VietnamJournalof Mathematics25.'2(1997)8l-90
Vil.etrmannjourrrna[
of
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q Springer-Verlag 1997
SeparateAnalyticity and RelatedSubiects
NguyenThanhVan
Paul Sabatier Unioersity,3106l Toulouse Cedex, France
March6. 1997
Received
Abshact.After recallingimportant historic steps,we give a surveyof recentresultson
andrelatedsubjects.
separately
analyticfunctions(andmappings)
1. Historical Preliminaries
Let D and G be open setsin C^ and C', respectively.Given a complex function
_f(",.) on D x G, we note by f"the function w +f(2, w) for everyfrxedzeD,
and by f'the functionz-f(2, w) for everyfixedweG.If / is continuousin
D x G and if f" and f* are analytic in G and D, respectively,for every fixed
(2, w) e D x G, then by Fubini's theoremand Cauchy'sintegral formula for polydiscs,one seeseasilythat f is analyticin D x G. Osgood[33,34] remarkedin 1899
that the continuity of / is superfluous;it sufficesthat f is locally bounded in
DxG.
The first important stepis Hartogs' 1906paper [16], wherelocal boundedness
is dropped. The proof is basedon the all important, so-calledHartogs lemma.
Another important resultwas performedby Bernstein.
Theoreml. If f is a complexfunction on l-a, al x l-b, bl suchthat
(i)Vx e l-a, al, f, has an analytic continuqtionto the open ellipseE(b, S) of
focis *b andmeanaxis bS,
(ii)Vy e l-b, bl, fv has an analytic continuationto the open ellipseE(a, R) of
focis -l a and meanalis aR,
(iii) theseseparatedcontinuationsare uniformly bounded,then f has an analytic
continuationto the openset
U n@, Rd; x.E1b,,sr-o;.
0<d<l
This is the first result related to global analyticity of separatelyanalytic functions of real variables. Unfortunately, it was practically unknown for the past fifty
years.Now we know of it thanks to Akhiezer and Ronkin [1]. It has beenrediscoveredby Cameronand Storvick [13] in a weakerforrn.
Nguyen Thanh Van
1961marked the appearanceof Lelong'swork 124]on separatereal analyticity.
Introducing new tools, in particular the Real Hartogs lemma, he proved the analyticity of separatelyreal analytic functionsof the classes12. This enabledhim to
prove the harmonic analog of the Hartogs theorem.At the sametime, motivated
by analyticity of somedistribution kernels,Browder [0] also consideredfunctions
of the sametype and Eavea weaker result which can be deducedfrom the Bernsteintheorem.Naturally, he was unawareof this.
Another generalizationof the Hartogs theorem was performed by Shimoda
[41] (1957)and Teradal45l (1967).ContinuingShimoda'swork [41],Teradahas
considerablyweakenedan assumptionin the Hartogs theorem:Instead of "f" is
analytic in G for everyfixed z e D", it sufficesto assumethis for everyfixedz e E,
where,E is nonpluripolar in eachconnectedcomponentof D. He showedalso that
this assumptionis optimal if E is a .g-set.
We now refer to the crucial works of Siciak and Zahaqfia. Unaware of the
Bernsteintheorem,Siciak[42] gavein 1969a more generalversionof this theorem
without any boundedness
assumption.one year later he put his resultin a general
context [43]; this is the well-known Siciak theorem.ln 1976,Zahajuta [49] generalized the Siciak theorem, resulting in the Siciak-Zahafuta theorem. We need
somepreliminariesfor the statementof this theoremand other results.
For an open set D c C* and E arbitrary subset of C^, we denote by
(D(., E, D) the upperregularizedof
sup{r.r
e PSH(D),u 1 l, u< 0 on E n D}.
E is calledlocally pluriregularat a point z if a(2, E, V) :0 for everyopenneighborhood V of z. This property is equivalentto the local polynomial condition (L)
of Leja. We pose
e o ( . ,E , D ) : l i m a r (- , E , D , ) ,
where (D") is an increasingsequenceof relatively compactopen subsetsof D such
that uD" : D (the secondmemberis independentof the choiceof (D")).
1.1. The Siciak-ZaharjutaTheorem
Let D and G be pseudoconvexdomainsin C' and C', respectively,and E and F
compact subsetsof D and G, respectively,each of them is locally pluriregular at
everyof its points. Let f be a complexfunction on the crossedset
X:(ExG)u(DxF).
If / is separatelyanalytic on X, i.e. f" (resp.f') is analyticin G (resp.D) for
everyz e E (resp.w e F), then f has an analytic continuationto
* : {Q, w) e D x G: a(2, E, n) + a(w, F, G) < 1}.
Remark.The theoremis also true for connectedSteinmanifolds D and G. Siciak
gave the p-separateanalyticityversionwith X: (D1x Ez x ... x E) u ...u
(h x "' x Eo-r x Dr), where$ c D1 c C.
Separate Analyticity
The present paper is divided into three parts:
. AgeneralversionoftheSiciak-Zahaqutatheorem(anditsdirectconsequences).
. Separate harmonicity and separate subharmonicity.
. Miscellaneous.
2. A General Version of the Siciak-Zahz4ata
Theorem
Theorem 2. Let E c. D c. C*, F c. G c C', where E and F are nonpluripolar, and
D and G are open. Let f be a separately analytic function on X : (E x G) u
(D x F). Denote by E' the set of locally pluriregular points of E.
0 If G is connected,then there exists a function f analytic in an open neighborhood
Qof Et x Gsuchthatf :f ondLnX whereEtis the setof locallypluriregular
points for E in D.
(ii) If D is pseudoconuex,then there exists a function f analytic in
* : {e, w) e D x G: co(2,E', D) * co(w,F, G) < r}
suchthati :f on X nX.
Indicationsfor the Proof.
(i) Schiffman[38] proved this part by Siciak'sinterpolationmethod.
(ii) We observethat without lossof generality,one can supposethe boundedness
of D and G (that implies @(' , E, D) : 6(' , E, D) and ro(' , F, G) :
eD(., F, G)). We needthe following lemma:(ii) is true when,E is tr-analytic
witha(2, E, D) insteadof o(2, E', D). This resultis provedin [29] (seealso
[30]) by seriesexpansionwith respectto a Bergman'sdoubly orthogonal system. Now we indicatehow to do without the hypothesis".8 is ff-analytic": it
sufficesto use (i) and to observethat E' is a G5-set,so it is ff-analytic and
(Bedfordand Taylor [8]),so @(' , E, D): a(. , E', D).
non-pluripolar
Remark. Sadullaev[36] has sketcheda proof of (ii) using tensor product of two
doubly orthogonal systems,with the assumptionthat D and G are pseudoconvex,
and that E and.F are Borel sets.
Very recently,Alehyane[4] gavea proof of the theoremusing only the method
ot [29].
The theorem is probably true without the assumption "D (or G) is
pseudoconvex".
2.1. Direct Consequences
. Consequence
of Part (i) , 138,Corollary 31.With the notationsand hypothesisof
the theorem,if D\E is of Lebesguemeasure0, then there exists/ analytic in
D x G suchthat/ :;f almost everywhere.
. Consequence
of Part (i),[38, Theoreml]. Let Q be an opensubsetofR-, G a
domain in Cn, and f (x, w) a complexfunction definedon C) x G suchthat /" is
analytic in G for every x e O and /' is (real) analytic in C) for everyz e F,
whereF is a nonpluripolar subsetof G. Then f is analyticin O x G.
84
Nguyen Thanh Van
' consequence
of Part (ii).with the notations and hypothesisof the theorem,if
we supposeD is connectedand aD(., F, G) :0, then thereexists/ analyticin
D x G suchthat/ :f on X.
Remark.The last result is proved in [52] for the caseG: c'. The generalcaseis
given by Zahajuta[49],however,the proof indicatedby Zaha{lfia cannotwork; it
usesthe localpluriregularity of compactsetsE and ,F.
3. SeparateHarmonicity and SeparateSubharmonicity
3.1. Harmonic Analogs of the Siciak and TeradaTheorems
Theorem3.1501Forj:1,2,...,
p,let Ei be acompqctsubsetofRz satisfying
the
local Harmonicpolynomialcondition(H) at eueryof itspoints, and Di be a domain
in R" containing\. If f is a separatelyharmonicfunction on
X : ( D rx E 2 x . . . x E p ) \ r . . u. ( h x . . . x E o _ 1
x Do),
thenf hasa harmoniccontinuationto
(.
* : l ( \ , . . ., xpe) D r x " ' x D o :D r1 w,,Ei,D) < ll.
L
;
:
r
)
i-
)
This result has been proved earlier [26] under strong assumptions.we recall
the following.
Definition of (H). E c R^ sqtisfies(H) at a point xs tf, for eueryneighborhoodv
of x0, eueryfamily g of harmonicpolynomialsof m real uariables,uerifuing
s u p { l / ( x )fl : e g } < c o , V x e V a E ,
and eueryb > l, there existsa neighborhoodw of xs and a positiueconstantM
suchthat
l " f ( " ) l< M . b d ",fY x e W , Y f e 9 .
very recentlythe author hasgiventhe following analogof the Teradatheorem.
Theorem4. [28] Let D be a domainin R^, E a subsetof D satisfying( H) at some
point of D, and G an opensubsetof R". If f(x, y) is a complexfunction in D x G
suchthat f, is harmonicin G for eueryx e E, and ft is harmonicin D for euery
y e G, thenf is harmonicin D x G.
We now give a result similar to Cor. 3 of Shiffman[37] (seeI.2. above).
Proposition5. In theprecedingtheoremif we supposethat ft is harmonicin D for
almost all y e G, and f, is harmonicin G for eueryx e E c D, whereE is nonpluripolar as a subsetof C^ : R- + iR', then there existsi harmonicin D x G
suchthat f :f almosteuerywhere.
85
Separate Analyticity
Proof. Let
i:
U { z e C * :llt - ,ll < 2-r/zdist(x,dD)),
xeD
G : U { w e C "ll,
: - yll < 2-tt2dist(y,dG)).
yeG
LetF e G suchthatmes(G\F):0 andfv is harmonicin D for everyy e F'Let
(Dr) be a sequenceof subdomainsof D such that D' - D"+r and vD': D. For
J ) Js, Es : E n D" is nonpluripolar in C^ . We remark that it sufficesto prove
that the conclusionis true for D" insteadof D (s > ss).Let D" be a pseudoconvex
open connectedneighborhoodof D" lwhich is a connectedpolynomially convex
compactof C^) in D. For y e,F (resp.x e E), /r (resp./") is analyticallycontinuableto D" (resp.G;, ,o w" can definea functionf separatelyanalytic on X" :
(D, x F) ,.,(4 G) and equal to f on (D, x F) u (E x G). By Theorem 2(ii),
"
in
there existsf analytic
{(x, w) e b, x G: aQ, E,, D,) I a(w, F, G) < 1}
and is equal to f on *rnXr. Following [15], F is locally pluriregularat every
point of G, thus, @(', F, G) :0 in G' On the other side,o( ', E,, Dr) < I in
Dr, because.E',is nonpluripolar in the domain D". Thus, D, x G - *, i i1 real;
analytic in D, x G, and f : f : f on D,x F (so /: f a.e. in D" x G). Lf
t,:
is a uniquenesssetfor real-analyticfunctionsin D" x G, we have A/ = 0.
Remark.Following the proof we havef harmonic in D x G if F : G. This result
of the precedingtheorem.In fact,
can be consideredas an immediateconsequence
following [8], E is locally pluriregular at a point xs e E, so E verifies(H) at xo.
The converseis not true.-Onecan find inl44]an exampleof a setE c R2,which is
pluripolar in C2 and verifies(H).
3.2. SeparateSubharmonicity
The following problem: "Let D and G be open setsin R' and R', respectively.Is
every separatelysubharmonicfunction in D x G subharmonic?"has been open
until 1988,when Wiegerinck[47] EaYea very simplecounter-example.
Subharmonicity of separatelysubharmonic functions was first studied by
Lelong l24l and his student Avanissian [7]. They have given a positive answer
under the hypothesisthat f is locally upper bounded.Arsove[6] assumedonly that
f hasa l,lo" majorant. More recentresultsof this kind are due to Riihentauss[53]
and Armitage and Gardiner [5]. Assumptionson the partial functions fx and 7t
are also considered:
(i) f(x, y) is real-analyticand subharmonicin x, and harmonic in y [17] (this
result can be consideredas an immediate consequenceof [38, Theorem l]
citedabovein Sec.2.1).
(ii) /(x, y) is C2 and subharmonicin x and harmonrcin y 1221.
Nguyen Thanh Van
We also cite a result of Cegrelland Sadullaev[14] usedinl22l.
Let 81 and 82 be open balls in R' and R', respectivelyand f a real function
definedin a neighborhoodof Etv6,
subharmonicin x and harmonicin y. Then
thereexist two closedsetswith empty interiors E1 e 81 and E2 c ,B2such that f
is subharmonicin (81 x B2)\@1x E).
We end this paragraphby a result of Wiegerinckand Zeinstra [48].
Everyseparately
(1,p)-subharmonic
functionin R'is subharmonic(0 < p < n).
We recallthat a functionf(x1,..., xn)definedin an opensetC)e R" is called
separately(1,p)-subharmoniciff, for every fixed xf , ..., xl-_-, the function
to O n {*i : rl, j : il,..., in-p}is subha"rmonic.
f(xr..., xn)restricted
4. Miscellaneous
4.1. SingularSetsof SeparatelyAnalytic Functionsof Real Variables
For a function f(x, y) separatelyanalyticin an open set O of Rf x R], we pose
A(f ) : {(*, y) e Q: f is analyticin a neighborhoodof (x, y)},
S(,f) : O\,4(,f).
S(/)is calledthe singularsetof /.
Theorem6 (Saint Raymond-siciak). Let s be a closedsubsetof an openset Q in
R- x R', and 51 and 52 theprojectionof S onR^ andR', respectiuely.
Then51 and
52 arepluripolar in C^ : R- + iR- and C' : R' * iRn,respectiuely,
if and only if
S is thesingularset of a separatelyanalyticfunction f (*, y) on tl (x e R', y e Rr).
This theoremis from siciak [55] who proved a more generalversion (seealso
[51]).It hasbeenfirst provedby SaintRaymond[54]for m: n: l.
4.2. SeparateAnalyticity in Infinite Dimension
The Hartogs theoremwas extendedto separatelyanalytic functions on complete
metrizabletopologicalvector spaces(r'-spaces)by Noverraz [32]. we give here an
extensionof the Terada theorem.
Theorem7. Let D and G be opensubsetsin F-spaceA and B, respectiuely,
D connected.Let E be a subsetof D satisfyingthe local PolynomialConditionof Leja at
somepoint of D. If f: D x G ---+C rs suchthat f, (resp.fv) is analytic in G (resp.
D) for eueryx e E (resp.y e G), thenf is analytic in D x G.
In[27], the author provedthis resultwith an additional assumptionon / which
was droppedrecentlyby Bui and Nguyen [12].
Separate Analyticity
87
4.3. SeparatelyAnalytic Mappings
Let X be a complexanalytic spacehaving the Hartogs ExtensionProperty (HEP).
If / is an analytic mapping from
H(r) : {(r, w) eCz: lzl < r or ltl > 1 - r}, 0 I r I I,
into X, thenf is the restrictionto H(r) of an analytic mappingfrom the bidisc A2
into X.
Analogs of the Terada theorem for analytic mappings into X e (HEP) are
given by Shiffrnan[39]. Recently,Alehyane [2] has extendedTheorem 2 to these
mappings:samestatement,with a complexanalytic spaceX e (HEP) insteadof C.
4.4. SeparatelyMeromorphic Functionsand Mappings
Firstly, the meromorphicanalogof the Hartogstheoremwas givenin the 1950sby
Rothstein[35]and Sakai[37].Inlg76,Kazatian [20]gavethe analogof the Siciak
theorem,ten yearsafterl2ll, the analogof the Siciak-Zaharjttatheorem. Recently,
Alehyane[3] hasgeneralizedTheorem2 to meromorphicmappingsinto a complex
analytic space having the p-Meromorphic Extension Property (p-MEP) with
p : m * n: samestatementwith X e (p-MEP) insteadof C.
We recall the definition: X e (p-MEP) signifiesthat everymeromorphicmapping from
Hp(r): {(z', zp)e Cp-l x C: lz'l< r or Itol> | - r}, 0 < r < I
into X is the restrictionof a meromorphicmapping from the polydisc N into X.
Every compactKiihler manifold verifies(p-MEP) for p 2 2 [18] (seealso [31] for
similar results).In 1994,Shiflrnan [40] gave at Dolbeault's colloquium various
resultson separatelymeromo{phicmappingsinto compactKtihler manifolds.
4.5. Applications
Resultson separatelyanalytic functionshave various applicationsin Partial Differential Equations and Theoretical Physics(Feynman Integrals, "Edge of the
wedge,'theorem). For the 1960s,see the Introduction of [11]. Akhiezer and
Ronkin [1] gave an application to the "Fine End of the Wedge". A more recent
applicationto the study of Anosov and geodesicflows is in [19].
References
variables
analyticfunctionsof several
l. N. I. AkhiezerandL. I. Ronkin,On separately
28 (1973)27-44.
Math.surueys
on thethin endof thewedge,Russian
andtheorems
s6par6ment
deHartogspourlesapplications
du th6ordme
Uneextension
2. O. Alehyane,
C. R Acad.Sci.Paris,Ser.I,323 (1996).
holomorphes,
analytiqtesBull.
danslesspaces
m6romorphes
3. O. Alehyane,Applicationss6par6ment
Soc. Math. France (stbnitted).
88
Nguyen Thanh Van
4. o. Alehyane,IJne remarquesur les fonction s6par6mentholomorphes,preprint, 1997.
5. D. H. Armitage and S. J. Gardiner,Conditionsfor separatelysubharmonicfunctionsto
be subharmonic,PotentialAnalysis2 (1993)255-261.
6. M. G. Arsove, On the subharmonicityof doubly subharmonicfunctions,Proc. Amer.
Math. Soc.l7 (1996)622-626.
7. V. Avanissian, Fonctions phurisousharmoniqueset fionctions doublement sousharmoniques,Ann. Sci.de I'EcoleNorm. Sup.78 (1961)l0l-161.
8. E. Bedford and B. A. Taylor, A new capacityfor phurisubharmonicfirrctions, Acta
Math. 149(1982)1-40.
9. S. N. Bernstein,Sur I'ordre de la meilleureapproximationdesfonctionscontinuespar
des polyndmesde degr6 donn6,Mimoires Acad. Roy. Belgique,Ct. des Sc.4 (19121922),Fasc.1(1912).
10. F. E. Browder, Real analytic functions on product spacesand separateanalyticity,
Canad.J. Math.13 (1961)650-656.
I I . F. E. Browder,on the edgeof the wedgetheorem, canad.J. Math. 15 ( 1963)lz5-131.
12. D. T. Bui and v. H. Nguyen, weakly holomorphic extension,and the condition (L),
Acta MathematicaVietnamica(to appear).
13. R. H. Cameroand D. A. Storvick, Analytic continuationfor functionsof severalvarir
ables,Trans.Amer. Math. Soc. 125 (1967\7-12.
14. u. cegrell and A. sadullaev, Separatelysubharmonicfunctions, uzbek. Math. J. I
(1993)78_83.
15. R. M. Dudley and B. Randel,Implicationsof pointwiseboundson polynomials,Duke
Math. J. 29 (1962)4ss-4s8.
16. F. Hartogs, Zur Theorie der analytischenFunktionen mehrererunabhiingigerVeriinderlichen insbesonderetiber der Darstellung derselbendurch Reihen welche nach
PotenzeneinerVeriinderlichenfortschereiten,Math. Ann.62 (1906)l-88.
17. S. A. Imemkulov,Separatelysubharmonicfunctions,Dokl Uz. SSR.2 (1990)8-10
(Russian).
18. s. M. Ivashkovitch,The Hartogs-typeextensiontheorem for the meromorphicmaps
into Kiihler manifolds,Inu. Math. 109 (1992)47-54.
19. A. Katok, G. Kniepper, M. Pollicott, and H. weiss, Differentiabilityand analyticity of
topologicalentropyfor Anosovand geodesic
flows,Inuent.Math. 98 (1981)581-597.
20. M. Y. Kazaian, On functions of severalcomplex variablesthat are separatelymeromorphic, Math. USSRSbornik2S (1976)481-489.
21. M. Y . Kazaiant, Meromorphiccontinuationwith respectto groupsof vaiables, Math.
USSRSbornik53 (1986)385-398.
22. S. Kolodzeiej and J. Thorbicirson,Separatelyharmonic and subharmonicfunctions,
PotentialAnalysis5 (1996)463-466.
23. P. Lelong, Les fonctionsphurisousharmoniques,
Ann. sci. EcoleNorm. sup. 62 (1945)
301-328.
24. P. Lelong,Fonctionsphurisousharmoniques
et fonctionsanalytiquesde variablesr6elles,
Ann. Inst. Fourier11 (1961)515-562.
25. NguyenThanh Van, Basede Schauderdanscertainsespaces
de fonctionsholomorphes,
Ann. Inst. Fourier22 (1972),169-253.
26. Nguyen Thanh Van, Basescommunespour certainsespaces
de fonctionsharmoniques,
Bull. Sci.Math. 97 (1973)33-49.
27. NguyenThanhVan, Fonctionss6par6ment
analytiqueet prolongementanalytiquefaible
en dimensioninfinte,Ann. Polon. Math. 32 (1976)7l-83.
28. Nguyen Thanh van, Separatelyharmonic functions, workshop on phuripotential
Theoryand Applications,Warsaw, 10-14 February 1997(manuscript).
Separate Analyticity
89
29. Nguyen Thanh Van and A. Zeiahi, Une extensiondu th6ordmede Hartogs sur les
fonctionss6par6mentanalytiques,AnalyseComplexeMultiuariables,RtcentsDiuelopments,A. Meril (ed.),Editel,Rene,1991,pp. 183-194.
30. Nguyen Thanh Van, Systdmesdoublementorthogonauxde fonctionsholomorpheset
applications,Topic in ComplexAnalysis,vol. 31, P. Jacobzakand W. Plesniak(eds.),
BanachCenterPublications,1995,pp. 281-297.
Ann. Inst. Fourier
31. NguyenVan Khue and Le Mau Hai, Meromorphicextensionspaces,
42 (r99D s01-515.
et analytiquesdansles spacesvectoriels
32. Ph. Noverraz,Fonctionsphurisosharmoniques
topologiques,Ann Inst. Fourier 19 (1969)419-494.
33. W. F. Osgood,Note tiber analytischeFunktionenmehererVeriinderlichen,Math. Ann.
s2 (1899)462-464.
34, W. F. Osgood, Zweite Note iiber analytischeFunktionen meherer Veriinderlichen,
Math. Ann.53 (1990)461-464.
35. W. Rothstein,Ein neuerBeweisdes HartogsschenHauptsazesund seineAusdehnung
anf meromorphieFunktionen,Math. Z. 53 (1950)84 95.
36. A. Sadullaev,Phurisubharmonicfunctions, SeueralComplex VariablesII, Geometric
FunctionTheory,Encycl.Math. Sci.8 (1989).
37. E. Sakai, A note on meromorphicfunctions in severalcomplexvariables,Mem. Fac.
Sci.KyushuUniu Ser.A Math.11 (1957)75-80.
38. B. Shiffman, Separateanalyticity and Hartogs theorems,Indiana uniu. Math. J. 38
1989\ 943-9s7.
39. B. Shiffman, Hartogs theoremsfor separatelyholomorphic mappings into complex
spaces,C. R. Acad. Sci.Paris,Ser.I,310 (1990)89-94.
40. B. Shiffman,Separatelymeromorphicmappingsinto compactKdhler manifolds, Contributionsto ComplexAnalysisand Analytic Geometry,H. Skoda and J. M. Tr6preau
Vol. E26,Yieweg,1994(eds.),Aspectsof Mathematics,
41. I. Shimoda,Note on the functions of two complex variables,I. GakugeiTokushima
U n i u . 8( 1 9 5 7l)- 3 .
42. J. Siciak, Analyticity and separateanalyticity of functions defined in lower dimensional subsetsof Cn, Zeszyty Nauk. Uniw. lagiello, Pace Math. Zeszyt 13 (1969)
57-70.
43. J. Siciak, Separatelyanalytic functions and envelopesof holomorphy of some lower
145-l7lof C", Ann. Polon Math. 22 (196911970)
dimensionalsubsets
on Phuripotential
operators,
Workshop
for
elliptic
theorem
Walsh
44. J. Siciak,Bernstein
Warsaw,10-14 February 1997,preprint, 1996(to appear).
and Applicatiorzs,
45. T. Tereda.Sur une certainecondition souslaquelleune fonction de plusieursvariables
complexesest holomorphe,Publ. Res.Inst. Math. Sci.,Kyoto Unio.,Ser. A, 2 (1967)
383-396.
46. T. Tereda, Analyticit6s relativesd chaquevariable, J. Math. Kyoto uniu. 12 (1972)
263-296.
47. J. Wiegerinck, Separatelysubharmonicfunctions need not be subharmonic,Proc.
Amer. Math. Soc. 104 (1988)770-771.
48. J. Wiegerinckand R. Zeinstra,Separatelysubharmonicfunctions,Proc. Symposiain
PureMath., Part l, 52 (1991)245-249.
of Hartogs theorem,and
49. Y . P. Zaharytta,Separatelyanalyticfunctions,generalizations
(1976)
.
30
5l-67
Sbornik
USSR
Math.
of
holomorphy,
envelopes
50.A. Zeiahi, Basescommunesdans certainsespacesde fonctionsharmoniqueset fonctions s6par6mentharmoniquessur certainsensemblesde C", Ann. Fac. Sci. Toulouse
Nouuelle4 0982\ 75-102.
Nguym Thanh Van
51. Z. Blocki, Singularsetsof separatelyanalytic functions,Ann. polon. Mattr Lylo992\
219-225.
52' Nguyen Thanh van and A. zeiahi, Famille de polyn6mes presque partout bom6es,
Bull. Sci.Math. 107(1983)8l-91.
53. J. Riihentaus,on a theoremof Avanissian-Arsove,Expo. Math.7 (lggg) 69-72.
54. J. Saint Raymond, Fonctions s6par6mentanalytiques,Ann. Inst. Fourier 40 6990)
79-101.
55. J. Siciak, Singular sets of separatelyanalytic functions, colloq. Math. d0/61 (1990)
281-290.