ON WEIGHTS WHICH ADMIT THE REPRODUCING KERNEL OF BERGMAN TYPE

In this paper we consider (1) the weights of integration for which the reproducing kernel of the Bergman type can be defined, i.e., the admissible weights, and (2) the kernels defined by such weights. It is verified that the weighted Bergman kernel has the analogous properties as the classical one. We prove several sufficient conditions and necessary and sufficient conditions for a weight to be an admissible weight. We give also an example of a weight which is not of this class. As a positive example we consider the weight μ(z)=(Imz)2 defined on the unit disk in ℂ.

boundary" of D. By Corollary 3.2, p is an a-weight.We find a complete orthonormal system in the space L2H(D,p) of all holomorphic p-square integrable functions on D. It makes possible the uniform approximation of the Bergman kernel K a by polynomials on any compact subset of D x D, see Theorem 2.1, (i).
Without any other explanations we use the following symbols: N the set of natural numbers; Z the set of integers; 7' + N U (0}; R the set of reals, C the complex plane.
Let D be an open nonempty set in C n and let W(D) be the set of weights on D, i.e., W(D) is the set of all Lebesque measurable real-valued positive functions on D under the convention that we consider two weights as equivalent if they are equal almost everywhere with respect to the Lebesque measure on D. If p G W(D), we denote by L2(D,p) the space of all Lebesque measurable complex-valued p-square integrable functions on D.
The space L2(D,p) is a separable complex Hilbert space with respect to the norm gve= by the scalar product < f g >#" I f(z) g(z) p(z)(dz)2n, D f,g C L2(D,p).
The map L2(D,#) f-, ,-.f C L2(D)is an isometry on L2(D,#) onto the space L2(D) of all complex-valued functions on D which are square integrable with respect to the Lebesque measure.
Let L2H(D,g) be the set of all holomorphic functions from L2(D,g).Then L2H(D,#) is a linear subspace of L2(D,g) perhaps equal to {0} called the #-Bergman space over D (see [11).For any z D we define the evaluation functional Ez on L2H(D,p) by the formula Ezf:= f(z), f C L2H(D,p).
(2.1) DEFINITION 2.1.A weight # W(D) is called an admissible weight, an a-weight for short, if L2H(D,p) is a closed subspace of L2(D,p) and for any z ( D the evaluation functional Ez is continuous on L2H(D,p).The set of all a-weights on D will be denoted by AW(D).
It is well known that the characteristic function of the set D satisfies conditions of the above definition (Krantz [6] and Maurin [7]).
If q AW(D) then by the Riesz representation theorem, for any z D there exists a unique function ez, L2H(D,p) such that for each f L2H(D,#) Ezf < e,,, f ># I ez'#(w) f(w) #(w)(dw) 2n D DEFINITION 2.2.The function Kg:Dx D --, C given by the formula (2.2) K,(z, w):= ez, l(w), z,w .D , is called the #-Bergman function of the set D (see [3]).Since the formula (2.2) can be written in the form f(z) I z,( z, w) f(w) #(w)(dw) 2n (2.3) D the function K, is also called the #-Bergman reproducing kernel of D. Similarly as in the classical case we can show the following properties of Kt, (see [3] and [6]).(2.5) (iii) the function Kt(z,w is antiholomorphic in w and holomorphic in z; (iv) Kp is analytic in the real sense; (v) if Pt* is the orthogonal projector on L2(D,#) upon L2H(D,p) then for every f e L2(D,p) and each z e D [Ptf](z)= I Kt*(z'w)f(w)#(w)(dw)2n D i.e., Kt, is the integral kernel of the operator P#.where (%(z)} is the sequence of complex numbers.If (} is an infinite sequence, the series on the rih had side of (2.7) converes in 1,2H(D,/).Since the evaluation functionals axe all continuous we obtain ha for every w D Kt,(z,w)= E "ak(z) Ck(w) On the other hand for any k we have /c(z) <ez, pl/> .aj(z)<jlCk,>=alc(z), see (2.2).Applying it to (2.8) we get (2.4).The point (ii)is an immediate consequence of (i).
By its very definition K,(z,w) is holomorphic in w.Hence K,(z,w) is antiholomorphic in w and, by (ii), it is holomorphic in z.
In order to prove (iv) let us consider the function DDOg(z,w) KO(z,w)'= Kt(z,) e C, where DO= {we C n" He D}.From (iii)it follows that K0 is holomorphic in each variable separately.Then by the Hartogs theorem on separate analyticity, it is holomorphic on D x D O (see For the proof of (v) let f e L2(D,) and let z e D. We have by (i) where {k} is an arbitrary complete orthonormal system in L2H(D,I).This completes the proof of our theorem.
PROPOSITION 2.1.Under the assumptions of Theorem 2.1.let {ffrn} be an orthonormal complete system in L2H(D, Iz).If {rn} is an infinite sequence then the series Crn(Z)Ckrn(W) converges to Kl(z,w uniformly on any compact set M C D x D. m PROOF.We can assume that M X x X, where X is an arbitrary compact subset of D, i.e., for any compact set M C D x D there exists a compact set X C D such that M C X x X.From Theorem 2.1, (i) it follows that for any z (5 D E I(z) K(z,z) < m=l i.., (,( e .tr(l.I,,(/1, e . We have that (Tn) is a decreasing sequence of continuous functions on D, which converges to f 0.
By the Dini theorem (Maurin [8]), Chapter V, Section 4) we obtain that (Tn) converges uniformly on any compact set X C D. The uniform convergence of Cm(z)m(w) on X x X follows now from the Schwarz inequality m 2 Cm(z)m(w) <_ Tn(z)Tn(w), (z,w) X xX.
m=l Now we shall give the necessary and sufficient conditions for a weight / W(D) to be an a- weight.
THEOREM 2.2.Let/ W(D).The following are equivalent: (i) / is an admissible weight; (ii) for any compact set X C D there exists a constant C X > 0 such that for any z X and each f L2H(D,#) Ezfl Cx f ; (2.10) (iii) for any z D there exists a compact set Y C D which contains z and has the following property: for each z' COY there exists a neighbourhood Vz, of z' in D and a constant Cz, > 0 such that for any w V z, and any f L2H(D, Iz) Efl C=, f . .(iv) for any z D there exists a neighborhood Vz of z in D and a constant Cz > 0 such that for any w V z and each f L2H(D,p) Ewf -Cz f ,. (2.12) PROOF.(i) = (ii).
theorem, for any z D Notice that, by Theorem 2.1, (i) and by the Riesz representation E II.Kl(z," )11.
If C X > 0 is such a constant that (2.10) holds then for any z' cOY we can take Vz,: B(z',r) and Cz, C X.
(iii) (iv).Let z D and let Y be chosen as in the statement (iii) above.If z OY then the existence of the neighborhood Vz and the constant Cz follows immediately form (iii). Suppose now that z int Y. Choose for each z' OY a neighborhood Vz, and a constant Cz such as in (iii).
Since OY is compact we can find a finite number of points z l, z2,... z k OY such that Then for any z' (F_ OY and any f L2H(D,) where Cz: max {Czl,Cz2,...,Czt,}.Using now the maximum principle for holomorphic functions we obtain that (2.12) holds for each w Vz: int Y.
(iv) =), (i).The continuity of evaluation functionals follows immediately from (2.12).Moreover, if   {fro} is a sequence of elements of L2H(D,I) which converges in L2(D,/) to a function f L2(D,/) then, by (iv), this sequence converges locally uniformly on D. Hence, by the Weierstrass theorem on the limit of a uniformly convergent sequence of holomorphic functions, the function f is .holomorphic,i.e., f L2H(D,).This implies that L2H(D,) is a closed subspace of L2(D,/).
REMARK.The notion of a /-Bergman function is not necessary for the proof of the implication (i) =), (ii) in the above theorem.One can prove this implication directly from Definition 2.1 using the Banach-Steinhaus theorem on sequences of linear continuous operators (Rudin [9] 5.8).

SUFFICIENT CONDITIONS FOR A WEIGHT TO BE AN ADMISSIBLE WEIGHT.
A verification of conditions (ii), (iii) or (iv) from Theorem 2.2 usually needs additional considerations and it is sometimes a sufficiently difficult task.In this section we give more effective sufficient conditions for a weight to be an admissible weight.THEOREM 3.1.Assume that/ W(D), U is an open subset of D and there exists a number a > 0 such that the function/-a is integrable on U with respect to the Lebesque measure.Then for any z U there exists a neighborhood Vz of z in D and a constant Cz > 0 such that for each w Vz and each f L2H(D, lt) PROOF.If xCn, R>0, define the open ball B(x,R)'={xCn'jx-xl <R}.Let z U and let a number r > 0 be such that B(z,2r) C U. Put Vz" B(z,r), p" (1 + a)/a and q'= 1 + a.We have that, for every w Vz, B(w,r) C U. (The symbol X denotes the closure of the set X.) If f L2H(D,g) then Ill 2/p is a subharmonic function (see [6]), Corollary 2.1.15or (Herv6 [10] 1.2.2)and therefore 2 2 If(w) l -< vol(w,) I f()l (dz)2n, weyz B(w,r) which is the mean value property, (see [6] Theorem 2.14).Since p,q > 1 and lip + 1]q 1 we have by the H61der inequality and we can take f(w) _< ( f It(:r,)-a(dz)l (vol B(w,r)) II/II.u COROLLARY 3.1.Let It E W(D).Assume that for each z E D there exists a compact set Y C D which contains z and has the following property: for any w 0Y there exists a neighborhood Uw of w in D and a number a w > 0 such that the function p-a, is integrable on Uw with respect to the Lebesque measure.Then It AW(D).If in particular, the function It-a is locally integrable on D for some a > 0 then It AW(D).
COROLLARY 3.2.Let It E W(D).If for any z 5 D there exist a neighborhood U of z in D, a constant a > 0 and a function f 6 C (U, R) such that: (i) 0 is a regular value of f; (ii) for almost all w U (w) _> f(w) , then It AW(D).
PROOF.Let z D and let U, a and f be associated with z as in the assumptions of the Corollary.If f(z) # 0 then there exists a bounded neighborhood U' of z in D and a constant b > 0 such that for any w E U' f(w) a > .
Hence p(w) -1 _< f(w) -< b-1, i.e., It-1 is an integrable function on U'.Suppose now that f(z) 0. The function f is regular in z which implies that there exists r > 0 and a bounded diffeomorphism F'B(O,r)U such that F(0)=z and for each w (Wl, w,) B(O, r) is an integrable function on U'.The theorem follows now from Corollary 3.1.EXAMPLE 3.1.Let D be the unit disk in C. Let for any z E D ft(z)'-" IImzl t, t(O,+o) 9(z)" IImzl -I1" (z).{ e Iz[-forz#O 0 for z=O Then the functions 9, h and ft, (0, +) are a-weights on D.

A WEIGHT WHICH IS NOT AN ADMISSIBLE WEIGHT.
In the sequel we shall use the following theorem.THEOREM 4.1.(Runge).Let X be a compact subset of C whose complement is connected.
Let f: X C be continuous on X and holomorphic on the interior of X.Then f is the uniform limit on X of holomorphic polynomials.(See [9], 13.9).Now we are in position to give the example of the weight which is not an a-weight.Let D: {z C: zl < } be the unit disk in C.
Let n N. Define An:=B(O, 2-n)u{zD:Rez> O, Ilmzl <2-n}, Mn" (D an) U 7t n + where and n + denotes the closure in C of D and A n + respectively.Let fn" Mn-* C be given the formula i +n 1-for z n+ fn(Z) Since the set C-Mn is connected we obtain by the Runge theorem that there exists a holomorphic polynomials tln such that for any z Mn f.(z)-gn(z) < 1In.
Since Ig.(0) > t we have that Ihn(z) < i/. for z e D-An and z A n + fl D.Moreover, ha(O)= 1.Now define a weight t W(D) as follows: Ih,(z) < 1 +2In for We have that for any z D, /(z)_< 1.It is clear that h n L2H(D,I) for n q I1 and Eohn 1.
Notice that for each z E D hn(z) 12it(z) <9 and lim hn(z) 2 It(z) O.
Then by the Lebesque theorem on majorized convergence lim Ilhnl]2 lim f ]hn(z) 12it(z)(dz)2n=O, D whereas lim Eoh n 1.This implies that the evaluation functional E 0 is not continuous on L2H(D,#) and therefore It is not an admissible weight on D. We axe going to find a complete orthonormal system in L2H(D, It).Let for any n E Z + g.(z): zn, z D.
Let us calculate the norm f t" By (5.2) and (5,8) we get Ilfoll =, Ilflll ( Since fm # 0 we obtain that a m # 0 for each m E Z+.Now define a sequence (m) of elements of L2H(D,p) by the formula: ( m+3 frn 2Tram am + 2' *,n fm We are going to show that the orthonormal system (m) is complete iu L2H(D,p).
Finally we can state that (5.11) is true for m n and by the induction principle it is true for any mEN.
completes the proof of our theorem.This

THEOREM 2 . 1 .
Let # e AW(D) and let Kt be the p-Bergman function.Then (i) for any complete orthonormal system {k} in L2H(D, p) and any (z, w) e D x D Kt,(z, w) E Ck(z) Ck(w) for any (z, w) e D D Kt,(w,z Kt,(z w);

5 .
THE WEIGHT It(z)= (Im z) 2 ON THE UNIT DISC IN C.