Existence domains for holomorphic Lp functions

Publicacions Matemàtiques, Vol 38 (1994), 207-212 . ExISTENCE I3OMAIN S FOR HOLOMORPHIC L p FUNCTION S NICHOLAS J . DARA S A bstract  If SZ is a domain of holomorphy in e n , having a compact topological closure into another domain of holomorphy U C CC n such tha t (9,U) is a Runge pair, we construct a function F holomorphic i n 1 - 1 which is singular at every boundary point of SZ and such tha t F is in L P (S2), for any p E (0, +oa ) . 1 . Statement of the proble m The following notation and terminology will be used without furthe r explanation . The open polydisc in e n with center a and radious r i s denoted by q n (cx ; r) ; if n = 1, then we use the notation q (a ; r) . Fo r every open set D in C C n , O(D) denotes the space of all holomorphic func - tions in D . If K is a compact subset of D, we define the B(D)-hull k D of K by K D := {z E D ; f (z)l ç Sup wEK ~ for all f E 8(D)} . Fo r p E (o, +oa], we set g L p (D) := B(D) n L p (D) . Obviously, O L°° (D) equal s the algebra H°° (D) of bounded holomorphic functions in D . If D carries a function F E OL P (D), which cannot be holomorphically extende d across the boundary of D, then D is said to be an existence domain fo r O L P or of type oL r . Asking for the conditions under which a bounded domain of holomorphy 1 - 2 is of type 9L P , we recali the following result : If SZ cc e n i s a domain of holomorphy with C°° boundary and (cx Y E SZ ; v E N) i s a sequence such that (limv~~ cx v } E aSZ, then there exists a functio n F E BL 2 (S2) satisfying limv_, 00 lF(av ) ~ = + oo ([4]) . The question we ar e interested is the following : Is any bounded domain of holomorphy in c n existence domain for OL P , for every p E (o, +oo ) ? In [1] Catlin showe d that any smoothly bounded domain of holomorphy in e n is of type BL° ° (, and consequently of type OL p , far every p E (o, +oa)) . However in [s] , Sibon y showe d tha t ther e is a bounde d Rung e complete Hartog s domai n of holomorphy SZs c q 2 (o ;1) (9s  q 2 (o ;1 ) ) such that all bounded 248  N . J . DARA S holomorphic functions in 9s extend holomorphically to the open uni t bidisc, thatis 9s is not of type BL°° . The concern of this note is to give an answer to the aboye question . Our approach illustrates a partial extension of Catlin ' s improvement . More precisely, we shall prove that any domain of holomorphy SZ cc e n , having a compact topological closure into another domain of holomorph y Usuch that (9, U) is a Runge pair, is of type OL P for any p E (o, +oo ) . 2 . Unbounded holomorphic functions in Runge domain s Let 9 cc U be domains of holomorphy in e n . Assume that S2 is a bounded Runge domain relative to U . Let (zm ; m E N) be a dense sequence in 9, such that every point of th e sequence is counted infinitely many times . Let r m be the largest numbe r with ,á n (zm ; rm ) C SZ . SZ can be exhausted by compact sets E j , so tha t ^ E 3 c E j + 1 . Letting K1 : = E1, we find a point w 1 E q n (z 1 ; r 1 ) — K 1,U . Obviously, there exists a j 1 > 1, with w i E E h . . Put K 2 := E .j1 . Now , ~ there is a point w 2 E q n (z2 ; r2) — k2 S U . If we set K3 := E~ 2 (, j 2 ~ j 1 ) , then w 2 E Continuing like this, we find an exhaustive sequenc e (Km ; m E N) of compact subsets of SZ and a sequence (w m ; m E N) o f points of  with the following properties : - w mEl~m +1 -I~m,U (, m E N ) , - whenever w E 09 n q n (l ; p) for a polydisc q n W p) and V is a connecte d component o f S ~ n q n ~~ ; p) clusterin g a t w, ther e exist s a subsequence of (wrn ; m E N) converging to w in V . To eac h w m ther e correspond s a holomorphi c functio n f m E O(U) , suc h that f rnl w m l~ > supz E x m ifm( z ) = 1 . If we let o < E m < I fm (wrn,) ~ - - 1 , then ~ l < ~ — em whenever z E Km . Hence, for suitabl y choosen numbers v,-, z > 0, the serie s F(z) = ([fm(z)r-l[f .(w . ) m = converges absolutely and compactly on and 1m, for any m . It follows that whenever w E c79, q n ( ; p) is a polydisc containing w and V is a connected component of 9 n q n ~~ ; p) clustering at w, F i s unbounde d in V . So, F is a functio n holomorphi c ~ n 9, whic h is singula r (unbounded) at every boundary point of 9 ([3]) . EXISTENCE DOMAINS FOR L p FUNCTIONS  209 3 . Runge domains of type 9L p Let the notations and assumptions be as in Section 2 . The principa l purpose of this paragraph is to announce the following : Theorem 1 . Let SZ CC U be domains of holomorphy in en such tha t (~~, U) is a Runge pair . Then, F E eL p (Q), far any p E( D, +oo) . Proa : The evaluation of more useful choice of v m is our first aim . Le t 6 > 2 . For each m E N, choose v m so that 1i — em ~ _ S m . It i s easily seen that the power serie s h(() = o o E [fm(wm) — E mi -vm ( m m= 1 converges into the disc q (0 ; 6) . Define a linear functiona l A h : P(C) -4 C ; x m --> A h (x m ) := {fm( W m) — Em] 'm , where 1P (C) is the vector space of complex polynomials in C . In order t o prove the theorem two lemmas play crucial role : Lemma 1 . ([2]) The functional A h is continuous and there is a continuous extension of A h into 6(0(0 ; 61 )) . Further, for each ( E O(0 ; 6 ) there holds A h ((1 — x() -1 ) = h(() (, x E O(0 ; 61 )) . Proof o# ' Lernrna 1 : Let r C S . If p(x) is a polynomial in x E e , the n by Cauchy's integral formula we hav e lAn .(p)f ~ M ( r ) ' sup I' sup I p ( x )l , ixI ~ T where the constant M(r) depends only on r . Hence, by density, there i s a continuous extension of A h on B ( q (0 ; 61 )) . If now ( E q (0 ; S} and i f ( is fixed, then the number A h ((1 — x()') is well defined ( : Ah acts o n the variable x E A ( 0 ;6 - 1 ) and ~ is regarded as a parameter) . By th e continuity of Ah, we obtain Ah(( 1 - - x() -1 ) = h(() . n The next lemma is a consequence of Lemma 1, but is much more usefu l since the choice of the functionalA h is eliminated . 210  N . J . DARA S Lemma 2 . lf z E 9, then there hold s 1F(z)i f (1) . E 00 .1[f .(z)r— I for any T E (2, S) and where the constant  depends only on T but i s independent of z . Proof of Lemma 2 : Assuming that z E 9, x E q (o ; 6 -1 ) and T E (2, S) , we have by Cauchy ' s integral formula and by Lemma 1 : E A h (x m ) [ f m (z)r m 1 A h (l/(( — x) ) 2~ri  C1= E [f .( z ) r - rrt = 1 < ~ <LO . s u p i A n,( l /(( — x))I  slip L I[frn( z )N} ) ~ l = T  ~~~ =T m= 1 ~ ~ that is IF(z)l  ( : )  *~  r . c0 .1[fm(z)]v-1  ■ End of Proof of Theorem 1 : Let 0 C p < +oo . By Lemma 2 andb y Fatou'sTheorem, it is enough to show tha t for sorne 7 E (2,6) . (d\ ( . ) is the Lebesgue measure in e n ) . Suppose T E(2, S) . For any v E N, choose a positive number 2~~ T 1 (,k z , E N), such tha t L (É I[fm( z )] v— 1) dA(z) ¡2k„—1l ° P m= l This choice permits us to obtain the following inequalitie s (É IdA(z) (2k~~v  ,  llp < 1 `  J m= l sUp  I[fm(Z)mI) P dA(z)j v EIN `< +oo , T m 'm .= 1 ~ ~ m-- 1 { ( l  \ v ~ EXISTENCE DOMAINS FOR L E FUNCTIONS  21 1 for any v E N . Therefore , (É I[fm(z)] v_ 1 ) P for any v E N and consequently , f(> ~ I[f rn (z)J vm j IP dA (z) 1 , 1 for any v E N, which completes the proof . ■ We are now in position to formulate the main result of this note, whic h is an immediate consequence of Theorem 1 : Theorem 2 . Let SZ CC U be domains of holomorphy in en . Assum e that S~ is a bounded Runge domain relative to U . Then, S2 is an existenc e domain for OL E , for any p E (0, +oo) . In particular, any bounded Rung e domain of holomorphy is of type BL E , for any p E (0,+oo) . W e finaily turn to the question whether Sibony's example SZs in [6 ] is an existence domain of L E holomorphic functions . The answer is a direct consequence of Theorem 2 : Since Sibony's example is a bounde d Runge domain of holomorphy, it is an existence domain for BL E , for an y p E (O,+oo) . Referente s 1. CATLIN, D ., Boundary behavior of holomorphic functions on pseudoconvex domains, J . Diff . Geometry 15 (1980), 605-625 . 2. DARAS, N . J ., The convergence of Padé-type approximants to holomorphic functions of severai complex variables, Appl . Num . Math . 6 (1990-91), 341-360 . 3. FORNAESS, J . E . AND STENSONES, B ., " Lectures on counterexampies in several complex variables, " Mathematical Notes, Princeto n University Press, 1987 . 4. PFLUG, P ., Quadratintegrable holomorphe funktionen und di e Serre - Vermutung, Math . Ann . 216 (1976), 285-288 . 5. RANGE, R . M ., "Holomorphic functions and integral representa - tions in several complex variables," Graduate Texts in Mathematic s 108, Springer - Verlag, 1986 . 212  N . J . DARA S 6 . SIBONY, N ., "Prolongement analytique des fonctions holomorphe s bornees, " Séminaire P . Lelong 1972-73, Lecture Notes in Mathematics 410, Springer - Verlag, 1974, pp . 44-66 . Department of Mathematic s University of Athen s Panepistemiopoli s 157 84 Athen s GREEC E Rebut el 18 d'Octubre de 1993