≤ m>k ∑ |am| ≤ (1 + c)|ak+1|. m>k From this we deduce easily the following estimate |h(z1 , z2 ) − 2pk (z1 ) − pk (z2 )| ≤ 3(c + 1)|ak+1 |. Therefore uk (z, h(z)) ≤ 1 jk+1 (log |ak+1 | + log(3(c + 1))), ∀k ≥ 1, so that u(z, h(z)) = −∞ by (3). Let z ∈ ∆2 and w ̸= h(z) we will show that u(z, w) > −∞. Since |w − 2pk (z1 ) − pk (z2 )| ≥ |w − h(z1 , z2 )| − 2| f (z1 ) − pk (z1 )| − | f (z2 ) − pk (z2 )|, we can choose δ > 0 and k0 ≥ 1 such that |w − 2pk (z1 ) − pk (z2 )| > δ , ∀k ≥ k0 . This implies ( u(z, w) ≥ −(k0 − 1) + (log δ ) ∑ k≥k0 1) > −∞. jk Next we prove that u(z, w) > −∞ if z = (z1 , z2 ) ̸∈ ∆2 and w ∈ C. For this, we claim that lim |2pk (z1 ) + pk (z2 )| = ∞, ∀(z1 , z2 ) ̸∈ ∆2 . k→∞ This is clear if either |z1 | ≤ 1 or |z2 | ≤ 1, since limk→∞ |pk (ξ )| = ∞ if |ξ | > 1. Now we consider the remaining cases. Case 1. |z2 | > |z1 | > 1. Choose ε > 0 such that 1+ε |z1 | > . (1 − ε )2 Choose m0 such that (1 − ε ) jm < |am | < (1 + ε ) jm ∀m > m0 . So for k > m0 we obtain the following estimates |pk (z1 )| ≤ A + k((1 + ε )|z1 |) jk , |pk (z2 )| ≤ A + k((1 + ε )|z2 |) jk , where A > 0 is independent of k. It follows that |2pk (z1 ) + pk (z2 )| ≥ |pk (z2 )| − 2|pk (z1 )| ≥ |ak ||z2 | jk − |pk−1 (z2 )| − 2|ak ||z1 | jk − 2|pk−1 (z1 )| |ak | |ak | (|z2 | jk − 4|z1 | jk ) + |z2 | jk − 3(k − 1)((1 + ε )|z2 |) jk−1 − 3A 2 2 1 |ak | ≥ (|z2 | jk − 4|z1 | jk ) + ((1 − ε )|z2 |) jk − 3(k − 1)((1 + ε )|z2 |) jk−1 − 3A 2 2 1 |ak | (|z2 | jk − 4|z1 | jk ) + ((1 + ε )|z2 |) jk /2 − 3(k − 1)((1 + ε )|z2 |) jk−1 − 3A. ≥ 2 2 Since |z2 | > |z1 | and jk / jk−1 → ∞ the last expression goes to ∞ as k tends to ∞. Case 2. |z1 | ≥ |z2 | > 1. By a similar and somewhat simpler argument than the above case we also get that limk→∞ |2pk (z1 ) + pk (z2 )| = ∞. The claim follows. So we can find k0 ≥ 1 such that ≥ |w − 2pk (z1 ) − pk (z2 )| ≥ |2pk (z1 ) + pk (z2 )| − |w| > 1 ∀k > k0 . This implies that u(z, w) > −∞. All in all, we have shown that u = −∞ exactly on Γh (∆2 ). This concludes the proof for the case n = 2. For general n, we modify the function h as follows. Let λ1 , · · · , λn be positive numbers such that if 1 ≤ i ≤ n − 1 then |λi | > ∑ |λ j |. i+1≤ j≤n Now we set ∑ λi f (zi ). 1≤i≤n on ∆n . Next we let h(z1 , · · · , zn ) = Then h is continuous on ∆n and holomorphic 1 u(z, w) = ∑ max{ log |w − ∑ λi pk (zi )|, −1}. jk+1 1≤i≤n k≥1 In a similar fashion as in the case n = 2 we can show that u is plurisubharmonic on Cn+1 , u = −∞ on Γh (∆n ). Using the condition (1) we can check that u(z, w) > −∞ if z ̸∈ ∆n and w ∈ C, or z ∈ ∆n and w ̸= h(z). We also need the following fact Lemma 3.3. Let E be an Fσ -pluripolar subset of Cn and let F be a holomorphic mapping from an open neighborhood U of EC∗ n to C such that F(E) ⊂ ∆ and F(EC∗ n ) ⊂ ∆. Then F(EC∗ n ) ⊂ ∆. Proof. This lemma can be deduced from from the proof of Theorem 4.6 in [EW]. For the reader convenience, we give some details. Assume that there exists some point z0 ∈ EC∗ n such that F(z0 ) ∈ ∂ ∆. Fix 0 < r < 1, R > 1. For δ > 0 we let ∆δ := {z ∈ C : |z| < δ } and Uδ := {z ∈ U : F(z) ∈ ∆1+δ }. The key observation is that if ε > 0 then Uε is an open neighborhood of EC∗ n . In particular ∂ Uε ∩ EC∗ n = 0. / Set BR := {z ∈ Cn : |z| < R}. By applying the localization principle of Edigarian and Wiegerinck (Theorem 4.1 in [EW]) we have for every ε > 0 0 ≤ ω (z0 , E ∩ F −1 (∆r ) ∩ BR , BR ) = ω (z0 , E ∩ F −1 (∆r ) ∩ BR , BR ∩Uε ) ≤ ω (F(z0 ), ∆r , ∆1+ε ). By letting ε → 0 we get So ω (z0 , E ∩ F −1 (∆r ) ∩ BR , BR ) = 0. z0 ∈ / (E ∩ F −1 (∆r ) ∩ BR , BR )− BR . Since F(E) ⊂ ∆, by letting r → 1 we infer that z0 ̸∈ (E ∩ BR , BR )− BR . Finally, since R > 1 is arbitrarily large, by Theorem 2.4 in [LP] we get z0 ̸∈ EC∗ n , a contradiction. The proof is complete. Proof of Theorem 3.1. We have D = {z ∈ Ω : | f1 (z)| < 1, · · · , | fk (z)| < 1}, where f1 , · · · , fk are holomorphic on Ω. According to Lemma 3.2, we can find h a continuous function on ∆, k holomorphic on ∆ such that Γh (∆ ) is complete pluripolar in Ck+1 . Let F := ( f1 , · · · , fk ) and g := h◦F. We will show that X := Γg (D) is complete pluripolar in Cn+1 . For this, we first claim that X = Γg (D) is complete pluripolar in Cn+1 . Indeed, let u be a plurisubharmonic function k on Ck+1 such that u = −∞ precisely on Γh (∆ ). Consider φ (z, w) := u(F(z), w). Then φ is plurisubharmonic on Ω × C and φ = −∞ exactly on X. Since X is closed in Cn+1 . By a result of Coltoiu about equivalence between locally complete pluripolarity and globally complete pluripolarity we see that X is complete pluripolar in Cn+1 . This proves the claim. Next, we show XC∗ n+1 = X. By the above reasoning XC∗ n+1 ⊂ X. For every 1 ≤ j ≤ k, consider the map Fj : Ω × C → C defined by Fj (z1 , · · · , zn , w) = f j (z1 , · · · , zn ). Then Fj is holomorphic on an open neighborhood of XC∗ n+1 and Fj (XC∗ n+1 ) ⊂ Fj (X) = ∆. Observe that Fj (X) ⊂ ∆, so we may apply Lemma 3.3 to conclude that Fj (XC∗ n+1 ) ⊂ ∆. Since it is true for every 1 ≤ j ≤ k we infer XC∗ n+1 = X. Since X = X \ Γg (∂ D) and since Γg (∂ D) is closed we deduce that X is a Gδ set. Finally, since X is also Fσ , we apply Zeriahi’s theorem to get that X is complete pluripolar in Cn+1 . The proof is complete. Remark. If D is a simply connected, proper subdomain in C with real analytic boundary then by Riemann’s mapping theorem we can find a holomorphic bijection map f from D onto ∆. Since D has real analytic boundary, the map f extends to a larger neighborhood Ω of D. Thus D is an analytic polyhedron in Ω, and so Theorem 3.1 applies in this case. IV. Graphs of holomorphic functions which can be rapidly approximated by rational functions It is proved in Theorem 2 of [Go] that if a sequence {rm }m≥1 of rational functions converges rapidly in measure on an open set X to a holomorphic function f defined on a domain D(X ⊂ D) then {rm }m≥1 must converge rapidly in measure to f on D. Much later, by using delicate techniques of pluripotential theory, Bloom was able to prove an analogous result in which rapidly convergence in measure is replaced by rapidly convergence in capacity and the set X is only required to be compact and non-pluripolar (see Theorem 2.1 in [Bl]). Along this line, we will prove the following refinement of the above results. Theorem 4.1. Let f be a holomorphic function on a domain D ⊂ Cn and X a Borel nonpluripolar subset of D. Suppose that there exists a sequence {rm }m≥1 of rational functions with poles off X that converges rapidly pointwise to f on X. Then the following assertions hold: (a) There exists a pluripolar subset (possibly empty) E of Cn and a sequence {qm } of polynomials such that E = {z ∈ Cn : limm→∞ m1 log |qm (z)| = −∞} and for every point z ∈ D \ E we have lim inf| f (z) − rm (z)|1/m = 0. m→∞ (b) The sequence {rm }m≥1 converges rapidly in capacity to f on D. (c) If the poles of {rm }m≥1 are all disjoint from D then {rm }m≥1 goes to f rapidly uniformly on compacta i.e., for every compact K ⊂ D 1/m lim ∥ f (z) − rm (z)∥K m→∞ = 0. The above theorem demonstrates a sharp contrast between uniform convergence and rapidly uniform convergence. Indeed, there exists a sequence of polynomials that converges on compact subsets of the unit disk ∆ ⊂ C but is not bounded on any open neighborhood of ∆ (e.g., rm (z) = zm ). Proof. (a) First, we claim that there exists a non-pluripolar subset X ′ of X such that {rm }m≥1 is uniformly bounded on X ′ . For this, observe that since the sequence {rm }m≥1 converges pointwise to f on X, we have supm≥1 |rm (z)| < ∞ for every z ∈ X. Now we set XN = {z ∈ X : |rm (z)| ≤ N, ∀m ≥ 1}, N ≥ 1. Since ∪N≥1 XN = X and X is non-pluripolar, we infer that there must exist N0 such that XN0 is non-pluripolar. The claim follows by choosing X ′ := XN0 . Now we write rm = pm /qm , where pm , qm are polynomials of degree ≤ m and ∥qm ∥X ′ = 1. This implies ∥pm ∥X ′ ≤ N0 for every m. Since X ′ is non-pluripolar, by Bernstein-Walsh inequality we deduce that the two sequences 1 1 n m log |pm |, m log |qm | are uniformly bounded from above on compact sets of C . The next step is to prove the following claim: For every compact subset K of D we have 1 log ∥qm f − pm ∥K → −∞. m To see this, for m ≥ 1 we set 1 pm (z) , ∀z ∈ D \ q−1 um (z) := log f (z) − m (0). m qm (z) Then for every z ∈ D we have 1 1 vm (z) := um (z) + log |qm (z)| = log |qm (z) f (z) − pm (z)|. m m Since m1 log |pm |, m1 log |qm | are uniformly bounded from above on compact sets of D, the functions vm are plurisubharmonic on D and uniformly bounded from above on compacta in D. By Lemma 2.1, either there exists a subsequence vm converges almost everywhere to v ∈ PSH(D), v ̸≡ −∞ or vm goes to −∞ uniformly on compact subsets of D. We will show that the first case does not occur. For the sake of seeking a contradiction we can assume that the sequence vm itself converges almost everywhere to v ∈ PSH(D), v ̸≡ −∞. Applying again Lemma 2.1 we have lim supvm = v outside a pluripolar set. Since limm→∞ um (z) = −∞ for every z ∈ X m→∞ and since 1 log |qm (z)| < ∞ ∀z ∈ Cn m m≥1 sup we infer lim vm (z) = −∞, ∀z ∈ X. m→∞ This implies v = −∞ on a non-pluripolar subset of D, a contradiction. It follows that vm converges uniformly to −∞ on compact subsets of D. This proves our assertion. Next, we note that the sequence m1 log |qm | is uniformly bounded from above on compact sets of Cn . So using Lemma 2.1 and the fact that supX ′ m1 log |qm | = 0 for every m, we deduce that 1 log |qm (z)| = −∞} m→∞ m is pluripolar. For z ̸∈ E, we can choose a subsequence {mk } such that 1 inf log |qmk (z0 )| > −∞. k≥1 mk On the other hand, by above arguments, we have E := {z ∈ Cn : lim lim vmk (z) = −∞, ∀z ∈ D \ E. k→∞ It follows that limk→∞ umk (z) = −∞. We are done. (b) Assume that {rm }m≥1 does not converge in capacity to f on D. Then there exists a compact subset K of D, a subsequence {m j } and positive constants ε , δ > 0 such that cap(K j , D) > δ ∀ j ≥ 1, where K j := {z ∈ K : | f (z) − rm j (z)|1/m j > ε }. It follows from the proof of (a) that for every M > 0 there exists jM ≥ 1 such that 1 log |qm j (z) f (z) − pm j (z)| < −M ∀ j ≥ jM , z ∈ K. vm j (z) = mj So for j ≥ jM we have the following inclusion K j ⊂ L j := {z ∈ K : h j (z) := 1 log |qm j (z)| < −M − log ε }. mj This implies cap(L j , D) ≥ cap(K j , D) > δ , ∀ j ≥ jM . 1 (D) to a Moreover, by passing to a subsequence we may assume that h j converges in Lloc plurisubharmonic function as j → ∞. Let ω be a small neighborhood of K which is relatively compact in D. Then we have sup ∥h j ∥L1 (ω ) < ∞. j≥1 Take u j ∈ PSH(D), −1 < u j < 0 such that ∫ Lj (dd c u j ) > δ . By a version of the Chern-Levine-Nirenberg’s inequality (see the proof of Proposition 2.2.3 in [Bl]) we obtain for every j ≥ jM the following estimate ∫ ∫ CK,ω 1 c n δ < (dd u j ) ≤ |h j |(dd c u j )n ≤ ∥h j ∥L1 (ω ) . M + log ε K M + log ε Lj Here CK,ω is a positive constant depends only on K, ω . By letting M → ∞ we get a contradiction. (c) If the polynomials qm are nowhere vanishing on D then the functions hm := m1 log |qm | are pluriharmonic on D. By the proof of (a), this sequence is also bounded from above on compacta in D. On the other hand, since supX ′ hm = 0 for every m, by Lemma 2.1’, we deduce that the sequence {hm } is uniformly bounded on compact sets of D. It then follows from (a) that 1/m ∥ f − rm ∥K goes to 0 for every compact K in D. This proves our theorem. Remarks. 1) Under the assumption of Theorem 4.1, we do not know if f can be rapidly pointwise approximated everywhere on D by a sequence of rational functions (possibly different from {rm }m≥1 ). 2) It is of interest to see if we can replace ”liminf” in Theorem 4.1(a) by ”lim”. 3) Combining Proposition 2.3 and Theorem 4.1(b) we obtain Theorem 2.1 in [Bl] which says that rapid convergence in capacity on some non-pluripolar Borel set X ⊂ D implies rapid convergence in capacity on D. On the other hand, in view of the remark following Proposition 2.3, it is not clear whether our result follows from Bloom’s theorem. The following result contains some information about pluripolar hulls of Γ f (D) where f is a holomorphic function satisfying the assumption of Theorem 4.1. Proposition 4.2. Let f , D, X, E be as in the theorem and (z0 , w0 ) be a point in (Γ f (D))∗Cn+1 . Then the following assertions hold: (a) If z0 ∈ D \ E then w0 = f (z0 ); (b) If z0 ∈ Cn \ (D ∪ E) then lim inf|w0 − rm (z0 )|1/m = 0. m→∞ In particular, if lim rm (z0 ) = λ , then w0 = λ . m→∞ Proof. (a) Define the plurisubharmonic functions 1 φm (z, w) = log |qm (z)w − pm (z)| m 1 1 n+1 on C . Since m log |pm |, m log |qm | are uniformly bounded from above on compact sets of Cn , by easy estimates we deduce that φm are uniformly bounded from above on compact subsets of Cn+1 . Fix a point (z0 , w0 ) ∈ (Γ f (D))∗Cn+1 with z0 ∈ D \ E. Suppose that w0 ̸= f (z0 ). We will construct a plurisubharmonic function v in Cn+1 such that v ≡ −∞ on Γ f (D) whereas v(z0 , w0 ) > −∞. Choose a subsequence mk ↑ ∞ and a constant C > −∞ such that 1 1 log |qmk (z0 )| > C, lim log | f (z0 ) − rmk (z0 )| = −∞. k→∞ mk k≥1 mk inf In particular limk→∞ rmk (z0 ) = f (z0 ). So we can find a constant δ > 0 and k0 ≥ 1 large enough such that |w0 − rmk (z0 )| > δ , ∀k ≥ k0 . Fix a closed ball B ⊂ D. By the proof of Theorem 4.1 (a) we can find a sequence of positive constants {dk }k≥k0 ↑ ∞ such that for z ∈ B the following estimates hold φmk (z, f (z)) ≤ −dk . Take a sequence {ck }k≥k0 of positive real numbers such that ∑ ck = 1 whereas ∑ ck dk = ∞. Let v(z, w) = ∑ ck max{φmk (z, w), −dk }. k≥k0 Since the sequence {φnk } is locally uniformly bounded form above, the function v is plurisubharmonic on Cn+1 or v ≡ −∞. Moreover, by the choice of dk we also have v ≡ −∞ on Γ f (B). Since B is non-pluripolar we obtain v ≡ −∞ on Γ f (D). Finally, we claim that v(z0 , w0 ) > −∞. Indeed, ck ck log |qmk (z0 )| + ∑ log |w0 − rmk (z0 )| v(z0 , w0 ) ≥ ∑ k≥k0 mk k≥k0 mk ck > C + (log δ )( ∑ ) > −∞. k≥k0 mk Thus v is the desired plurisubharmonic function as claimed. We are done. (b) Suppose for the sake of seeking a contradiction that there exists δ > 0 and m0 ≥ 1 such that |w0 − rm (z0 )|1/m > δ ∀m ≥ m0 . Since z0 ̸∈ E, we can choose a sequence {mk }k≥1 ↑ ∞ such that 1 log |qmk (z0 )| > C > −∞. k≥1 mk inf Now, using a similar argument as in the previous part, we may construct a plurisubharmonic function v on Cn+1 such that v ≡ −∞ on Γ f (D) while v(z0 , w0 ) > −∞. This is absurd since (z0 , w0 ) ∈ (Γ f (D))∗Cn+1 . Remarks. 1) If the poles of {rm }m≥1 are all disjoint from D then using Theorem 4.1(c) we get D ∩ E = 0. / So by Proposition 4.2 (a) we have (Γ f (D))∗Cn+1 ∩ (D × C) = Γ f (D). Without this hypothesis, we still have (Γ f (D))∗Cn+1 ∩ (D × C) ⊂ Γ f (D) ∪ (E × C). 2) In the case where z0 ∈ Cn \ (D ∪ E) even if there exists the limit limm→∞ rm (z0 ) = λ , it is not clear to us whether the point (z0 , λ ) belongs or not to (Γ f (D))∗Cn+1 . 3) It is natural to see if the singular set E can actually occurs in (b). Using some work of Edigarian and Wiegerinck, we can give a partial answer in the case n = 1. Corollary 4.3. Let E be a analytic hypersurface of Cn and f be a holomorphic function on Cn \ E. Then we have (Γ f (Cn \ E))∗Cn+1 ∩ ((Cn \ E) × C) = Γ f (Cn \ E). Proof. By Theorem 3 in [Ch], there exists a sequence {rm }m≥1 of rational functions that converges rapidly and uniformly to f on compact subsets of D. Applying Proposition 4.2 (a) we get the desired conclusion. Proposition 4.4. Let D be a domain in C and f be holomorphic on D. Assume that there exists a countable subset A of D satisfying (Γ f (D))∗C2 ∩ (D × C) ⊂ Γ f (D) ∪ (A × C). Then (Γ f (D))∗C2 ∩ (D × C) = Γ f (D). For the proof of this result we first need the following Lemma 4.5. Let f be a holomorphic function on a bounded domain D in C. Assume that there exists a polar subset E of D such that (Γ f (D))∗C2 ∩ (D × C) ⊂ Γ f (D) ∪ (E × C). Let (z0 , w0 ) ∈ (Γ f (D))∗C2 \ Γ f (D) with z0 ∈ E, w0 ̸= f (z0 ). Then for every neighborhood U of z0 in C we can find z′ ∈ E ∩U, z′′ ∈ U \ {z′ } such that (z′ , f (z′′ ) ∈ (Γ f (D))∗C2 and f (z′ ) ̸= f (z′′ ). Proof. Choose a closed disk S in D such that z0 ̸∈ S. For j ≥ 1 we set ∆2j := {(z, w) ∈ C2 : |z| < j, |w| < j}. Then we get from Theorem 2.4 in [LP] the following relations, (z0 , w0 ) ∈ (Γ f (D))∗C2 = (Γ f (S))∗C2 = ∪ . (Γ f (S) ∩ ∆2j )− ∆2 j≥1 j So we can choose j0 so big that the following properties are fulfilled: 1) Γ f (S) ⊂ ∆2j0 , D {|z| < j0 }; 2) (z0 , w0 ) ∈ (ΓS ( f ))− . ∆2 j0 Since z0 ̸∈ S, E is polar and since f is non-constant, we can choose an open disk D′ around z0 satisfying the following conditions: 3) D′ ∩ S = 0, / w0 ̸∈ f (D′ ); 4) ∂ D′ ∩ E = 0; / 5) f (z0 ) ̸∈ f (∂ D′ ). In view of (5) we can choose another disk D′′ D′ such that z0 ∈ D′′ , f (∂ D′ ) ∩ f (D′′ ) = 0/ and ∂ D′′ ∩ E = 0. / Now we consider the open set V := D′′ × {{w ∈ C : |w| < j0 + 1} \ f (D′ )}. / Now we may apply Lemma 3.4 in [LP] It follows from (3) that (z0 , w0 ) ∈ V and V ∩ Γ f (S) = 0. (see also Corollary 2.6 in [EW]) to obtain a point ξ = (ξ1 , ξ2 ) ∈ (∂ V ) ∩ ∆2j0 ⊂ (∂ V ) ∩ ∆2j0 such that 0 < ω ((z0 , w0 ), ΓS ( f ), ∆2j0 ) ≤ ω (ξ , ΓS ( f ), ∆2j0 +1 ). We may apply Theorem 2.4 in [LP] and get ξ ∈ (Γ f (S))− ∆2 j0 +1 ⊂ (Γ f (S))∗C2 = (Γ f (D))∗C2 . Since ξ1 ∈ D, in view of our assumption we get ( ) ( ) ξ ∈ (∂ V ) ∩ ∆2j0 ∩ Γ f (D) ∪ (E × C) . It is easy to see that ∂ V = A1 ∪ A2 ∪ A3 where A1 = ∂ D′′ × {{w ∈ C : |w| < j0 + 1} \ f (D′ )}; A2 = ∂ D′′ × {{w ∈ C : |w| = j0 + 1} ∪ f (∂ D′ )}; A3 = D′′ × {{w ∈ C : |w| = j0 + 1} ∪ f (∂ D′ )}. Since ∂ D′′ ∩ E = 0, / we get that A1 ∩ (E × C) = 0. / This implies ( ) A1 ∩ Γ f (D) ∪ (E × C) = 0. / In the same way, we have ) ( A2 ∩ ∆2j0 = ∂ D′′ × f (∂ D′ ) ∩ ∆2j0 . Using the facts that f (∂ D′ ) ∩ f (D′′ ) = 0/ and ∂ D′′ ∩ E = 0/ we obtain ( ) A2 ∩ Γ f (D) ∪ (E × C) = 0. / Now we observe that ( ) A3 ∩ ∆2j0 = D′′ × f (∂ D′ ) ∩ ∆2j0 . ( ) Since f (∂ D′ ) ∩ f (D′′ ) = 0, / we have D′′ × f (∂ D′ ) ∩ Γ f (D) = 0. / Putting all this together we deduce that ( ) ξ ∈ D′′ × f (∂ D′ ) ∩ (E × C). This means that ξ1 ∈ D′′ ∩ E, ξ2 = f (η ), where η ∈ ∂ D′ . By letting D′ shrink towards z0 we complete the proof. The next result (Theorem 3.4 in [DH]) might be useful in investigating more deeply the problem of determining the exceptional set E appearing in Proposition 4.2. Proposition 4.6. Let Ω be a domain in Cn and E be a pluripolar subset of Ω such that E ∩ H = ∗ ∩ H is pluripolar (relative to H). 0, / where H is the hyperplane z1 = 0. Then EΩ Proof of Proposition 4.4. Suppose that there exists a point (a, b) ∈ (Γ f (D))∗C2 with b ̸= f (a). It follows that f is non-constant and a ∈ A. Let S be a closed ball contained in D, A ∩ S = 0/ and G := Γ f (S). Then G ∩ (A × C) = 0/ and G∗C2 = (Γ f (D))∗C2 . Since A is countable, using Proposition 4.6, we deduce that the set T := {w ∈ C : (z, w) ∈ G∗C2 , z ∈ A} is polar. Since f is non-constant we deduce f −1 (T ) is polar in D. So we can choose a disk D′ ⊂ D around a such that / D′ ∩ S = 0, / b ̸∈ f (D′ ), f (a) ̸∈ f (∂ D′ ), ∂ D′ ∩ f −1 (T ) = 0. ′′ ′ ′′ We also take a smaller disk D relatively compact in D such that a ∈ D , f (∂ D′ ) ∩ f (D′′ ) = 0. / ′ ′ ∗ ′ ′′ Then, following the proof of Lemma 4.5, we can find a point (a , b ) ∈ GC2 where a ∈ D ∩ A, b′ ∈ f (∂ D′ ). In particular b′ ∈ T . This contradicts our choice of the set D′ . We are done. Remark. If A is polar and uncountable then our proof of Proposition 4.4 breaks down because we do not know if the set T is polar. Finally we come to the promised explicit example of a sequence of rational functions that converges rapidly uniformly on compacta to a holomorphic function. For this, we first fix a non-polar compact subset F of the unit circle ∂ ∆ = {z ∈ C : |z| = 1} which is nowhere dense in ∂ ∆. A standard construction of such F is the following. Let C be the 1/3− Cantor set in [0, 1] and P be the orthogonal projection from ∂ ∆ onto the real axis. Then we can choose F := P−1 (C ). ¯ a sequence {rm }m≥1 of Proposition 4.7. There exist a countable subset A of C \ ∆ with F ⊂ A, rational functions on C and a function f : C \ A → C which is holomorphic on C \ A such that the following properties holds true: (a) The poles of {rm }m≥1 are included in A for every m ≥ 1; (b) {rm }m≥1 converges rapidly uniformly on compact sets of C \ A to f ; (c) {rm }m≥1 converges rapidly pointwise on F = A \ A to f . Remarks. (a) It follows from Proposition 4.2 that in the situation of Proposition 4.7, there exists a polar subset F ′ ⊂ F such that the pluripolar hull in C2 of Γ f (C \ A) over every point z ∈ C \ (A ∪ F ′ ) contains at most the point (z, f (z)). (b) It would be nice if we could construct a similar example as Proposition 4.7 but with the additional property the set E arises in Theorem 4.1 becomes empty. Then, by applying again Proposition 4.2 (b), the set F ′ is in fact empty. The proof proceeds through two lemmas. For the first one, we need to introduce some notation. We index (N∗ )2 = {(m, n) : m, n ≥ 1} by the graded lexicographic order, that is (m, n) ≺ (p, q) if and only if m + n < p + q, or m + n = p + q but m < p. Let ind(m, n) is the index of (m, n) in the ordered sequence. Simple computation gives ind(m, n) = √ (m + n − 2)(m + n − 1) + m. 2 Hence, max(m, n) ≥ ind(m,n) for all m, n ≥ 1. 2 Lemma 4.8. There exists a double-indexed sequence {rmn } ⊂ (0, 1) such that the corresponding graded lexicographic sequence {sn }n≥1 satisfies the condition ( ∞ )1 n lim ∑ (1 − sn ) = 0. n→∞ j=n Proof. Fix a real number a > 1. We will prove that the sequence defined by√r jk = 1 − a− j ind( j,k) . 2 satisfies the condition. Indeed, we know that r jk = sind( j,k) and max( j, k) ≥ ∞ ∑ (1 − sn) = ∑ j=n ∑√ (1 − r jk ) ≤ ind( j,k)≥n j≥⌊ ∑√ (1 − r jk ) + n 2⌋ k≥⌊ (1 − r jk ). n 2⌋ Next, we estimate the each terms in the last expression. We see that ∑√ j≥⌊ ∑√ (1 − r jk ) = n 2⌋ j≥⌊ ≤ a Hence, a− j n 2⌋ √n −(⌊ ∞ j=n =( ∑√ j≥⌊ 4 2 ⌋) ∑ (1 − sn) ≤ 4 −k4 a− j )( ∑ a−k ) 4 n 2⌋ 4 k≥1 √ a2 −(⌊ n2 ⌋)4 (∑ a ) = a . (a − 1)2 j=0 ∞ −j 2 √ 2a2 −(⌊ n2 ⌋)4 a . (a − 1)2 The last estimate implies the desired limit. Lemma 4.9. Let a = eiθ and b = reiξ with 0 ≤ θ , ξ ≤ π2 . Then ¯ ≥ 2 |θ − ξ |. |1 − ba| π 4 −k4 We have Proof. We write √ ¯ |1 − ba| = |1 − rei(θ −ξ ) | = (1 − r cos(θ − ξ ))2 + (r sin(θ − ξ ))2 √ 2 = 1 + r2 − 2r cos(θ − ξ ) ≥ | sin(θ − ξ )| ≥ |θ − ξ |, π where in the last estimate we use the inequality sint ≥ π2 t for all 0 ≤ t ≤ π2 . Lemma 4.10. There exists a sequence B = {α j } j≥1 ⊂ ∆ such that (a) F is the set of accumulation points of B; ( )1 (b) limn→∞ ∑∞j=n+1 (1 − |α j |) n = 0; 1−|α j | jξ | (c) For all ξ ∈ F, ∑∞j=1 |1−α¯ < ∞. Moreover, lim k→∞ ( ∞ 1 − |α j | ) 1k = 0. ¯ j=k+1 |1 − α j ξ | ∑ Proof. We learn the arguments of Colwell. The set ∂ ∆ \ F is countable union of disjoint open arcs. Let A be the set of end-points of these arcs. We write A = {am , bm : m ≥ 1}, |am | = |bm | = 1, φm = arg am < arg bm = ψm . For each m, n ≥ 1, we define two double-indexed sequences by 1 ( ψm − φm )n 1 ( ψ m − φm )n ξmn = φm + , ηmn = ψm − . 2 2π 2 2π ( )n Notice that, for fixed m, ξmn ↓ φm and ηmn ↑ ψm . We let tmn := 1 − (1 − rmn ) ψm2−πφm < 1 − rmn where the rmn are defined in Lemma 4.8. Let us set cmn = tmn eiξmn , dmn = tmn eiψmn . We order two sequences {cmn } and {dmn } by using the graded lexicographic order to get two new sequences {c˜n }n≥1 and {d˜n }n≥1 respectively. Let B := {αn }n≥1 be the following sequence c˜1 , d˜1 , c˜2 , d˜2 , . . . , c˜n , d˜n , . . . . It is easily seen that F is the set of accumulation points of B and B ⊂ {z : |z| = 1, 0 ≤ arg z ≤ π2 }. ( )1 From Lemma 4.8 we also have limn→∞ ∑∞j=n+1 (1 − |α j |) n = 0, since ∞ ∑ (1 − |α j |) ≤ j=n+1 ∞ ∑n (1 − |c˜ j | + 1 − |d˜j |) j=⌊ 2 ⌋ = ∑ (1 − |c jk | + 1 − |d jk |) ≤ ind( j,k)≥⌊ n2 ⌋ ∑ 2(1 − r jk ). ind( j,k)≥⌊ 2n ⌋ To prove the last property, we remark that if ξ = eiθ ∈ F, then θ ̸= arg α for all α ∈ B. More precisely, 1 ( ψm − φm )n 1 ( ψm − φm )n |θ − arg cmn | ≥ , |θ − arg dmn | ≥ , ∀m, n ≥ 1. 2 2π 2 2π ( )n On the other hand, 1 − |cmn | = 1 − |dmn | = (1 − rmn ) ψm2−πφm . We obtain from Lemma 4.9 the following estimates 1 − |cmn | π 1 − |cmn | ≤ ≤ π (1 − rmn ). |1 − c¯mn ξ | 2 |θ − arg cmn | The same inequality as above holds true for dmn . Thus, ∞ ( ∞ ∞ 1 − |α | 1 − |cmn | 1 − |dmn | ) j = ∑ + ≤ 2π ∑ (1 − rmn ) < ∞. ∑ ¯ |1 − d¯mn ξ | j=1 |1 − α j ξ | m,n=1 |1 − c¯mn ξ | m,n=1 For the last assertion, note that ∞ 1 − |α j | ∑ |1 − α¯ j ξ | ≤ ∑ j=k+1 ind(m,n)≥⌊ 2k ⌋ ( 1 − |c | 1 − |dmn | ) mn ≤ 2π + |1 − c¯mn ξ | |1 − d¯mn ξ | ∞ ∑ (1 − rmn ). ind(m,n)≥⌊ 2k ⌋ By Lemma 4.8, the k-th root of the last term tends to 0 as k → ∞. This completes the proof of the lemma. Proof of Proposition 4.7. Let B := {α j } j≥1 ⊂ ∆ be the sequence in Lemma 4.10. Now we define for m ≥ 1 the finite Blaschke product {rm }m≥1 |α j | α j − z . ¯ j=1 α j 1 − α j z m rm (z) = ∏ Let A := { α¯1j : j ≥ 1}. Then the set of accumulation points of A is F. We claim the infinite Blaschke product ∞ |α | α − z j j f (z) := ∏ α 1 − α¯ j z j j=1 is well-defined on C \ A. To prove this assertion, we use Lemma 4.10. We fix a compact ¯ For j ≥ 1 and z ∈ K we have K ⊂ C \ A. |α j | α j − z (α j + |α j |z)(1 − |α j |) −1 = ≤ MK (1 − |α j |), α j 1 − α¯ j z α j (1 − α¯ j z) where MK > 0 depends only on K and A. So {rm }m≥1 goes to f uniformly on K. Moreover, for ξ ∈ F, above identity also gives |α j | α j − ξ 1 − |α j | − 1 ≤ M′ , j ≥ 1. α j 1 − α¯ j ξ |1 − α¯ j ξ | Here M ′ is a constant depending on A. Hence there exists limm→∞ rm (ξ ) =: f (ξ ). For rapidly uniformly convergence, we write ∞ |α j | α j − z | f (z) − rm (z)| = |rm (z)|| ∏ − 1|. ¯ j=m+1 α j 1 − α j z Using above equations we obtain ∞ |α j | α j − z ∏ α j 1 − α¯ j z − 1 = j=m+1 ( (α j + |α j |z)(1 − |α j |) ) −1 ∏ 1− α j (1 − α¯ j z) j=m+1 ) ∞ ( (α + |α |z)(1 − |α |) j j j ≤ ∏ | |+1 −1 α j (1 − α¯ j z) j=m+1 ) ∞ ( ≤ ∏ MK (1 − |α j |) + 1 − 1 ∞ j=m+1 ≤ exp( ≤ 2MK ∞ ∑ MK (1 − |α j |)) − 1 ∑ (1 − |α j |), j=m+1 ∞ j=m+1 et ≤ 1 + 2t, 0 ≤ t ≪ 1, when m is sufficiently enough with the remark that ∑∞j=m+1 (1−|α j |) → 0 as m → ∞. Notice that {rm }m≥1 converges to f uniformly on K, so there exists a constant CK > 0 such that ∥rm ∥K < CK for every m ≥ 1. We get | f (z) − rm (z)| ≤ 2CK MK ∞ ∑ (1 − |α j |), ∀z ∈ K. j=m+1 Now the conclusion follows from Lemma 4.10(b). For rapid convergence at ξ ∈ F, we write |α j | α j − ξ (α j + |α j |ξ )(1 − |α j |) 1 − |α j | −1 = ≤4 , α j 1 − α¯ j ξ α j (1 − α¯ j ξ ) |(1 − α¯ j ξ )| when j is big enough. Using the same arguments as above we obtain ∞ |α j | α j − ξ 1 − |α j | − 1 ≤ 8 ∑ |1 − α¯ j ξ | . ∏ α j 1 − α¯ j ξ j=m+1 j=m+1 ∞ Since limn→∞ rn (ξ ) = f (ξ ), we can find Mξ > 0 such that |rn (ξ )| ≤ Mξ for all n ≥ 1. Hence, | f (ξ ) − rm (ξ )| ≤ 8Mξ ∞ 1 − |α j | . ¯ j=m+1 |1 − α j ξ | ∑ The assertion follows from Lemma 4.10(c). This completes the proof. References [Bl] T. Bloom, On the convergence in capacity of rational approximants, Constr. Approx. 17 (2001), 91-102. [Bl] Z. Blocki, Lecture notes in pluripotential theory, preprint, http://gamma.im.uj.edu. pl/ blocki /publ/ln/wykl.pdf. [Ch] E. Chirka, Expansions in series and the rate of rational approximations for holomorphic functions with analytic singularities, Math. USSR Sbornik 22 (1974), 323-332. [Co] M. Coltoiu, Complete locally pluripolar sets, J. Reine Angew. Math. 412 (1990), 108112. [Col] P. Colwell, On the boundary behavior of Blaschke products in the unit disk, Proc. Amer. Math. Soc. 17 (1966), 582-587. [DH] Nguyen Quang Dieu and Pham Hoang Hiep, Pluripolar hulls and complete pluripolar sets, Potential Anal. 29 (2008), no. 4, 409-426. [EW] A. Edigarian and J. Wiegerinck, Determination of the pluripolar hull of graphs of certain holomorphic functions, Ann. Inst. Fourier (Grenoble) 54 (2004), 2085-2104. [Go] A. Gonchar, A local condition for the single valuedness of analytic functions of several variables, Math. USSR Sbornik 22 (1974), 305-322. [Ho] L. H¨omander, Notions of Convexity, Birkh¨auser Press,1993. [Kl] M. Klimek, Pluripotential Theory, Oxford Academic Press 1991. [LMP] N. Levenberg, G. Martin and E. Poletsky, Analytic disks and pluripolar sets, Indiana Math. J., 41 (1992), 515-532. [LP] N. Levenberg and E. Poletsky, Pluripolar hulls, Michigan Math. J., 46 (1999), 151-162. [PW] E. Poletsky and J. Wiegerinck, Graphs with multiple sheeted pluripolar hulls, Ann. Polon. Math. 88 (2006), 161-171. [Sad] A. Sadullaev, Plurisubharmonic measures and capacities on complex manifolds, Uspekhi Mat. Nauk 36 (1981), 53-105. [Ze] A. Zeriahi, Ensembles pluripolaires exceptionnels pour la croissance partielle des fonctions holomorphes, Ann. Polon. Math. 50 (1989), no. 1, 81-91. H ANOI NATIONAL U NIVERSITY OF E DUCATION , 136 X UAN T HUY STREET, C AU G IAY, H ANOI , V IETNAM E-mail address: dieu vn@yahoo.com E-mail address: manhlth@gmail.com [...]... locally pluripolar sets, J Reine Angew Math 412 (1990), 108112 [Col] P Colwell, On the boundary behavior of Blaschke products in the unit disk, Proc Amer Math Soc 17 (1966), 582-587 [DH] Nguyen Quang Dieu and Pham Hoang Hiep, Pluripolar hulls and complete pluripolar sets, Potential Anal 29 (2008), no 4, 409-426 [EW] A Edigarian and J Wiegerinck, Determination of the pluripolar hull of graphs of certain... the proof of Lemma 4.5, we can find a point (a , b ) ∈ GC2 where a ∈ D ∩ A, b′ ∈ f (∂ D′ ) In particular b′ ∈ T This contradicts our choice of the set D′ We are done Remark If A is polar and uncountable then our proof of Proposition 4.4 breaks down because we do not know if the set T is polar Finally we come to the promised explicit example of a sequence of rational functions that converges rapidly... some work of Edigarian and Wiegerinck, we can give a partial answer in the case n = 1 Corollary 4.3 Let E be a analytic hypersurface of Cn and f be a holomorphic function on Cn \ E Then we have (Γ f (Cn \ E))∗Cn+1 ∩ ((Cn \ E) × C) = Γ f (Cn \ E) Proof By Theorem 3 in [Ch], there exists a sequence {rm }m≥1 of rational functions that converges rapidly and uniformly to f on compact subsets of D Applying... compact subset F of the unit circle ∂ ∆ = {z ∈ C : |z| = 1} which is nowhere dense in ∂ ∆ A standard construction of such F is the following Let C be the 1/3− Cantor set in [0, 1] and P be the orthogonal projection from ∂ ∆ onto the real axis Then we can choose F := P−1 (C ) ¯ a sequence {rm }m≥1 of Proposition 4.7 There exist a countable subset A of C \ ∆ with F ⊂ A, rational functions on C and a function... holomorphic functions, Ann Inst Fourier (Grenoble) 54 (2004), 2085-2104 [Go] A Gonchar, A local condition for the single valuedness of analytic functions of several variables, Math USSR Sbornik 22 (1974), 305-322 [Ho] L H¨omander, Notions of Convexity, Birkh¨auser Press,1993 [Kl] M Klimek, Pluripotential Theory, Oxford Academic Press 1991 [LMP] N Levenberg, G Martin and E Poletsky, Analytic disks and pluripolar. .. deeply the problem of determining the exceptional set E appearing in Proposition 4.2 Proposition 4.6 Let Ω be a domain in Cn and E be a pluripolar subset of Ω such that E ∩ H = ∗ ∩ H is pluripolar (relative to H) 0, / where H is the hyperplane z1 = 0 Then EΩ Proof of Proposition 4.4 Suppose that there exists a point (a, b) ∈ (Γ f (D))∗C2 with b ̸= f (a) It follows that f is non-constant and a ∈ A Let S... from Lemma 4.10(c) This completes the proof References [Bl] T Bloom, On the convergence in capacity of rational approximants, Constr Approx 17 (2001), 91-102 [Bl] Z Blocki, Lecture notes in pluripotential theory, preprint, http://gamma.im.uj.edu pl/ blocki /publ/ln/wykl.pdf [Ch] E Chirka, Expansions in series and the rate of rational approximations for holomorphic functions with analytic singularities,... ind(m,n)≥⌊ 2k ⌋ By Lemma 4.8, the k-th root of the last term tends to 0 as k → ∞ This completes the proof of the lemma Proof of Proposition 4.7 Let B := {α j } j≥1 ⊂ ∆ be the sequence in Lemma 4.10 Now we define for m ≥ 1 the finite Blaschke product {rm }m≥1 |α j | α j − z ¯ j=1 α j 1 − α j z m rm (z) = ∏ Let A := { α¯1j : j ≥ 1} Then the set of accumulation points of A is F We claim the infinite Blaschke... properties holds true: (a) The poles of {rm }m≥1 are included in A for every m ≥ 1; (b) {rm }m≥1 converges rapidly uniformly on compact sets of C \ A to f ; (c) {rm }m≥1 converges rapidly pointwise on F = A \ A to f Remarks (a) It follows from Proposition 4.2 that in the situation of Proposition 4.7, there exists a polar subset F ′ ⊂ F such that the pluripolar hull in C2 of Γ f (C \ A) over every point... Analytic disks and pluripolar sets, Indiana Math J., 41 (1992), 515-532 [LP] N Levenberg and E Poletsky, Pluripolar hulls, Michigan Math J., 46 (1999), 151-162 [PW] E Poletsky and J Wiegerinck, Graphs with multiple sheeted pluripolar hulls, Ann Polon Math 88 (2006), 161-171 [Sad] A Sadullaev, Plurisubharmonic measures and capacities on complex manifolds, Uspekhi Mat Nauk 36 (1981), 53-105 [Ze] A Zeriahi, ... D such that f is the limit of a sequence {rm }m≥1 of rational functions which converged rapidly to f on D, see the next section for the definition of these types of convergence Here, we always... constructed as a square root of a holomorphic function g which can be rapidly approximated by a sequence gm of rational functions whose zeros and poles are related to the endpoints of the intervals removed... to offer a simpler (D) together with the Chern-Levineproof using compactness of PSH(D) in the topology Lloc Nirenberg inequality instead of the more delicate comparison result of Alexander and