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A Removability Result for Holomorphic Functions of Several Complex Variables.

Byline: Juhani Riihentaus

Abstract: Suppose that fi is a domain of On , n S 1 , E c f i closed in fi , the Hausdorff measure H2 n -1 (E )= 0 , and f is holomorphic in fi \ E . It is a classical result of Besicovitch that if n =1 and f is bounded, then f has a unique holomorphic extension to fi . Using an important result of Federer, Shiffman extended Besicovitch's result to the general case of arbitrary number of several complex variables, that is, for n S 1 . Now we give a related result, replacing the boundedness condition of f by certain integrability conditions of f and of (EQ.).

Keywords: Holomorphic function, subharmonic function, Hausdorff measure, exceptional sets.

1. INTRODUCTION

1.1. Previous Results

The following result of Besicovitch is well-known:

Theorem 1. ([1], Theorem 1, p. 2) Let D be a domain in O. Let E c D be closed in D and let H1 (E) = 0. If f: D \ E c O is holomorphic and bounded, then f has a unique holomorphic extension to D.

Above and below Ha is the a - dimensional Haus- dorff (outer) measure in k, kS2.

Much later Shiffman gave the following general result:

Theorem 2. ([2], Lemma 3, p. 115) Let fi be a domain in On , n S 1 . Let E c f i be closed in fi and let H2 n-1 (E )= 0 . If f : fi\E c O is holomorphic and bounded, then f has a unique holomorphic extension to fi.

Shiffman's proof was based on Besicovitch's result, Theorem 1 above, on coordinate rotation, on the use of Cauchy integral formula and on the following result of Federer:

Lemma 1. ([3], Theorem 2.10.25, p. 188, and [2], Corollary 4, Lemma 2, p. 114) Suppose that E c k , k S 2 , is such that Hk -1 (E )= 0 . Then for all j , 1 S j S k , and for Hk -1 -almost all Xj c k -1 the set E( X j ) is empty. For slightly more general versions of Shiffman's result with different proofs, see [4], Theorem 3.1, p. 49, Corollary 3.2, p. 52, and [5], Theorem 3.1, p. 333, Corollary 3.3, p. 336.

1.2. Notation

Our notation is more or less standard, see [6-8]. However and for the convenience of the reader, we recall here the following. If x = (x ,..., x ) c n , n > 2 and j j c D , 1 S j S n , then we write x = (x j , X j ) , where X j = ( x1 ,..., x j -1 , x j +1 ,..., xn ). Moreover, if E c R, (Eq) and (Eq) , we write

(EQUATIONS)

If (EQ) and p > 0 , then (EQ) , p > 0 , is the space of functions u in (Eq.) for which (Eq.) is locally integrable on fi . We identify (EQ) , with (EQ). We use the common convention (Eq.).

For the definition and properties of subharmonic functions, see e.g. [9-12], for the definition of holomorphic functions see e.g. [13-15].

2. AN EXTENSION RESULT FOR HOLOMORPHIC FUNCTIONS

2.1. Our result is related to Theorem 2 above, and reads as follows:

Theorem 3. Suppose that (Eq.) is a domain in On , n S 1 . Let (Eq.) be closed in fi and let H2 n-1 (E )= 0. Let (Eq.) be holomorphic and such that the following conditions are satisfied:

(EQUATIONS)

Then f has a holomorphic extension to (EQ)

2.2. The proof will be based, in addition to Federer's cited Lemma 1 above, also on the following recent result:

Lemma 2. ([8], Theorem, p. 568) Suppose that (EQ) is a domain in (EQ) Let E c (EQ) be closed in fi and let Hn-1 (E )less than (Eq.) . Let u : (Eq.) be such that the following conditions are satisfied:

(EQUATIONS)

Proof of Theorem 3. Write f = u + iv . It is sufficient to show that u and v have subharmonic extensions to (EQ). As a matter of fact, then f will be locally bounded in (EQ) , and thus the claim will follow from Theorem 2 or also from the already cited slightly more general results from [4, 5]. To see that u and v have indeed subharmonic extensions to (EQ) , we use our Lemma 2 as follows.

It is sufficient to show that the assumption (iv) of Lemma 2 is satisfied. For that purpose take j , arbitrarily. By Federer's result, Lemma 1 above, we know that for H2n-1 almost all (EQ) the set (EQ) is empty. Thus for H2n-1 almost all (EQ) the functions (EQ) and (EQ) are (EQ) functions. Therefore, the assumption (iv) is satisfied both for u and for v, concluding the proof.

REFERENCES

[1] Besicovitch AS. On sufficient conditions for a function to be analytic, and on behavior of analytic functions in the neighborhood of non-isolated singular point. Proc London Math Soc (2) 1931; 32: 1-9.

[2] Shiffman B. On the removal of singularities of analytic sets.Michigan Math J 1968; 15: 111-120. http://dx.doi.org/10.1307/mmj/1028999912

[3] Federer H. Geometric measure theory. Berlin: Springer 1969.

[4] Riihentaus J. Removable singularities of analytic functions of several complex variables. Math Z 1978; 158: 45-54. http://dx.doi.org/10.1007/BF01214564

[5] Riihentaus J. Removable singularities of analytic and meromorphic functions of several complex variables. Colloquium on Complex Analysis, Joensuu, Finland, August 24-27, 1978 (Complex Analysis, Joensuu 1978). In: Proceedings (eds. Ilpo Laine, Olli Lehto, Tuomas Sorvali), Lecture Notes in Mathematics 747; 1978: 329-342, Springer- Verlag, Berlin 1979.

[6] Riihentaus J. Subharmonic functions, mean value inequality, boundary behavior, nonintegrability and exceptional sets. Workshop on Potential Theory and Free Boundary Flows; August 19-27, 2003: Kiev, Ukraine. In: Transactions of the Institute of Mathematics of the National Academy of Sciences of Ukraine; 2004: Kiev; 1 (no. 3): 169-91.

[7] Riihentaus J. An inequality type condition for quasinearly subharmonic functions and applications. Positivity VII, Leiden, July 22-26, 2013, Zaanen Centennial Conference. In: Trends in Mathematical Series, Birkhauser, to appear.

[8] Riihentaus J. Exceptional sets for subharmonic functions. J. Basic and Applied Sciences 2015; 11: 567-571. http://dx.doi.org/10.6000/1927-5129.2015.11.75

[9] Helms LL. Introduction to potential theory. New York: Wiley-Interscience 1969.

[10] Herve M. Analytic and plurisubharmonic functions in finite and infinite dimensional spaces. Lecture Notes in Mathematics 198. Berlin: Springer 1971.

[11] Lelong P. Plurisubharmonic functions and positive differential forms. New York: Gordon and Breach 1969.

[12] Rado T. Subharmonic functions. Berlin: Springer 1937.

[13] Chirka, EM. Complex Analytic Sets. Dordrecht: Kluwer Academic Publisher 1989.

[14] Jarnicki M., Pflug P. Extension of Holomorphic Functions. Berlin: Walter de Gruyter 2000.

[15] Jarnicki M., Pflug P. Separately Analytic Functions. Zurich: European Mathematical Society 2011.
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Author:Riihentaus, Juhani
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Date:Dec 31, 2016
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