The range of Toeplitz operators on the ball

Boris Korenblum State University of New York, Albany, New York 12222, U.S.A. John E. McCarthy ^{∗} Washington University, St. Louis, Missouri 63130, U.S.A.

October 24 1994

0 Introduction

Let B_{d} be the unit ball in ℂ^{d}, S_{d} be the boundary of B_{d}, and σ_{d} be normalized Lebesgue measure on S_{d}. The Hardy space H^{2}(B_{d}) is the closure in L^{2}(S_{d}, σ_{d}) of the analytic polynomials. The space H^{∞}(B_{d}) of bounded functions in H^{2}(B_{d}) is precisely the space of functions that are radial limits (σ_{d}-almost everywhere) of bounded analytic functions on B_{d}. Let P denote the orthogonal projection from L^{2}(S_{d}, σ_{d}) onto H^{2}(B_{d}). If m is in H^{∞}(B_{d}), the co-analytic Toeplitz operator T H^{2}(B_{d}) ; m̄ is defined by

T H^{2}(B_{d}) ; m̄f = P m̄f.

The purpose of this paper is to study the common range of all the co-anlytic Toeplitz operators T H^{2}(B_{d}) ; m̄.

For the case d = 1, it was shown in [2] that a function f is in the range of every non-zero co-analytic Toeplitz operator T H^{2}(B_{1}) ; m̄ if and only if the Taylor coefficients of f at zero satisfy

f̂(n) = O(e^{−c\sqrt{n}})

for some c > 0. It was also shown that, for the case d > 1, if the Taylor coefficients of some f in H^{2}(B_{d}) satisfy

f̂(α) = O(e^{−c|α|^{d/(d+1)}})

∗Partially supported by the National Science Foundation grant DMS 9301508

for some c > 0, then the function f will be in the range of every non-zero co-analytic Toeplitz operator on H^{2}(B_{d}). It was asked if this sufficient condition were also necessary. Our main theorem answers this question in the negative:

Theorem 1 Let f(z_{1}, …, z_{d}) = f_{1}(z_{1}) = ∑_{n=0}^{∞} a_{n}z_{1}^{n}, let ε > 0, and suppose that a_{n} = O(e^{−cn^{\frac{1}{2} +ε}}) for some c > 0. Then f is in the range of the Toeplitz operator T H^{2}(B_{d}) ; m̄ for every non-zero m in H^{∞}(B_{d}).

The exponent n^{\frac{1}{2} +ε} is not optimal - using results of [3] it can be improved to \sqrt{n} log n. We do not know what necessary and sufficient conditions are for a function to be in the range of every non-zero co-analytic Toeplitz operator.

In dimension d = 1, Szegö’s theorem [9] states that a necessary and sufficient condition for a positive bounded function g on the circle to be the modulus of a non-zero function in H^{∞}(B_{1}) is

∫_{S_{d}} log(g)dσ_{d} > −∞  (0.1)

For d > 1, condition (0.1) is necessary and sufficient for g to be the modulus of a function in the larger Nevanlinna class N(B_{d}), consisting of those holomorphic functions f on the ball for which

T (f, 1) := sup_{0<r<1} ∫_{S_{d}} log^{+} |f(rζ)|dσ_{d}(ζ) < ∞

[7, Theorem 10.11]. It is no longer sufficient, however, for g to be the modulus of a bounded analytic function, because the function

ζ ↦ ess sup_{−π≤θ≤π} |m(e^{iθ}ζ)|

must be lower semi-continuous on S_{d} if m is in H^{∞}(B_{d}) [7]. In [7, Theorem 12.5], Rudin proves that if g is log-integrable, and there exists some non-zero f in H^{∞}(B_{d}) with g ≥ |f| a.e. and g/|f| lower semi-continuous, then there does exist m in H^{∞}(B_{d}) with g = |m| a.e. We show

Theorem 2 Let d ≥ 2. There is a non-negative continuous function g on S_{d}, with ∫_{S_{d}} log(g)dσ_{d} > −∞, and which vanishes at only one point, but such that for no non-zero function m in H^{∞}(B_{d}) is |m| ≤ g almost everywhere [σ_{d}].

This answers question 15 of [7] in the negative.

When the original version of this paper was circulated in pre-print form (see the announcement in [1]), H. Alexander [private communication] produced a very simple constructive example of a function g satisfying the conclusion of Theorem 2, obviating the complicated construction in our proof. However, as we think our construction may be of some use in solving the problem of characterising exactly which functions are moduli of H^{∞}(B_{d}) functions, we elected to retain the proof of Theorem 2 in this paper.

1 Preliminary Lemmata

We need to know explicitly the projection from L^{2}(B_{d}) onto H^{2}(B_{d}). Let α = (α_{1}, …, α_{d}) be a multi-index and ζ = (z_{1}, …, z_{d}) a point in ℂ^{d}. The function ζ^{α} then maps ζ to z_{1}^{α_{1}} … z_{d}^{α_{d}}. The notation |α| stands for α_{1} + … α_{d}, and α! = α_{1}! … α_{d}!.

Lemma 1.1

∫_{S_{d}} ζ^{α}\overline{ζ^{β}}dσ_{d} = δ_{α,β} \frac{(d − 1)! α!}{(d − 1 + |α|)!}.  (1.2)

Moreover, if P_{H^{2}(B_{d})} denotes the projection from L^{2}(σ_{d}) onto H^{2}(B_{d}), then

P_{H^{2}(B_{d})}|z_{2}^{α_{2}}|^{2} … |z_{d}^{α_{d}}|^{2}\overline{z_{1}^{j}}z_{1}^{i} = \{ \begin{matrix}0 & if i < j \\ \frac{(d−1+i−j)! i! α_{2}! ...α_{d}!}{(i−j)! (d−1+i+α_{2}+...+α_{d})!}z_{1}^{i−j} & if i ≥ j. \end{matrix}  (1.3)

Proof: Formula (1.2) is proved in [6]. The expression on the left-hand side of (1.3) is orthogonal to every monomial except z_{1}^{i−j}; taking inner products gives the constant. □

We need to consider co-analytic Toeplitz operators on different spaces. If μ is a compactly supported measure on ℂ^{d}, let P^{2}(μ) denote the closure of the polynomials in L^{2}(μ), and let P_{P^{2}(μ)} denote the orthogonal projection from L^{2}(μ) onto P^{2}(μ). If m is a bounded analytic function on the support of μ, the co-analytic Toeplitz operator T P^{2}(μ) ; m̄ is defined by

T P^{2}(μ) ; m̄f = P_{P^{2}(μ)} m̄f.

When μ is σ_{d}, the space P^{2}(μ) is the Hardy space H^{2}(B_{d}), and we recover our original definition.

In order to transfer information about co-analytic Toeplitz operators with the same symbol on different spaces, we use the following lemma, whose proof is immediate:

Lemma 1.4 Let H be a Hilbert space of holomorphic functions on B_{d} in which the monomials are mutually orthogonal. Let m(z_{1}, …, z_{d}) = ∑_{β∈ℕ^{d}} b_{β}ζ^{β}. Then

T H ; m̄ \frac{ζ^{α}}{∥ζ^{α}∥_{H}^{2}} = ∑_{β≤α} b̄_{α−β} \frac{ζ^{β}}{∥ζ^{β}∥_{H}^{2}}  (1.5)

This lemma also allows us to define Toeplitz operators with an unbounded conjugate analytic symbol. The formal definition (1.5) defines an upper triangular operator, with respect to the orthonormal basis of normalized monomials. It therefore has a domain which contains all the polynomials; we extend its domain to include all functions on which Tm̄, thought of as a formal operator on the power series, gives a power series whose coefficients are the Taylor coefficients of some function in H.

Lemma 1.6 Let g be in the Nevanlinna class N(B_{1}), with g(0) ≠ 0, and 1 ≤ α < 2. Then

∫_{B_{1}}(log^{−} |g|)^{α}dA ≤ K,

where K is some constant depending only on T (g, 1), |g(0)| and α.

Proof: The proof is in two parts. First we prove it for g zero-free, then we prove it for g a Blaschke product.

(a) Suppose g has no zeroes in B_{1}, and without loss of generality assume ∥g∥_{∞} < 1. Then there is a singular measure μ_{s} such that, for any 0 < r < 1,

log^{−} |g(re^{iθ})| = \frac{1}{2π} ∫_{0}^{2π} P_{re^{iθ}}(e^{iϕ})[log^{−} |g(e^{iϕ})|dϕ + dμ_{s}(ϕ)]

where P_{re^{iθ}}(e^{iϕ}) is the Poisson kernel. Therefore

∫_{B_{1}} | log^{−} |g(re^{iθ})||^{α}rdrdθ
= ∫_{0}^{1} rdr ∫_{0}^{2π} dθ{∫_{0}^{2π} P_{re^{iθ}}(e^{iϕ})[log^{−} |g(e^{iϕ})|dϕ + dμ_{s}(ϕ)]}^{α}
≤ \{∫_{0}^{2π}(∫_{0}^{1} rdr ∫_{0}^{2π} dθ[P_{re^{iθ}}(e^{iϕ})]^{α})^{\frac{1}{α}} [log^{−} |g(e^{iϕ})|dϕ + dμ_{s}(ϕ)]\}^{α}

by Minkowski’s inequality. As the L^{α}(A) norm of the Poisson kernel for a fixed boundary point is at most (\frac{8}{2−α})^{\frac{1}{α}}, we get

∫_{B_{1}}(log^{−} |g(re^{iθ})|)^{α}rdrdθ ≤ (\frac{8}{2 − α})(\frac{1}{log |g(0)|})^{α}.

(b) Suppose now g is a Blaschke product with zero-set {w_{n}}. Then

[∫_{B_{1}} | log |g(z)||^{α}dA(z)]^{\frac{1}{α}} = [∫_{B_{1}}(∑_{n=0}^{∞} log |\frac{1−w̄_{n}z}{z−w_{n}}|)^{α}dA(z)]^{\frac{1}{α}}
≤ ∑_{n=0}^{∞}[∫_{B_{1}}(log |\frac{1−w̄_{n}z}{z−w_{n}}|)^{α}dA(z)]^{\frac{1}{α}}  (1.7)

Now let us estimate ∫_{B_{1}}(log |\frac{1−w̄_{n}z}{z−w_{n}}|)^{α}dA(z).

The terms for |w_{n}| ≤ \frac{1}{2} are dominated by T (g, 1) + log^{−} |g(0)|, by Jensen’s formula. For convenience, assume w is positive, and make the change of variables ζ = re^{iθ} = \frac{z−w}{1−wz}. Then

∫_{B_{1}}(log |\frac{1 − wz}{z − w} |)^{α}dA(z) = \frac{1}{π} ∫_{B_{1}}(log \frac{1}{r})^{α} \frac{(1 − w^{2})^{2}}{|1 − wre^{iθ}|^{4}} rdrdθ
= 2(1 − w^{2})^{2} ∫_{0}^{1}(log \frac{1}{r})^{α} \frac{1 + (rw)^{2}}{(1 − (rw)^{2})^{3}} rdr  (1.8)

Break the integral (1.8) into two pieces: from 0 to \frac{1}{e}, where the integrand is bounded by some constant C_{1} independent of w, and from \frac{1}{e} to 1. For the latter integral, use the inequality log \frac{1}{r} ≤ 2(1 − wr). One gets that (1.8) is bounded by C_{2} \frac{1}{2−α}(1 − w)^{α}, where C_{2} depends on neither w nor α. So (1.7) is dominated by (\frac{C_{2}}{2−α})^{\frac{1}{α}} ∑_{n=0}^{∞}(1 − |w_{n}|), and Jensen’s formula again means we can dominate everything by a constant depending on α, log |g(0)| and T (g, 1). □

Let A^{−n} consist of all holomorphic functions m in the unit disk that satisfy |m(z)| = O((1 − |z|)^{−n}). The space A^{0} is H^{∞}(B_{1}).

Lemma 1.9 Let f be in A^{−n} for some n, and 0 < α < 2. Then

∫_{B_{1}}(log^{−} |f|)^{α}dA < ∞.

Proof: We can assume that f(0) ≠ 0. As f need not be in N(B_{1}) we cannot apply Lemma 1.6 directly; but f is in the Nevanlinna class of certain smaller domains that touch the boundary of B_{1} at only one point, and we shall average over these.

Fix p strictly between 1 and \frac{2}{α}, let a = αp < 2, let q = \frac{p}{p−1} and let N > q. Let D_{1} be a smoothly bounded convex domain inside the disk, containing \{z : |z| < \frac{1}{2}\}, whose closure touches the unit circle only at 1, and which has a high degree of tangency at 1: let the boundary of D_{1} be {ρ(θ)e^{iθ} : −π ≤ θ ≤ π}, and assume 1 − ρ(θ) ∼ |θ|^{N}. For any other point ζ = e^{iθ_{0}} on the boundary of the unit disk, let D_{ζ} = e^{iθ_{0}}D_{1}.

Let ψ_{ζ} be the Riemann map of D_{ζ} onto B_{1} that takes 0 to 0 and ζ to ζ. As the boundary of D_{ζ} is smooth, it follows from the Kellog-Warschawski theorem (see e.g. [5]) that ψ_{ζ} and its derivatives extend continuously to the closure of D_{ζ}, so distances before and after the conformal mapping are comparable.

If r < \frac{1}{N}, then f is in H^{r}(D_{ζ}), and sup_{ζ∈S_{1}} ∥f ∘ ψ_{ζ}^{−1}∥_{H^{r}} < ∞. Thus

∫_{D_{e^{iθ}}} (log^{−} |f|)^{a}dA ≤ C, for all e^{iθ}.

Integrating with respect to θ and changing the order of integration yields

∫_{B_{1}}(log^{−} |f(re^{iϕ})|)^{a}(1 − r)^{\frac{1}{N}} rdrdϕ < ∞.

Now

∫_{B_{1}}(log^{−} |f|)^{α}dA ≤ [∫_{B_{1}}(log^{−} |f|)^{αp}(1 − r)^{\frac{p}{N}} dA]^{\frac{1}{p}} [∫_{B_{1}}(1 − r)^{− \frac{q}{N}} dA]^{\frac{1}{q}}
< ∞.
□

Let μ_{n} be the measure on the unit disk given by dμ_{n}(z) = \frac{1}{π}(1 − |z|^{2})^{n}dA(z), and let H_{n} be P^{2}(μ_{n}). It is routine to verify that in H_{n} the monomials are mutually orthogonal, and

∥z^{k}∥_{H_{n}}^{2} = \frac{n!}{(k + 1) … (k + n + 1)}.

The space H_{0} is the usual Bergman space for the disk. The following lemma is proved in [3] (in fact a slightly sharper form is proved). We include the following proof, which is sufficient for our purposes, for completeness:

Lemma 1.10 Let n ≥ 0, and m be a function in A^{−n}, not identically zero. Suppose f(z) = ∑_{k=0}^{∞} a_{k}z^{k} where a_{k} = O(e^{−ck^{\frac{1}{2} +ε}}) for some ε and c greater than 0. Then for any s ≥ 2n there exists g in H_{s} such that T H_{s} ; m̄g = f.

Proof: First, observe that f = T H_{s} ; m̄g for some g if and only if there is a constant C such that for all polynomials p

|⟨p, f⟩_{H_{s}}| ≤ C\sqrt{∫ |p|^{2}|m|^{2}dμ_{s}}.

So it is sufficient to prove that

| ∑_{k=0}^{∞} ā_{k}p̂(k)\frac{1}{(k + 1) … (k + s + 1)}| ≤ C\sqrt{∫ |p|^{2}|m|^{2}dμ_{s}}.

This in turn will follow from the Banach-Steinhaus theorem if we can show that for any function h in P^{2}(|m|^{2}μ_{s}),

ĥ(k) = O(e^{ck^{\frac{1}{2} +ε}}).  (1.11)

Now Stoll showed in [8] that if h satisfies

∫_{B_{1}}(log^{+} |h|)^{α}dA < ∞

for some α > 0 then ĥ(k) = O(e^{o( \frac{2}{2+α})}). We can assume ε is small, and take α = \frac{2−4ε}{1+2ε}. As h is in P^{2}(|m|^{2}μ_{s}), h(z)m(z)(1 − |z|^{2})^{s/2} = k(z) is in L^{2}(dA), and

log^{+} |h| ≤ log^{+} |k| + log^{−} |(1 − |z|^{2})^{s/2}| + log^{−} |m|.

The first two terms on the right are clearly integrable to the α^{th} power, and so is the third by Lemma 1.9; therefore h satisfies (1.11) as desired. □

We want to be able to restrict functions in the ball to planes and factor out zeros without losing control of the size of the function; the next lemma allows us to do this.

Lemma 1.12 Let m be holomorphic on B_{d} and satisfy

|m(z_{1}, …, z_{d})| ≤ C(1 − \sqrt{|z_{1}|^{2} + … |z_{d}|^{2}})^{−s}.

Suppose also that

m(z_{1}, …, z_{d}) = z_{d}^{t}m_{2}(z_{1}, …, z_{d}) + z_{d}^{t+1}m_{3}(z_{1}, …, z_{d}),

where m_{2} and m_{3} are analytic. Let m_{1}(z_{1}, …, z_{d−1}) = m_{2}(z_{1}, …, z_{d−1}, 0). Then

|m_{1}(z_{1}, …, z_{d−1})| ≤ (3d)^{s+t}C(1 − \sqrt{|z_{1}|^{2} + … |z_{d−1}|^{2}})^{−(s+t)}.

Proof: Let (z_{1}, …, z_{d−1}) be in B_{d−1}, and let ε = \frac{1}{3d}(1 − \sqrt{|z_{1}|^{2} + … + |z_{d−1}|^{2}}). Then the polydisk centered at (z_{1}, …, z_{d−1}, 0) with multi-radius (ε, …, ε) is contained in (1 − ε)B_{d}.

Integrating on the distinguished boundary of the polydisk we get

|m_{1}(z_{1}, …, z_{d−1})| = |m_{2}(z_{1}, …, z_{d−1}, 0)|
= |  ∫_{(z_{1},...,z_{d−1},0)+εT^{d}} \frac{m(ζ_{1},...,ζ_{d})}{ζ_{3}^{t}} |
≤  \frac{C}{ε^{s+t}} .
□

2 Common Range of Tm̄

We can now prove that a function that depends on only one variable is in the range of every T H^{2}(B_{d}) ; m̄ if its Taylor coefficients decay like e^{−ck^{\frac{1}{2} +ε}}.

Theorem 1 Let f(z_{1}, …, z_{d}) = f_{1}(z_{1}) = ∑_{n=0}^{∞} a_{n}z_{1}^{n}, let ε > 0, and suppose that a_{n} = O(e^{−cn^{\frac{1}{2} +ε}}) for some c > 0. Then f is in the range of the Toeplitz operator T H^{2}(B_{d}) ; m̄ for every non-zero m in H^{∞}(B_{d}).

Proof: For d = 1, this is proved (without the ε) in [2], so assume d ≥ 2. Fix m in H^{∞}(B_{d});

m(z_{1}, …, z_{d}) = ∑_{i_{1},...,i_{d}=0}^{∞} b_{i_{1},...,i_{d}}z_{1}^{i_{1}} … z_{d}^{i_{d}}.

Let

S = {(i_{2}, …, i_{d}) : for some i_{1}, b_{i_{1},...,i_{d}} ≠ 0}.

Define

t_{d} = inf{i_{d} : for some i_{2}, …, i_{d−1}, (i_{2}, …, i_{d−1}, i_{d}) ∈ S},

and define t_{k} inductively by

t_{k} = inf{i_{k} : for some i_{2}, …, i_{k−1}, (i_{2}, …, i_{k−1}, i_{k}, t_{k+1}, …, t_{d}) ∈ S}.

Let n = t_{2} + … + t_{d}.

Case (a): n = 0.

Then the function

m_{1}(z_{1}) = m(z_{1}, 0, …, 0)

is not identically zero, and is in H^{∞}(B_{1}). By Lemma 1.1,

T H^{2}(B_{d}) ; m̄z_{1}^{i} = ∑_{j} b̄_{j,0,...,0} \frac{(i − j + 1) … (i − j + d − 1)}{(i + 1) … (i + d − 1)}z_{1}^{i−l}.

So by Lemma 1.4, if one can solve the equation

T H_{d−2} ; m̄_{1}g_{1} = f_{1}  (2.1)

for some g_{1} in H_{d−2}, then g(z_{1}, …, z_{d}) = g_{1}(z_{1}) solves

T H^{2}(B_{d}) ; m̄g = f,

and by Equation (1.2) ∥g∥_{H^{2}(B_{d})} = \sqrt{(d − 1)!}∥g_{1}∥_{H_{d−1}} < ∞. By Lemma 1.10, equation (2.1) has a solution.

Case (b): n > 0.

One can decompose m as

m(z_{1}, …, z_{d}) = z_{2}^{t_{2}} … z_{d}^{t_{d}}m_{2}(z_{1}, …, z_{d}) + m_{3}(z_{1}, …, z_{d}),

where each term in the expansion of m_{3} is divisible by some z_{k}^{t_{k}+1}. Applying Lemma 1.12 inductively, m_{1}(z) = m_{2}(z, 0, …, 0) is in A^{−n}, and by the choice of t_{2}, …, t_{d}, it is not identically zero. Consider the function

f_{2}(z) = ∑_{k=0}^{∞} a_{k}(k + d)(k + d + 1) … (k + dn + 1)z^{k}.

As d ≥ 2, we can apply Lemma 1.10 with s = dn, so there is

g_{2}(z) = ∑_{k=0}^{∞} γ_{k}(k + 1)(k + 2) … (k + dn + 1)z^{k}

in H_{dn} with

T H_{dn} ; m̄_{1}g_{2} = f_{2}.  (2.2)

Define g by

g(z_{1}, …, z_{d}) = \frac{1}{t_{2}! … t_{d}!}z_{2}^{t_{2}} … z_{d}^{t_{d}} ∑_{k=0}^{∞} γ_{k}(k + 1)(k + 2) … (k + n + d − 1)z_{1}^{k}.

The function g is in H^{2}(B_{d}) because

∥g∥_{H^{2}(B_{d})}^{2} = \frac{(d − 1)!}{t_{2}! … t_{d}!} ∑_{k=0}^{∞} |γ_{k}|^{2}(k + 1) … (k + n + d − 1)
≤ \frac{(d − 1)!}{t_{2}! … t_{d}!} ∑_{k=0}^{∞} |γ_{k}|^{2}(k + 1) … (k + dn + 1)
= \frac{(d − 1)!}{t_{2}! … t_{d}!}∥g_{2}∥_{H_{(d−1)n}}^{2}
< ∞.

Moreover

T H^{2}(B_{d}) ; m̄g = T_{z̄_{2}^{t_{d}}...z̄_{d}^{t_{d}} \overline{m_{1}(z_{1})}}^{H^{2}(B_{d})}g

is a function of z_{1} only; it is, in fact, f. For if T H^{2}(B_{d}) ; m̄g = ∑_{k=0}^{∞} e_{k}z_{1}^{k}, and m_{1}(z) = ∑_{k=0}^{∞} c_{k}z^{k}, then taking the inner product with z_{1}^{j} we get

\frac{(d − 1)!}{(j + 1) … (j + d − 1)}e_{j} = ⟨T H^{2}(B_{d}) ; m̄g, z_{1}^{j}⟩_{H^{2}(B_{d})}
= ⟨g, z_{2}^{t_{2}} … z_{d}^{t_{d}}m_{1}z_{1}^{j}⟩_{H^{2}(B_{d})}
= (d − 1)! ∑_{k=j}^{∞} γ_{k}c̄_{k−j}  (2.3)

Taking the inner product with z^{j} in Equation(2.2), we get

\frac{1}{(j + 1) … (j + d − 1)}a_{j} = ⟨T H_{dn} ; m̄_{1}g_{2}, z^{j}⟩_{H_{dn}}
= ⟨g_{2}, m_{1}z^{j}⟩_{H_{dn}}
= ∑_{k=j}^{∞} γ_{k}c̄_{k−j}  (2.4)

Comparing Equations(2.3) and (2.4), we see that T H^{2}(B_{d}) ; m̄g = f, as desired. □

3 Boundary moduli

Define F_{c,w} by

F_{c,w}(z) = exp(c\frac{1 − |w|^{2}}{(1 − ⟨z, w⟩)^{d+1}} )  (3.1)

We need the following two results. The first was proved by Drewnowski; a proof is given in [4, Lemma 3.2].

Lemma 3.2 (Drewnowski)

lim_{c→0} sup_{w∈B_{d}} ∫_{S_{d}} log(1 + |cF_{c,w}|)dσ_{d} = 0.

The second result, due to Nawrocki, estimates the growth of the Taylor coefficients of F_{c,w}. We are interested in w = re_{1} = (r, 0, …, 0); in this case all the Taylor coefficients of F_{c,re_{1}} are positive, and the following follows easily from the proof of [4, Lemma 3.3]:

Lemma 3.3 (Nawrocki) For each c > 0 there exists ε > 0 such that

inf_{i∈ℕ} sup_{0<r<1} \sqrt{\frac{(d − 1 + i)!}{(d − 1)! i!}}F̂_{c,re_{1}}(i, 0, …, 0)e^{−εi^{\frac{d}{d+1}}} > 0.

We can now use our knowledge of the common range of co-analytic Toeplitz operators to prove:

Theorem 2 Let d ≥ 2. There is a continuous non-negative function g on S_{d}, vanishing only at the point e_{1}, and satisfying ∫_{S_{d}} log(g)dσ_{d} > −∞, with the property that the only function m in H^{∞}(B_{d}) with |m| ≤ g almost everywhere [σ_{d}] is the zero function.

Proof: Let V_{n} = \{ζ ∈ S_{d} : |ζ − e_{1}| ≥ \frac{1}{n}\}. By Lemma 3.3, for any sequence c_{n} tending to zero, one can choose i_{n} and r_{n} such that

F̂_{c_{n},r_{n}e_{1}}(i_{n}, 0, …, 0) > \frac{n}{c_{n}} e^{(i_{n})^{\frac{4}{7}}}  (3.4)

(because \frac{4}{7} < \frac{d}{d+1}). Moreover, by passing to a subsequence, one can assume that

sup_{ζ∈V_{n}} c_{n}|F_{c_{n},r_{n}e_{1}}(ζ)| ≤ \frac{1}{2^{n}} ,  (3.5)

because ζ ∈ V_{n} implies that |1 − ⟨ζ, r_{n}e_{1}⟩| ≥ \frac{1}{2n^{2}} , and that

∫_{S_{d}} log(1 + |c_{n}F_{c_{n},r_{n}e_{1}}|)dσ_{d} ≤ \frac{1}{2^{n}} ,

by Lemma 3.2. Define g by

g(ζ) = \sqrt{\frac{1}{1 + ∑_{n=1}^{∞} |c_{n}F_{c_{n},r_{n}e_{1}}(ζ)|^{2}}} .

It follows from (3.5) that g is continuous and vanishes only at e_{1}. Moreover

∫_{S_{d}} log gdσ_{d} = −\frac{1}{2} ∫_{S_{d}} log(1 + ∑_{n=1}^{∞} |c_{n}F_{c_{n},r_{n}e_{1}}|^{2})dσ_{d}
> − ∫_{S_{d}} log ∏_{n=1}^{∞}(1 + |c_{n}F_{c_{n},r_{n}e_{1}}|)^{2}dσ_{d}
= −2 ∑_{n=1}^{∞} ∫_{S_{d}} log(1 + |c_{n}F_{c_{n},r_{n}e_{1}}|)dσ_{d}
≥  −2.

Now suppose there is a non-zero m in H^{∞}(B_{d}) with |m| ≤ g a.e. Then each of the functions c_{n}F_{c_{n},r_{n}e_{1}}, being analytic in the ball of radius \frac{1}{r_{n}}, is in P^{2}(|m|^{2}σ); moreover they are all of norm less than one in this space, because

∫_{S_{d}} |c_{n}F_{c_{n},r_{n}e_{1}}|^{2}|m|^{2}dσ ≤ ∫_{S_{d}} |c_{n}F_{c_{n},r_{n}e_{1}}|^{2}g^{2}dσ < 1.

Let

f(z_{1}, …, z_{d}) = ∑_{k=0}^{∞} e^{−k^{\frac{4}{7}}} \frac{(k + d − 1)!}{(d − 1)!k!}z_{1}^{k}.

By Theorem 1, there is a function h in H^{2}(B_{d}) with

T H^{2}(B_{d}) ; m̄h = f.

It follows that the linear map

Γ : p ↦ ⟨p, f⟩_{H^{2}(B_{d})},

defined a priori on the polynomials, extends by continuity to a bounded linear map on P^{2}(|m|^{2}σ), as

|Γ(p)| = |⟨p, P (m̄h)⟩| = | ∫ pmh̄dσ_{d}| ≤ ∥h∥_{H^{2}(B_{d})}∥p∥_{P^{2}(|m|^{2}σ)}.

Moreover, each function c_{n}F_{c_{n},r_{n}e_{1}} is uniformly approximated on S_{d} by the partial sums of its Taylor series; hence

Γ(c_{n}F_{c_{n},r_{n}e_{1}}) = ∑_{k=0}^{∞} c_{n}F̂_{c_{n},r_{n}e_{1}}(k)e^{−k^{\frac{4}{7}}} .  (3.6)

But all the terms on the right-hand side of (3.6) are positive, and the i_{n}^{th} term is at least n by Equation (3.4). This contradicts the boundedness of Γ. □

References

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