Analytic continuation in several complex variables

Transcription

1 Analytic continuation in several complex variables An M.S. Thesis Submitted, in partial ulillment o the requirements or the award o the degree o Master o Science in the Faculty o Science, by Chandan Biswas Department o Mathematics Indian Institute o Science Bangalore April, 2012

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3 Declaration I hereby declare that the work in this thesis has been carried out by me in the Integrated Ph.D. program under the supervision o Proessor Gautam Bharali, and in partial ulillment o the requirements o the Master o Science degree o the Indian Institute o Science, Bangalore. I urther declare that this work has not been the basis or the award o any degree, diploma, ellowship or any other title elsewhere. Indian Institute o Science, Bangalore, April, Chandan Biswas S. R. No Proessor Gautam Bharali (Thesis adviser)

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5 Acknowledgements It is my pleasure to thank Proessor Gautam Bharali or many valuable suggestions, interesting discussions and his help in the preparation o this thesis, especially or the ininite patience that he has shown in correcting the mistakes in my writings. I must also thank Proessor Kaushal Verma or the many discussions he had with me during my coursework. Finally, I wish to thank Purvi Gupta and Jonathan Fernandes or riendly advice and tips in the course o typing thesis.

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7 Abstract We wish to study those domains in C n, or n 2, the so-called domains o holomorphy, which are in some sense the maximal domains o existence o the holomorphic unctions deined on them. We demonstrate that this study is radically dierent rom that o domains in C by discussing some examples o special types o domains in C n, n 2, such that every unction holomorphic on them extends to strictly larger domains. Given a domain in C n, n 2, we wish to construct the maximal domain o existence or the holomorphic unctions deined on the given domain. This leads to Thullen s construction o a domain (not necessarily in C n ) spread over C n, the so-called envelope o holomorphy, which ulills our criteria. Unortunately this turns out to be a very abstract space, ar rom giving us a sense in general how a domain sitting in C n can be constructed which is strictly larger than the given domain and such that all the holomorphic unctions deined on the given domain extend to it. But with the help o this abstract approach we can give a characterization o the domains o holomorphy in C n, n 2. The aorementioned characterization is as ollows: a domain in C n is a domain o holomorphy i and only i it is holomorphically convex. However, holomorphic convexity is a very diicult property to check. This calls or other (equivalent) criteria or a domain in C n, n 2, to be a domain o holomorphy. We survey these criteria. The proo o the equivalence o several o these criteria are very technical requiring methods coming rom partial dierential equations. We provide those proos that rely on the irst part o our survey: namely, on analytic continuation theorems. I a domain Ω C n, n 2, is not a domain o holomorphy, we would still like to explicitly describe a domain strictly larger than Ω to which all unctions holomorphic on Ω continue analytically. Aspects o Thullen s approach are also useul in the quest to construct an explicit strictly larger domain in C n with the property stated above. The tool used most oten in such constructions is called Kontinuitätssatz. It has been invoked, without a clear statement, in many works on analytic continuation. The basic (unstated) principle that seems to be in use in these works appears to be a olk theorem. We provide a precise statement o this olk Kontinuitätssatz and give a proo o it. 7

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9 Contents 1 Introduction and basic results 11 2 The Hartogs phenomenon Dierentiating under the integral sign The -problem The Hartogs continuation theorem Envelopes and domains o holomorphy Envelopes o holomorphy Domains o holomorphy More characterizations o domains o holomorphy in C n Further characterizations o domains o holomorphy Domains o holomorphy that have smooth boundary A schlichtness theorem or envelopes o holomorphy A schlichtness theorem Proo o the main theorem A simple Kontinuitätssatz Recalling notations and terminology A undamental proposition The proo o the main theorems

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11 Chapter 1 Introduction and basic results The initial part o this chapter will be devoted to understanding what it means or a unction : Ω C, where Ω is an open subset o C n, n 2, to be holomorphic. For this purpose, let us introduce some convenient notations (which we shall ollow in the later chapters also). We write z C n as z := (z 1, z 2,..., z n ) and urther, z j = x j + iy j, j = 1, 2,..., n. Given α := (α 1, α 2,..., α n ) N n, and or any z C n, we set α := α α n, α! := α 1!... α n!, z α := z α 1 1 zα zα n n. We deine the dierential operators := 1 ( i ), z j 2 x j y j := 1 ( + i ), z j 2 x j y j D α := α z 1 α 1... zn α n. Given a point a C n and r = (r 1,..., r n ) R n +, the set D(a, r) := {z C n : z j a j < r j, j = 1,..., n} is called a polydisc with centre at a and polyradius r. We will have the occasion where all the r j s are equal. In that case we will use the notation D n (a, r) := D(a, (r,..., r)), where r > 0. Here are our plausible deinitions or holomorphicity, each o which gives us the usual deinition o holomorphicity when n = 1. Let Ω below be a domain (open connected set) in C n. 11

12 Deinition 1.1. A unction : Ω C is holomorphic i or each j = 1,..., n, and each ixed z 1,..., z j 1, z j+1,..., z n, the unction ζ (z 1,..., z j 1, ζ, z j+1,..., z n ) is holomorphic in the classical one variable sense on the set Ω(z 1,..., z j 1, z j+1,..., z n ) := {ζ C : (z 1,..., z j 1, ζ, z j+1,..., z n ) Ω}. Deinition 1.2. A unction : Ω C that is continuously (real) dierentiable with respect to each pair o variables (x j, y j ) separately on Ω is said to be holomorphic i satisies the Cauchy- Riemann equations in each variable separately. Deinition 1.3. Let : Ω C be continuous in each variable separately and locally bounded.the unction is said to be holomorphic i or each w Ω there is an r = r(w) > 0 such that D n (w, r) Ω and (z) = or all z D n (w, r) (2πi) n ζ n w n =r ζ 1 w 1 =r (ζ 1,..., ζ n ) (ζ 1 z 1 )... (ζ n z n ) dζ 1... dζ n Deinition 1.4. A unction : Ω C is holomorphic i or each a Ω there is an r = r(a) > 0 such that D n (a, r) Ω and can be written as an absolutely convergent power series (z) = a α (z a) α or all z D n (a, r), α N n that converges uniormly on compact subsets o D n (a, r). It turns out that all our deinitions are equivalent.to see this let us provisionally assume that : Ω C is continuous. That is trivial. Showing that involves iterating the one variable Cauchy Integral Formula. To carry out this argument we must worry about the order o integration. Also we need to know, or instance that the unctions {t C : t w k = r} t (ζ (k), t, z k+1,..., z n ) are integrable or each ixed ζ (k) D 1 (w j, r) and (z k+1,..., z n ) D n k ((w k+1,..., w n ), r), 1 j k 1 k = 1, 2,..., n 1. We do not have to worry about these points because we have taken to be continuous on Ω. Now using the same trick o expanding out a geometric series that we use in one complex variable. Finally, because uniorm convergence allows us to evaluate the C-derivative with respect to a given z j, j = 1, 2,..., n, term by term. Just as Goursat showed that a univariate unction admitting all irst order partial derivatives (not necessarily continuous) and satisying the Cauchy-Riemann equation is automatically C- dierentiable, it can be shown that the a priori assumption that is continuous is superluous.this was shown by Hartogs.We direct the reader to [ [10]][chapter 3] or a proo o Hartogs result. 12

13 We must mention that many undamental results rom the one-variable theory, e.g. the Cauchy Integral Formula, Liouville s Theorem, the Open Mapping Theorem, Weierstrass Theorem, Montel s Theorem, etc., have obvious generalizations to several variables. Their proos, in most cases, involve subscripts by multi-indices in suitable places and iterating the one-variable argument, so we will skip these proos. One theorem that has a slightly dierent orm in higher dimensions is the principle o analytic continuation (which we shall use in proving the theorems presented below). Theorem 1.5 (Principle o analytic continuation). Let Ω be a domain in C n and let be holomorphic on Ω. I vanishes on a non-empty open subset o Ω then 0. The second part o this chapter ocuses on how, in contrast to the above-named theorems, remarkably dierent the behaviour o holomorphic unctions in multiple variables can be rom the one-variable case. For instance, it is easy to describe domains with the property that every unction holomorphic on them extend holomorphically to a strictly larger domain. It is less easy to describe completely the larger domain to which the above unctions extend holomorphically. But we can provide a complete description or an important special class o domains. These are the motivations or the ollowing results. Theorem 1.6. Let V be a neighbourhood o D n (0, 1), n 2 such that V D n (0, 1) is connected. Then, or any holomorphic unction on V, there is a holomorphic unction F on D n (0, 1) V so that F V. Proo. Let ɛ > 0 be such that i A := {z C n : 1 ɛ < z 1 < 1, z j < 1, j 2} {z C n : 1 ɛ < z 2 < 1, z j < 1, j 2}, A V. Existence o such a ɛ is guaranteed by the compactness o D n (0, 1). Let z = (z 2,..., z n ). We note here that in this proo, and elsewhere in this report, i v = (v 1,..., v n ) is a vector, then we shall, by a slight abuse o notation, denote v := max j n v j (to distinguish it rom the Euclidean norm ). I z < 1, the unction z 1 (z 1, z ) is holomorphic on the annulus {z 1 C : 1 ɛ < z 1 < 1}, so that (z 1, z ) = a ν (z )z ν 1, ν Z or any z D n 1 (0, 1). For each ν Z a ν (z ) = 1 2πi z 1 =1 (z 1, z ) z ν+1 1 dz 1, z D n 1 (0, 1). Dierentiating under the integral sign, we get a ν (z ) = 1 1 (z z j 2πi z 1 =1 z ν+1 1, z ) dz 1 = 0, j = 2, 3,..., n, z 1 j 13

14 or each z D n 1 (0, 1). O course one must justiy whether we can validly dierentiate under the integral sign above. We shall give a criterion or being able to do this in the irst part o Chapter 2. In view o Deinition 1.2, a ν (z ) is holomorphic in D n 1 (0, 1) or each ν Z. I now 1 ɛ < z 2 < 1, z 3 < 1,..., z n < 1, the unction z 1 (z 1, z ) is holomorphic in the disc z 1 < 1, so that its Laurent series contains no negative powers o z 1 ; i.e. a ν (z ) = 0 or ν < 0. As D n 1 (0, 1) is connected, by the principle o analytic continuation a ν (z ) = 0 or ν < 0, z D n 1 (0, 1). We deine { (z), z V, F(z) := ν N a ν (z )z ν 1, z Dn (0, 1). This latter series converges uniormly on compact subsets o D n (0, 1) and so is holomorphic there; urther, it coincides with on {z C n : 1 ɛ < z 1 < 1, z j < 1, j 2}, a nonempty open subset o V D n (0, 1) and so on the whole o V D n (0, 1), as this set is open and connected. We need a deinition in order to ormulate our next theorem. Deinition 1.7. Let Ω be a domain in C n. We say that Ω is a Reinhardt domain i whenever z = (z 1, z 2,..., z n ) Ω and θ 1, θ 2,..., θ n R, we have (e iθ 1 z 1, e iθ 2 z 2,..., e iθ n z n ) Ω. Theorem 1.8. Let Ω be a Reinhardt domain in C n. Then or any holomorphic unction on Ω, there is a Laurent series α Z n a α z α which converges uniormly to on compact subsets o Ω. Moreover, the a α s are uniquely determined by. Proo. We begin by proving the uniqueness. Let w Ω be a point with coordinates (w 1,..., w n ), w j 0 or all j. Then, since the series converges uniormly to on compact subsets o Ω, we may set z j = w j e iθ j, multiply by e i(α 1θ 1 +α 2 θ 2 + +α n θ n ) and integrate term by term. Now we observe that π { 0, i n 0, e inθ dθ = This gives, or α Z 2π, i n = 0. n, π π a α = w α (2π) n π π π (w 1 e iθ 1, w 2 e iθ 2,..., w n e iθ n )e i(α 1θ 1 +α 2 θ 2 + +α n θ n ) dθ 1... dθ n. It seems that a α depends on w, so that the coeicient a α is not unique. So suppose (z) = α N n b αz α be a dierent representation o in Ω. Then 0 = α Z n(a α b α )z α, z Ω. Then, by the previous ormula : π a α b α = w α (2π) n π π π 0 e i(α 1θ 1 +α 2 θ 2 + +α n θ n ) dθ 1... dθ n = 0. To prove the existence o an expansion as above, we irst remark that i D = {z C n : r j < z j < R j, < r j < R j, j = 1, 2,..., n} and is holomorphic on D, then, by iteration o the Laurent 14

15 expansion o one complex variable unctions, it ollows that has an expansion in a Laurent series. Let w Ω. I ɛ(w) > 0 is small enough, since Ω is a Reinhardt domain, the set D(w, ɛ(w)) = {z C n : w j ɛ(w) < z j < w j + ɛ(w)} is contained in Ω. Since this is a set o the orm D above, there is a Laurent expansion α Z n a α (w)z α = (z), z D(w, ɛ(w)), converges to uniormly in a neighbourhood o w. Now, i w 1 D(w, ɛ(w)) and α Z n a α(w 1 )z α is the expansion corresponding to w 1 in a set D(w 1, ɛ(w 1 )) Ω, then the uniqueness assertion above shows that a α (w) = a α (w 1 ). Hence the unction w a α (w) is locally constant on Ω or any α Z n and since Ω is connected, a α (w) a α, independent o w. Hence any point z Ω has a neighbourhood N(z) Ω so that the series α Z n a α z α converges uniormly to (z) on N(z), hence uniormly to (z) or z in any compact subset o Ω. Corollary 1.9. Let Ω be a Reinhardt domain such that or each j, 1 j n, there is a point z Ω whose j-th coordinate is 0. Then any holomorphic unction on Ω admits an expansion (z) = α N n a α z α which converges uniormly to on compact subsets o Ω. Proo. For n = 1 as the domain Ω is a Reinhardt domain in C containing 0, Ω = D 1 (0, r) or some r > 0 or Ω = C. Thereore in the Laurent series expansion o all the coeicients with negative index vanish. Hence we get the result. For n 2, let w 1 represent an element in Ω whose irst coordinate is 0. So w 1 = (0, w 2..., w n ) Ω. As Ω is a Reinhardt domain, there is a ɛ > 0 such that D(w 1, ɛ) Ω, where D(w 1, ɛ) := D 1 (0, ɛ) Ann(0, w 2 ɛ, w 2 + ɛ) Ann(0, w n ɛ, w n + ɛ). Fix z = (z 2, z 3,..., z n ) Ann(0, w 2 ɛ, w 2 +ɛ) Ann(0, w n ɛ, w n +ɛ). Then the unction z 1 (z 1, z ) is holomorphic on D 1 (0, ɛ). Using Theorem 1.8 (z) = (z 1, z ) = a (α1,α )z α z 1 α 1, z 1 D 1 (0, ɛ). α 1 Z α Z n 1 But as z 1 (z 1, z ) is holomorphic on D 1 (0, ɛ), a (α1,α ) z α = 0, α 1 < 0 and z Ann(0, w 2 ɛ, w 2 + ɛ) Ann(0, w n ɛ, w n + ɛ). α Z n 1 15

16 Ann( w 2 ɛ, w 2 + ɛ) Ann( w n ɛ, w n + ɛ) is a Reinhardt domain in C n 1. Hence rom the uniqueness assertion in Theorem 1.8 a (α1, α ) = 0, or all α Z n : α 1 < 0. Similarly we can show that a (α1,α 2,...,α n ) = 0, i α j < 0 or some j. Hence (z) = α N n a αz α. Corollary Let Ω be a Reinhardt domain such that or each j, 1 j n, there is a point z Ω whose j-th coordinate is 0. Then any holomorphic unction on Ω has an unique holomorphic extension F to the set Ω := {(λ 1 z 1, λ 2 z 2,..., λ n z n ) : 0 λ j 1, (z 1, z 2,..., z n ) Ω}. Proo. By Corollary 1.9, (z) = α N a αz α or z Ω. Now let z = (λ n 1 z 1, λ 2 z 2,..., λ n z n ) Ω, where or all j, λ j [0, 1] and (z 1, z 2,..., z n ) Ω. As the series α N a αz α converges at the point n (z 1, z 2,..., z n ), by Abel s Lemma the series also converges at the point z = (λ 1 z 1, λ 2 z 2,..., λ n z n ). Thereore F(z) := α N n a α z α, z Ω is a well deined holomorphic unction on Ω. Clearly F extends since Ω Ω. Now 0 Ω. Let z = (z 1, z 2,..., z n ) Ω. Then λz = (λz 1, λz 2,..., λz n ) Ω or all λ [0, 1] by deinition o Ω. Thereore Ω is connected. Now Ω is a nonempty open subset o Ω. Hence by the principle o analytic continuation i there exists another holomorphic unction G on Ω such that it also extends then F(z) = G(z) or all z Ω. 16

17 Chapter 2 The Hartogs phenomenon We saw in Chapter 1 that we can easily give examples o domains in C n, n 2, with the property that every unction holomorphic on them extends holomorphically to a strictly larger domain. However, the proo o the above act depended on the domains having a lot o symmetry. All the domains in Theorem 1.8 o Chapter 1, and in its corollaries, are Reinhardt. However, Hartogs showed that the phenomenon demonstrated by Theorem 1.6 holds true or arbitrary domains in C n, n 2. This will be demonstrated in Section 2.3 below. But irst, we will need to go through a ew technicalities. 2.1 Dierentiating under the integral sign Many theorems in Complex Analysis involve dierentiating under the integral sign. However, this needs a justiication. In many cases, the classical Leibniz Theorem does not suice. The ollowing proposition provides suicient conditions that are general enough to be usable in many contexts. Proposition 2.1. Let K : R n \ {0} C be a measurable unction that satisies lim x 0 K(x) = +, but which diverges suiciently slowly at 0 R n that K L 1 loc (Rn ). 1 Suppose : R n R n R is a continuous unction having the ollowing two properties: There exists R > 0 such that, or each x R n, (x, y) = 0 or all y R n \ B(x, R); The amily { (., y) : y R n } is equicontinuous at each x R n. Then, the unction g(x) := K(y) (x, y) dv(y) R n (2.1) is well-deined and continuous. 17

18 2 Now suppose that, given N 1, α urthermore, that the amilies N. exists or all α N n such that α N. Assume, { x α } α (., y) : y R n x α Then, g C N (R n ), and we can, in act, dierentiate under integral sign. Proo. Observe that, by hypothesis: K(y) (x, y) dv(y) = R n B(x,R) are equicontinuous at each x R n, α K(y) (x, y) dv(y). The integral on the right-hand side is convergent and hence g is well deined. Now note; or any x 0 R n : g(x) g(x 0 ) K(y) (x, y) (x 0, y) dv(y). (2.2) R n By the equicontinuity hypothesis, given ɛ > 0, there exists δ 1 δ 1 (x 0, ɛ) > 0 such that x x 0 < δ 1 implies (x, y) (x 0, y) < ɛ or all y R n. Hence, by (2.2), g(x) g(x 0 ) K(y) (x, y) (x 0, y) dv(y) ɛ K L1 (B(x 0,R+1)), B(x 0,R+1) whenever x x 0 < min(1, δ 1 (x 0, ɛ)). This establishes continuity and hence part 1. For clarity o explanation, we shall carry out the proo o part 2 in the ollowing distinct steps: Step 1: Illustrating a special case. Pick a j n. Then, rom the irst condition in part 1, we see that the analogue o that condition is true or x j also. Hence, the integral I j (x) := K(y) (x, y) dv(y) (2.3) R x n j is well deined. For any t R \ {0}, g(x 0 + te j ) g(x 0 ) I j (x 0 ) { t = (x0 + te j, y) (x 0, y) K(y) } (x 0, y) dv(y) R t x n j { 1 [ = K(y) (x 0 + ste j, y) ] } (x 0, y) ds dv(y) R n 0 x j x j.(2.4) Here e j denotes the j-th vector o the standard basis o R n. The last equation is obtained by applying the Fundamental Theorem o Calculus to the univariate unctions s (x 0 + ste j, y). 18

19 Once again, by the equicontinuity hypothesis, given any ɛ > 0 we can ind a δ 2 δ 2 (x 0, ɛ) such that t < δ 2 = x j (x 0 + ste j, y) x j (x 0, y) < ɛ or all y Rn and or all s [0, 1]. Applying the above inequality to (2.4) gives us g(x 0 + te j ) g(x 0 ) I j (x 0 ) t ɛ K L 1 (B(x 0,R+1)), whenever 0 < t < min(1, δ 2 (x 0, ɛ)). We have established that, or each j n : g (x 0 ) = [ ] K(y) (x 0, y) dv(y) x j x j R n = R n K(y) x j (x 0, y) dv(y). The continuity o g x j ollows by repeating the proo o part 1 with g x j replacing in (2.1). Hence g C 1( R n). Step 2: Completing the proo. We use induction to complete the proo. We are given N 1. Assume we have the desired result or all α N n such that α m or some m < N. Now we notice that step 1 was the base case o the induction. Given an α N n such that α = m + 1, there exists β such that β = m and a j n such that (α 1, α 2,..., α n ) = (β 1,..., β j 1, β j + 1, β j+1,..., β n ) =: β, j. We now repeat the argument in step 1 with the ollowing replacements: β x β replacing in the relevant places ; x j ( β x β ) replacing x j in the deinition o I j in (2.3) ; which establishes the result or all α N n such that α = m + 1. By induction, g C N (R n ). 2.2 The -problem Let D be a domain in C n. For n 2, the -problem denotes the ollowing system o PDE s: subject to: u z j = j on D j = 1,..., n, (2.5) j z k = k z j or all j k, (2.6) 19

20 where 1,..., n L 1 loc (D). When n = 1, the compatibility condition (2.6) is vacuous, and the -problem is just the PDE (2.5). We note that i the j s are not smooth, then (2.6) is interpreted in the sense o distributions. We must demand (2.6) when n 2 because it is a necessary condition or (2.5) to have a smooth solution. To see this, irst assume that the j s are o class C 1. I a u smooth solution u exists, then as it satisies the PDE (2.5) we have or each j, z j = j. As u is a smooth unction, thereore all the mixed second-order partial derivatives o u exist and they are independent o the order o the variables. In particular as j C 1, (2.6) is satisied. The same argument can be made in the weak sense i the j s are not smooth. Let us provide a solution to the -problem when n = 1, D = C and the right-hand side is in C k (C) and has compact support. For this, we need an elementary act rom Advanced Calculus. Theorem 2.2 ( The Generalized Cauchy Integral Formula). Let Ω be a bounded domain in C such that Ω is a disjoint union o initely many piecewise smooth simple closed curves. Let U be a neighbourhood o Ω and let : U C be o class C 1 (U). Then or any w Ω, (w) = 1 2πi Ω (z) z w dz 1 π Ω 1 z z w da(z). Theorem 2.3. Let k 1 and let φ C k (C), and suppose φ is compactly supported. Then, the unction u(z) := 1 φ(w) π w z da(w) satisies the PDE Furthermore, u C k (C). C u z = φ. Proo. Since φ is compactly supported, there exists r 0 > 0 such that supp(φ) B(0; r 0 ). Now, given z C, let R > r 0 and such that z B(0, R). By Theorem 2.2 φ(z) = 1 1 φ π w z z (w) da(w) = 1 1 φ (w) da(w). (2.7) π w z z B(0,R) Now, rom (2.7), and the act that R can be arbitrarily large, the last integral represents φ(z) or all z C. Now we observe that, by a change o variable u(z) = 1 1 φ(w + z) da(w). π w We observe that i we set C K(w) : = 1 w, w C \ {0}, (z, w) : = φ(w + z), 20 C

21 then the hypotheses o Proposition 2.1, are satisied. Thus, u C k (C), and we can dierentiate under the integral sign to get u z (z) = 1 1 (z, w) da(w) π w z = 1 π C C 1 φ (w) da(w) [ change o variable ] w z z = φ(z) or all z C. [ by (2.7) ] The above plays a very important part in the solution o the -problem in C n, n 2, with compactly-supported data. Theorem 2.4. Let j C 0 k (C n ), j=1,...,n, where k > 0, and assume that (2.6) is ulilled (n > 1). Then there is a unction u C 0 k (C n ) satisying (2.5). Now note that the part about u having compact support in this theorem is alse or n = 1 (take an arbitrary 1 C 0 with Lebesgue integral dierent rom 0). Proo. We set u(z) := 1 π C 1 (w, z 2,..., z n ) w z 1 da(w) = 1 π C 1 (z 1 w, z 2,..., z n ) w da(w). The second orm o the deinition shows that we can apply Proposition 2.1 to get that u C k (C n ). Now u(z) = 0 i z z n is large enough. By Theorem 2.3 it ollows that u z 1 = 1. I k > 1, by dierentiating under the sign o integration and using the act that 1 z k u (z) = 1 1 k (w, z 2,..., z n ) da(w) = k (z). z k π w z 1 z 1 = k z 1, we obtain Hence u satisies all the equations in (2.5), which means in particular u is analytic outside a compact set. As this complement is connected, and u(z) = 0 when z z n is large, we deduce rom the principle o analytic continuation, that u has compact support. 2.3 The Hartogs continuation theorem Now we have suicient background to prove a theorem that is nowadays called Hartogs phenomenon. Clearly, as mentioned in the beginning o Chapter 2, this theorem gives a generalization o Theorem 1.6. Notation: H(Ω) denotes the set o all holomorphic unctions rom Ω into C. 21

22 Theorem 2.5. Let Ω be an open set in C n, n > 1, and let K be a compact subset o Ω such that Ω \ K is connected. For every u H ( Ω \ K ) one can then ind U H(Ω) so that u = U in Ω \ K. Proo. Using the partition o unity we construct a unction ϕ C 0 (Ω), such that ϕ is identically equal to 1 in a neighbourhood o K. Now we deine 0 i z K, u 0 (z) := (1 ϕ(z))u(z) i z Ω \ K. Then u 0 C (Ω), and we want to ind v C (C n ), so that U := u 0 v is holomorphic on Ω. Now the U will be analytic i U = 0, i.e. v = u 0 = u ϕ =, (2.8) where is deined by 0 i z K, (z) := u(z) ϕ(z) i z Ω \ K, 0 i z C n \ Ω. (2.9) Now we note that as ϕ is compactly supported the components o are in C 0 (Cn ). Hence, i we write = n j=1 j dz j, by (2.9) we have j z k = k z j or all j k. Thereore by Theorem 2.4, (2.8) has a C 0 (Cn ) solution. Furthermore, a careul look at the proo o Theorem 2.4 tells us that the solution given by the ormula in Theorem 2.4 vanishes in the unbounded component o the complement o the support o ϕ. Now the boundary o this set belongs to Ω \ K, so there exists an open set in Ω \ K where v = 0 and u = u 0. Hence the analytic unction U in Ω which coincides with u on some open subset o Ω \ K, and since this is a connected set, we have u U on Ω \ K. 22

23 Chapter 3 Envelopes and domains o holomorphy In view o the phenomenon seen in Chapters 1 and 2, we would now like to discuss, given a domain Ω, the largest domain, in the appropriate sense, to which all unctions in H(Ω) simultaneously extend. As our experience with the logarithm in 1-variable complex analysis shows, we need to take the help o germs to construct this largest domain. But because we will need to construct such objects or the simultaneous extension o a whole amily o unctions, we must introduce the idea o S -germs. But irst, we provide a deinition. Deinition 3.1. Let X be a Hausdor topological space. I there exists a map p : X C n such that p is a local homeomorphism then (X, p, C n ) is called an (unramiied) domain over C n. Let S be some ixed set. We want to deine the shea O(S ), the shea o S -germs o holomorphic unctions on C n. I U is a non-empty open set, then the pair (U, { s } s S ) will denote an arbitrary amily o unctions holomorphic on U indexed by S. For a C n and (U, { s } s S ), (V, {g s } s S ) with a U V, we say that these two pairs are equivalent i there exists a neighbourhood W o a, W U V such that, or all s S, s W g s W. An equivalence class with respect to this relation is called an S -germ o holomorphic unctions at a. We denote the set o all S -germs at a by O a (S ). We set O(S ) := a C O a(s ). We have a natural projection p = p n s : O S C n deined by p(g a ) := a i g a O a (S ). We deine a topology on O(S ) as ollows: Let g a O a (S ) and let (U, { s } s S ) be a representative o g a. Let [U, s : s S ] b denote the germ at b determined by the above pair or any b U. Now we write N(U, { s } s S ) := [U, s : s S ] b. b U We topologize O(S ) by demanding that the sets N(U, { s } s S ) orm a undamental system o neighbourhoods o g a. The proo o the ollowing proposition is very routine and mechanical, so we shall skip its proo. Proposition 3.2. The map p : O(S ) C n is continuous and is a local homeomorphism o O(S ) onto C n. Further, O(S ) is a Hausdor topological space. The triple (O(S ), p, C n ) is an (unramiied) domain over C n. 23

24 3.1 Envelopes o holomorphy Let p : X C n be a domain over C n. A continuous unction : X C is said to be holomorphic i or each open set U X such that p U is a homeomorphism, (p U ) 1 H(p(U)). We can extend the notion o holomorphicity to maps between domains in an analogous ashion. Now let p : X C n, p : X C n be domains over C n. A continuous map u : X X is called a local isomorphism i every point a X has a neighbourhood U such that u U is a homeomorphism onto an open set U X and u U, (u U ) 1 are holomorphic (on U and U respectively). I, in addition, u is a homeomorphism o X onto X, we say that u is an isomorphism. I u : X X is a continuous map such that p u = p, then u is automatically a local isomorphism. Deinition 3.3. Let p 0 : Ω C n be a connected domain and S H(Ω). Let p : X C n be a connected domain and ϕ : Ω X a continuous map with p ϕ = p 0. We say (X, p, ϕ) is an S -extension o p 0 : Ω C n i, to every S, there exists F H(X) such that F ϕ =. Note that F is uniquely determined (irst on Ω since F ϕ =, hence on X by the principle o analytic continuation). F is called the extension (or continuation) o to X. Deinition 3.4. Let p 0 : Ω C n be a connected domain and S H(Ω). An S -envelope o holomorphy is an S -extension (X, p, ϕ) such that the ollowing holds: ( ) For any S -extension (X, p, ϕ ) o p 0 : Ω C n, there is a holomorphic map u : X X such that p u = p, u ϕ = ϕ and F = F u or all S, where F, F are the extensions o S to X, X respectively. Note that u in ( ) is unique (since it is determined on ϕ (Ω) by the equation u ϕ = ϕ). Remark 3.5. The S -envelope o holomorphy, i it exists, is unique up to isomorphism. In act, let (X, p, ϕ), (X, p, ϕ ) be two S -envelopes o holomorphy. Then, by ( ) o Deinition 3.4 there are holomorphic maps u : X X, v : X X such that p = p v, p = p u, ϕ = u ϕ, ϕ = v ϕ. Then u v ϕ = u ϕ = ϕ, so that u v is the identity on ϕ(ω) which is open in X. Hence, by the principle o analytic continuation u v = identity on X. Similarly, v u = identity on X. Thus, u is an isomorphism o X onto X with p = p u, ϕ = u ϕ. Theorem 3.6 (Thullen). The S -envelope o holomorphy o any S H(Ω) exists. Proo. For any p 0 : Ω C n and S H(Ω), we deine a map ϕ = ϕ(p o, S ) o Ω into O(S ) as ollows. Let a Ω and a 0 = p 0 (a) C n. Let U be an open neighbourhood o a such that p 0 U is an isomorphism onto an open set U 0 C n. Let [U 0, s : s S ] a0 be the S -germ at a 0 deined by the pair (U 0, { s } s S ), where s := s (p 0 U ) 1, s S. We set ϕ(a) := [U 0, s : s S ] a0. Easy to veriy that ϕ is a continuous and that p ϕ = p 0, where p : O(S ) C n is the natural projection. In particular, ϕ is a local isomorphism. Since Ω is connected, so is ϕ(ω). Let X be the connected component o O(S ) containing ϕ(ω), and denote again by p the restriction to X o the map p : O(S ) C n. We claim that (X, p, ϕ) is an S -envelope o holomorphy o Ω. 24

25 First, we observe that, or all s S, we have a holomorphic unction F s on O(S ) deined as ollows. We set F s ([U 0, s : s S ] a0 ) := s (a 0 ). Easy to veriy that F s is holomorphic on O(S ). We denote the restriction o F s to X again by F s. Now, by the very deinition o ϕ, it ollows that F s ϕ = s or all s S. Let p : X C n, ϕ : Ω X be given with p ϕ = p 0, and suppose that or all s S, there exists F s H(X ) so that s = F s ϕ. Let S = {F s : s S }. Let u : X O(S ) be the map ϕ(p, S ) (deined at the beginning o this proo). Since F s ϕ = s and p ϕ = p 0, we have ϕ = u ϕ (locally, F s p 1 = F s ϕ ϕ 1 p 1 = s p 1 0 ). Clearly, p = p u. Deinition 3.7. Let p 0 : Ω C n be a connected domain over C n. I S = H(Ω), the S -envelope o holomorphy o Ω is called the envelope o holomorphy o Ω. Proposition 3.8. Let p 0 : Ω C n be a connected domain over C n and H(Ω). let F be its extension to the envelope o holomorphy (X, p, ϕ). Then (Ω) = F(X). In particular, i is bounded, (x) M or all x Ω, then F is bounded and F(x) M or all x X. Proo. Since = F ϕ, we have (Ω) F(X). Suppose there is a c F(X) \ (Ω). Then 1 H(Ω). Let G be its extension to X. Then G.(F c) is the extension o 1 = ( c c) 1 ( c) to X, so that G.(F c) 1 on X. This implies that F(x) c or all x X, a contradiction. At this stage, one might like to know how two envelopes o holomorphy are related to each other i the underlying domains are related by a holomorphic mapping. To answer such questions, we require two lemmas. Their proos are routine and so we shall skip their proos. Lemma 3.9. Let p 0 : Ω C n be a domain and Ψ : Ω C n be a holomorphic map such that det(dψ)(a) 0 or some a Ω. Then, there exists a neighbourhood U a such that Ψ U is an analytic isomorphism. Lemma Let p 0 : Ω C n, p 0 : Ω C n be connected domains over C n and (X, p, ϕ ) be the T-envelope o holomorphy o p 0 : Ω C n, T H(Ω ). Let u : Ω Ω be a local isomorphism, and let S = { u : T}. Then (X, p, ϕ u) is the S -envelope o q o : Ω C n where q o = p 0 u. Proposition Let p 0 : Ω C n, p 0 : Ω C n be connected domains over C n, and (X, p, ϕ), (X, p, ϕ ) their envelopes o holomorphy. Let u : Ω Ω be a local isomorphism. Then there exists a holomorphic map ũ : X X such that ϕ u = ũ ϕ. Proo. Let v := ϕ u : Ω X. Then v is holomorphic and a local isomorphism. We have to show that there is u : X X so that u ϕ = v. Consider the map ψ := p v : Ω C n. Then ψ is again a local isomorphism. I ψ = (ψ 1, ψ 2,..., ψ n ), the ψ j s are holomorphic. Let η : Ω C be given by η(x) := det(dψ(x)). Then, since ψ is a local isomorphism, η(x) 0 or all x Ω. Let Ψ j be the extension o ψ j to X, and let Ψ = (Ψ 1, Ψ 2,..., Ψ n ). Let H be the extension o η to X. Then, by the principle o analytic continuation, H = det(dψ). Moreover, by Proposition 3.8 we have H(x) 0 or all x X. Hence by Lemma 3.9, Ψ : X C n is a local isomorphism. Moreover, Ψ ϕ = ψ. 25

26 Consider now the domains ψ : Ω C n, p : X C n, and v : Ω X. Let S = { u : H(Ω )} = {F v : F H(X )}. By Lemma 3.10 (X, p, v) is the S -envelope o holomorphy o ψ : Ω C n. Now any holomorphic unction on Ω can be extended to X, so that (X, p, Ψ) is an S -extension o ψ : Ω C n. Thereore there exists a holomorphic map ũ : X X such that p ũ = Ψ and ũ ϕ = v. Deinition Let p 0 : Ω C n be a connected domain over C n and S H(Ω). Ω is called an S -domain o holomorphy i the natural map o Ω ϕ : x [p 0 (U x ), (p 0 Ux ) 1 ] p0 (x) into its S -envelope o holomorphy is an analytic isomorphism. I S = H(Ω), Ω is called a domain o holomorphy. The reader would intuit that the envelope o holomorphy o a connected domain p 0 : Ω C n is a domain o holomorphy. We now give the proo behind this intuition. Corollary I (X, p, ϕ) is the envelope o holomorphy o p 0 : Ω C n then p : X C n is a domain o holomorphy. Proo. In Lemma 3.10 i we replace Ω by X, T by {F : H(Ω)} and u by ϕ then we get S = H(Ω). Thereore by Lemma 3.10 we get the desired result. Consider a domain Ω C n (in which case p 0 is just the inclusion map). I such an Ω is a domain o holomorphy, then combining the above corollary with the details o Thullen s construction in Theorem 3.6, it is easy to deduce the ollowing equivalent deinition or Ω C n to be a domain o holomorphy. Deinition Let Ω C n. We say Ω is a domain o holomorphy i there do not exist nonempty open sets U 1, U 2, with U 2 connected, U 2 Ω, U 1 U 2 Ω such that to every H(Ω), there exists F H(U 2 ) satisying F U1 = U Domains o holomorphy Now that we have introduced the concept o a domain o holomorphy, it would be interesting to see i one can characterize when a connected domain p 0 : Ω C n is a domain o holomorphy. This is quite technical, even when Ω C n. In the latter case, there are several equivalent characterizations. A characterization that is easy to check would require more space than is available in this report. But we shall aim to provide one characterization. Deinition Let p 0 : Ω C n be a connected domain over C n, H(Ω) and A Ω. We write A := sup x A (x). I A Ω, and S H(Ω), we set  S := {x Ω : (x) A or all S }. 26

27 I S = H(Ω), we simply write  :=  S. We shall need the ollowing technical, but elementary, lemma. Lemma Let A Ω such that A < or any H(Ω). Then, there is a compact set K Ω such that A K. Lemma The ollowing two statements are equivalent. (a) For any K Ω, K compact, K is also compact. (b) For any (ininite) sequence (x ν ) Ω which has no limit point in Ω there exists H(Ω) such that { (x ν )} is unbounded. Proo. (a) (b). Let (x ν ) be a sequence without limit point in Ω. Then {x ν } K or any compact set K. By Lemma 3.16 there exists H(Ω) such that { (x ν )} is not bounded. (b) (a). I K is not compact, there exists a sequence {x ν } K, which has no limit point in k and thereore in Ω as K is closed in Ω. Let H(Ω) such that { (x ν )} is unbounded. Then K =. But, it ollows rom the deinition o K that K = K <. I the conditions o Lemma 3.17 are satisied we say that Ω is holomorphically convex. We now extend the notion o a polydisc to arbitrary domains over (i.e. not necessarily contained in) C n. Such a notion would enable us to deine, or any point a Ω, a notion o distance rom the boundary even when Ω is not a domain contained in C n. Readers will note that the extended notion o a polydisc will result in the polydiscs D n (a, r) when Ω C n, and that the notion o distance rom the boundary will coincide with the distance rom Ω with respect to the -norm when Ω C n. To distinguish these new objects rom the classical polydiscs, we shall use the notation P(a, r). To be precise, we make a deinition. Deinition Let p 0 : Ω C n be a connected domain over C n, a Ω. A polydisc o radius r about a is a connected open set U containing a such that p 0 U is an analytic isomorphism onto the set {z C n : z j b j < r}, where p 0 (a) = (b 1, b 2,..., b n ). We denote the set U by P(a, r). The maximal polydisc around P(a, r 0 ) is the union o all polydiscs about a. Lemma P(a, r 0 ) is a polydisc about a o radius r 0 = sup r where the supremum is over all polydiscs P(a, r) about a. Proo. It suices to show that the map p 0 : P(a, r 0 ) D n (b, r 0 ) is bijective. By deinition p 0 (P(a, r 0 )) D n (b, r 0 ). p 0 injective: I x, x P(a, r 0 ), there is a polydisc P(a, r) containing both x, x, so that p 0 (x) p 0 (x ). p 0 is surjective: I z D n (b, r 0 ), then max j z j b j < r 0, hence there is a polydisc P(a, r), such that z j b j < r r 0, so that there is a point x P(a, r) with p 0 (x) = z. 27

28 Deinition The radius o the maximal polydisc about a is called the distance o a rom the boundary o Ω and is denoted by d(a). Lemma I there is a point a Ω with d(a) =, then p 0 is an isomorphism o Ω onto C n. Proo. Let S = {x Ω : d(x) = }. By assumption S. By deinition S is open as d(a) = implies that there is an open set U containing a such that p o U is an isomorphism onto C n. Let x n S, x n x. Let P(x, r) be a polydisc about x. Now we consider x n0 such that x n0 P(x, r). Now as x n0 S there exists U containing x n0 such that p o U is isomorphism onto C n. Thereore P(x, r) U, and hence x U. Thereore x S, and hence S is closed. As Ω is a domain we have S = Ω. Thereore p 0 is a covering. As C n is simply connected we have that p 0 : Ω C n is an isomorphism. Deinition I A Ω, we set d(a) := in a A d(a). Remark Since d is continuous (which ollows rom deinition), i K is a compact subset o Ω then d(k) > 0. The ollowing is a very important result that we shall make use o in this and the next chapter. Proposition Let p o : Ω C n be a domain. Let K be a compact subset o Ω and x 0 K. Let a = p 0 (x 0 ), and let V be a polydisc about x 0, and D = p 0 (V). Then, or any H(Ω), i g := ( ) p O 1 V, the series Dα g(a) (z a) α α! α N n converges in the polydisc D n (a, d(k)). Proo. Note that clearly g H(D). Let 0 < r < d(k). For any x K, we consider the compact set, K = x K P(x, r). Let M = K. By Cauchy s inequality applied to P(x,r), we have D α (x) M.α!.r α, x K, so that D α K M.α!.r α. Hence, by deinition o K, we have D α k M.α!.r α. Since x 0 K, this implies that D α g(a) M.α!.r α. Thereore the series α N n Dα g(a) (z a) α α! converges on D n (a, r). Since r < d(k) is arbitrary, the result ollows. Beore we prove the main result o this section, we state a simple Lemma Let p 0 : Ω C n be a domain and let S H(Ω) has the property that or each S and α N n, D α S. Let (X, p, ϕ) be the S -envelope o holomorphy o p 0 : Ω C n. Then, ϕ is injective i and only i S separates points o p 1 0 {p 0(a)} or every a Ω. In particular, i p 0 : Ω C n is a domain o holomorphy, then H(Ω) separates points o Ω. The above is an elementary consequence o identiying X with a special connected component o O(S ) as done in Thullen s construction. 28

29 Theorem 3.26 (H. Cartan-P. Thullen). Suppose that p 0 : Ω C n is a domain o holomorphy. Then, or any compact set K Ω, we have d(k) = d( K). Proo. Clearly d(k) d( K) as K K. Suppose that strict inequality holds. Then, there is x 0 K such that d(x 0 ) = r 0 < ρ = d(k). Let a = p 0 (x 0 ) and let D 0 = {z C n : z a < d(k) = ρ}. Let Ω 0 be the connected component o p 1 0 (D 0) containing x 0. We assert that p 0 Ω0 is injective. In act, it ollows rom Proposition 3.24 that or any H(Ω), there is g H(D 0 ) such that Ω0 = g p 0. On the other hand, since p 0 : Ω C n is a domain o holomorphy, by Lemma 3.25, since g p 0 takes the same value at any two point x, y Ω 0 with p 0 (x) = p 0 (y), it ollows that p 0 Ω0 is injective. Let r 0 < r < ρ and Ω 1 = {x Ω 0 : p 0 (x) a < r}, and let Y be the disjoint union o Ω and D, where D = {z C n : z a < r}. We deine an equivalence relation on Y by the requirement that z D is equivalent to at most one point x Ω, and that, i and only i x Ω 1 and p 0 (x) = z. Let X be the quotient o Y by this equivalence relation. Then X is Hausdor, and the map o Y into C n which is p 0 on Ω and the inclusion o D in C n induces a local homeomorphism p : X C n. Moreover, the inclusion o Ω in Y induces a map ϕ : Ω X such that p ϕ = p 0. We claim that or any H(Ω), there is F H(X) such F ϕ =. In act, since there is g H(D 0 ) such that Ω = g p o by Proposition 3.24, we deine a unction G on Y by G Ω =, G D = g D. This induces F H(X) and we clearly have F ϕ =. Hence (x, p, ϕ) is an H(Ω)-extension o p 0 : Ω C n. Since p 0 : Ω C n is, by assumption, a domain o holomorphy, it ollows that ϕ is an analytic isomorphism. In particular, since X contains a polydisc o radius r about x 0, Ω contains a polydisc o radius r about x 0, contradicting our assumption that r > r 0 = d(x 0 ). The same reasoning can be used to prove the ollowing: Theorem 3.27 (Cartan-Thullen). Let p 0 : Ω C n be a domain. Let S H(Ω) be a subalgebra o H(Ω) containing the unctions p 1,..., p n, (p 0 = (p 1,..., p n )) and closed under dierentiation (i.e., S implies D α S or all α N n ). Then, i the natural map o Ω into its S -envelope o holomorphy is an isomorphism, we have d(k) = d( K S ) or any compact K Ω. Corollary I Ω is an open set in C n which is a domain o holomorphy and p 0 is the inclusion o Ω in C n, then or any compact set K Ω, K is also compact. Proo. K is clearly closed in Ω. Moreover, since d(k) = d( K), it ollows that K is closed in C n (since the closure o K in C n cannot meet Ω). Moreover, K is contained in the polydisc {z C n : z ρ} where ρ = max z j K, and so is bounded. Hence K is compact. j We are now quite close to characterizing domains o holomorphy when the domains lie in C n. Proposition Suppose Ω C n is a connected open subset and has the property that or any compact set K Ω, K is also compact. Then Ω is a domain o holomorphy. 29

30 Proo. Since Ω C n it is possible to construct a countable subset Z = {x ν : ν Z + } o Ω that contains no limit points in Ω and such that Ω Z. Now by Lemma 3.17 there exists F H(Ω) such that F is unbounded on Z. Now, or each z Ω, there is a subsequence {x νk } k N such that x νk z. Thus, as z Ω was arbitrary, we conclude that or any domain U such that U Ω, F is unbounded on U Ω. Now I U 2 is a domain in C n such that U 2 Ω and U 2 Ω then U 2 Ω. Thereore by the last two assertions and by Deinition 3.14, Ω is a domain o holomorphy. Summarizing rom Corollary 3.28 and the above proposition: Theorem Let Ω C n be a connected open subset. Then, Ω is a domain o holomorphy i and only i it is holomorphically convex. 5This is because o (ii) above. This allows us to use Proposition 30

31 Chapter 4 More characterizations o domains o holomorphy in C n Chapter 3 ended with a characterization o a proper subdomain o C n to be a domain o holomorphy. Unortunately, when n 2, the condition o holomorphic convexity is very hard to check. It would thus be desirable to have other (equivalent) characterizations that are more amenable to computation. It is with this aim that we present a long theorem in Section 4.1 that presents several equivalent characterizations or a domain Ω C n to be a domain o holomorphy. The proo o this theorem, i.e. Theorem 4.8, is not easy. The proo o one o the implications requires the technical and very sophisticated result o Hörmander that the -problem has a solution, with not necessarily compactly-supported data, on a pseudoconvex domain. This will be clariied in greater detail just ater the statement o Theorem 4.8. Hörmander s work is a bit beyond the scope o this survey. Thereore, we shall present proos o only some o the parts o Theorem Further characterizations o domains o holomorphy We begin this section with an application o Theorem Theorem 4.1. Let Ω be a convex domain in C n. Then, Ω is a domain o holomorphy. Proo. Without loss o generality, assume 0 Ω. Deine: K ν := (1 1 ν ) Ω Bn (0, ν) or each ν ( 2) Z +. Then K ν s are convex sets. Now pick a compact subset L Ω. Then there exists ν(l) Z + such that L K ν or all ν ν(l). Let z 0 K ν(l). Then, there exists a real linear unctional on R 2n, Λ, such that Λ(z 0 ) = 1, sup z Kν(L) Λ(z) = α < 1. (4.1) 31

32 Now write z j = x j + iy j or j = 1,..., n. We can write Λ(z) = n j=1 a j x j + b j y j or some a j, b j R. Now let n λ(z) := (a j ib j )z j. Note that Re λ(z) = Λ(z). Now deine F(z) := e λ(z), z C n. By (4.1) we have j=1 F(z 0 ) = e > e α = sup Kν(L) F sup L F. Hence i z 0 K ν(l), then z 0 L Ω. Thereore Ω K ν(l) L Ω. This implies L Ω is compact. Hence, by Theorem 3.30, Ω is a domain o holomorphy as L is an arbitrary compact subset o Ω. Deinition 4.2. A unction µ : C n R is called a distance unctional i: i) µ 0 and µ(z) = 0 i and only i z = 0; ii) µ(tz) = t µ(z) or all t C and or all z C n ; and ii) µ is continuous. Given a distance unctional µ, we can talk about the distance rom Ω o a point z Ω measured in terms o µ. This motivates the next deinition. Deinition 4.3. Let Ω be a domain in C n, and let µ : C n R be a distance unctional. For each z Ω we deine µ Ω (z) := in{µ(z w) : w Ω C }. (4.2) I X is a subset o Ω, then we write µ Ω (X) := in{µ Ω (x) : x X}. In the above deinition, we do not distinguish between proper subdomains and the case when Ω = C n. From the right-hand side o (4.2), we see that or some z Ω, µ Ω (z) = Ω = C n. This is reminiscent o Lemma In act, i the Ω o Lemma 3.21 were a domain in C n (with p 0 just being the inclusion map), then the entire discussion in Chapter 3 could be viewed as one where the l -norm on C n is the distance unctional o choice. In a sense, parts o the theorem below arise rom the lexibility we get by considering various distance unctionals and not just the l -norm. Perhaps the most important concept in studying domains o holomorphy is that o plurisubharmonic unctions. We recall that a C 2 -smooth unction h in an open set Ω( C) is called harmonic i h = 4 2 h/ z z = 0 in Ω. Deinition 4.4. A unction u deined in an open set Ω C and with values in [, + ) is called subharmonic i i) u is upper semicontinuous, that is, {z : z Ω, u(z) < s} is open or every real number s; 32

33 ii) For every compact set K Ω and every continuous unction h on K which is harmonic in the interior o K and is u on the boundary o K, we have u h in K. Deinition 4.5. A unction u deined in an open set Ω C n and with values in [, + ) is called plurisubharmonic i i) u is upper semicontinuous; ii) For every arbitrary z and w C n, the unction τ u(z + τw) is subharmonic in its domain o deinition. We need a couple o inal deinitions beore we can state Theorem 4.8. The deinitions give two notions o convexity or domains in C n. One o the outcomes o Theorem 4.8 is that the two notions coincide or domains with C 2 -smooth boundaries. Deinition 4.6. Let Ω be a domain in C n. We say that Ω is Hartogs pseudoconvex i there exists a distance unctional µ : C n [0, ) such that the unction logµ Ω is plurisubharmonic on Ω. Deinition 4.7. Let Ω C n be a domain having C 2 -smooth boundary and let ρ be a deining unction or Ω, i.e. ρ : V R is a C 2 -smooth unction, where V is an open neighbourhood o Ω, such that Ω = {z V : ρ(z) < 0}; and ρ(z) 0 z Ω. We say that Ω is Levi pseudoconvex i n j,k=1 2 ρ z j z k (z)v j v k 0 or all V = (v 1,..., v n ) T z ( Ω) it z ( Ω), and or all z Ω. (4.3) Note that, in (4.3), we are viewing the tangent spaces to Ω extrinsically as real subspaces in C n R 2n. Thus, T z ( Ω) it z ( Ω) denotes the largest R-linear subspace o T z ( Ω) that is C- linear in the ambient C n. We shall abbreviate T z ( Ω) it z ( Ω) := H z ( Ω). We now have all the deinitions needed to state the main theorem o this chapter. Theorem 4.8. Let Ω be a domain in C n. Then, the ollowing statements are equivalent. 1) Ω is a domain o holomorphy; 2) For any distance unctional µ : C n [0, ), the unction logµ Ω is plurisubharmonic on Ω; 3) Ω is Hartogs pseudoconvex; 4) Ω admits a continuous exhaustion unction u, i.e. given any c R, {z Ω : u(z) < c} is a relatively compact subset o Ω; 33

34 5) Ω is holomorphically convex. Furthermore i Ω has C 2 -smooth boundary, then the ollowing is equivalent to the ive statements above: (6) Ω is Levi pseudoconvex. A sketch o the proo. The scheme o the proo o the equivalance o the irst ive statements above is summarized by the ollowing diagram o implications: (1) (4) (5) (2) (3) The proos o some o the above implications are very technical. O these, the proo that (3) (5) actually relies on the very sophisticated result o Hörmander that i Ω is Hartogs pseudoconvex, then or every smooth (0, q)-orm F on Ω (F not necessarily compactly supported), q = 1,..., n 1, such that F = 0, there exists a u C (0,q 1)(Ω) such that u = F. The details are given in [7, Theorem ]; to the best o my knowledge, there is no other approach to proving (3) (5). As stated above, the solvability o the -problem in this generality is a deep result, which is beyond the scope o this survey. Hence, we shall present proos o only a ew selected implications. We shall discuss some aspects o the equivalence o Levi pseudoconvexity, in case Ω has C 2 - smooth boundary, with the statements (1) (5) in greater detail in Section 4.2. There are several dierent proos o this last equivalence. We will present a proo that is in keeping with the theme i.e. analytic continuation o this survey. (2) (3): Given (2), (3) ollows rom the deinition o Hartogs pseudoconvexity. (3) (4): By (3) we know that there exists a distance unctional µ such that logµ Ω (z) is plurisubharmonic on Ω. Now i Ω is unbounded then it is possible that logµ Ω (z) is not a exhaustion unction (although, by hypothesis, it is plurisubharmonic). Note that i {z ν } ν Z+ is a sequence that approaches Ω, then logµ Ω (z ν ) as ν, but, i Ω is unbounded, then the sets {z Ω : logµ Ω (z) < c} could be unbounded. Thereore i we set u(z) := z 2 log µ Ω (z) where µ is given by (3) we get the desired implication (it is easy to check that z 2 is plurisubharmonic). (5) (1): We already have proved this in Theorem