Skiba, Alicja

Acta Mathematica

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Jagiellonian Univeristy Repository

UNIVERSITATIS IAGELLONICAE ACTA MATHEMATICA, FASCICULUS XLI2003BERNSTEIN QUASIANALYTIC FUNCTIONSON ALGEBRAIC SETSby Alicja SkibaAbstract. We extend the notion of Bernstein quasianalytic functions toalgebraic sets in Cn. We prove a uniqueness principle for such functions.1. Introduction. Let E ⊂ R be a compact interval. It is the well-knownresult of Bernstein that a function f ∈ C(E) can be extended to a holomorphicfunction in a neighbourhood U ⊂ C of the set E if and only if(1.1) lim supk→∞k√distE(f,Pk) < 1,where Pk = Pk(C) denotes the space of all polynomials of one complex variableof degree at most k and distE(f,Pk) = inf{‖f − p‖E ; p ∈ Pk}. If a functionf satisfies (1.1) then obviously the following identity principle holds:(IP) f = 0 on a subinterval of E implies that f vanishes on E.As was observed by Bernstein, to establish the above identity principle itis enough to assume that the function f satisfies the weaker condition(1.2) lim infk→∞k√distE(f,Pk) < 1.If a function f has property (1.2) it is called quasianalytic in the senseof Bernstein. Condition (1.2) can be reformulated as follows. There exist aconstant % ∈ (0, 1) and a strictly increasing sequence of positive integers {nj}such thatlim supnj→∞nj√distE(f,Pnj ) = %.Szmuszkowiczo´wna [17] and, independently, Lelong [5] extended (IP) byproving that a quasianalytic function f 6= 0 defined on E can vanish only on a216subset of E with a transfinite diameter equal to 0 (or, equivalently, on a polarsubset of E).An important property of quasianalytic functions was found by Mazurkie-wicz [8]: the set B(E) of all quasianalytic functions defined on E is residualin the space C(E). Let us recall that a subset A of a topological space X isresidual if the set X \ A is a union of a countable number of nowhere densesubsets of X. Another interesting result was obtained by Markuszewicz [7]:for any function f ∈ C(E) there exist f1, f2 ∈ B(E) such that f = f1 + f2.The notion of quasianalyticity has been extended to the n–dimensionalcase in [9]. The theory of quasianalytic functions of several variables in thesense of Bernstein has been developed by Ples´niak in [11].The aim of this paper is to show that the notion of a quasianalytic functionin the sense of Bernstein can be extended to algebraic subsets of Cm. Inparticular, it has been proved (Theorem 3.7) that an identity principle forsuch functions also holds on some algebraic sets. We complete the paper bySection 4, in which we give some examples of compact sets E ⊂ Cm preservingthe Szmuszkowiczo´wna–Lelong type identity principle (see Definition 3.6).2. Preliminaries. A subset A of Cm is said to be pluripolar if there existsa plurisubharmonic function u on Cm such that A ⊂ {u = −∞}. If for eachpoint a ∈ A there exist an open neighbourhood V of a and a plurisubharmonicfunction v on V such that A ∩ V ⊂ {v = −∞}, then the set A is said to belocally pluripolar. Josefson [4] proved that both notions are equivalent.Let us recall that a set A ⊂ Cm is called locally analytic if for each point a ∈A there are an open neighbourhood U of a and functions f1, ..., fs holomorphicin U such that A ∩ U = {z ∈ U : f1(z) = ... = fs(z) = 0}. Let M be alocally analytic subset of Cm whose subset Mreg of regular points is a complexsubmanifold of Cm of pure dimension k (k 6 m). A function defined on M issaid to be plurisubharmonic onM if it is plurisubharmonic on Mreg and locallybounded from above onM. We say that a set N ⊂M is pluripolar inM if thereexists a plurisubharmonic function u on M such that N ∩Mreg ⊂ {u = −∞}.Let E be a subset of the space Cm. The functionVE(z) = sup{u(z) : u ∈ L(Cm), u|E 6 0 },where L(Cm) = {u ∈ PSH(Cm) ; supz∈Cm [u(z) − log(1 + |z|)] < ∞} is theLelong class of plurisubharmonic functions with minimal growth, is called theextremal function of the set E. Let P = P(Cm) be the space of all polynomialsof m complex variables. For a compact set E ⊂ Cm Siciak [15] has introducedthe functionΦE(z) = {|p(z)|1/deg p : p ∈ P(Cm), deg p > 1, ‖p‖E= supz∈E|p(z)| 6 1}217by now called Siciak’s extremal function. It is known (Zakharyuta [18], Siciak[16]) that VE(z) = log ΦE(z) for z ∈ Cm. It follows directly from the defi-nition of ΦE that for any compact set E and any polynomial p the followingBernstein–Walsh–Siciak inequality holds:(2.1) |p(z)| 6‖p‖E [ΦE(z)]deg p , z ∈ Cm.A set D ⊂ Cm is called negligible if there exists a family of functions{uι} ⊂ PSH(Cm) locally bounded from above such thatD ⊂ {z ∈ Cm ; supuι(z) < (supuι(z))∗},where h∗(z) = lim supu→z h(u).An essential role in this paper is played by the following Bedford–Taylorcounterpart of the classical Kellogg lemma (see e.g. [13, Theorem 4.2.5]).Theorem 2.1. ([2, Theorem 7.1]) Negligible sets in Cm are exactly pluripo-lar sets.Let now E be a compact subset of Cm. Observe thatF := {z ∈ E ; VE is not continuous in z} = {z ∈ E ; V ∗E(z) > VE(z) = 0}.Therefore F is a negligible subset of Cm and, by the above Bedford–Taylorversion of the Kellogg lemma, F must be pluripolar.3. Bernstein quasianalytic functions on algebraic sets.Definition 3.1. Let M ⊂ Cm be an algebraic set, and let K be a compactsubset of M. A function f defined on K, with values in C, is said to bequasianalytic on K in the sense of Bernstein if there exist a strictly increasingsequence of positive integers {nj}∞j=1 and a sequence of polynomials pnj ∈Pnj (Cm), j = 1, 2, ..., such that(3.1) lim supj→+∞nj√‖f − pnj ‖K < 1.The set of all quasianalytic functions on K is denoted by B(K).This definition extends the notion of quasianalyticity of functions definedon a compact subset of the space Ck to those defined on pieces of algebraicsets.Lemma 3.2. Let h be a holomorphic mapping defined on an open set U⊂Ck,with values in a locally analytic set M in Cm of pure dimension k (k 6 m). As-sume that h is non-degenerate, which means that rankUh = maxz∈U rankzh =k. Let E be a compact subset of U and F = h(E). If a set N ⊂ F is non-pluripolar in M, then the set h−1(N) ∩ E is non-pluripolar in Ck.218Proof. Let Msing be the set of singular points of M. Then Msing is ananalytical subset of M and dim Msing < k (see e.g. [6, Chapter IV.2.4]).Hence, in view of Josefson’s theorem, the set Msing is pluripolar in M. The setA = {z ∈ U : rankzh < k} is analytic. By the well-known theorem (see e.g.[3, Chapter 1.3.8]) the set h(A) is contained in at most countable family oflocally analytic sets inM of dimension less than k. Consequently, by Josefson’stheorem, h(A) is a pluripolar subset ofM. Hence the set N˜ := N\(h(A)∪Msing)is non-pluripolar in M. So applying again Josefson’s theorem we show thatthere exists a point a ∈ N˜ such that(3.2) for each ε > 0 the set B(a, ε) ∩ N˜ is non-pluripolar in M.Since a is a regular point of the set M, we can find a constant ε0 > 0 and abiholomorphism φ of the set B(a, ε0)∩M onto the unit ball B in Ck. Let us takea point b ∈ h−1(a) ∩ E. Then b /∈ A, so one can find an open neighbourhoodV ⊂ U of the point b such that h|V is a biholomorphism of V onto h(V ) anda ∈ h(V ) ⊂ B(a, ε0). By (3.2) the set φ(h(V ) ∩ N˜) is non-pluripolar. Hence,since the mapping φ ◦ h|V is a biholomorphism of V onto φ(h(V )), the set(φ ◦ h|V )−1(φ(h(V ) ∩ N˜)) = h−1|V (h(V ) ∩ N˜) is non-pluripolar in Ck. SinceN˜ ⊂ F = h(E) and h|V is a biholomorphism, we have h−1|V (h(V ) ∩ N˜) ⊂ E.Consequently, the set E ∩ h−1(N) must be non-pluripolar in Ck.Lemma 3.3. Let E ⊂ Ck be a non-pluripolar, polynomially convex compactset. Let h be a non-degenerate holomorphic mapping defined on an open neigh-bourhood U of E, with values in an algebraic set M ⊂ Cm (k 6 m). Let K bea compact subset of M such that h(E) ⊂ K. If a function f is quasianalyticon K, then the function g = f ◦ h is quasianalytic on E.To prove this lemma we shall need a characterization of algebraic sets inCm given by Sadullaev [14].Theorem 3.4. (Sadullaev’s criterion) An analytic subset A of Cm isalgebraic if and only if Siciak’s extremal function ΦE is locally bounded on Afor some (and hence for each) non-pluripolar compact subset E of A.An important role in the following proof is played by a uniform version ofthe Bernstein–Walsh–Siciak theorem [10, Lemma 1].Theorem 3.5. Let A(U) be the space of bounded holomorphic functionsdefined in an open set U ⊂ Ck, and let ‖ f ‖U := supz∈U |f(z)|. For everypolynomially convex compact subset E of U there exist constants M > 0 anda ∈ (0, 1) such that219distE(f,Pl) := inf{‖f − p‖E ; p ∈ Pl} 6M ‖f ‖U alfor f ∈ A(U) and l ∈ N.Proof of Lemma 3.3. The definition of a quasianalytic function (3.1)implies that for such a function f one can choose a sequence of polynomialspnj ∈ Pnj of m variables and a constant % ∈ (0, 1) such that(3.3) ‖f − pnj ‖K6 %nj for j > j0.Since K ⊃ h(E), we have(3.4) ‖f ◦ h− pnj ◦ h‖E=‖f − pnj ‖h(E)6‖f − pnj ‖K6 %nj for j > j0.We may assume that h is bounded on U . Then taking a constant R > 0sufficiently large, we have h(U) ⊂ B(0, R) ∩ M. Hence, by the Bernstein–Walsh–Siciak inequality (2.1), we get(3.5) supz∈U|pnj ◦ h(z)| 6 supw∈h(U)|pnj (w)| 6‖pnj ‖K[supw∈h(U)ΦK(w)]nj .By [1, Lemma 0.1], h(E) is a non-pluripolar subset of the algebraic set M,whence by Sadullaev’s criterion we get(3.6) C1 := supw∈h(U)ΦK(w) <∞.Now, since‖pnj ‖K6‖f − pnj ‖K + ‖f ‖K6 1+ ‖f ‖Kfor all j > j0, by (3.5) and (3.6) one can find a constant C > 0 such thatsupz∈U|pnj ◦ h(z)| 6 Cnj , j > j0.Owing to this we can apply Theorem 3.5 to the family of functions {pnj ◦ h}and we getdistK(pnj ◦ h, Pl(Ck)) 6M ‖pnj ◦ h‖U al 6MCnjalfor each j, l ∈ N with suitably chosen constants M > 0 and a ∈ (0, 1). Conse-quently, for each j ∈ N and each l ∈ N there exists rl ∈ Pl(Ck) such that‖pnj ◦ h− rl ‖K6MCnjal.Choosing an integer t such that Cat 6 a and putting l = tnj gives(3.7) ‖pnj ◦ h− rtnj ‖K6Manj .By the triangle inequality, (3.4) and (3.7), for j > j0 we get‖f ◦ h− rtnj ‖E6 %nj + Manj 6 2Mηnj = 2M(η1t )tnj ,where η = max{%, a} < 1.220Finally, setting uj := tnj giveslim supj→∞uj√‖f ◦ h− ruj ‖E 6 η1t < 1,whence f ◦ h is a quasianalytic function on E.Definition 3.6. A compact set E ⊂ Ck is said to satisfy condition (NB)if for every f ∈ B(E) and every non-pluripolar set F ⊂ E, f = 0 on F impliesf = 0 on E.Theorem 3.7. Let E be a polynomially convex compact set in Ck satisfyingcondition (NB). Leth : U ⊃ E 7→ M ⊂ Cmbe a non-degenerate holomorphic mapping in an open neighbourhood U of E,with values in an algebraic set M of pure dimension k (k 6 m), and K = h(E).If f is quasianalytic on K and f(z) = 0 for z ∈ N ⊂ K, where N is a non-pluripolar subset of M, then f ≡ 0 on K.Proof. In view of Lemma 3.2 the set F = h−1(N)∩E is a non-pluripolarsubset of E on which the function f ◦ h vanishes. By Lemma 3.3 the functionf ◦ h is quasianalytic on E. Consequently, by condition (NB), f ◦ h ≡ 0 on E,and therefore f ≡ 0 on K = h(E).4. Sets satisfying condition (NB). By the Szmuszkowiczo´wna–Lelongtheorem every closed subinterval of C satisfies condition (NB). In [9] it has beenproved that this condition is satisfied by subsets of Cn of type E = E1×. . .×Enwhere each set Ei is a continuum in C. By Theorem 2.1 and (IP) we derivethat if E is a convex compact subsets of Cn then it satisfies (NB). Now we shallprove essentially more, viz. that the sets whose two arbitrary points can beconnected by an analytic curve belong to the class of NB–sets. We shall need acounterpart of Lemma 3.3 in the case where h is a holomorphic mapping withvalues in Cm.Lemma 4.1. Let E ⊂ Ck be a non-pluripolar, polynomially convex compactset, and let K ⊂ Cm be a non-pluripolar compact set. Let h be a holomorphicmap defined in an open neighbourhood U of E, with values in Cm, such thath(E) ⊂ K. Then for every quasianalytic function f on K the function f ◦ his quasianalytic on E.Remark 4.2. The dimensions k and m can be arbitrary.The proof of Lemma 4.1 is similar to that of Lemma 3.3. Now, the constantC1 in (3.6) is finite, since Siciak’s extremal function associated with a non-pluripolar compact set is locally bounded (see [16, Lemma 3.4, Corollary 3.9and Theorem 3.10]).221Theorem 4.3. Let E be a non-pluripolar compact subset of Cn. Assumethat for any two different points of E there exists an analytic mapping la,bdefined in a neighbourhood of [0, 1] such that la,b([0, 1]) ⊂ E, la,b(0) = a andla,b(1) = b. Then the set E satisfies (NB).Proof. Let us take a function f ∈ B(E) and a non-pluripolar set F ⊂ Esuch that f(z) = 0 for z ∈ F . By the continuity of the function f it can beassumed that the set F is compact. The definition of a quasianalytic functionimplies that for some % ∈ (0, 1) and a sequence of polynomials {pnj} withdeg pnj 6 nj we have‖f − pnj ‖F6‖f − pnj ‖E6 %nj for j > j0.Since f|F ≡ 0, it follows that ‖pnj ‖F6 %nj . Due to the above estimate andinequality (2.1), we get(4.1) |pnj (z)| 6 %nj [ΦF (z)]nj .By Theorem 2.1 the set of all points of F at which Siciak’s extremal functionΦF is not continuous must be pluripolar. So there exists a point a ∈ F at whichΦF is continuous. If we take η ∈ (0, 1%) then we can choose a constant ε > 0such that |ΦF (z)| 6 η for z ∈ B(a, ε). Applying this estimate to inequality(4.1) gives ‖ pnj ‖B(a,ε)6 (% η)nj for j > j0. Since (% η) < 1, f(z) = 0 forz ∈ B(a, ε) ∩ E. Now let us choose an arbitrary point b ∈ E \ {a} andan analytic map la,b satisfying the assumptions of the theorem. Let us notethat the function f ◦ la,b defined in a neighbourhood of the interval [0, 1] isquasianalytic on [0, 1] (Lemma 4.1) and that there exists α > 0 such thatla,b([0, α]) ⊂ B(a, ε)∩E. Hence f ◦ la,b ≡ 0 on [0, α]. By the classical Bernsteintheorem it follows that h ◦ la,b ≡ 0 on [0, 1]. Hence, in particular, f(b) = 0. Bythe arbitrariness of the choice of b ∈ E we derive that f ≡ 0 on E.By Lemma 3.2 and Lemma 3.3, property (NB) is invariant under biholo-morphic mappings. Under certain conditions, we can prove more, namelyTheorem 4.4. Let W ⊂ Ck be a compact set satisfying (NB). Assumemoreover that for every point a ∈W and for every constant ε > 0(4.2) the set B(a, ε) ∩W is non-pluripolar.Let h be a non-degenerate holomorphic mapping defined in an open neighbour-hood U of the set W , with values in Cm (k > m). Then h(W ) satisfies (NB).Proof. Condition (4.2) implies that the set W is non-pluripolar. Sinceh is a non-degenerate holomorphic mapping, the set h(W ) is non-pluripolar(see [12, Lemma 2.5]). Let us take a quasianalytic function q on h(W ) anda non-pluripolar set N ⊂ h(W ) such that q = 0 on N . We can proceed likein the first part of the proof of Theorem 4.3. Namely we may assume that N222is compact and we can choose a sequence of polynomials pnj ∈ Pnj such that‖q − pnj ‖h(W )6 %nj for a certain % ∈ (0, 1) and a sequence of positive integersnj ↗∞.Then, due to the non-pluripolarity of N , by Theorem 2.1 and the Bernstein–Walsh–Siciak inequality (2.1), there exist a point b ∈ N and constants r > 0,γ ∈ (%, 1) such that ‖ pnj ‖B(b,r)6 γnj for j > j0. Since h is a continuousmapping, the set h−1(B(b, r)) is an open subset of U . Let a ∈ h−1(b) ∩W .Obviously, there is a constant δ > 0 such that B(a, δ) ⊂ h−1(B(b, r)). Byassumption (4.2) the set F := B(a, δ) ∩W is not pluripolar. Let us note thath(F ) = h(B(a, δ) ∩W ) ⊂ h(h−1(B(b, r)) ∩W )⊂ h(h−1(B(b, r))) ∩ h(W ) = B(b, r) ∩ h(W ).So we have‖q ◦ h‖F=‖q‖h(F )6‖q − pnj ‖h(F ) + ‖pnj ‖h(F )6 %nj + γnj 6 2γnj .Hence the function q ◦h vanishes on a non-pluripolar subset F of W . Since Wsatisfies (NB) and, by Lemma 3.3, the function q ◦ h is quasianalytic, we haveq ◦ h ≡ 0 on W . Consequently, q ≡ 0 on h(W ).Other examples of NB–sets are yielded by the followingProposition 4.5. Let E1 and E2 be compact subsets of Ck satisfying (NB).If the set E1 ∩ E2 is non-pluripolar then the set E := E1 ∪ E2 also satisfies(NB).Proof. Let q ∈ B(E), and let F ⊂ E be a non-pluripolar set such thatq vanishes on F . We may assume that F ∩ E1 is a non-pluripolar set. SinceE1 ∈ (NB), we get q ≡ 0 on E1. Hence q = 0 on the non-pluripolar subsetE1 ∩ E2 of E2. Since E2 ∈ (NB), q = 0 on E2.Acknowledgments. The author is indebted to Professor Wies law Ples´niakfor his valuable suggestions and remarks.References1. Baran M., Ples´niak W., Polynomial inequalities on algebraic sets, Studia Math., 141(3)(2000), 209–219.2. Bedford E., Taylor B.A., A new capacity for plurisubharmonic functions, Acta Mathe-matica, 149 (1982), 1–40.3. Chirka E.M., Complex Analytic Sets, Mathematics and Its Applications, Vol. 46, 1985.4. Josefson B., On the equivalence between locally and globally polar sets for plurisubhar-monic functions in Cn, Arkiv fo¨r Matematik, 16 (1978), 109–115.5. Lelong P., Sur une proprie´te´ simple des polynoˆmes, R. Acad. Sci. Paris, 224 (1947),883–885.2236.  Lojasiewicz S., Introduction to Complex Analytic Geometry, Birkha¨user, 1991.7. Markuszewicz A.I., On the best approximation, Doklady AN USSR, 44 (1944), 290–292(in Russian).8. Mazurkiewicz S., Les fonctions quasi-analytiques dans l’espace fonctionnel, Mathematica(Cluj), 13 (1937), 16–21.9. Ples´niak W., Quasianalytic functions of several complex variables, Zeszyty Nauk. Uniw.Jagiell., 15 (1971), 135–145.10. ,On superposition of quasianalytic functions, Ann. Polon. Math., 26 (1972), 73–84.11. ,Quasianalytic functions in the sense of Bernstein, Dissertationes Mathematicae,147 (1977).12. ,Invariance of the L–regularity of compact sets in CN under holomorphic map-pings, Trans. Amer. Math. Soc., 246 (1978), 373–383.13. Ransford T., Potential theory in the complex plane, Cambridge Univ. Press, 1995.14. Sadullaev A., An estimate for polynomials on analytic sets, Math. USSR Izv., 20 (1983),493–502.15. Siciak J., On some extremal functions and their applications in the theory of analyticfunctions of several complex variables, Trans. Amer. Math. Soc., 105 (1962), 322–357.16. ,Extremal plurisubharmonic functions in Cn, Ann. Polon. Math., 39 (1981), 175–211.17. Szmuszkowiczo´wna H., Un the´ore`me sur les polynoˆmes et son application a` la the´orie desfonctions quasi-analytiques, C.R. Acad. Sci. Paris, 198 (1934), 1119–1120.18. Zakharyuta V.P., Extremal plurisubharmonic functions, orthogonal polynomials andBernstein–Walsh theorem for analytic functions of several complex variables, Ann. Polon.Math., 33 (1976), 137–148 (in Russian).Received June 17, 2003Jagiellonian UniversityInstitute of MathematicsReymonta 430-059 Krako´w, Polande-mail : skiba@im.uj.edu.pl

Bernstein quasianaytic functions on algebraic sets

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Bernstein quasianaytic functions on algebraic sets

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