aa r X i v : . [ m a t h . C V ] M a y PLURIPOLARITY OF GRAPHS OF ALGEBROID FUNCTIONS
ZAFAR S. IBRAGIMOV
Let K ⊂ C n be an arbitrary compact set and let f ( z ) be a continuous functionon K . By ρ m ( f, K ) we denote the least deviation of f ( z ) on K from the rationalfunctions of degree less than or equal to m . That is, ρ m ( f, K ) = inf r m || f − r m || K , where || − || K is the uniform norm and the infimum is taken over all rationalfunctions of the form r m ( z ) = Σ | α |≤ m a α z α Σ | α |≤ m b α z α , where α = ( α , α , . . . , α n ) is a multiindex . As usual, we denote by e m ( f, K ) the least deviation of function f ( z ) on K fromthe polynomials of degree less than or equal to m . Obviously, ρ m ( f, K ) ≤ e m ( f, K ) for each m = 1 , , . . . . In ([1], [2]) Gonchar proved that if K = [ a, b ] ⊂ R ⊂ C , then the class of functions R ([ a, b ]) = { f ∈ C [ a, b ] : lim m →∞ m p ρ m ( f, K ) < } possesses one of the important properties of the class of analytic functions. Namely,if lim m →∞ m p ρ m ( f, K ) < } and if f ( x ) = 0 on a set E ⊂ [ a, b ] of positive logarithmic capacity, then f ( x ) ≡ a, b ] (see also [5]).By analogy with the class B ( K ) = { f ∈ C ( K ) : lim m →∞ m p e m ( f, K ) < } , Date : May 3, 2010.2000
Mathematics Subject Classification.
Primary 32U. which is called the class of quasianalytic functions of Bernstein ([3],[4],[6]), we call R ( K ) = { f ∈ C ( K ) : lim m →∞ m p ρ m ( f, K ) < } the class of quasianalytic functions of Gonchar.The classes B ( K ) and R ( K ) are not linear spaces; the sum of two quasianalyticfunctions are not, in general, quasianalytic (see [4]). We consider the followingsubclass R ( K ) of the class R ( K ): R ( K ) = { f ∈ C ( K ) : lim m →∞ m p ρ m ( f, K ) < } It is not hard to see that, if f and f belong to R ( K ), then so are c f + c f and f · f , where c and c are arbitrary complex numbers.In ([7]) K. Diederich and J.E. Fornass constructed an example of a smooth(infinitely differentiable) function, whose graph is not pluripolar in C . Recently,D. Coman, N. Levenberg and E.A. Poletskiy have proved, that if f ∈ B ([ a, b ]),then its graph Γ f is pluripolar in C .In ([8]) A. Edigarian studied the following analogue of a theorem of N. Shcherbina[9]. Let D be a domain in C n and let Γ ⊂ D × C be a graph of some algebroidfunction, i.e.,Γ = n ( z, w ) ∈ D × C : w k + a ( z ) w k − + · · · + a k ( z ) = 0 o , where a ( z ) , a ( z ) , . . . , a k ( z ) are continuous functions on D . Then Γ is pluripolarin C n +1 if and only if the functions a ( z ) , a ( z ) , . . . , a k ( z ) are holomorphic in D .In this paper we prove a similar theorem on pluripolarity of graphs of algebroidfunctions in the class of quasianalytic functions. Theorem 0.1.
Let [ a, b ] ⊂ R ⊂ C and let Γ ⊂ C be a graph of some algebroidfunction, i.e. Γ = n ( z, w ) ∈ D × C : w k + a ( z ) w k − + · · · + a k ( z ) = 0 o , where a l ∈ R ([ a, b ]) , l = 1 , , . . . , k . Then Γ is pluripolar in C Proof.
We consider the following function on [ a, b ] × C f ( z, w ) = w k + a ( z ) w k − + · · · + a k ( z ) . LURIPOLARITY OF GRAPHS OF ALGEBROID FUNCTIONS 3
Since a l ∈ R ([ a, b ]) for l = 1 , , . . . , k , there exist a sequence of rational functions r j ( z ) , r j ( z ) , . . . , r jm ( z ) , . . . such that m q ρ m ( a j , [ a, b ]) = m q || a j ( z ) − r jm ( z ) || [ a,b ] ≤ δ j < . The function f ( z, w ) is quasianalytic in the sense of Gonchar on the compact set[ a, b ] × {| w | ≤ h } ⊂ C k +1 , where h is an arbitrary positive number. Indeed, ρ m + k (cid:0) f, [ a, b ] × {| w | ≤ h } (cid:1) ≤ || f ( z, w ) − w k − k X j =1 r jm ( z ) w k − j || [ a,b ] ×{| w |≤ h } ≤ k X j =1 || a j ( z ) − r jm ( z ) || [ a,b ] · h j ≤ k X j =1 h j δ m + kj ≤ k max { h, h k } δ m + k , where δ = max { δ j : j = 1 , , . . . , k } . It follows thatlim m →∞ ρ / ( m + k ) m + k (cid:0) f, [ a, b ] × {| w | ≤ h } (cid:1) ≤ δ < . Consequently, the graphΓ f = (cid:8) ( z, w, f ( z )) : ( z, w ) ∈ [ a, b ] × C (cid:9) of the function f ( z, w ) is pluripolar in C . Now we consider sectionsΓ f ( λ ) = (cid:8) ( z, w, f ( z, w )) : f ( z, w ) = λ (cid:9) . For each λ , the section Γ f ( λ ) is pluripolar in C . (Indeed, if the graph Γ f ( λ )is nonpluripolar for some λ ∈ C , then according to the uniqueness propertyof quasianalytic functions, the function f ( z, w ) is identically equal to λ , whichcontradicts the definition of f ( z, w )). In particular, we obtain the pluripolarityof the graphΓ = n ( z, w ) ∈ D × C : w k + a ( z ) w k − + · · · + a k ( z ) = 0 o of algebroid functions. The proof of the theorem is complete. (cid:3) References [1] A.A. Gonchar,
On best approximations by rational functions , Dokl. Akad. Nauk SSSR, 100(1955), 205–208, (Russian).[2] A.A. Gonchar,
Quasianalytic class of functions, connected with best approximated by rationalfunctions , Izv. of the Acad. of Sci. Armenia SSR. 1971, VI . No. 2-3, 148–159.[3] A.F. Timan,
Theory of approximation of functions of a real variable , Pergaon Press, Macmil-lian, New York, 1963.
Z. S. IBRAGIMOV [4] W. Plesniak,
Quasianalytic functions of several complex variables , Zeszyty Nauk. Uniw.Jagiell. 15 (1971), pp. 135–145.[5] W. Plesniak,
Characterization of quasianalytic functions of several variables by means ofrational approximation , Ibid. 27 (1973), pp. 149-157.[6] D. Coman, N. Levenberg, E.A. Poletskiy,
Quasianalyticity and Pluripolarity , J. Amer. Math.Soc. 18 (2005) No. 2, pp. 9–16.[7] K. Diederich, J.E. Fornass,
A smooth curve in which is not a pluripolar set , Duke Math. J.49 (1982), pp. 931–936.[8] A. Edigarian,
Graphs of multifunctions , Math. Z., 250 (2005) 145–147.[9] N. Shcherbina,
Pluripolar graphs are holomorphic , Acta Math., 194 (2005), pp. 203–216.
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