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Coefficient Estimates on Weighted Bergman Spaces

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Coefficient Estimates on Weighted Bergman Spaces Coefficient estimates on weighted Bergman spaces John E. McCarthy ∗ Washington University, St. Louis, Missouri 63130, U.S.A. 0 Introduction Let A denote normalized area measure for the unit disk D in C. The Bergman 2 2 space La is the sub-space of the Hilbert space L (A) consisting of functions n 2 that are also analytic in D. The monomials z are orthogonal in La, and p 1 P1 n 2 have norm n+1 ; so a holomorphic function f(z) = n=0 anz is in La if and P1 2 1 PN n only if n=0 janj n+1 < 1, and if this is so, the partial sums n=0 anz are polynomials converging to f. The weighted Bergman spaces normally studied are obtained by replacing the measure dA(z) by the radial measure dAα(z) = (1 − jzj2)αdA(z); in these spaces the monomials are again orthogonal, and a function is approximable in norm by the partial sums of its power series at zero. In this paper we are interested in studying non-radial weights of the 2 1 form jm(z)j dAα(z), where m is the modulus of a function in H , the space of bounded analytic functions on D. There are two different ways of generalizing the Bergman space to these 2 2 weights. We shall use La(jmj Aα) to denote the space of analytic functions 2 2 2 2 on D that also lie in L (jmj Aα), and P (jmj Aα) to denote the closure of the ∗The author was partially supported by the National Science Foundation grant DMS 9296099. 1 2 2 polynomials in L (jmj Aα). These two spaces are, in general, different. If m 2 2 2 has no zeroes, L (jmj Aα) is just the set of quotients ff=m : f 2 Lag; but this 2 2 2 2 coincides with P (jmj Aα) only when 1=m is in P (jmj Aα), which in turn is 2 equivalent to requiring that 1 lie in the La closure of fmp : p a polynomialg, i.e. that m be a cyclic vector for multiplication by z (this is discussed in Section 3 below). When m has an infinite number of zeroes, we do not know 2 2 2 2 when P (jmj Aα) is all of La(jmj Aα). 2 2 2 2 We are interested in obtaining, for functions in P (jmj Aα) and La(jmj Aα), estimates on the size of the Taylor coefficients, and on the rate of growth as the boundary is approached. Our principal results are the following. 1 2 2 Theorems 2.1, 2.6, 1.4 Let m be in H . Let f be in La(jmj Aα). Then p jf^(n)j = eO( n log n): 2 2 If f is in La(jmj A), then p jf^(n)j = eO( n): If f is in P 2(jmj2A), then p jf^(n)j = eo( n): Theorem 3.6 Assume m in H1 is zero-free, and suppose the Taylor series 1 1 P1 n expansion of m is given by m(z) = n=0 αnz . Then v u n ^ 2 2 uX 2 supfjf(n)j : f 2 La(jmj A); kfk ≤ 1g = t (n + 1 − j)jαjj j=0 2 1 P 2(jmj2A) 2 2 2 Let PLa denote the orthogonal projection from L (A) onto La, and for m 2 2 1 La 2 2 La in H define the co-analytic Toeplitz operator Tm¯ : La ! La by Tm¯ f = 2 PLa (mf ¯ ). If σ is Lebesgue measure on the circle, the classical Hardy space 2 2 2 2 H is the closure of the polynomials in L (σ) (so H = P (σ)), and if PH2 denotes projection from L2(σ) onto H2, then define the co-analytic Toeplitz H2 2 2 H2 operator Tm¯ : H ! H by Tm¯ f = PH2 (mf ¯ ). Just as in the Hardy space case [2, 5], the function f is in the range of the 2 La Toeplitz operator Tm¯ if and only if there exists some constant C such that, for all polynomials p, j R pfdA¯ j2 ≤ C R jpj2jmj2dA, i.e. if the functional D p 7! R pfdA¯ is bounded on P 2(jmj2A). If this functional is bounded, we D shall say f is in the dual of P 2(jmj2A). The important point is that if P1 ^ n 2 2 P1 n f(z) = n=0 f(n)z is in the dual of P (jmj A), and g(z) = n=0 g^(n)z is in P 2(jmj2A), then the sum 1 1 X g^(n)f^(n) n=0 n + 1 must converge; so knowing a function f is in the dual of P 2(jmj2A) means that all functions g satisfy n + 1 g^(n) = o( ): f^(n) The common range of co-analytic Toeplitz operators on H2 was charac- terized in [5] in terms of the Fourier coefficients f^(n) of the function f: Theorem 1.1 The function f is in the range of every co-analytic Toeplitz operator on the Hardy space if and only if there exists a constant c > 0 such p that f^(n) = O(e−c n). In the expository article [7] I stated that Theorem 1.1 was also true in the Bergman space. However, whereas the proof of the necessity of the decay 3 condition on the Taylor coefficients is valid, the dismissal of the proof of sufficiency seems unjustified. Using a result of Lotto and Sarason [4], we can prove the theorem. Lotto and Sarason's result is that the set Y , of analytic p functions f satisfying f^(n) = O(e−c n) for some c, is what they call Toeplitz stable, i.e. it is a subspace with the properties that 1 H2 (i) If f is in Y and m is an outer function in H , then Tm¯ f is in Y . 1 H2 (ii) If f is in Y and m is an outer function in H , then f = Tm¯ h for some h in Y . P1 n 2 Theorem 1.2 Let f(z) = n=0 anz be in La. A necessary and sufficient condition that f lie in the range of every non-zero co-analytic Toeplitz oper- p 2 −c n ator on La is that, for some c > 0, an = O(e ). Proof: Necessity: We use the following relation between the Toeplitz op- 2 2 La 2 H 2 erator Tm¯ on La and the operator Tm¯ on H : If 1 1 H2 X n X n Tm¯ cnz = dnz n=0 n=0 then 1 1 2 La X n X n Tm¯ cn(n + 1)z = dn(n + 1)z ; (1.3) n=0 n=0 P1 2 whenever n=0 jcnj (n + 1) < 1. Now for each m there is some function 2 P1 n P1 2 1 La g(z) = n=0 bnz , where n=0 jbnj n+1 < 1, with Tm¯ g = f. By (1.3), H2 P1 1 n P1 1 n P1 1 n Tm¯ n=0 bn n+1 z = n=0 an n+1 z . So n=0 an n+1 z is in the range of every 1 co-analytic operator on the Hardy space, and so by Theorem 1.1 fan g, p n+1 −c n and hence also fang, is O(e ). 1 P1 an n Sufficiency: Fix m in H . Let f1(z) = n=0 n+1 z . Then f1 is in Y , P1 n H2 so there is some function g1(z) = n=0 bnz in Y with Tm¯ g1 = f1. Let 2 P1 n 2 La g(z) = n=0 bn(n + 1)z ; then g is in La and by (1.3) Tm¯ g = f. 2 Corollary 1.4 Let m be in H1 and g be in P 2(jmj2A). Then p jg^(n)j = eo( n): 4 2 2 2 La(jmj Aα) 2 2 We first estimate the growth of the Taylor coefficients of functions in La(jmj Aα). We shall use C to denote a generic constant, that may change from one line to the next. Instead of requiring m to be in H1 we shall allow it to range over the larger class A−α consisting of all functions g that are holomorphic iθ C on D and satisfy jg(re j ≤ (1−r)α . −α 2 2 Theorem 2.1 Let m be in A , and f be in La(jmj Aα). Then iθ O( 1 log( 1 )) jf(re )j = e 1−r 1−r ; 0 < r < 1 (2.2) and p jf^(n)j = eO( n log n); n > 1: (2.3) 1+r 3+r Proof: Let ρ be between 2 and 4 , and let it iθ it ρe + re P iθ (ρe ) = < re ρeit − reiθ be the Poisson kernel. Then Z 2π iθ 1 it it 1 log jf(re )j ≤ Preiθ (ρe )[log jfm(ρe )j + log ]dt: 2π 0 jm(ρeit)j Without loss of generality, m(0) 6= 0 (otherwise, this zero can be factored out). Then by Jensen's formula Z 2π 1 log− jm(ρeit)jdt ≤ C + α log 0 1 − ρ so Z 2π iθ 4 1 1 it it log jf(re )j ≤ [C + α log ] + Preiθ (ρe ) log jfm(ρe )jdt; 1 − r 1 − r 2π 0 5 and hence Z 2π iθ C 4α log 1 it it jf(re )j ≤ e 1−r e 1−r 1−r Preiθ (ρe ) log jfm(ρe )jdt: (2.4) 0 Multiply both sides of (2.4) by (1 − ρ2)α and integrate with respect to ρ 1+r 3+r from 2 to 4 .
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