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 √ 1 P∞ n 2 have norm n+1 ; so a holomorphic function f(z) = n=0 anz is in La if and P∞ 2 1 PN n only if n=0 |an| n+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 − |z|2)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 ∞ form |m(z)| 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(|m| Aα) to denote the space of analytic functions 2 2 2 2 on D that also lie in L (|m| Aα), and P (|m| 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 (|m| Aα). These two spaces are, in general, different. If m 2 2 2 has no zeroes, L (|m| Aα) is just the set of quotients {f/m : f ∈ La}; but this 2 2 2 2 coincides with P (|m| Aα) only when 1/m is in P (|m| Aα), which in turn is 2 equivalent to requiring that 1 lie in the La closure of {mp : p a polynomial}, 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 (|m| Aα) is all of La(|m| Aα). 2 2 2 2 We are interested in obtaining, for functions in P (|m| Aα) and La(|m| 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.

∞ 2 2 Theorems 2.1, 2.6, 1.4 Let m be in H . Let f be in La(|m| Aα). Then

√ |fˆ(n)| = eO( n log n).

2 2 If f is in La(|m| A), then

√ |fˆ(n)| = eO( n).

If f is in P 2(|m|2A), then

√ |fˆ(n)| = eo( n).

Theorem 3.6 Assume m in H∞ is zero-free, and suppose the Taylor series 1 1 P∞ n expansion of m is given by m(z) = n=0 αnz . Then

v u n ˆ 2 2 uX 2 sup{|f(n)| : f ∈ La(|m| A), kfk ≤ 1} = t (n + 1 − j)|αj| j=0

2 1 P 2(|m|2A)

2 2 2 Let PLa denote the orthogonal projection from L (A) onto La, and for m 2 2 ∞ 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, | R pfdA¯ |2 ≤ C R |p|2|m|2dA, i.e. if the functional D p 7→ R pfdA¯ is bounded on P 2(|m|2A). If this functional is bounded, we D shall say f is in the dual of P 2(|m|2A). The important point is that if P∞ ˆ n 2 2 P∞ n f(z) = n=0 f(n)z is in the dual of P (|m| A), and g(z) = n=0 gˆ(n)z is in P 2(|m|2A), then the sum

∞ 1 X gˆ(n)fˆ(n) n=0 n + 1 must converge; so knowing a function f is in the dual of P 2(|m|2A) means that all functions g satisfy n + 1 gˆ(n) = o( ). fˆ(n)

The common range of co-analytic Toeplitz operators on H2 was charac- terized in [5] in terms of the Fourier coeﬃcients fˆ(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 √ that fˆ(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 √ functions f satisfying fˆ(n) = O(e−c n) for some c, is what they call Toeplitz stable, i.e. it is a subspace with the properties that ∞ H2 (i) If f is in Y and m is an outer function in H , then Tm¯ f is in Y . ∞ H2 (ii) If f is in Y and m is an outer function in H , then f = Tm¯ h for some h in Y .

P∞ 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- √ 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 ∞ ∞ H2 X n X n Tm¯ cnz = dnz n=0 n=0 then ∞ ∞ 2 La X n X n Tm¯ cn(n + 1)z = dn(n + 1)z , (1.3) n=0 n=0 P∞ 2 whenever n=0 |cn| (n + 1) < ∞. Now for each m there is some function 2 P∞ n P∞ 2 1 La g(z) = n=0 bnz , where n=0 |bn| n+1 < ∞, with Tm¯ g = f. By (1.3), H2 P∞ 1 n P∞ 1 n P∞ 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 {an }, √ n+1 −c n and hence also {an}, is O(e ). ∞ P∞ an n Suﬃciency: Fix m in H . Let f1(z) = n=0 n+1 z . Then f1 is in Y , P∞ n H2 so there is some function g1(z) = n=0 bnz in Y with Tm¯ g1 = f1. Let 2 P∞ 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 H∞ and g be in P 2(|m|2A). Then √ |gˆ(n)| = eo( n).

4 2 2 2 La(|m| Aα)

2 2 We ﬁrst estimate the growth of the Taylor coeﬃcients of functions in La(|m| 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 H∞ 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 |g(re | ≤ (1−r)α .

−α 2 2 Theorem 2.1 Let m be in A , and f be in La(|m| Aα). Then

iθ O( 1 log( 1 )) |f(re )| = e 1−r 1−r , 0 < r < 1 (2.2)

and √ |fˆ(n)| = 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

Without loss of generality, m(0) 6= 0 (otherwise, this zero can be factored out). Then by Jensen’s formula

Z 2π 1 log− |m(ρeit)|dt ≤ C + α log 0 1 − ρ so

Z 2π iθ 4 1 1 it it log |f(re )| ≤ [C + α log ] + Preiθ (ρe ) log |fm(ρe )|dt, 1 − r 1 − r 2π 0

5 and hence

Z 2π iθ C 4α log 1 it it |f(re )| ≤ e 1−r e 1−r 1−r Preiθ (ρe ) log |fm(ρe )|dt. (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 . This gives

iθ C C 4α log 1 |f(re )| ≤ e 1−r e 1−r 1−r , (2.5) (1 − r)α+1

which yields (2.2). Cauchy’s formula tells us that for any r < 1,

ˆ 1 iθ |f(n)| ≤ n sup |f(re )|. r 0≤θ≤2π

√1 Choosing r = 1 − n in (2.2) yields (2.3). 2

Note that if α = 0 the bound in (2.5) is sharper, which enables us to eliminate the log-term in the coeﬃcient estimate:

∞ 2 2 Corollary 2.6 If m is in H and f is in La(|m| A), then √ |fˆ(n)| = eO( n).

Let F + be the space of functions f satisfying

√ |fˆ(n)| = eo( n). (2.7)

This space was ﬁrst studied by Yanagihara [13]; he showed that a function f is in the algebra F + if and only if

( o(1) ) |f(z)| = e 1−|z| . (2.8)

6 Yanagihara also showed that the Smirnov class is contained in F + [14] (see also [6]), so in particular it contains the reciprocal of every outer function. It also contains, by a result of J. Shapiro and A. Shields [10], those singular inner functions whose associated measures are non-atomic; we include a proof for completeness. (For definitions and basic properties of the Smirnov class, inner and outer functions see e.g. the book [1].) Proposition 2.9 Suppose the H∞ function m is the product of an outer function, h, and a singular inner function, u, whose associated measure is 1 + non-atomic. Then m is in F . Proof: Assume |m| is given by Z iθ − log |u(z)| = Pz(e )dµ(θ). (2.10) T Define l(s) = sup{µ(I): Ian interval, σ(I) = s}, and notice that lims→0 l(s) = 0 precisely when µ is non-atomic. Without loss of generality, take z = r to be positive, and break the integral (2.10) into the integrals over the sets where 1 (i) |θ| < (1 − r) 4 , and 1 (ii) |θ| ≥ (1 − r) 4 . 1 1+r 4 The first integral is less than {l[2(1 − r) ]}{ 1−r }, and the second is less 1 1 i(1−r) 4 i(1−r) 4 √ 1 than kµkPr(e ). For r close to 1, Pr(e ) is approximately 1−r . 1 Therefore m satisfies (2.8), as required. 2 2 2 We want to consider conditions on m which imply that La(|m| A) is contained in F +. Theorem 2.11 Let m in H∞ be the product of an outer function, a singular inner function whose associated measure is non-atomic, and a Blaschke prod- ∞ P∞ − 1 uct whose zero-set {wn}n=1 satisfies n=0(1 − |wn|)(log (1 − |wn|)) 2 < ∞. 2 2 Let f be in La(|m| A). Then √ |fˆ(n)| = eo( n).

7 Proof: Let B be the Blaschke product with the same zero-set as m, so m = Bm1 where m1 is the product of a non-atomic singular inner function 2 2 + and an outer function. Let g = fm1 ∈ La(|B| A). If we can show g is in F , then so is f, because 1 is in F + by the preceding remarks. To show g is in m1 F +, it is enough to show that Z (log− |B|)2dA < ∞, (2.12) D for this implies that Z (log+ |g|)2dA < ∞, D and this in turn implies that g is in F + by a theorem of Stoll [12].

So, suppose B is a Blaschke product with zero-set {wn}. Then

Z 1 1 2 R P∞ 1−w¯nz 2 [ | log |B(z)|| dA(z)] 2 = [ ( log | |) dA(z)] 2 D n=0 z−wn D 1 P∞ R 1−w¯nz 2 ≤ [ (log | |) dA(z)] 2 (2.13) n=0 D z−wn

Now let us estimate R (log | 1−w¯nz |)2dA(z). D z−wn 1 As each term in (2.13) is ﬁnite, we can assume each |wn| ≥ 2 . For convenience, assume w is positive, and make the change of variables ζ = iθ z−w re = 1−wz . Then

Z Z 2 2 1 − wz 2 1 1 2 (1 − w ) (log | |) dA(z) = (log ) iθ 4 rdrdθ D z − w π D r |1 − wre | Z 1 1 1 + (rw)2 = 2(1 − w2)2 (log )2 rdr(2.14) 0 r (1 − (rw)2)3

1 Break the integral (2.14) into two pieces: from 0 to e , where the integrand 1 is bounded by some constant C1 independent of w, and from e to 1. For the 1 latter integral, use the inequality log r ≤ 2(1 − wr). One gets that (2.14) is bounded by C(1 − w)2(1 + log−(1 − w)), so by hypothesis 2.13 is ﬁnite, as required. 2

8 3 Kernel Functions

Let us ﬁx some function m in H∞. Every function in P 2(|m|2A) is analytic on D, and evaluation at each point of D is a bounded linear functional. Therefore m for each w in D, there exists some kernel function kw satisfying Z m 2 f(w) = f(z)kw (z)|m| dA(z). D m m As |f(w)| ≤ kfkP 2(|m|2A)kkw kP 2(|m|2A), we want to estimate kkw kP 2(|m|2A) = q m kw (w). When m ≡ 1, the kernel function, which we shall then call kw, is 1 known to be kw(z) = (1−wz¯ )2 . Therefore for any polynomial p, Z Z m p(w) = p(z)kw(z)dA(z) = p(z)kw (z)m(z)m(z)dA(z). (3.1) D D m 2 2 m 2 m As kw is in P (|m| A), mkw is in La, and (3.1) says that kw andmmk ¯ w have the same inner product with polynomials, i.e.

2 1 m La m = k = P 2 mmk¯ = T mk (3.2) (1 − wz¯ )2 w La w m¯ w

m Thus to ﬁnd kw , we must solve (3.2). Consider the function 1 1 1 Km(z) = , (3.3) w m(w) m(z) (1 − wz¯ )2

2 2 m and let us suppose for the time being that it lies in P (|m| A); then Kw m must in fact be kw , for

Z Z 1 1 1 2 1 1 p(z) 2 |m(z)| dA(z) = p(z)m(z) 2 dA(z) D m(w) m(z) (1 − wz¯ ) m(w) D (1 − wz¯ ) = p(w).

m 2 2 1 2 2 When is Kw in P (|m| A)? This occurs if and only if m is in P (|m| A), which in turn happens if and only if m is a cyclic vector for the Bergman 2 ∞ shift (the operator of multiplication by z on La). For an H function m,

9 a necessary and suﬃcient condition that it be cyclic for the Bergman shift is that it not vanish on D and that the singular measure associated with its singular inner factor put no mass on any BCH set (necessity was proved by H. Shapiro [9], suﬃciency independently by B. Korenblum [3] and J. Roberts [8]; see the survey article by A. Shields [11]). A BCH set (which stands for Beurling-Carleson-Hayman set) is a compact subset of the circle, of Lebesgue P∞ measure zero, for which n=0 σ(In) log(σ(In)) > −∞, where {In} are the disjoint open arcs of the complement. m Thus, for m a cyclic vector for the Bergman shift, (3.3) gives us kw explicitly, and hence the norm of evaluation at w. If m is not cyclic, then m m 2 2 2 2 kw is the projection of Kw from L (|m| A) onto P (|m| A); unfortunately, we do not know how to calculate this explicitly. Summarising the above, we have proved:

Theorem 3.4 Let m be in H∞, and suppose m(w) 6= 0. Then the reproducing kernel on P 2(|m|2A) for evaluation at w is given by

m m kw (z) = PP 2(|m|2A)[Kw (z)] 1 1 1 = P 2 2 [ ]. P (|m| A) m(w) m(z) (1 − wz¯ )2

m 2 2 If m is cyclic for the Bergman shift then Kw lies in P (|m| A), and the norm 1 1 of evaluation at the point w is |m(w)| 1−|w|2 ; otherwise the norm is strictly smaller than this.

2 2 A similar discussion applies to La(|m| A). The reproducing kernel at w m m 2 2 2 2 is PLa(|m| A)Kw , and Kw lies in La(|m| A) if and only if m is zero-free. m If m vanishes at w, it is easy to work out what Kw should be. A calcu- lation gives:

Scholium 3.5 Suppose m has a zero of order n at w. Then

n m (n + 1)! 1 z k (z) = P 2 2 . w P (|m| A) m¯ (n)(w) m(z) (1 − wz¯ )n+2

10 m Once one knows the reproducing kernels kw for evaluation at w, one also knows the kernels that, when integrated against a polynomial, give the values of some derivative of the polynomial; for if Z m 2 p(w) = p(z)kw (z)|m(z)| dA(z), then Z (n) (n) m 2 Dw p(w) = p(z)Dw kw (z)|m(z)| dA(z),

(n) where Dw denotes diﬀerentiating n times with respect to w. This observa- tion enables us to calculate explicitly the growth rate of the Taylor coeﬃcients of a function in P 2(|m|2A) when m is cyclic for the Bergman shift:

Theorem 3.6 Assume m in H∞ is cyclic for the Bergman shift, and suppose 1 1 P∞ n the Taylor series expansion of m is given by m(z) = n=0 αnz . Then q ˆ 2 2 2 2 2 sup{|f(n)| : f ∈ P (|m| A), kfk ≤ 1} = |αn| + 2|αn−1| + ... + (n + 1)|α0| (3.7)

m 2 2 Proof: Let kw,n be the function in P (|m| A) that gives the value of the nth derivative at w: Z (n) m 2 p (w) = p(z)kw,n(z)|m(z)| dA(z).

m th m As remarked above, kw,n is just the n derivative, with respect tow ¯, of kw , so from Theorem 3.4 we have 1 1 1 km (z) = D(n) w,n w¯ m(z) m(w) (1 − wz¯ )2 ! 1 n (n) 1 1 = [ Dw¯ ( )( ) m(z) 0 m(w) (1 − wz¯ )2 ! n (n−1) 1 1 + Dw¯ ( )Dw¯( ) + ... 1 m(w) (1 − wz¯ )2 ! n 1 (n) 1 + ( )(Dw¯ )] n m(w) (1 − wz¯ )2

11 In particular, when w = 0, we get n ! m 1 X n (n−j) 1 j k0,n(z) = [ Dw¯ ( )|w¯=0(j + 1)!z ] m(z) j=0 j m(w) n 1 X j = [n! (j + 1)¯αn−jz ]. m(z) j=0

1 m The left-hand side of (3.7) is n! kk0,nk. Calculating:

m 2 n m kk0,nk = Dz (k0,n(z))|z=0 n n 1 X j = n!Dz [ (j + 1)¯αn−jz ]|z=0 m(z) j=0 n ! X n (n−j) 1 = n! [Dz ( )(j + 1)¯αn−jj!]|z=0 j=0 j m(z) n 2 X 2 = (n!) (j + 1)|αn−j| . j=0 Taking the square-root, and dividing by n!, one gets that the norm of the functional that sends a function to its nth Taylor coeﬃcient at 0 is the right- hand side of (3.7), as desired. 2 The same technique as above enables one to estimate the growth rate of Taylor coeﬃcients of functions in H2(|m|2σ). One gets:

1 P∞ n Theorem 3.8 Let m be an outer function, and suppose m(z) = n=0 αnz . Then n ˆ 2 2 X 2 1 sup{|f(n)| : f ∈ H (|m| σ), kfk ≤ 1} = [ |αj| ] 2 . j=0

Note that this is a qualitative improvement on the “asymptotic Szeg¨o theorem,” Theorem 3.1 in [6], which just states that, for any c > 0, the √ c n left-hand side is less than or equal to C1e .

Acknowledgement: The author would like to thank the referee for the key ideas in the proof of Theorem 2.1, which allowed the theorem to be signiﬁcantly strengthened from its original form.

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