Vector-Valued Holomorphic and Harmonic Functions

Concr. Oper. 2016; 3: 68–76

Concrete Operators Open Access

Research Article

Wolfgang Arendt* Vector-valued holomorphic and harmonic functions

DOI 10.1515/conop-2016-0007 Received November 25, 2015; accepted March 8, 2016.

Abstract: Holomorphic and harmonic functions with values in a Banach space are investigated. Following an approach given in a joint article with Nikolski [4] it is shown that for bounded functions with values in a Banach space it sufﬁces that the composition with functionals in a separating subspace of the dual space be holomorphic to deduce holomorphy. Another result is Vitali’s convergence theorem for holomorphic functions. The main novelty in the article is to prove analogous results for harmonic functions with values in a Banach space.

Keywords: Holomorphic functions, Banach space, Harmonic functions, Dirichlet problem, Vitali’s Theorem

MSC: 46E40, 46T25, 31C99, 31B20

1 Introduction

Holomorphic functions with values in a Banach space E play an important role in operator theory but also for applications (cf. [2, 11]). Here we present a concise presentation of two results which seem of particular interest. The ﬁrst is the fact that bounded very weakly holomorphic functions are already holomorphic; i.e. if f E is W ! a bounded function such that ' f is holomorphic for all ' in a separating subset of the dual space of E, then f ı is holomorphic. Thus, it sufﬁces to test holomorphy on a minimal set of functionals. The second is Vitali’s Theorem on the propagation of convergence for a sequence of holomorphic functions. Even if one is merely interested in holomorphic functions with values in C, it is worth it to associate to the sequence a holomorphic function with values in `1. In fact, then Vitali’s Theorem is nothing else than the uniqueness theorem for holomorphic functions. These results of Section 2 and 3 are actually taken from the article [4] (see also [5]) in collaboration with Nikolski. But we organize things in a different way. This allows us to prove analogous results for harmonic functions in Section 5. Harmonic functions with values in a Banach space are most natural and beautiful objects. They share many properties with holomorphic functions and in particular the two results mentioned above are valid. Vitali’s Theorem and a version of Montel’s compactness theorem are of importance also in the vector-valued case. For real-valued functions Harnack’s inequality can be used to prove a convergence theorem. But in the vector-valued case there is no ordering and Harnack’s inequality does not make sense. Thus the `1-technique we use here is very different from the usual arguments. Harmonic functions with values in locally convex spaces (and in particular their extension properties) are investigated systematically by Bonet, Frerick and Jordá [7]. Banach-space-valued solutions of elliptic equations (such as Laplace’s equation which we consider here) play a role for applications, see Amann [1]. So it might be welcome that we show in Section 6 that the Dirichlet problem is well posed for vector-valued functions if and only if it is so for real-valued ones (even though this is a simple consequence of the maximum principle). Talking about pde we cannot resist showing in Section 4 how to pass from the real Lumer-Phillips theorem to a complex version. This is indeed a typical situation covered by Vitali’s Theorem.

*Corresponding Author: Wolfgang Arendt: Institute of Applied Analysis, University of Ulm, 89069 Ulm, Germany, E-mail: [email protected]

Giuseppe Vitali (1875–1932) was professor at the University of Bologna. It was a special honour for the author to talk on Vitali’s surprising theorem in the framework of the most interesting and charming Twelfth Advanced Course in Operator Theory and Complex Analysis 2015 in Bologna.

2 Very weakly holomorphic functions

Let C be an open set and E a complex Banach space. A function f E is called holomorphic if f .z h/ f .z/ W ! f 0.z/ lim C h exists for all z . As in the scalar case holomorphic functions are the same as D h 0 2 analytic functions.! It is well known that a function f E is holomorphic if and only if it is weakly holomorphic; W ! i.e. ' f is holomorphic for all ' E . This condition can be relaxed considerably testing only with a minimal ı 2 0 subset of functionals. A subset W of E is called separating if for all v E; '; v 0 for all ' W implies 0 2 h i D 2 v 0. We say that a function f E is very weakly holomorphic if there exists a separating subset W of E D W ! 0 such that ' f is holomorphic for all ' W . Note that each holomorphic function is locally bounded (i.e. bounded ı 2 on each compact subset of ). For locally bounded functions we now have the following.

Theorem 2.1. A locally bounded very weakly holomorphic function is holomorphic.

This result is somehow surprising because a priori it is not even clear why the function should be measurable, or separably valued. So at first sight there seems no clue how to get Cauchy’s integral formula into the game. Theorem 2.1 answers a problem by Wrobel published in 1982 [16]. It was Grosse-Erdmann who gave a quite complicated proof in his Habilitationsschrift [9] 1992 which in a modified way was published in 2004 [10]. Meanwhile a different, very short proof had been given in [4]. It is this proof which we reproduce here with a different way of organizing things - a way which makes work the arguments also for harmonic functions as we will see in the second part of the paper. The clue is of course Cauchy’s integral formula, or the Poisson formula in the case of harmonic functions. Let B B.z0; r0/ z C z z0 < r0 be a disc. If g @B E is Bochner D WD f 2 W j j g W ! integrable, then 1 Z g.t/ u.z/ dt (1) WD 2i t z t z0 r0 j jD defines an analytic (hence holomorphic) function u B E. This shows us already that weakly holomorphic W ! functions are holomorphic. In fact, weakly continuous function are always Bochner integrable. More precisely, we have the following result.

Lemma 2.2. Let be a compact metric space, a ﬁnite Borel measure on and g E be weakly continuous. W ! Then g is Bochner integrable.

Proof. Since is compact and metric there exists a countable dense subset D of . Thus E0 span g.D/ is WD separabel. By the Hahn-Banach Theorem the norm closure and weak closure of subspaces of E coincide. Thus g./ E0. We have shown that g is separably valued. Since g is weakly measurable the claim follows from Pettis’ Theorem [2, Theorem 1.1.1], [8, Chapter II].

As consequence we obtain that weakly holomorphic functions are holomorphic.

Lemma 2.3. Let f E be weakly holomorphic. Then f is holomorphic. W !

Proof. Choose a disc B such that B . Deﬁne u by (1) where g f @B . Then Cauchy’s integral formula for N D j ' f shows that ı '.u.z// '.f .z// for all z B and all ' E : D 2 2 0 Thus f u and so f is holomorphic. D 70 W. Arendt

The pointwise limit of a sequence of holomorpic functions is not holomorphic, in general. But it is if the sequence is bounded. We need this result even for nets instead of sequences.

Proposition 2.4. Let .fi /i I be a net of holomorphic functions from to E such that 2 (a) fi .z/ c for all z ; i I ; k k Ä 2 2 (b) f .z/ lim fi .z/ exists for all z . WD I 2

Then f is holomorphic and the convergence is uniform on compact subsets of .

Proof. Cauchy’s integral formula shows that the set fi i I is equicontinuous. This implies that fi converges f W 2 g uniformly on compact subsets to f . Thus f is continuous and Cauchy’s integral formula on discs is valid for f .

A subset W of E0 is separating if and only if it is .E0;E/-dense. If this is the case, then we may approximate each ' E by a net in W . But in general there is no bounded net that does this. We make this more precise in the 2 0 following remark.

Remark 2.5. Assume that W E0 has the property that for each ' E0 there exists a bounded net .'i /i I in W 2 2 such that 'i .v/ '.v/ for all v E. Then W determines boundedness. This is the following property: if A E ! 2 such that sup '.a/ < for all ' W , then A is bounded. This follows from [4, Theorem 1.5] and Proposition 2.4. a Aj j 1 2 Note that determining2 boundedness is a property which is much stronger then separating.

The Krein-Šmulyan Theorem gives more reﬁned information on the weak*-topology which will allow us to use bounded nets in spite of the difﬁculty mentioned above. Denote by B the closed dual unit ball. E0

Theorem 2.6 (Krein-Šmulyan). Let G be a subspace of E such that G BE is .E ;E/-closed. Then G is 0 \ 0 0 .E0;E/-closed.

Proof of Theorem 2.1. Let f E be a bounded, very weakly holomorphic function. Then the space G ' W ! WD f 2 E ' f is holomorphic is .E ;E/-dense in E . We want to show that G E and then Lemma 2.3 shows the 0W ı g 0 0 D 0 result. By the Krein-Šmulyan Theorem it sufﬁces to show that G BE is .E0;E/-closed. Let .'i /i I be a net in \ 0 2 G BE such that \ 0 '.v/ lim 'i .v/ for all v E: D I 2

Then '.f .z// lim 'i .f .z// for all z . It follows from Proposition 2.4 that ' f is holomorphic, i.e. D I 2 ı ' G. 2

3 Vitali’s Theorem

Vitali’s Theorem showing that pointwise convergence propagates from small sets to the entire domain surprised mathematicians. G. Pólya and G. Szegö compare it with the spread of infectious diseases [14, p. 134] and Stieltjes who proved the result in 1894 under stronger hypotheses writes in a letter to Hermite in 1894 concerning his result: ”J’ai dû l’examiner avec d’autant plus de soin qu’a priori il me semblait que le théorème énoncé ne pouvait pas exister et devait être faux.” Remmert writes in his book from 1992 [14, p. 134] ”Der Satz von Vitali wird besonders gut verstanden, wenn man ihn in Analogie zum Identitätssatz sieht.” Thus Remmert emphasizes the analogy of Vitali’s Convergence Theorem and the Identity Theorem. Our intention is to show that Vitali’s Theorem is really just the Identity Theorem if we apply it to a sequence space. And indeed we show that convergence propagates from any set of uniqueness to the entire domain. Let C be an open connected set. A subset N is called a set of uniqueness if

f N 0 implies f 0 j D D Vector-valued holomorphic and harmonic functions 71 for all holomorphic f C. For example if N is inﬁnite and contained in a compact subset of , then it is W ! a set of uniqueness. For D z C z < 1 a set N an n N D is of uniqueness if an am for D WD f 2 W j j g D f W 2 g ¤ n m and ¤ X1 .1 an / : j j D 1 n 1 D Let E be a Banach space and let

`1.E/ x .xn/n N E sup xn < ; WD f D 2 W n Nk k 1g 2 which is a Banach space for the norm x sup xn . The space c.E/ of all convergent sequences is a closed k k1 WD n Nk k 2 subspace of `1.E/. We denote by q ` .E/ ` .E/=c.E/ W 1 ! 1 the quotient map.

Theorem 3.1 (Vitali 1903). Let C be open and connected and let N be a set of uniqueness. Let fn E be holomorph such that W !

fn.z/ c for all n N; z : k k Ä 2 2

Assume that lim fn.z/ exists for all z N . Then fn.z/ converges uniformly on compact subsets of to n 2 a holomorphic!1 function f E: W !

Proof. Consider the mapping F `1.E/ given by F .z/ .fn.z//n N. It follows from Theorem 2.1 that F W ! D 2 is holomorphic. Thus also q F ` .E/=c.E/ ı W ! 1 is holomorphic. Since q F N 0 and N is a set of uniqueness it follows that q F .z/ 0 for all z . Thus ı j D ı D 2 F .z/ c.E/ for all z . This proves pointwise convergence; uniform convergence on compact subsets then 2 2 follows by Proposition 2.4. Vitali also proved a compactness result, a subject later elaborated further by Montel (see [15, p. 71] for Vitali’s contributions). Also this becomes very transparent by passing via `1.E/ as before. We formulate a vector-valued version.

Theorem 3.2 (cf. [4, Corollary 2.3])). Let C be open and connected and let N be a set of uniqueness. Let fn E be holomorphic such that W ! (a) fn.z/ c for all n N; z , k k Ä 2 2 (b) fn.z/ K for all n N; z N , 2 2 2 where K is a compact subset of E. Then there exist a subsequence and a holomorphic function f E such that W ! fn .z/ f .z/ uniformly on compact subsets of . k !

Proof. Choose a countable dense subset D of N . By Cantor’s diagonal argument there is a subsequence .fnk / of .fn/ such that fn .z/ converges for all z N . Since the sequence is equicontinuous it converges also for all z D. k 2 2 Consider the map F `1.E/ given by F .z/ .fnk .z//k N. Then F is holomorphic by Theorem 2.1. Since W ! D 2 q.F .z// 0 for all z N it follows that q F 0. Thus F .z/ c.E/ for all z . Uniform convergence on D 2 ı Á 2 2 compact subsets follows as before. 72 W. Arendt

4 The complex Lumer-Phillips Theorem

Vitali’s Theorem has many applications to evolution equations (cf. [2, p. 33, 41, 58, 87, 177, 300, 321]). We explain here how the basic generation result by Hille-Yosida can be extended to a generation result for holomorphic semigroups. Let A be an operator on a complex Hilbert space H; i.e. A D.A/ H is linear, where D.A/ is the domain of W ! A which is by assumption a subspace of H.

Theorem 4.1 (Lumer-Phillips). The following are equivalent:

(i) A generates a C0-semigroup .T .t//t 0 of contractions. (ii) a) Re .Ax x/ 0 for all x D.A/ (accretivity); j 2 b) for all y H there exists x D.A/ such that x Ax y (range-condition). 2 2 C D In that case I tA is invertible for all t > 0; .I tA/ 1 1 and C k C k Ä t n T .t/x lim .I A/ x (2) D n C n !1 for all x H; t > 0. 2 Actually, Hille’s proof of the Hille-Yosida Theorem from 1948 consists in showing that (2) converges (see [11, IX §1 Sec. 2]). Lumer and Phillips introduced in 1961 the very convenient notion of accretivity even on Banach spaces. Note that condition (ii) implies automatically that the domain D.A/ of A is dense [2, Proposition 3.3.8]. Next we want to formulate a complex version of this theorem. Our proof corresponds to the one in [3], but here we use Euler’s formula (2) instead of the Yosida approximation. For 0 let Ä Ä 2 i˛ †Â re r > 0; ˛ < : (3) WD f W j j g Theorem 4.2 (Complex Lumer-Phillips Theorem). Let 0 < ; . The following are equivalent: Ä 2 0 WD 2 i A generates a C0-semigroup which has a holomorphic extension T †Â L.X/ such that T .z/ 1 for W ! k k Ä all z †Â ; T .z1 z2/ T .z1/T .z2/ for all z1; z2 †Â and lim T .z/x x for all x H. 2 C D 2 z 0 D 2 z !†Â 2 ii (a) .Ax x/ †Â for all x D.A/ and j 2 0 2 (b) y H x D.A/ such that x Ax y. iÂ8 2 9 2 C D iii e˙ A generate a contractive C0-semigroup. In that case z n T .z/x lim .I A/ x for all z †Â ; x H: (4) D n C n 2 2 !1 i˛ Proof. We only show (ii) (i). Condition (a) says that e A is accretive for ˛ . Thus if z †Â and I zA ) 1 j j Ä 2 C is surjective, the operator is invertible and .I zA/ 1. Since †Â is connected (b) implies that I zA is k C k Ä t n C invertible for all z †Â . From Theorem 4.1 we know that T .t/x lim .I A/ x exists for all x H . Thus 2 D n C n 2 !1 Vitali’s Theorem implies (4) and the holomorphy of T. / on †Â . The semigroup property propagates from .0; / 1 to †Â by the Uniqueness Theorem. Also the strong continuity at the origin follows easily from the semigroup property.

It is most convenient to have at hand Vitali’s Theorem for the proof of Theorem 4.2. However, we could also apply Theorem 4.1 to ei˛A with a to deduce convergence of (4) and merely use Proposition 2.4. j j Ä We are grateful to I. Chalendar for the following interesting remark.

Remark 4.3. We may view the semigroup generated by e iÂ A on the rays re iÂ r > 0 . Then the theorem says ˙ f ˙ W g that given these boundary values, there is a bounded holomorphic function having these boundary values as strong limit (cf. also [2, Section 3.9]). This extension has the semigroup property. However, there are always other (non- holomorphic) semigroup extensions. In fact let .T .t//t 0 be an arbitrary C0-semigroup on H with generator A. Deﬁne T .z/ T.Re z/. Then T †=2 L.H / has the semigroup property and is strongly continuous, however Q WD Q W ! not holomorphic. The generator of .T .eiÂ t// is .cos /A (and not eiÂ A/. Q t>Â Vector-valued holomorphic and harmonic functions 73

5 Harmonic functions

Motivated by the easy proofs given in Section 2 and 3 we will now establish similar results for harmonic functions with values in a Banach space. n 1 Let R be open and let E be a real Banach space. By C . E/ we denote the space of all functions I f E such that the partial derivatives W ! f .x hej / f .x/ @j f .x/ lim C WD h 0 h h!R 2 n exist for each x and such that @j f E is continuous for j 1; : : : ; n. Here ej R is the j -th unit 2 W ! D 2 vector. We let 2 1 1 C . E/ f C . E/ @j f C . E/ for j 1; : : : ; n : I WD f 2 I W 2 I D g n 2 P 2 2 For f C . E/ we let f @j f . A function f E is harmonic if f C . E/ and f .x/ 0 2 I D j 1 W ! 2 I D for all x . By Har. E/ weD denote the space of all harmonic functions from into E and write simply 2 I Har./ if E R. D A function f E is called weakly harmonic if ' f is harmonic for all ' E0. Now let B B.x0; r0/ n W ! ı 2 WD WD x R x x0 < r0 be a ball such that B . On @B we consider the surface measure . f 2 W j j g N Lemma 5.1 (Poisson formula). Let u E be a weakly harmonic function. Then W ! Z 2 2 1 r0 x x0 u.x/ j n j u.t/d.t/ (5) D nr0 x t @B j j for all x B. 2

Here n is the surface of the unit sphere. The right-hand side is the Bochner integral which exists by Lemma 2.2. Proof. Applying a functional ' E on both sides of (5) and interchanging the integral and ' we obtain a true 2 0 identity by the classical result for scalar valued harmonic functions. Thus the identity (5) follows from the Hahn- Banach Theorem.

From this we obtain our ﬁrst main result.

Theorem 5.2. Each weakly harmonic function is harmonic.

In fact, expression (5) shows that u is a real-analytic function on with values in E; i.e. for each x0 there exist 2 c˛ E such that 2 X ˛ u.x/ c˛.x x0/ D n ˛ N0 2 where the series converges absolutely in a neighborhood of x0. This is proved as in the scalar case, see e.g. [6, Theorem 1.24]. In particular, u is inﬁnitely partially differentiable and u.x/ 0 for all x . Next we D 2 show that the pointwise limit of harmonic functions is harmonic. For later purposes we formulate the result not only for sequence but more generally for nets.

Proposition 5.3. Let .fi /i I be a net of harmonic functions fi E. Assume that fi .x/ c for all 2 W ! k k Ä x ; i I and that lim fi .x/ exists for all x in a dense subset of . Then there exists a harmonic function 2 2 I f E such that f .x/ lim fi .x/ uniformly on compact subsets of . W ! D I

Proof. From the Poisson formula (5) one sees that each bounded family of harmonic functions is equicontinuous at each x . Thus convergence of .fi .x//i I on a dense subset of implies uniform convergence on each compact 2 2 subset of to a function f E. Given a ball B with B we may pass to the limit in the Poisson formula W ! N for fi . Thus f is continuous and given by the Poisson formula on B. This implies that f is harmonic. 74 W. Arendt

We next show that for a bounded function f E it sufﬁces to test harmonicity for functionals in a separating W ! subset of E0.

n Theorem 5.4. Let R be an open set, E a Banach space and f E a bounded function. Assume that W ! ' f is harmonic for all ' W where W is a separating subset of E . Then f is harmonic. ı 2 0 Proof. By Theorem 5.2 it sufﬁces to show that f is weakly harmonic. Let

W ' E ' f is harmonic : Q WD f 2 0W ı g Since W W , the set W is .E ;E/-dense in E . We show that W is .E ;E/-closed. By the Krein-Šmulyan Q Q 0 0 Q 0 Theorem it sufﬁces to show that W BE is .E0;E/-closed. For that let .'i /i I be a net in W BE such that Q \ 0 2 Q \ 0 ' lim 'i in .E0; .E0; E//. Thus 'i 1 for all i I and 'i .v/ '.v/ for all v E. Then 'i .f .x// D I k k Ä 2 ! 2 ! '.f .x// for all x . It follows from Proposition 5.3 that ' f is harmonic. Thus ' W BE . 2 ı 2 Q \ 0 n Next we prove a Vitali-type theorem. Let R be open and connected. We say that a subset N of is a set of uniqueness for harmonic functions if for each u Har./; u N 0 implies that u 0. By the Hahn-Banach 2 j D D Theorem this remains true also for functions in Har. E/. I Examples 5.5. a) Each ball B in is a set of uniqueness for harmonic functions. In fact, it is a set of uniqueness for all analytic functions. b) Let B be an open, bounded set such that B . Then N @B is a set of uniqueness for harmonic functions. N WD In fact, if f Har./ satisﬁes f .z/ 0 for all z @B, then f .x/ 0 for all x B by the maximum 2 D 2 D 2 principle. It follows from a) that f 0 on . Á c) See [12] for further examples.

n Theorem 5.6. Let R be open and connected and let N be a set of uniqueness for harmonic functions. Let fn E be harmonic such that W !

fn.x/ c for all x : k k Ä 2

Assume that .fn.x//n N converges for all x N . Then .fn.x//n N converges uniformly on compact subsets to 2 2 2 a harmonic function f E. W !

Proof. Let F `1.E/ be given by F .z/ .fn.z//n N. It follows from Theorem 5.4 that F is harmonic. W ! D 2 Consider the quotient map q ` .E/ ` .E/=c.E/ which is linear and continuous. Thus q F is harmonic. By W 1 ! 1 ı assumption q F N 0. Because N is a set of uniqueness it follows that q F 0. Thus F .z/ c.E/ for all ı j D ı Á 2 z . It follows that .fn.x//n N converges for all x E. Uniform convergence on compact subsets of follows 2 2 2 from equicontinuity. Finally we formulate a Montel-type Theorem whose proof is analogous to the one of Theorem 3.2.

Theorem 5.7. Let C be open and connected and N a set of uniqueness for harmonic functions. Let fn E be harmonic such that fn.x/ c for all n N; x . Assume that there exists a compact set W ! k k Ä 2 2 K E such that fn.x/ K for all x N; n N. Then there exists a subsequence fnk converging uniformly on 2 2 2 compact subsets to a harmonic function f E. W ! The pointwise limit of harmonic functions is not harmonic, in general (see [14] for the holomorphic case). However, similar to the well-known Theorem of Osgood (see [14, p. 130]) it is harmonic on a dense open set.

n Theorem 5.8. Let R be open, fm E be harmonic such that f .x/ lim fm.x/ exists for all W ! WD m !1 x . Then there exists an open dense set 0 of such that f is harmonic. 2 j 0 Vector-valued holomorphic and harmonic functions 75

S Proof. Let Ak x fn.x/ k n N . Since Ak it follows from Baire’s Theorem that D f 2 W k k Ä 8 2 g D k N 2 [ Ak 0 WD V k N 2 is dense in . By Proposition 5.3 f is harmonic. j 0

6 The Dirichlet problem

n Finally, we consider the Dirichlet problem. A bounded open set R is called Dirichlet regular, if for all g C.@/ there exists u Har./ C./ such that u @ g. 2 2 \ N j D Examples 6.1. a) If n 2, then is Dirichlet regular whenever is simply connected, [13, Theorem 4.2.1]. D b) If has Lipschitz boundary then is Dirichlet regular. 2 c) D 0 R is not Dirichlet regular. n f g Now we consider the vector-valued Dirichlet problem. Let E be a real Banach space.

Theorem 6.2. Assume that is Dirichlet regular and let g C.@ E/. Then there exists a unique u 2 I 2 Har. E/ C. E/ such that u @ g. I \ N I j D

Proof. 1. Let u Har. E/ C. E/; g u @ . Then by the classical maximum principle for ' E0; ' 1, 2 I \ N I D j 2 k k Ä one has for all x 2 '.u.x// sup '.u.z// g C.@;E/ : j j Ä z @j j Ä k k 2 Thus u C.;E/ g C.@;E/. This implies uniqueness of the solution. k k N Ä k k 2. Let v E; k C.@/; g.z/ k.z/v. Since is Dirichlet regular there exists h Har./ C./ such that 2 2 D 2 \ N h @ k. Let u.x/ h.x/v. Then u Har. E/ C./ E/, and u @ g. j D D 2 I \ N I j D 3. Call a function g C.@; E/ simple, if it is a linear combination of functions of the form 2. Then by 2. for simple 2 functions g C.@ E/ there is a solution u. 2 I 4. Let g C.@ E/. Then there exist simple functions gn such that gn g in C.@ E/. By 3. there exist 2 I ! I un Har. E/ C. E/ such that un gn. By 1. @ 2 I \ N I j D

un um C.;E/ gn gm C.@;E/ : k k N Ä k k

Thus u lim un exists in C.; E/. It follows that u @ g. By Proposition 5.3 the function u is harmonic D n N j D on . !1

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