CARLESON MEASURES FOR BESOV SPACES ON THE BALL WITH APPLICATIONS

H. TURGAY KAPTANOĞLU

Abstract. Carleson and vanishing Carleson measures for Besov spaces on the unit ball of ℂ^{N} are characterized in terms of Berezin transforms and Bergmanmetric balls. The measures are defined via natural imbeddings of Besov spaces into Lebesgue classes by certain combinations of radial derivatives. Membership in Schatten classes of the imbeddings is considered too. Some Carleson measures are not finite, but the results extend and provide new insight to those known for weighted Bergman spaces. Special cases pertain to Arveson and Dirichlet spaces, and a unified view with the usual Hardy-space Carleson measures is presented by letting the order of the radial derivatives tend to 0. Weak convergence in Besov spaces is also characterized, and weakly 0-convergent families are exhibited. Applications are given to separated sequences, operators of Forelli-Rudin type, gap series, characterizations of weighted Bloch, Lipschitz, and growth spaces, inequalities of Fejér-Riesz and Hardy-Littlewood type, and integration operators of Cesàro type.

1. Introduction

We let 𝔹 be the unit ball of ℂ^{N} and H(𝔹) the space of holomorphic functions on 𝔹. When N = 1, we have the unit disc 𝔻. Unless otherwise specified, our main parameters and their range of values are

q ∈ ℝ,  0 < p < ∞,  s ∈ ℝ,  t ∈ ℝ,  0 < r < ∞;

and given q and p, we often choose t to satisfy

q + pt > −1. (1)

Let ν be the volume measure on 𝔹 normalized with ν(𝔹) = 1. We define on 𝔹 also the measures

dν_{q}(z) = (1 − |z|^{2})^{q} dν(z), (2)

which are finite only for q > −1, where |z|^{2} = ⟨z, z⟩ and ⟨z, w⟩ = z_{1}\overline{w}_{1}+⋯+z_{N} \overline{w}_{N}. The corresponding Lebesgue classes are L_{q}^{p}. We also let dμ_{q}(z) = (1 − |z|^{2})^{q} dμ(z) for a general measure μ on 𝔹.

Consider the linear transformation I_{s}^{t} defined for f ∈ H(𝔹) by

I_{s}^{t}f(z) = (1 − |z|^{2})^{t} D_{s}^{t}f(z),

2000 Mathematics Subject Classification. Primary 32A37, 47B38; Secondary 32A36, 32A35, 32A18, 46E15, 46E22, 46E35, 47B07, 47B10, 30B10.

Key words and phrases. Carleson measure, Berezin transform, Bergman metric, Bergman projection, weak, ultraweak convergence, Schatten-von Neumann ideal, Besov, Bergman, Dirichlet, Hardy, Arveson, Bloch, Lipschitz, growth space, separated sequence, Forelli-Rudin-type operator, lacunary series, Fejér-Riesz, Hardy-Littlewood inequality, Cesàro-type operator.

The research of the author is partially supported by a Fulbright grant.

where D_{s}^{t} is a bijective radial differential operator on H(𝔹) of order t for any s, and every I_{s}^{0} is the identity I. The following definition is known to be independent of s, t, where the term norm is used even when 0 < p < 1; see [K2, Theorem 4.1] or [BB, Theorem 5.12 (i)], for example.

Definition 1.1. The Besov space B_{q}^{p} consists of all f ∈ H(𝔹) for which the function I_{s}^{t}f belongs to L_{q}^{p} for some s, t satisfying (1). The L_{q}^{p} norms of I_{s}^{t}f are all equivalent. We call any one of them the B_{q}^{p} norm of f and denote it by ∥f∥_{B_{q}^{p}}.

So I_{s}^{t} is an imbedding of B_{q}^{p} into L_{q}^{p}. The necessary background for B_{q}^{p} spaces is given in Section 3. They are all complete, Banach spaces for p ≥ 1, and Hilbert spaces for p = 2. They include many known spaces as special cases.

Definition 1.2. We call a positive Borel measure μ on 𝔹 a Carleson measure for B_{q}^{p} provided some I_{s}^{t} maps B_{q}^{p} into L^{p}(μ) continuously.

Now we are ready to state our main result. Here, the b(w, r) is the ball in the Bergman metric with center w ∈ 𝔹 and radius r, and an r-lattice is defined by Lemma 2.5. As is commonly used, C is a finite positive constant whose value might be different at each occurrence. The context makes it clear what each C depends on, but C never depends on the functions in the formula in which it appears.

Theorem 1.3. Let q be fixed but unrestricted. Let p and r, and also s be given. The following conditions are equivalent for a positive Borel measure μ on 𝔹.

(i) There is a C such that

sup_{w∈𝔹} \frac{μ(b(w, r))}{ν_{q}(b(w, r))} ≤ C.

(ii) There is a C such that if {a_{n}} is an r-lattice in 𝔹, then

sup_{n∈ℕ} \frac{μ(b(a_{n}, r))}{ν_{q}(b(a_{n}, r))} ≤ C.

(iii) There is a C such that if t satisfies (1), then

∫_{𝔹} |I_{s}^{t}f|^{p} dμ ≤ C ∥f∥_{B_{q}^{p}}^{p} (f ∈ B_{q}^{p}).

(iv) There is a C such that if t satisfies (1), then

sup_{w∈𝔹} (1 − |w|^{2})^{N+1+q+pt} ∫_{𝔹} \frac{(1 − |z|^{2})^{pt}}{|1 − ⟨z, w⟩|^{(N+1+q+pt)2}} dμ(z) ≤ C.

Condition (iii) is the statement that μ is a Carleson measure for B_{q}^{p}.

As is common with Carleson-measure theorems, the property of being a Carleson measure is independent of p or r, and now also of s, t as long as (1) holds, because (i) is true for any p, s, t and (iii) is true for any r. However, all conditions depend on q. So for a fixed q, a Carleson measure for one B_{q}^{p} with one suitable s, t is a Carleson measure for all B_{q}^{p} with the same q with any other such s, t. And we conveniently call such a μ also a q-Carleson measure. So setting

qμ̂_{r}(w) = \frac{μ(b(w, r))}{ν_{q}(b(w, r))} (w ∈ 𝔹),

a q-Carleson measure is a positive Borel measure on 𝔹 for which the averaging function _{q}μ̂_{r} is bounded on 𝔹 for some r. Thus Theorem 1.3 gives a full characterization of q-Carleson measures for all real q.

We can draw some immediate conclusions from Theorem 1.3. Clearly the model q-Carleson measure is ν_{q}. So Carleson measures need not be finite for q ≤ −1. By Lemma 2.2 below, ν_{q}(b(w, r)) is of order (1−|w|^{2})^{N+1+q}. Thus by (i), any ν_{q_{1}} with q_{1} > q is also a q-Carleson measure while no ν_{q_{2}} with q_{2} < q is. Further, by (i) again, any finite Borel measure is a q-Carleson measure for q ≤ −(N + 1). And for q = −(N + 1), q-Carleson measures are precisely those Borel measures that are finite on Bergman balls of a fixed radius. On the question of finiteness, with w = 0 and b = pt, (iv) immediately implies the following.

Corollary 1.4. If μ is a q-Carleson measure, then the measure μ_{β} is finite for any β with β + q > −1.

Theorem 1.3 is better appreciated when we restrict to q > −1. Then t = 0 satisfies (1) for any p, and by Definition 1.1, the space B_{q}^{p} coincides with the weighted Bergman space A_{q}^{p}. In this case Theorem 1.3 becomes a well-known result, and Corollary 1.4 implies that a Carleson measure must then be finite; see [CM, Theorem 2.36] for N = 1. But it is possible to take t ≠ 0 also with q > −1 as long as t satisfies (1); then Theorem 1.3 extends known results for weighted Bergman spaces by giving equivalences also with I_{s}^{t} in place of the inclusion map.

Moreover, the space B_{−1}^{2} is the Hardy space H^{2}. Now (1) requires a t > 0, no matter how small. It follows that Definition 1.2 and Theorem 1.3 are about Carleson measures different from the usual Carleson measures on H^{2}. However, as t → 0^{+}, we show that we indeed obtain the usual Carleson measures on H^{2}, and hence on H^{p}. Therefore we unify the theory of Carleson measures on weighted Bergman, Besov, and Hardy spaces simultaneously.

Theorem 1.3 depends on an imbedding of B_{q}^{p} into a Lebesgue class via I_{s}^{t} which involve certain combinations of radial derivatives of functions in B_{q}^{p}. Using derivatives to imbed holomorphic function spaces into Lebesgue classes is not uncommon; see [AFP, Theorem 13], [L] and its references, and [CKY]. On the other hand, descriptions of Carleson measures defined using the inclusion map on Besov spaces are limited to certain values of q and p and to N = 1. For example, q = −(N +1) = −2 in [ARS] although their Besov spaces are defined with a more general weight than 1 − |z|^{2}. In other places, the equivalent conditions are not uniform over the values of q, p considered; for example, see [W] for q + p > −1 with N = 1.

It is still possible to strengthen the characterization of q-Carleson measures by relaxing their dependence on Besov spaces and weakening the condition in Theorem 1.3 (iv) after a relabeling of the parameters. The following result seems to be new in its generality also for Bergman-space Carleson measures and even in the most classical case q = 0. Recalling that ν_{q} is the model q-Carleson measure and in view of [R, Proposition 1.4.10], its conditions are as natural as can be hoped for.

Theorem 1.5. Let μ be a positive Borel measure on 𝔹. If

U_{α,β,q}μ(w) = (1 − |w|^{2})^{α} ∫_{𝔹} \frac{(1 − |z|^{2})^{β}}{|1 − ⟨z, w⟩|^{N+1+α+β+q}} dμ(z) (w ∈ 𝔹)

is bounded for some real α, β, and q, then μ is a q-Carleson measure. If μ is a q-Carleson measure, α > 0, and β + q > −1, then U_{α,β,q}μ is bounded on 𝔹.

The idea of this theorem leads to a characterization of Hardy-space Carleson measures which also seems new in its generality.

Theorem 1.6. Let μ be a positive Borel measure on 𝔹. If U_{α,0,−1}μ(w) is bounded on 𝔹 for some real α, then μ is a Hardy-space Carleson measure. If μ is a Hardyspace Carleson measure and α > 0, then U_{α,0,−1}μ(w) is bounded on 𝔹.

Some of the results in this paper have been announced in [K3].

All our results are valid when s and t are complex numbers too; we just need to replace them with their real parts in inequalities as done in [FR] and [K2].

The proof of Theorem 1.3 is in Section 5 along with a discussion of related Berezin transforms. The little oh version of this theorem that connects the compactness of I_{s}^{t} to vanishing Carleson measures is Theorem 5.3. This section contains also the proof of Theorem 1.5 and its little oh version. An immediate application is given to separated sequences in 𝔹. We further give equivalent conditions for the imbedding I_{s}^{t} : B_{q}^{2} → L^{2}(μ) to belong to the Schatten ideal S^{c} with c ≥ 2 in Theorem 5.12. Compact operators require a characterization of ultraweak convergence in Besov spaces which is in Section 4. We next give examples of ultraweakly convergent families in Besov spaces in Example 4.7 that are instrumental in the proof of the implication (iii) ⟹ (iv) of Theorem 1.3. We gather some basic facts about Bergman geometry in Section 2, and review Besov spaces in Section 3. Later in Section 6, we show how the Hardy-space Carleson measures come into the picture as the order t of the derivative D_{s}^{t} tends to 0 when q = −1. The proof of Theorem 1.6 is also here.

The remaining sections are for applications. In Section 7, we apply Theorem 1.5 to an analysis of integral operators on L^{∞} inspired by Forelli-Rudin estimates. In Section 8, we characterize functions in weighted Bloch and little Bloch spaces B^{α} and B_{0}^{α} for all α ∈ ℝ, which include the Lipschitz classes and the growth spaces. In Section 9, we develop a finiteness criterion for positive Borel measures imbedding Bloch spaces into Lebesgue classes using I_{s}^{t}, and we construct Carleson measures from functions in Besov spaces, using gap series for both. In Section 10, we generalize to Besov spaces two classical inequalities of Fejér-Riesz and Hardy-Littlewood for Hardy spaces, which are reobtained in a limiting case. In Section 11, we investigate integration operators companion to a Cesàro-type operator.

As for notation, if X is a set, then \overline{X} denotes its closure and ∂X its boundary. The surface measure on ∂𝔹 is denoted σ and normalized with σ(∂𝔹) = 1. Bounded measurable and bounded holomorphic functions on 𝔹 are denoted by L^{∞} and H^{∞}, and ∥f∥_{H^{∞}} = sup_{∂𝔹} |f|. Note that L_{q}^{∞} = L^{∞} for any q. We let C be the space of continuous functions on \overline{𝔹} and C_{0} its subspace whose members vanish on ∂𝔹.

We use the convenient Pochhammer symbol defined by

(x)_{y} = \frac{Γ(x + y)}{Γ(x)}

when x and x + y are off the pole set −ℕ of the gamma function Γ. For fixed x, y, Stirling formula gives

\frac{Γ(c + x)}{Γ(c + y)} ∼ c^{x−y}  and  \frac{(x)_{c}}{(y)_{c}} ∼ c^{x−y}  (c → ∞), (3)

where x ∼ y means that |x/y| is bounded above and below by two positive constants that are independent of any parameter present (c here).

We use multi-index notation in which λ = (λ_{1}, …, λ_{N}) ∈ ℕ^{N} is an N-tuple of nonnegative integers, |λ| = λ_{1} + ⋯ + λ_{N}, λ! = λ_{1}! ⋯ λ_{N}!, z^{λ} = z_{1}^{λ_{1}} ⋯ z_{N}^{λ_{N}}, and 0^{0} = 1.

2. Bergman Geometry

We collect here some standard facts on balls in the Bergman metric, and prove some subharmonicity results with respect to these balls.

The biholomorphic automorphism group Aut(𝔹) of the ball is generated by unitary mappings of ℂ^{n} and the involutive Möbius transformations φ_{a} that exchange 0 and a ∈ 𝔹. A most useful property of φ_{a} is

1 − ⟨φ_{a}(z), φ_{a}(w)⟩ = \frac{(1 − |a|^{2}) (1 − ⟨z, w⟩)}{(1 − ⟨z, a⟩) (1 − ⟨a, w⟩)} (a, z, w ∈ 𝔹); (4)

the real Jacobian of the transformation w = φ_{a}(z) is

J_{ℝ}φ_{a}(z) = (\frac{1 − |a|^{2}}{|1 − ⟨z, a⟩|^{2}} )^{N+1} ; (5)

see [R, §2.2]. The Bergman metric on 𝔹 is

d(z, w) = \frac{1}{2} log \frac{1 + |φ_{z}(w)|}{1 − |φ_{z}(w)|} = tanh^{−1} |φ_{z}(w)|,

where |φ_{z}(w)| = d_{ψ}(z, w) is the pseudohyperbolic metric on 𝔹. These metrics are invariant under the automorphisms of 𝔹; that is, d(ψ(z), ψ(w)) = d(z, w) and d_{ψ}(ψ(z), ψ(w)) = d_{ψ}(z, w) for ψ ∈ Aut(𝔹).

The balls centered at w of radius r in the Bergman (hyperbolic), pseudohyperbolic, and Euclidean metrics are denoted by b(w, r), b_{ψ}(w, r), and b_{e}(w, r), respectively. A pseudohyperbolic ball is a Bergman ball rescaled by the hyperbolic tangent, and a Euclidean ball is a pseudohyperbolic ball translated by an automorphism of 𝔹, as explicitly displayed by the relations

(6) b(w, r) = b_{ψ}(w, tanh r) = φ_{w}(b_{e}(0, tanh r)),

where 0 < tanh r < 1. The automorphism invariance of the two metrics d and d_{ψ} shows that φ_{a}(b(w, r)) = b(φ_{a}(w), r) and φ_{a}(b_{ψ}(w, r)) = b_{ψ}(φ_{a}(w), r).

Lemma 2.1. Given r_{1} > 0 and w ∈ 𝔹, we have

1 − ⟨z_{1}, z_{2}⟩ ∼ 1 − |w|^{2}

for all z_{1}, z_{2} ∈ b(w, r) and r ≤ r_{1}. Hence

1 − |z|^{2} ∼ 1 − |w|^{2}  and  1 − ⟨z, w⟩ ∼ 1 − |w|^{2}

for all z ∈ b(w, r) and r ≤ r_{1}.

Proof. If z_{j} ∈ b(w, r), then z_{j} = φ_{w}(v_{j}) for some v_{j} with |v_{j}| < tanh r for j = 1, 2 by (6). Then

1 − ⟨z_{1}, z_{2}⟩ = 1 − ⟨φ_{w}(v_{1}), φ_{w}(v_{2})⟩ = \frac{(1 − |w|^{2}) (1 − ⟨v_{1}, v_{2}⟩)}{(1 − ⟨v_{1}, w⟩) (1 − ⟨w, v_{2}⟩)} ∼ 1 − |w|^{2},

because the other factors are ∼ 1 since |v_{j}| < tanh r ≤ tanh r_{1}, j = 1, 2.

Lemma 2.2. Given q and r_{1} > 0, we have

ν_{q}(b(w, r)) ∼ (1 − |w|^{2})^{N+1+q} (w ∈ 𝔹, r ≤ r_{1}).

Proof. Computing using (6), (4), and (5),

ν_{q}(b(w, r)) = ∫_{b(w,r)}(1 − |z|^{2})^{q} dν(z) = ∫_{φ_{w}(b_{e}(0,tanh r))}(1 − |z|^{2})^{q} dν(z)

= ∫_{b_{e}(0,tanh r)}(1 − |φ_{w}(z)|^{2})^{q} J_{ℝ}φ_{w}(z) dν(z)

= (1 − |w|^{2})^{N+1+q} ∫_{|z|<tanh r} \frac{(1 − |z|^{2})^{q}}{|1 − ⟨z, w⟩|^{2(N+1+q)}} dν(z).

The last integral is equivalent to 1 since tanh r ≤ tanh r_{1}.

Corollary 2.3. Given q and r_{1}, r_{2}, r_{3}, r_{4}, r_{5} > 0, we have

\frac{ν_{q}(b(z, r))}{ν_{q}(b(w, ρ))} ∼ 1

for all r ≤ r_{1}, ρ ≤ r_{2}, r_{3} ≤ r/ρ ≤ r_{4} and z, w ∈ 𝔹 with d(z, w) ≤ r_{5}.

Definition 2.4. A sequence {a_{n}} in 𝔹 is called separated (or uniformly discrete) if there is a constant τ > 0, called the separation constant, such that d(a_{n}, a_{m}) ≥ τ for all n ≠ m.

The disc version of the following covering lemma is in [Ax, Lemma 3.5]. A sequence {a_{n}} satisfying its conditions is called an r-lattice in 𝔹 in the literature.

Lemma 2.5. There is a positive integer M such that for any given r, there exists a sequence {a_{n}} in 𝔹 with |a_{n}| → 1 satisfying the following conditions.

(i) 𝔹 = ⋃_{n=1}^{∞} b(a_{n}, r); (ii) {a_{n}} is separated with separation constant r/2; (iii) any point in 𝔹 belongs to at most M of the balls b(a_{n}, 2r).

It is common to use Carleson windows in theorems and proofs on Carleson measures. These windows are extensions to 𝔻 of arcs on ∂𝔻, and their higherdimensional generalizations. The arcs are the balls of the natural metric on ∂𝔻, which is the natural domain for the Hardy spaces. However, when considering Bergman or Besov spaces on 𝔻 and especially on 𝔹, it is much more natural to use balls of the relevant metric, which is the Bergman metric. Certain details of proofs using Carleson windows involve a decomposition of 𝔻 into windows that get smaller as they get closer to ∂𝔻. As a matter of fact, they do so in such a way that their size in the Bergman metric remain roughly fixed. In Lemma 2.5 instead, we use a decomposition of 𝔹 into balls of a fixed radius that does the same job in a much less complicated manner.

Lastly we obtain two generalized subharmonicity properties with respect to each of the measures ν_{Q} on Bergman balls. The proofs given in [Zh1] for Q = 0 work equally well for other Q too. A final use of Jensen inequality in the second extends the result to p ≠ 1.

Lemma 2.6. Given Q ∈ ℝ and r_{1} > 0, there is a constant C such that for all p, g ∈ H(𝔹), w ∈ 𝔹, and r ≤ r_{1}, we have

|g(w)|^{p} ≤ \frac{C}{ν_{Q}(b(w, r))} ∫_{b(w,r)} |g|^{p} dν_{Q}.

Lemma 2.7. Given q and r_{1} > 0, there is a constant C such that for all p, positive Borel measure μ on 𝔹, w ∈ 𝔹, and r ≤ r_{1}, we have

μ(b(w, r))^{p} ≤ \frac{C}{ν_{q}(b(w, r))} ∫_{b(w,r)} μ(b(z, r))^{p} dν_{q}(z).

3. Besov Spaces

There are several different ways to define Besov spaces on 𝔹. All require one kind of derivative or another, but the easiest one to use is the radial derivative. The particular description started in [K1] and continued in [K2] suits best our interests. We review their relevant points here. Another major source of information is [BB]. For comparison, our B_{q}^{p} space is their A_{1+q+pt,t}^{p} space.

Let f ∈ H(𝔹) be given by its homogeneous expansion f = ∑_{k=0}^{∞} F_{k}, where F_{k} is a homogeneous polynomial of degree k. Then its radial derivative at z is Rf(z) = ∑_{k=1}^{∞} k F_{k}(z). In [K2, Definiton 3.1], for any s, t, the radial differential operator D_{s}^{t} is defined on H(𝔹) by D_{s}^{t}f = ∑_{k=0}^{∞} _{s}^{t}d_{k} F_{k}, where

_{s}^{t}d_{k} = \begin{cases}\frac{(N + 1 + s + t)_{k}}{(N + 1 + s)_{k}} , & if s > −(N +1), s+t > −(N +1); \\ \frac{(N +1+s+t)_{k} (−(N +s))_{k+1}}{(k!)^{2}} , & if s ≤ −(N +1), s+t > −(N +1); \\ \frac{(k!)^{2}}{(N +1+s)_{k} (−(N +s+t))_{k+1}} , & if s > −(N +1), s+t ≤ −(N +1); \\ \frac{(−(N + s))_{k+1}}{(−(N + s + t))_{k+1}} , & if s ≤ −(N +1), s+t ≤ −(N +1). \end{cases}

What is important is that

_{s}^{t}d_{k} ≠ 0 (k = 0, 1, 2, …)  and  _{s}^{t}d_{k} ∼ k^{t} (k → ∞) (7)

for any s, t. It turns out that each D_{s}^{t} is a continuous invertible operator of order t on H(𝔹) with two-sided inverse

(D_{s}^{t})^{−1} = D_{s+t}^{−t}. (8)

Other useful properties are that D_{s}^{0} is the identity for any s, D_{−N}^{1} = I + R, D_{s+t}^{u}D_{s}^{t} = D_{s}^{u+t}, D_{s}^{t}(1) = _{s}^{t}d_{0} > 0, and D_{s}^{t}(z^{λ}) = _{s}^{t}d_{|λ|}z^{λ}.

The next result, reproduced from [K2, Theorem 4.1], justifies Definition 1.1. It is equivalent to [BB, Theorem 5.12 (i)].

Proposition 3.1. The space B_{q}^{p} is independent of the particular choice of s, t as long as (1) holds. The L_{q}^{p} norms of I_{s_{1}}^{t_{1}}f and I_{s_{2}}^{t_{2}}f are equivalent as long as (1) is satisfied by t_{1} and t_{2}.

So the norm in Definition 1.1 represents a whole family of equivalent norms. The same is true in B_{q}^{2} for the inner product

[f, g]_{q} = ∫_{𝔹} I_{s}^{t}f \overline{I_{s}^{t}g} dν_{q} (9)

with s, t satisfying (1) with p = 2.

Each B_{q}^{2} space is a reproducing kernel Hilbert space with reproducing kernel

K_{q}(z, w) = \begin{cases}\frac{1}{(1 − ⟨z, w⟩)^{N+1+q}} = ∑_{k=0}^{∞} \frac{(N + 1 + q)_{k}}{k!} ⟨z, w⟩^{k}, & if q > −(N + 1); \\ \frac{_{2}F_{1}(1, 1; 1 − N − q; ⟨z, w⟩)}{−N − q} = ∑_{k=0}^{∞} \frac{k! ⟨z, w⟩^{k}}{(−N − q)_{k+1}} , & if q ≤ −(N + 1); \end{cases}

where _{2}F_{1} is the hypergeometric function; see [BB, p. 13]. Thus B_{q}^{2} spaces are nothing but Dirichlet spaces, B_{−1}^{2} the Hardy space H^{2}, B_{−N}^{2} the Arveson space A (see [Ar] and [AK1]), and B_{−(N+1)}^{2} the classical Dirichlet space D, the last due to the fact that K_{−(N+1)}(z, w) = −⟨z, w⟩^{−1} log(1 − ⟨z, w⟩).

Monomials {z^{λ}} form a dense orthogonal set in B_{q}^{2}. Moreover, by (3),

K_{q}(z, w) ∼ ∑_{k=0}^{∞} k^{N+q} ⟨z, w⟩^{k} = ∑_{λ} \frac{|λ|^{N+q} |λ|!}{λ!} z^{λ} \overline{w}^{λ} (10)

for any q, because

⟨z, w⟩^{k} = ∑_{|λ|=k} \frac{k!}{λ!} z^{λ} \overline{w}^{λ}. (11)

This shows that K_{q} is bounded for q < −(N + 1), and that for all q,

∥z^{λ}∥_{B_{q}^{2}}^{2} ∼ \frac{λ!}{|λ|^{N+q} |λ|!} (λ ∈ ℕ^{N}).

The reproducing property of K_{q} is that [f, K_{q}(·, w)]_{q} = Cf(w) with any s, t satisfying (1). Since K_{q}(·, w) ∈ B_{q}^{2} for any w ∈ 𝔹, we also have

∥K_{q}(·, w)∥_{B_{q}^{2}}^{2} = [K_{q}(·, w), K_{q}(·, w)]_{q} = CK_{q}(w, w) (12)

with any s, t satisfying (1). Although the results on reproducing property follow directly from considerations in reproducing kernel Hilbert spaces, they can be checked as well by using the integral forms of the inner product and the norm and [R, Proposition 1.4.10]. Differentiation in the holomorphic variable z and the series expansion of K_{q} show that always

D_{q}^{t}K_{q}(z, w) = K_{q+t}(z, w).

Almost as easily, we have the following for the spaces; but a proof can be found in [KÜ, Proposition 3.1].

Proposition 3.2. For any q, p, s, t, D_{s}^{t}(B_{q}^{p}) = B_{q+pt}^{p} is an isometric isomorphism under the equivalence of norms.

Lemma 3.3. Given q, p, s, t, there is a constant C such that if f ∈ B_{q}^{p}, then for z ∈ 𝔹,

|D_{s}^{t}f(z)| ≤ C ∥f∥_{B_{q}^{p}} \begin{cases}(1 − |z|^{2})^{−(N+1+q+pt)/p}, & if q > −(N + 1 + pt); \\ log(1 − |z|^{2})^{−1}, & if q = −(N + 1 + pt); \\ 1, & if q < −(N + 1 + pt). \end{cases}

Proof. See [BB, Lemma 5.6].

Definition 3.4. Extended Bergman projections are the linear transformations

P_{s}f(z) = ∫_{𝔹} K_{s}(z, w) f(w) dν_{s}(w) (z ∈ 𝔹)

defined for suitable f and all s.

The following result is contained in [K2, Theorem 1.2].

Theorem 3.5. For 1 ≤ p < ∞, P_{s} is a bounded operator from L_{q}^{p} onto B_{q}^{p} if and only if

q + 1 < p(s + 1). (13)

Given an s satisfying (13), if t satisfies (1), then

(P_{s} ∘ I_{s}^{t})f = \frac{N!}{(1 + s + t)_{N}} f (f ∈ B_{q}^{p}).

Together (13) and (1) imply s + t > −1 so that 1 + s + t does not hit a pole of Γ. If q > −1, we can take t = 0, and Theorem 3.5 reduces to the classical result on Bergman spaces. When p = ∞ for fixed q, the inequalities (1) and (13) turn into

t > 0  and  s > −1. (14)

Then the spaces B_{q}^{∞} are all the same and called the Bloch space B, which is the space of all f ∈ H(𝔹) for which some I_{s}^{t}f with t > 0 is bounded on 𝔹. Its subspace the little Bloch space B_{0} consists of those f ∈ B for which some I_{s}^{t}f with t > 0 vanishes on ∂𝔹. The norm on these spaces is the Bloch norm

∥f∥_{B} = sup_{𝔹} |I_{s}^{t}f|

valid for any t > 0.

Theorem 3.6. The operator P_{s} maps L^{∞} boundedly onto B if and only if s > −1; and it maps either of C or C_{0} boundedly onto B_{0} if and only if s > −1. Given such an s, if also t > 0, then (P_{s} ∘ I_{s}^{t})f = Cf for f ∈ B, and hence for f ∈ B_{0}.

Proof. See [KÜ, Theorem 5.3].

A consequence of Bergman projections is that for 1 ≤ p < ∞, the dual of B_{q}^{p} can be identified with B_{q}^{p′}, where p′ = p/(p − 1) is the exponent conjugate to p, under each of the pairings

q[f, g]_{s,q+t}^{t,−q+s} = ∫_{𝔹} I_{s}^{t}f \overline{I_{q+t}^{−q+s}g} dν_{q}, (15)

where s, t satisfy (13) and (1), or (14), and f ∈ B_{q}^{p}, g ∈ B_{q}^{p′}. Similarly, the dual of B_{0} can be identified with any B_{q}^{1} under each of the same pairings with f ∈ B_{0}, g ∈ B_{q}^{1}. The details can be found in [K2, §7].

4. Compact Operators and Ultraweak Convergence

This section has dual purpose. First we give a characterization of compact I_{s}^{t} acting on Besov spaces that leads to a little oh version of Theorem 1.3 for all p. Then we construct (ultra)weakly convergent families in Besov spaces that makes the proof of Theorem 1.3 possible. These are still normalized reproducing kernels, but the kernel and the normalization are of different spaces.

Definition 4.1. Let X and Y be F -spaces, that is, topological vector spaces whose topologies are induced by complete translation-invariant metrics. A linear operator T : X → Y is called compact if the images of balls of X under T have compact closures in Y.

Compactness of T is equivalent to that the image under T of a bounded sequence in X has a subsequence convergent in Y. We also know that if X and Y be Banach spaces and X is reflexive, a linear operator T : X → Y is compact if and only if f_{k} → 0 weakly in X implies ∥T f_{k}∥_{Y} → 0.

The only F -spaces we consider that are not Banach spaces are L^{p}(μ) and B_{q}^{p} for 0 < p < 1. For the latter, we have a family of equivalent invariant metrics ∥f − g∥_{B_{q}^{p}}^{p} for each s, t satisfying (1).

Extending a concept defined in [Zh4, p. 61], we make the following definition.

Definition 4.2. Let s, t satisfy (1). A sequence {f_{k}} converges (s, t)-ultraweakly to 0 in B_{q}^{p} if {∥f_{k}∥_{B_{q}^{p}}} is bounded and {I_{s}^{t}f_{k}} converges to 0 uniformly on compact subsets of 𝔹.

The next result is essential for the proof of Theorem 5.3. A similar result holds for composition operators on similar spaces too; see [CM, Proposition 3.11] and [Tj, Lemmas 3.7 and 3.8].

Theorem 4.3. Let μ be a positive Borel measure on 𝔹, and let s, t satisfy (1). The operator I_{s}^{t} : B_{q}^{p} → L^{p}(μ) is compact if and only if for any sequence {f_{k}} in B_{q}^{p} converging (s, t)-ultraweakly to 0, we have ∥I_{s}^{t}f_{k}∥_{L^{p}(μ)} → 0.

Proof. Suppose I_{s}^{t} is compact, and let {f_{k}} converge (s, t)-ultraweakly to 0 in B_{q}^{p}. Assume that there is an ε > 0 and a subsequence {f_{k_{j}}} such that ∥I_{s}^{t}f_{k_{j}}∥_{L^{p}(μ)}^{p} ≥ ε for all j. By the compactness of I_{s}^{t}, there is another subsequence {f_{k_{jm}} } such that I_{s}^{t}f_{k_{jm}} → h in L^{p}(μ). And there is a further subsequence {f_{k_{jm_{l}}} } such that I_{s}^{t}f_{k_{jm_{l}}} (z) → h(z) a.e. in 𝔹. But I_{s}^{t}f_{k}(z) → 0 for all z ∈ 𝔹 by uniform convergence on compact subsets. Thus h(z) = 0 a.e. in 𝔹 and I_{s}^{t}f_{k_{jm}} → 0 in L^{p}(μ). This contradicts the assumption.

Conversely, suppose {∥f_{k}∥_{B_{q}^{p}}^{p}} is bounded. By Lemma 3.3, for all k and R with 0 < R < 1, we have sup{ |D_{s}^{t}f_{k}(z)|^{p} : |z| ≤ R } ≤ C ∥f_{k}∥_{B_{q}^{p}}^{p} ≤ C. Hence {D_{s}^{t}f_{k}} is a normal family and has a subsequence {f_{k_{j}}} such that D_{s}^{t}f_{k_{j}} converges uniformly on compact subsets of 𝔹 to a function in H(𝔹) which we can take as D_{s}^{t}f for some f ∈ H(𝔹). Then also I_{s}^{t}f_{k_{j}} → I_{s}^{t}f uniformly on compact subsets of 𝔹. Then by Fatou lemma,

∫_{𝔹} |I_{s}^{t}f|^{p} dν_{q} = ∫_{𝔹} lim_{j→∞} |I_{s}^{t}f_{k_{j}}|^{p} dν_{q}

≤ lim inf_{j→∞} ∫_{𝔹} |I_{s}^{t}f_{k_{j}}|^{p} dν_{q} = lim inf_{j→∞} ∥f_{k_{j}}∥_{B_{q}^{p}}^{p} ≤ C,

which implies f ∈ B_{q}^{p}. Thus {f_{k_{j}} −f} is a sequence converging (s, t)-ultraweakly to 0 in B_{q}^{p}. It follows that ∥I_{s}^{t}(f_{k_{j}} − f)∥_{L^{p}(μ)}^{p} → 0 and {I_{s}^{t}f_{k_{j}}} converges in L^{p}(μ).

Considering the characterization of compactness on reflexive spaces, the following result is no surprise. It applies to weighted Bergman spaces by taking q > −1 and t = 0. But we can take other s, t as long as they satify (13) and (1) also with q > −1. Thus we obtain some new conditions for weak convergence on weighted Bergman spaces equivalent to the known ones.

Theorem 4.4. For 1 < p < ∞, a sequence {f_{k}} converges to 0 weakly in B_{q}^{p} if and only if it converges (s, t)-ultraweakly to 0 in B_{q}^{p} for some s, t satisfying (13) and (1).

Proof. Suppose {f_{k}} converges (s, t)-ultraweakly to 0 with s, t of the form given. Then D_{s}^{t}f_{k} → 0 uniformly on compact subsets of 𝔹. Since functions with compact support are dense in L_{q}^{p}, it suffices to consider the following. Let 0 < R < 1, χ be the characteristic function of the Euclidean ball b_{e}(0, R), and g = P_{q+t}χ. Now (13) is satisfied with q + t and p′ replacing s and p because of (1); hence g ∈ B_{q}^{p′} by Theorem 3.5. Then by (15), differentiation under the integral, and Fubini theorem, we obtain

q[f_{k}, g]_{s,q+t}^{t,−q+s} = ∫_{𝔹} I_{s}^{t}f_{k} \overline{I_{q+t}^{−q+s}g} dν_{q}

= ∫_{𝔹} I_{s}^{t}f_{k}(z) (1 − |z|^{2})^{s} ∫_{𝔹} \frac{(1 − |w|^{2})^{q+t} χ(w)}{(1 − ⟨w, z⟩)^{N+1+s+t}} dν(w) dν(z)

= ∫_{𝔹}(1 − |w|^{2})^{q+t} χ(w) ∫_{𝔹} \frac{(1 − |z|^{2})^{s+t}}{(1 − ⟨w, z⟩)^{N+1+s+t}} D_{s}^{t}f_{k}(z) dν(z) dν(w)

= ∫_{|w|<R}(1 − |w|^{2})^{q+t} P_{s+t}D_{s}^{t}f_{k}(w) dν(w).

Now by Proposition 3.2, D_{s}^{t}f_{k} ∈ B_{q+pt}^{p} = A_{q+pt}^{p}, and hence P_{s+t}(D_{s}^{t}f_{k}) = C D_{s}^{t}f_{k} by Theorem 3.5. Then

q[f_{k}, g]_{s,q+t}^{t,−q+s} = C ∫_{|w|<R}(1 − |w|^{2})^{q+t} D_{s}^{t}f_{k}(w) dν(w) = C ∫_{|w|<R} I_{s}^{t}f_{k}(w) dν_{q}(w),

Thus |_{q}[f_{k}, g]_{s,q+t}^{t,−q+s}| ≤ C sup{ |I_{s}^{t}f_{k}(w)| : |w| ≤ R }, and f_{k} → 0 weakly.

Suppose f_{k} → 0 weakly in B_{q}^{p}, and s, t satisfy (13) and (1). Then {∥f_{k}∥_{B_{q}^{p}}} is bounded. Lemma 3.3 yields that sup{ |D_{s}^{t}f_{k}(z)| : |z| ≤ R } ≤ C ∥f_{k}∥_{B_{q}^{p}} ≤ C for all k and R with 0 < R < 1. Then {D_{s}^{t}f_{k}} is a normal family and has a subsequence {D_{s}^{t}f_{k_{j}}} that converges uniformly on compact subsets. Putting h_{k} = I_{s}^{t}f_{k}, this forces {h_{k_{j}}} also to converge uniformly on compact subsets, say, to h. But f_{k_{j}} then converges weakly to f ≡ 0 and h = I_{s}^{t}f. Hence h ≡ 0. If {h_{k}} had another subsequence {h_{k_{l}}} that stayed bounded away from 0, then since f_{k_{l}} → 0 weakly, this subsequence would in turn yield a subsubsequence {h_{k_{lm}} } as above that would converge uniformly on compact subsets to 0, contradicting the defining property of {h_{k_{l}}}. Therefore the full sequence h_{k} → 0 uniformly on compact subsets of 𝔹.

Theorem 4.5. A sequence {g_{k}} converges to 0 weak-∗ in B_{q}^{1} = (B_{0})^{∗} if and only if it converges (s, t)-ultraweakly to 0 in B_{q}^{1} for some s, t satisfying (13) and (1) with p = 1. A sequence {g_{k}} converges to 0 weak-∗ in B = (B_{q}^{1})^{∗} if and only if it converges (s, t)-ultraweakly to 0 in B for some s, t satisfying (14).

Proof. The only differences from the proof of Theorem 4.4 are that we use a continuous χ for the first statement and Theorem 3.6 for the second statement.

Example 4.6. It is well-known ([Zh2, §6.1]) that if q > −1, then the normalized reproducing kernels g_{w}(z) = K_{q}(z, w)/∥K_{q}(·, w)∥_{A_{q}^{2}} converge to 0 weakly as |w| → 1 in the Bergman spaces A_{q}^{2}. More generally, g_{w}^{2/p} converges to 0 as |w| → 1 weakly in A_{q}^{p} for p > 1 and weak-∗ in A_{q}^{1}.

In Besov spaces B_{q}^{p} with −(N + 1) < q ≤ −1 when the associated reproducing kernel is binomial, the same idea gives ultraweakly 0-convergent families. We show the details, because derivatives have to be taken care of in the computations of norms. By (12) we have

g_{w}(z) ∼ (\frac{1 − |w|^{2}}{(1 − ⟨z, w⟩)^{2}} )^{(N+1+q)/p}

∼ (1 − |w|^{2})^{(N+1+q)/p} ∑_{k=0}^{∞} k^{(N+1+q)2/p−1}⟨z, w⟩^{k} (|w| → 1).

If s, t satisfy (1), then by (7) and (10),

I_{s}^{t}g_{w}(z) ∼ (1 − |w|^{2})^{(N+1+q)/p} (1 − |z|^{2})^{t} ∑_{k=0}^{∞} k^{(N+1+q)2/p−1+t} ⟨z, w⟩^{k}

∼ \frac{(1 − |w|^{2})^{(N+1+q)/p} (1 − |z|^{2})^{t}}{(1 − ⟨z, w⟩)^{(N+1+q)2/p+t}} (|w| → 1).

So if |z| ≤ R < 1, then |I_{s}^{t}g_{w}(z)| ≤ C (1 − |w|^{2})^{(N+1+q)/p} → 0 as |w| → 1. Further

∥g_{w}∥_{B_{q}^{p}}^{p} = ∫_{𝔹} |I_{s}^{t}g_{w}|^{p} dν_{q} ∼ (1 − |w|^{2})^{N+1+q} ∫_{𝔹} \frac{(1 − |z|^{2})^{q+pt}}{|1 − ⟨z, w⟩|^{(N+1+q)2+pt}} dν(z)
∼ 1 (|w| → 1)

by [R, Proposition 1.4.10]. Thus g_{w} → 0 as |w| → 1 (s, t)-ultraweakly.

In particular, the normalized reproducing kernels of the Hardy space H^{2} = B_{−1}^{2} and the Arveson space A = B_{−N}^{2} are weakly 0-convergent families in their own spaces.

Even when q = −(N + 1), when the associated reproducing kernel is logarithmic and hence unbounded, the same procedure gives weakly 0-convergent families in B_{−(N+1)}^{p}, but seems unlikely to work for q < −(N + 1) when the reproducing kernels are bounded. We need a modification.

Example 4.7. We now explicitly construct ultraweakly 0-convergent families in all Besov spaces B_{q}^{p}. Our construction works in Bergman spaces too and gives us such families that are not necessarily normalized reproducing kernels.

Fix q. Let t satisfy (1); then also N + 1 + q + pt > 0. Pick complex numbers c_{k} such that c_{k} ∼ k^{(N+1+q+pt)2/p−1−t} as k → ∞, and put

f_{w}(z) = ∑_{k=0}^{∞} c_{k} ⟨z, w⟩^{k}.

Similar to Example 4.6,

I_{s}^{t}f_{w}(z) ∼ \frac{(1 − |z|^{2})^{t}}{(1 − ⟨z, w⟩)^{(N+1+q+pt)2/p}} (|w| → 1). (16)

If |z| ≤ R < 1, then |I_{s}^{t}f_{w}(z)| ≤ C for any w ∈ 𝔹. Again similar to Example 4.6,

∥f_{w}∥_{B_{q}^{p}}^{p} ∼ \frac{1}{(1 − |w|^{2})^{N+1+q+pt}} . (17)

Set g_{w}(z) = f_{w}(z)/∥f_{w}∥_{B_{q}^{p}} so that each ∥g_{w}∥_{B_{q}^{p}} = 1. Moreover, if |z| ≤ R, then we have |I_{s}^{t}g_{w}(z)| ≤ C (1 − |w|^{2})^{(N+1+q+pt)/p} → 0 as |w| → 1. The (s, t)-ultraweak convergence follows.

Remark 4.8. Consider the case of a Hilbert space, p = 2, in Example 4.7. Let s satisfy (13) in which case t = −q + s satisfies (1) since −q + 2s > −1. Then c_{k} ∼ k^{N+s}, and by (10) we can take f_{w}(z) = K_{s}(z, w) = D_{q}^{−q+s}K_{q}(z, w) in B_{q}^{2}. Thus

g_{w}(z) = \frac{K_{s}(z, w)}{∥K_{s}(·, w)∥_{B_{q}^{2}}} = \sqrt{\frac{(1 − q + 2s)_{N+1}}{N!}} (1 − |w|^{2})^{(N+1−q+2s)/2} K_{s}(z, w)

∼ \frac{K_{s}(z, w)}{\sqrt{K_{−q+2s}(w, w)}} ∈ B_{q}^{2}

is a normalized reproducing kernel indeed, but it is the kernel of B_{s}^{2} normalized so that its B_{q}^{2} norm is 1, and is considered an element of B_{q}^{2}. The second equality above follows from the proof of [R, Proposition 1.4.10] using t = −q + s. It is interesting that

g_{w}(z) = D_{−q+2s}^{q−s} \frac{K_{−q+2s}(z, w)}{∥K_{−q+2s}(·, w)∥_{B_{−q+2s}^{2}}} .

It is possible to take s = q if and only if q > −1, the Bergman-space case. For q ≤ −1, (13) requires s > q. For such q, s = −q works for p ≥ 1 and s = 0 works for all p.

Specifically, the Bergman kernel K_{0}(·, w) is a weakly 0-convergent family in H^{2} or A as |w| → 1 when it is normalized by dividing it by its norm in H^{2} or A.

It is easy to see that f_{w} is the kernel in B_{q}^{2} for the evaluation of the derivative D_{s}^{−q+s}f of f ∈ B_{q}^{2} at w ∈ 𝔹 in the sense that [f, f_{w}]_{q} = C D_{s}^{−q+s}f(w), where C = N!/(1−q+2s)_{N} when [·, ·]_{q} = _{q}[·, ·]_{s,s}^{−q+s,−q+s}, and g_{w} is this kernel normalized in B_{q}^{2}. A similar weak-convergence result can be found in [CM, Proposition 7.13].

Example 4.9. We lastly obtain weak-∗ 0-convergent families in the Bloch space B. Let s, t satisfy (14), pick c_{k} ∼ k^{t−1} as k → ∞, and define f_{w} as in Example 4.7. Then we have ∥f_{w}∥_{B} ≤ C (1 − |w|^{2})^{−2t}. Setting g_{w}(z) = f_{w}(z)/∥f_{w}∥_{B}, we obtain that g_{w} → 0 weak-∗ in B as |w| → 1 by Theorem 4.5 as in Example 4.7. By taking t close to 0, we find families {g_{w}} in B that converge weak-∗ to 0 arbitrarily slowly.

5. Carleson Measures and Separated Sequences

In this section we prove Theorems 1.3 and 1.5 and their little oh counterparts on vanishing Carleson measures, and relate their conditions to Berezin transforms and averaging functions. We also prove an associated result on Schatten ideal criteria for I_{s}^{t}. Our results naturally extend well-known results on q-Carleson measures for q > −1 on weighted Bergman spaces in two directions; q ≤ −1, and for q > −1, imbeddings that are not inclusion. They are readily applied to separated sequences.

Lemma 5.1. Let q, r, also α, β ∈ ℝ, and a positive Borel measure μ on 𝔹 be given. Then

qμ̂_{r}(w) ≤ U_{α,β,q}μ(w) (w ∈ 𝔹),

where U_{α,β,q}μ is defined in Theorem 1.5.

Proof. By Lemmas 2.2 and 2.1,

qμ̂_{r}(w) ∼ \frac{1}{(1 − |w|^{2})^{N+1+q}} ∫_{b(w,r)} dμ

∼ (1 − |w|^{2})^{α} ∫_{b(w,r)} \frac{(1 − |z|^{2})^{β}}{|1 − ⟨z, w⟩|^{N+1+α+β+q}} dμ(z)

≤ (1 − |w|^{2})^{α} ∫_{𝔹} \frac{(1 − |z|^{2})^{β}}{|1 − ⟨z, w⟩|^{N+1+α+β+q}} dμ(z) = U_{α,β,q}μ(w)

for every w ∈ 𝔹.

After all the preparation, the proof of our main theorem goes very smoothly.

Proof of Theorem 1.3. (i) ⟹ (ii): There is nothing to prove.

(ii) ⟹ (iii): Suppose (ii) holds for some r. We start by applying Lemma 2.5 (i) to an integral of I_{s}^{t}f.

∫_{𝔹} |I_{s}^{t}f|^{p} dμ ≤ ∑_{n=1}^{∞} ∫_{b(a_{n},r)} |I_{s}^{t}f|^{p} dμ

≤ ∑_{n=1}^{∞} μ(b(a_{n}, r)) sup{ |I_{s}^{t}f(w)|^{p} : w ∈ b(a_{n}, r) }.

If w ∈ b(a_{n}, r), we apply Lemma 2.6 with g = D_{s}^{t}f and Q = q + pt bearing in mind Lemma 2.2 to obtain

(1 − |w|^{2})^{pt} |D_{s}^{t}f(w)|^{p} ≤ \frac{C}{ν_{q}(b(w, r))} ∫_{b(w,r)}(1 − |z|^{2})^{pt} |D_{s}^{t}f(z)|^{p} dν_{q}(z).

Then by Corollary 2.3,

|I_{s}^{t}f(w)|^{p} ≤ \frac{C}{ν_{q}(b(a_{n}, r))} \frac{ν_{q}(b(a_{n}, r))}{ν_{q}(b(w, r))} ∫_{b(w,r)} |I_{s}^{t}f|^{p} dν_{q}

≤ \frac{C}{ν_{q}(b(a_{n}, r))} ∫_{b(a_{n},2r)} |I_{s}^{t}f|^{p} dν_{q},

because b(w, r) ⊂ b(a_{n}, 2r). The right side is now independent of w ∈ b(a_{n}, r), and we can take the supremum of the left side on all such w. By assumption (ii) and Lemma 2.5 (iii), it follows that

∫_{𝔹} |I_{s}^{t}f|^{p} dμ ≤ C ∑_{n=1}^{∞} \frac{μ(b(a_{n}, r))}{ν_{q}(b(a_{n}, r))} ∫_{b(a_{n},2r)} |I_{s}^{t}f|^{p} dν_{q}

≤ C ∑_{n=1}^{∞} ∫_{b(a_{n},2r)} |I_{s}^{t}f|^{p} dν_{q} ≤ CM ∫_{𝔹} |I_{s}^{t}f|^{p} dν_{q},

and we obtain (iii).

(iii) ⟹ (iv): Suppose (iii) holds for some p. We obtain this implication by picking a special f, one for each w ∈ 𝔹, namely, f = g_{w} of Example 4.7. Equations (16) and (17), and assumption (iii) imply that

(1 − |w|^{2})^{N+1+q+pt} ∫_{𝔹} \frac{(1 − |z|^{2})^{pt}}{|1 − ⟨z, w⟩|^{(N+1+q+pt)2}} dμ(z) ≤ C ∫_{𝔹} |I_{s}^{t}g_{w}(z)|^{p} dμ(z)
≤ C ∥g_{w}∥_{B_{q}^{p}}^{p} ≤ C,

which is (iv).

(iv) ⟹ (i): This is covered by Lemma 5.1 by picking α = N + 1 + q + pt and β = pt.

Definition 5.2. We call a Carleson measure for B_{q}^{p} a vanishing Carleson measure for B_{q}^{p} whenever some I_{s}^{t} mapping B_{q}^{p} into L^{p}(μ) is further compact.

Theorem 5.3. Let q be fixed. Let p and r, and also s be given. The following are equivalent for a positive Borel measure μ on 𝔹.

(i) It holds that

lim_{|w|→1} \frac{μ(b(w, r))}{ν_{q}(b(w, r))} = 0.

(ii) If {a_{n}} is an r-lattice in 𝔹, then

lim_{n→∞} \frac{μ(b(a_{n}, r))}{ν_{q}(b(a_{n}, r))} = 0.

(iii) The measure μ is a vanishing Carleson measure for B_{q}^{p} with respect to I_{s}^{t},

where t satisfies (1). (iv) If t satisfies (1), then

lim_{|w|→1} (1 − |w|^{2})^{N+1+q+pt} ∫_{𝔹} \frac{(1 − |z|^{2})^{pt}}{|1 − ⟨z, w⟩|^{(N+1+q+pt)2}} dμ(z) = 0.

Proof. (ii) ⟹ (iii): Suppose (ii) holds for some r. Let {f_{k}} be a sequence in B_{q}^{p} converging ultraweakly to 0. Then {∥f_{k}∥_{B_{q}^{p}}} is bounded, and I_{s}^{t}f_{k} → 0 uniformly on compact subsets of 𝔹. Let ε > 0. By assumption (ii), there is an n_{0} such that

\frac{μ(b(a_{n}, r))}{ν_{q}(b(a_{n}, r))} < ε (n > n_{0}).

Then as in the proof of the corresponding implication of Theorem 1.3, for all k,

∑_{n=n_{0}+1}^{∞} ∫_{b(a_{n},2r)} |I_{s}^{t}f_{k}|^{p} dμ ≤ Cε ∑_{n=n_{0}+1}^{∞} ∫_{b(a_{n},2r)} |I_{s}^{t}f_{k}|^{p} dν_{q}

≤ CMε ∫_{𝔹} |I_{s}^{t}f_{k}|^{p} dν_{q} ≤ CMε ∥f_{k}∥_{B_{q}^{p}}^{p} ≤ CMε.

On the other hand,

lim_{k→∞} ∑_{n=1}^{n_{0}} ∫_{b(a_{n},2r)} |I_{s}^{t}f_{k}|^{p} dμ = 0

by uniform convergence on compact subsets. Then

lim_{k→∞} ∫_{𝔹} |I_{s}^{t}f_{k}|^{p} dμ = lim_{k→∞} ∥I_{s}^{t}f_{k}∥_{L^{p}(μ)}^{p} ≤ CMε

by Lemma 2.5 (i). Since ε > 0 is arbitrary, this is (iii) by Theorem 4.3.

The proofs of the implications (iii) ⟹ (iv) ⟹ (i) are entirely similar to the proofs of the corresponding implications of Theorem 1.3 and are omitted.

So for fixed q, a vanishing Carleson measure is independent of p, r, the r-lattice, and s, t satisfying (1). And we call such a μ also a vanishing q-Carleson measure. So a vanishing q-Carleson measure is a positive Borel measure on 𝔹 for which the averaging function _{q}μ_{r}(w) has limit 0 as |w| → 1 for some r. No ν_{q} or ν_{q_{2}} with q_{2} < q is a vanishing q-Carleson measure, but any ν_{q_{1}} with q_{1} > q is by Lemma 2.2.

Remark 5.4. The discussion after Theorem 1.3 yields that I_{s}^{t} : B_{q}^{p} → L_{q_{1}}^{p} with s, t satisfying (1) is bounded if and only if q_{1} ≥ q. The previous discussion now yields that I_{s}^{t} : B_{q}^{p} → L_{q_{1}}^{p} with s, t satisfying (1) is compact if and only if q_{1} > q. That I_{s}^{t} : B_{q}^{p} → L_{q}^{p} cannot be compact is also clear from the fact that it is an isometry and L_{q}^{p} is not finite-dimensional.

Corollary 5.5. A positive Borel measure μ on 𝔹 is a q-Carleson (respectively, vanishing q-Carleson) measure if and only if μ_{Q} is a (q +Q)-Carleson (respectively, vanishing (q + Q)-Carleson) measure.

Corollary 5.6. If μ is a q-Carleson (respectively, vanishing q-Carleson) measure, then μ is a Q-Carleson (respectively, vanishing Q-Carleson) measure too for any Q ≤ q. Equivalently, if μ is a q-Carleson (respectively, vanishing q-Carleson) measure and Q ≥ 0, then μ_{Q} is also a q-Carleson (respectively, vanishing q-Carleson) measure.

Proof. Both corollaries follow from Theorems 1.3 (i) and 5.3 (i) and Lemma 2.2.

Definition 5.7. We call an operator B_{q}^{u} that takes a function f on 𝔹 to

B_{q}^{u}f(w) = (1 − |w|^{2})^{N+1+u} ∫_{𝔹} \frac{(1 − |z|^{2})^{−q+u}}{|1 − ⟨z, w⟩|^{(N+1+u)2}} f(z) dν_{q}(z) (w ∈ 𝔹)

or a measure μ on 𝔹 to

B_{q}^{u}μ(w) = (1 − |w|^{2})^{N+1+u} ∫_{𝔹} \frac{(1 − |z|^{2})^{−q+u}}{|1 − ⟨z, w⟩|^{(N+1+u)2}} dμ(z) (w ∈ 𝔹)

a Berezin transform.

The Berezin transform B_{q}^{u}f makes sense for f ∈ L_{u}^{1}, for example. The B_{α} defined on functions for α > −1 in [HKZ, §2.1] is the B_{α}^{α} defined here. We use two parameters in order to accomodate measures and values of q ≤ −1.

Now Theorems 1.3 and 5.3 on Carleson measures can be reformulated in terms of Berezin transforms and averaging functions.

Theorem 5.8. Fix q. Let r and an r-lattice {a_{n}} in 𝔹, p and s, t satisfying (1), and also u > −1 be given. The following conditions are equivalent for a positive Borel measure μ on 𝔹.

(i) The measure μ is a q-Carleson (vanishing q-Carleson, respectively) measure; that is, the averaging function _{q}μ̂_{r} is bounded on 𝔹 (in C_{0}, respectively).

(ii) The sequence {_{q}μ̂_{r}(a_{n})} is bounded (has limit 0, respectively). (iii) The operator I_{s}^{t} : B_{q}^{p} → L^{p}(μ) is bounded (compact, respectively). (iv) The Berezin transform B_{q}^{u}μ is bounded on 𝔹 (in C_{0}, respectively).

Theorem 1.5 at first sight does not seem to offer anything new other than rewriting the equivalence of (iv) and (i) of Theorem 1.3 using the new parameters α = N + 1 + q + pt and β = pt. However, Theorem 1.3 (iv) has the restriction α > N, and lowering this to α > 0 needs some work.

Proof of Theorem 1.5. One direction is again covered by Lemma 5.1.

Conversely, suppose μ is a q-Carleson measure, that is, _{q}μ̂_{r} is bounded for some r. We follow the proof of (ii) ⟹ (iii) of Theorem 1.3; only now it is simpler. We take p = N + 1 + α + β + q, Q = β + q, and recall that Lemma 2.6 applies to g_{w}(z) = (1 − ⟨z, w⟩)^{−1} ∈ H(𝔹) too. Note that p > 0. By Corollary 5.5, μ_{β} is a (β + q)-Carleson measure. After doing the usual trick of replacing this measure by ν_{β+q} with the help of the open cover in Lemma 2.5, we obtain

∫_{𝔹} |g_{w}|^{p} dμ_{β} ≤ CM ∫_{𝔹} \frac{(1 − |ζ|^{2})^{β+q}}{|1 − ⟨ζ, w⟩|^{N+1+α+β+q}} dν(ζ) ∼ \frac{1}{(1 − |w|^{2})^{α}}

since α > 0, where we have used [R, Proposition 1.4.10], which requires β +q > −1. This is equivalent to the boundedness of U_{α,β,q}μ.

Corollary 5.9. Let μ be a positive Borel measure on 𝔹. If U_{α,β,q}μ(w) has limit 0 as |w| → 1 for some real α, β, and q, then μ is a vanishing q-Carleson measure. If μ is a vanishing q-Carleson measure, α > 0, and β + q > −1, then U_{α,β,q}μ(w) has limit 0 as |w| → 1.

Theorem 1.5 not only provides a general description of q-Carleson measures, but also generalizes the case c > 0 of [R, Proposition 1.4.10] from the Lebesgue measure to arbitrary positive Borel measures on 𝔹. See [S] for some other generalizations. Versions of Theorem 1.5 and Corollary 5.9 using Carleson windows when N = 1 are in [PR, Proposition 2.1] with additional restrictions such as β > −1 and q > 0.

We give an early application to separated sequences. Here the counting function n_{r}^{Z}(w) counts the number of points of a sequence Z = {z_{n}} that happens to fall in the ball b(w, r), and δ_{a} denotes the unit point mass at a.

Theorem 5.10. Let q, r, α > 0, β with β + q > −1 be given. The following are equivalent for a sequence Z = {z_{n}} of distinct points in 𝔹.

(i) The sequence Z is a disjoint union of finitely many separated sequences. (ii) The counting function n_{r}^{Z}(w) is bounded in 𝔹. (iii) The measure μ = ∑_{n=1}^{∞}(1 − |z_{n}|^{2})^{N+1+q} δ_{z_{n}} is a q-Carleson measure. (iv) There is a constant C such that

sup_{w∈𝔹} (1 − |w|^{2})^{α} ∑_{n=1}^{∞} \frac{(1 − |z_{n}|^{2})^{β}}{|1 − ⟨z_{n}, w⟩|^{α+β}} ≤ C.

(v) There is a constant C such that

sup_{m} (1 − |z_{m}|^{2})^{α} ∑_{m≠n=1}^{∞} \frac{(1 − |z_{n}|^{2})^{β}}{|1 − ⟨z_{n}, z_{m}⟩|^{α+β}} ≤ C.

This theorem is contained in [JMT, Lemma 4.1] to a large extent. What is new here is that we remove restrictions such as q > −1 and weaken others such as β > N present in this reference. We also aim to show that the proof requires almost no extra effort once we have strong results for q-Carleson measures for all q. As usual, the equivalences are independent of the values of q, r, α, β under the conditions in the statement of the theorem, and also of the values of parameters like p, s, t under (1) that would come from the equivalences in Theorem 1.3.

Proof. (i) ⟹ (iii): Let first Z be one separated sequence with separation constant τ. Let f ∈ B_{q}^{p}. As in the proof of (ii) ⟹ (iii) of Theorem 1.3, applying Lemma 2.6 with the choice g = D_{s}^{t}f and Q = q + pt > −1 on the disjoint balls b(z_{n}, τ/2) gives

(1 − |z_{n}|^{2})^{N+1+q+pt} |D_{s}^{t}f(z_{n})|^{p} ≤ C ∫_{b(z_{n},τ/2)}(1 − |z|^{2})^{pt} |D_{s}^{t}f(z)|^{p} dν_{q}(z).

Summing on n yields

∑_{n=1}^{∞} |I_{s}^{t}f(z_{n})|^{p} μ(z_{n}) ≤ C ∥f∥_{B_{q}^{p}}^{p}.

In general, if Z is a union of finitely many separated sequences, we add them on the left and reach the same conclusion.

(iii) ⟹ (iv): This is one direction of Theorem 1.5 applied to our measure μ. (iv) ⟹ (v): This is obvious.

(v) ⟹ (ii): Take an arbitrary b(w, r). By assumption and Lemma 2.1, we have

C ≥ sup_{z_{m}∈b(w,r)}(1 − |z_{m}|^{2})^{α} ∑_{z_{n}∈b(w,r)} \frac{(1 − |z_{n}|^{2})^{β}}{|1 − ⟨z_{n}, z_{m}⟩|^{α+β}} ∼ ∑_{z_{n}∈b(w,r)} 1 = n_{r}^{Z}(w).

(i) ⟺ (ii): This is in [DS, §2.11].

Separated sequences can be viewed as one way of constructing q-Carleson measures for any q. Here is another construction.

Example 5.11. Suppose μ is a positive Borel measure on 𝔹 and μ_{β} is finite for some real β. Assume that the monomials z^{λ_{1}} and z^{λ_{2}} are orthogonal in the space L^{2}(μ_{β}) for λ_{1} ≠ λ_{2}. Rotation invariance of μ would imply this for example. Put κ = (N + 1 + α + β + q)/2 and assume also κ > 0 for some real α and q. (The case κ ≤ 0 can likewise be investigated.) Then using an expansion like the one in (10), the orthogonality assumption, and (11), we have

\frac{U_{α,β,q}μ(w)}{(1 − |w|^{2})^{α}} = ∫_{𝔹} \frac{dμ_{β}(z)}{|(1 − ⟨z, w⟩)^{κ}|^{2}} = ∑_{λ} \frac{((κ)_{|λ|})^{2}}{(λ!)^{2}} |w^{λ}|^{2} ∫_{𝔹} |z^{λ}|^{2} dμ_{β}(z)

= ∑_{k=0}^{∞} \frac{((κ)_{k})^{2}}{(k!)^{2}} ∑_{|λ|=k} \frac{(k!)^{2}}{(λ!)^{2}} |w^{λ}|^{2} ∫_{𝔹} |z^{λ}|^{2} dμ_{β}(z)

≤ ∑_{k=0}^{∞} \frac{((κ)_{k})^{2}}{(k!)^{2}} |w|^{2k} ∑_{|λ|=k} \frac{k!}{λ!} ∫_{𝔹} |z^{λ}|^{2} dμ_{β}(z) = ∑_{k=0}^{∞} \frac{((κ)_{k})^{2}}{(k!)^{2}} m_{k} |w|^{2k}, (18)

where m_{k} = ∫_{𝔹} |z|^{2k} dμ_{β}(z) is the (k, k) “moment” of μ_{β}. Assume further, for some real η, that m_{k} ≤ C k^{−(N+β+q+η)}. Then by (3), (18) is

∼ ∑_{k=1}^{∞} k^{2κ−2} k^{−(N+β+q+η)} |w|^{2k} ∼ ∑_{k=1}^{∞} k^{α−η−1} |w|^{2k}. (19)

If one of the three pairs of inequalities 0 ≤ η < α, 0 < η = α, 0 ≤ η > α is satisfied, then U_{α,β,q}μ is bounded by (10). Note that (19) is binomial in the first case, logarithmic in the second, and bounded in the third. Therefore μ is a q-Carleson measure by Theorem 1.5.

For the model q-Carleson measure ν_{q}, we have η = 0 by [K2, Proposition 2.1], and we could choose α = β + q = −1/2 to get κ > 0. So this example is instructive in showing that the sufficiency part of Theorem 1.5 can hold as stated without the conditions on α or β required by the proof of the necessity part.

Our final purpose in this section is to investigate the conditions under which the Hilbert space operator I_{s}^{t} : B_{q}^{2} → L^{2}(μ) belongs to the Schatten-von Neumann ideal S^{c}. We refer to [MV, Chapter 16] for definitions and basic properties of singular numbers and Schatten ideals of operators from one Hilbert space into another. Recall that ν_{−(N+1)} is the Möbius-invariant measure on 𝔹; see [R, §2.2].

Theorem 5.12. Fix q. Let r and an r-lattice {a_{n}} in 𝔹, t satisfying (1) with p = 2, u > −1, and also 1 ≤ c < ∞ be given. The following conditions are equivalent for a q-Carleson measure μ on 𝔹.

(i) The averaging function _{q}μ̂_{r} belongs to L_{−(N+1)}^{c}. (ii) The sequence {_{q}μ̂_{r}(a_{n})} belongs to ℓ^{c}. (iii) The operator I_{q+t}^{t} : B_{q}^{2} → L^{2}(μ) belongs to S^{2c}. (iv) The Berezin transform B_{q}^{u}μ belongs to L_{−(N+1)}^{c}.

Proof. Let the singular number sequence of I_{q+t}^{t} be {s_{n}}. By [MV, Proposition 16.3], this is equivalent to saying that {s_{n}^{2}} is the singular number sequence of the operator (I_{q+t}^{t})^{∗}I_{q+t}^{t} on B_{q}^{2}. But this composite operator is the generalized Toeplitz operator _{q+t}T_{μ} on B_{q}^{2} as shown in [AK2, Theorem 4.6]. Thus I_{q+t}^{t} belongs to S^{2c} if and only if _{q+t}T_{μ} belongs to S^{c}. Then the equivalence of (i), (iii), and (iv) follows immediately from the corresponding equivalence involving positive Toeplitz operators proved independently in [AK2, Theorem 6.13]. It is now clear why we need I_{q+t}^{t} in S^{2c} rather than in S^{c}.

(i) ⟹ (ii): Lemma 2.7 with w = a_{n}, p = c, and q = −(N + 1), and once again Lemma 2.1 yield

(_{q}μ̂_{r}(a_{n}))^{c} ≤ C ∫_{b(a_{n},r)}(_{q}μ̂_{r})^{c} dν_{−(N+1)}.

Then

∑_{n=1}^{∞}(_{q}μ̂_{r}(a_{n}))^{c} ≤ C ∑_{n=1}^{∞} ∫_{b(a_{n},r)}(_{q}μ̂_{r})^{c} dν_{−(N+1)} ≤ CM ∫_{𝔹}(_{q}μ̂_{r})^{c} dν_{−(N+1)}

by Lemma 2.5 (iii).

(ii) ⟹ (i): Repeated use of Lemma 2.1, Lemma 2.5, and that b(w, r) ⊂ b(a_{n}, 2r) for w ∈ b(a_{n}, r) yield

∫_{𝔹}(_{q}μ̂_{r})^{c} dν_{−(N+1)} ≤ C ∑_{n=1}^{∞} ∫_{b(a_{n},r)} \frac{μ(b(w, r))^{c}}{(1 − |w|^{2})^{N+1+(N+1+q)c}} dν(w)

≤ C ∑_{n=1}^{∞} \frac{1}{(1 − |a_{n}|^{2})^{N+1+(N+1+q)c}} ∫_{b(a_{n},r)} μ(b(a_{n}, 2r))^{c} dν

∼ ∑_{n=1}^{∞} \frac{(1 − |a_{n}|^{2})^{N+1}}{(1 − |a_{n}|^{2})^{N+1+(N+1+q)c}} μ(b(a_{n}, 2r))^{c}

∼ ∑_{n=1}^{∞} \frac{μ(b(a_{n}, 2r))^{c}}{ν_{q}(b(a_{n}, 2r))^{c}} ∼ ∑_{n=1}^{∞}(_{q}μ̂_{r}(a_{n}))^{c},

because _{q}μ̂_{r}(a_{n}) and _{q}μ̂_{2r}(a_{n}) are equivalent by the proof of Theorem 1.3.

It is clear that the equivalences in Theorems 5.8 and 5.12 are independent of p, u, s, t, r, and {a_{n}} under the stated conditions. These theorems for q = 0 and t = 0 are in [HKZ, Theorems 2.15 and 2.16] and partly in [Zh2, Exercise 6.7].

Remark 5.13. By Lemma 5.1 once again, if U_{α,β,q}μ lies in L_{−(N+1)}^{c} for some d with 0 < d < ∞, then so does _{q}μ̂_{r}. It is also easy to show the reverse implication for d = 1, but we do not have it for d > 1.

6. Hardy-Space Limit

Our aim is to show that (−1)-Carleson measures reduce to Hardy-space Carleson measures in an appropriate limit. But let’s first highlight the important case of (−N)-Carleson measures covered by Theorem 1.3.

Corollary 6.1. Let q = −N and p = 2, whence B_{−N}^{2} = A, the Arveson space. The following conditions are equivalent for a positive Borel measure μ on 𝔹, called a Carleson measure for A.

(i) Given r > 0, there is a C such that

μ(b(w, r)) ≤ C (1 − |w|^{2}) (w ∈ 𝔹).

(ii) Given r > 0, there is a C such that if {a_{n}} is an r-lattice in 𝔹, then

μ(b(a_{n}, r)) ≤ C (1 − |a_{n}|^{2}) (n ∈ ℕ).

(iii) There is a C such that for all s and t with −N + 2t > −1, we have

∫_{𝔹}(1 − |z|^{2})^{2t} |D_{s}^{t}f(z)|^{2} dμ(z) ≤ C ∫_{𝔹}(1 − |z|^{2})^{−N+2t} |D_{s}^{t}f(z)|^{2} dν(z) (f ∈ A).

(iv) There is a C such that for all t with −N + 2t > −1, we have

∫_{𝔹} (\frac{(1 − |z|^{2})^{t}}{|1 − ⟨z, w⟩|^{1+2t}} )^{2} dμ(z) ≤ C (1 − |w|^{2})^{−(1+2t)} (w ∈ 𝔹).

Analogous statements can be obtained for the Dirichlet space D = B_{−(N+1)}^{2} or for vanishing Carleson measures.

Now let N = 1, when the Arveson space is the Hardy space H^{2} on 𝔻. Then Corollary 6.1 seems contrary to what is known for usual Carleson measures for Hardy spaces, because powers in (iv) do not seem right, and we have a characterization of Carleson measures on a Hardy space using Bergman discs in (i); cf. [Zh2, §8.2.1]. But (iii) depends on an imbedding of H^{2} in L_{−1}^{2} by way of I_{s}^{t} as described in Definition 1.1 in contrast to the usual imbedding of H^{2} in L^{2}(∂𝔻) by way of inclusion. The imbedding I_{s}^{t} and the equivalent norm

∥f∥_{B_{−1}^{2}}^{2} = ∫_{𝔻}(1 − |z|^{2})^{−1+2t} |D_{s}^{t}f(z)|^{2} dν(z) (t > 0)

for H^{2} in (iii) require a positive-order radial derivative, where ν now is the area measure. Thus the Carleson measures defined here are different from the usual Carleson measures for Hardy spaces.

Remark 6.2. However, we indeed obtain the usual Hardy-space Carleson measures by taking limits as t → 0^{+} in the case q = −1 for all N and p in Theorem 1.3 (iii) and (iv). The limit of the norm on the right of (iii) does not exist even for polynomials. But let’s replace the right side of (iii) by the equivalent quantity

\frac{(pt)_{N}}{N!} ∥ · ∥_{B_{−1}^{p}}^{p}, (20)

where the role of the coefficient is to normalize the measure ν_{−1+pt} in ∥ · ∥_{B_{−1}^{p}}^{p} with weight 1. Using weak-∗ convergence of measures, it is noted in [BB, §0.3] and a detailed proof is given in [KÜ, §3] that

lim_{t→0^{+}} \frac{(pt)_{N}}{N!} ∥f∥_{B_{−1}^{p}}^{p} = ∥f∥_{H^{p}}^{p} = ∫_{∂𝔹} |f|^{p} dσ (f ∈ B_{−1}^{p}).

Hence the limit of (iii) is the definition of a usual Carleson measure on H^{p}. More importantly, by Fatou lemma, (iv) becomes equivalent to being a usual Carleson measure on H^{p} as t → 0^{+}; see [Zh2, Corollary 8.2.3] for N = 1. Although B_{−1}^{p} ≠ H^{p} for p ≠ 2 (see [BGP, p. 840]), the above holds also for f ∈ H^{p}, because B_{−1}^{2} = H^{2}, and Carleson measures of either type do not depend on p.

Analogously, the limiting case of parts (iii) and (iv) of Theorem 5.3 as t → 0^{+} is [Zh2, Theorem 8.2.5].

We finish this section by a proof of Theorem 1.6.

Proof of Theorem 1.6. The basic ideas are in the standard proofs of theorems on Hardy-space Carleson measures. We outline the few differences for N ≥ 1 and more general α.

If μ is a Carleson measure, then the inclusion map imbeds H^{2} into L^{2}(μ). As in the proof of the implication (iii) ⟹ (iv) of Theorem 1.3, we use the function

g_{w}(z) = \frac{(1 − |w|^{2})^{α/2}}{(1 − ⟨z, w⟩)^{(N+α)/2}} ,

which lies in H^{2} with norm ∼ 1 by [R, Proposition 1.4.10] for α > 0.

For the converse, we recall that the Carleson windows in 𝔹 are the nonisotropic balls W (ζ, ρ) = \{ z ∈ \overline{𝔹} : |1 − ⟨z, ζ⟩| < ρ \} for |ζ| = 1 and 0 < ρ < 1 whose intersections with ∂𝔹 have surface measure ∼ ρ^{N}; see [CM, pp. 42–43]. It is easy to see that if w_{0} ∈ W (ζ, ρ), then 1 − |w_{0}|^{2} ∼ ρ and 1 − ⟨z, w_{0}⟩ ∼ ρ for z ∈ W (ζ, ρ), which gives rise to a result much like Lemma 5.1.

Corollary 6.3. Let μ be a positive Borel measure on 𝔹. If U_{α,0,−1}μ(w) has limit 0 as |w| → 1 for some real α, then μ is a Hardy-space vanishing Carleson measure. If μ is a Hardy-space vanishing Carleson measure and α > 0, then U_{α,0,−1}μ(w) has limit 0 as |w| → 1.

7. Forelli-Rudin Operators

Theorem 1.5 and Corollary 5.9 suggest a consideration of the operators

V_{α,β,γ}^{μ}f(w) = (1 − |w|^{2})^{α} ∫_{𝔹} \frac{(1 − |z|^{2})^{β}}{(1 − ⟨w, z⟩)^{N+1+α+β+γ}} f(z) dμ(z) (w ∈ 𝔹).

Although U_{α,β,q} and V_{α,β,γ}^{μ} for γ = q are almost the same operators, the emphasis on each is different. The former, U_{α,β,q}, is more a transform of the variable measure μ, while the latter, V_{α,β,γ}^{μ}, for fixed μ is more an operator that acts on a suitable variable function f.

Theorem 7.1. Let α > 0 and β + q > −1. The operator V_{α,β,q}^{μ} : L^{∞}(μ) → L^{∞} is bounded if and only if μ is a q-Carleson measure. Further, if μ is a vanishing q-Carleson measure, then V_{α,β,q}^{μ} : L^{∞}(μ) → L^{∞} is compact.

Proof. The type of kernels studied in [BB, §4] shows that replacing the integrand of V_{α,β,q}^{μ} by its modulus has no effect on our results. Then the claim on boundedness is an immediate consequence of Theorem 1.5.

Next suppose μ is a vanishing q-Carleson measure. Clearly V_{α,β,q}^{μ} is bounded by the first part. Take a bounded sequence {f_{k}} in L^{∞}(μ). By Corollary 5.9,

|V_{α,β,q}^{μ}f_{k}(w)| ≤ C (1 − |w|^{2})^{α} ∫_{𝔹} \frac{(1 − |z|^{2})^{β}}{|1 − ⟨z, w⟩|^{N+1+α+β+q}} dμ(z),

and the right side tends to 0 as |w| → 0 uniformly in k; that is, given ε > 0, there is an R < 1 such that for |w| > R and all k, we have |V_{α,β,q}^{μ}f_{k}(w)| < ε. In particular, every f_{k} ∈ C_{0}. Define Y f(w) = (1 − |w|^{2})^{−α} V_{α,β,q}^{μ}f(w). Then Y f_{k} ∈ H(𝔹) and Y f_{k}(w) = o((1 − |w|^{2})^{−α}) as |w| → 1 uniformly in k by the above discussion. So {Y f_{k}} is uniformly bounded on each compact subset of 𝔹 and a normal family. Thus there exists a subsequence {Y f_{k_{j}}} converging uniformly on compact subsets of 𝔹 to g ∈ H(𝔹). Since α > 0, also V_{α,β,q}^{μ}f_{k_{j}}(w) → h(w) = (1−|w|^{2})^{α}g(w) uniformly on compact subsets of 𝔹, and h ∈ C. So given a compact E ⊂ 𝔹 and ε > 0, there is a j_{0} such that for j > j_{0} and all w ∈ E, we have |V_{α,β,q}^{μ}f_{k_{j}}(w) − h(w)| < ε. If it were the case that h(w) ≠ o(1) as |w| → 1, there would be points {w_{l}} in 𝔹 with |w_{l}| → 1 and an η > 0 such that |h(w_{l})| ≥ η for all l. Taking ε = η/2, |w_{l_{0}}| > R, E = {w_{l_{0}}}, and j > j_{0}, we would get

|h(w_{l_{0}})| ≤ |h(w_{l_{0}}) − V_{α,β,q}^{μ}f_{k_{j}}(w_{l_{0}})| + |V_{α,β,q}^{μ}f_{k_{j}}(w_{l_{0}})| < ε + ε = η,

contradicting what was just assumed on the order of growth of g. Then h ∈ C_{0}, and |h(w)| < ε for |w| > R, picking a larger R than the one above if necessary. Now let E = { w : |w| ≤ R } and j > j_{0}. Therefore

sup_{w∈𝔹} |V_{α,β,q}^{μ}f_{k_{j}}(w) − h(w)| ≤ sup_{w∈𝔼} |V_{α,β,q}^{μ}f_{k_{j}}(w) − h(w)|

+ sup_{|w|>R} |V_{α,β,q}^{μ}f_{k_{j}}(w)| + sup_{|w|>R} |h(w)| < 3ε,

meaning that the subsequence {V_{α,β,q}^{μ}f_{k_{j}}} converges to h in L^{∞}. Thus V_{α,β,q}^{μ} is compact.

Special cases of V_{α,β,γ}^{μ} include the Berezin transforms (take α = N + 1 + u, β = −q + u, γ = q, μ = ν_{q}, and compare to Definition 5.7), certain cases of Bergman projections (take α = 0, β = 0, γ = s, μ = ν_{s} with s > −(N + 1), and compare to Definition 3.4), and their commonly used simpler version V_{α,β,0}^{ν} with γ = 0 and μ = ν.

The boundedness of the operators V_{α,β,γ}^{ν} with μ = ν on L_{q}^{p} for 1 ≤ p < ∞ are characterized in [KZ], but the case p = ∞ is missing. We fill in this gap now.

Theorem 7.2. The operator V_{α,β,γ}^{ν} : L^{∞} → L^{∞} is bounded if and only if either α > 0, β > −1, γ ≤ 0, or, α = 0, β > −1, γ < 0. Further, if α > 0, β > −1, and −(β + 1) < γ < 0, then V_{α,β,γ}^{ν} : L^{∞} → L^{∞} is compact.

Proof. As noted in the proof of Theorem 7.1, replacing the integrand by its modulus makes no difference.

The measure μ = ν is a 0-Carleson measure. If α > 0, β > −1, and γ ≤ 0, then |1−⟨z, w⟩|^{N+1+α+β+γ} ≥ |1−⟨z, w⟩|^{N+1+α+β}. This reduces the problem to the case q = 0 of Theorem 7.1, and V_{α,β,γ}^{ν} is bounded. If α = 0, β > −1, and γ < 0, then

|V_{α,β,γ}^{ν}f(w)| ≤ ∥f∥_{L^{∞}} ∫_{𝔹} \frac{(1 − |z|^{2})^{β}}{|1 − ⟨z, w⟩|^{N+1+β+γ}} dν(z),

which is bounded by [R, Proposition 1.4.10].

Conversely suppose V_{α,β,γ}^{ν} is bounded. First take f ≡ 1. For the integral in V_{α,β,γ}^{ν} to converge, β > −1 is necessary. With the same f, using the same machinery as in the proof of the if part of Theorem 1.5, we have

∥V_{α,β,γ}^{ν}∥ ≥ (1 − |w|^{2})^{α} ∫_{b(w,r)} \frac{(1 − |z|^{2})^{β}}{|1 − ⟨z, w⟩|^{N+1+α+β+γ}} dν(z) ∼ \frac{1}{(1 − |w|^{2})^{γ}}

for any w ∈ 𝔹 and r by Lemmas 2.1 and 2.2. This forces γ ≤ 0. Next take f(z) = (1 − |z|^{2})^{η} with η > 0 so large that α + γ − η < 0. Then ∥f∥_{L^{∞}} = 1, and

∥V_{α,β,γ}^{ν}∥ ≥ (1 − |w|^{2})^{α} ∫_{𝔹} \frac{(1 − |z|^{2})^{β+η}}{|1 − ⟨z, w⟩|^{N+1+α+β+γ}} dν(z) ∼ (1 − |w|^{2})^{α}

by [R, Proposition 1.4.10]. We must have α ≥ 0. Finally, using f ≡ 1 once again in the case α = 0, we are led to conclude that γ < 0 by [R, Proposition 1.4.10].

Under the conditions stated in the last claim, −γ > 0 so that μ = ν_{−γ} is a vanishing 0-Carleson measure, and β + γ > −1. Then the operator V_{α,β+γ,0}^{μ} is compact on L^{∞} by the case q = 0 of Theorem 7.1 since now L^{∞}(μ) = L^{∞}. This operator is just V_{α,β,γ}^{ν}.

Corollary 7.3. The operator V_{α,β,0}^{ν} : L^{∞} → L^{∞} is bounded if and only if α > 0 and β > −1.

This corollary also provides the converse to the if part that has already been shown in [Zh3, Theorem 9]. For 0 < p < 1, there is a partial result in [K2, Theorem 2.4 (b)]. We do not know of any earlier results in the literature on the compactness of the operators V_{α,β,γ}^{μ} on Lebesgue classes.

8. Weighted Bloch, Lipschitz, and Growth Spaces

This section is for descriptions of weighted Bloch, Lipschitz, and growth spaces in terms of Carleson measures. As corollaries, we obtain that these spaces can be described using any radial derivative whose order is sufficiently high, and how differentiation transforms one of these spaces to another. We start with the usual Bloch space.

Theorem 8.1. A function h ∈ H(𝔹) lies in the Bloch space B if and only if dμ = |I_{s}^{u}h|^{p} dν_{q} is a q-Carleson measure for some q, p, s, and u > 0. Such an h lies in the little Bloch space B_{0} if and only if μ is a vanishing q-Carleson measure.

Proof. First suppose h ∈ B. Then for any s and u > 0, |I_{s}^{u}h|^{p} is bounded, say, by C by (14). Choose α > 0 and β with β + q > −1. Then

U_{α,β,q}μ(w) ≤ C (1 − |w|^{2})^{α} ∫_{𝔹} \frac{(1 − |z|^{2})^{β+q}}{|1 − ⟨z, w⟩|^{N+1+α+β+q}} dν(z) (w ∈ 𝔹).

Then U_{α,β,q}μ is bounded by [R, Proposition 1.4.10], and consequently μ is a q-Carleson measure by Theorem 1.5.

Conversely, suppose μ is a q-Carleson measure for some u > 0. Then U_{α,β,q}μ is bounded on 𝔹 by Theorem 1.5. Taking w = 0 gives

∫_{𝔹}(1 − |z|^{2})^{β} |I_{s}^{u}h(z)|^{p} dν_{q}(z) < ∞

for some β with β+q > −1, that is, (D_{s}^{u}h)^{p} belongs to A_{Q}^{1} since Q = q+pu+β > −1. The reproducing property of the Bergman kernel K_{Q} on A_{Q}^{1} yields

(D_{s}^{u}h)^{p}(w) = C ∫_{𝔹} \frac{(1 − |z|^{2})^{q+pu+β}}{(1 − ⟨w, z⟩)^{N+1+q+pu+β}} (D_{s}^{u}h)^{p}(z) dν(z) (w ∈ 𝔹).

Thus

|I_{s}^{u}h(w)|^{p} ≤ C (1 − |w|^{2})^{pu} ∫_{𝔹} \frac{(1 − |z|^{2})^{β}}{|1 − ⟨z, w⟩|^{N+1+pu+β+q}} dμ(z) (w ∈ 𝔹).

The right side is bounded by assumption. This shows h ∈ B.

The second statement is proved similarly using Corollary 5.9 instead.

The special case q > 0, p = 2, N = 1 of Theorem 8.1 is contained in [ASX, Theorem 2.2] with u = 1 and in [ANZ, Theorem 5] with u a positive integer. The special case q > −1, p = 2, u = 1 of Theorem 8.1 with any N is essentially in [Yan1] and [Yan2]. The p of all four sources corresponds to our (N + 1 + q)/N, and the Bloch space is obtained when it exceeds 1. See also [CKP] for a similar result.

Theorem 8.1 can be generalized to cover functions in weighted Bloch spaces with parameter α. These spaces of holomorphic functions f are usually defined on 𝔻 for α > 0 by requiring that (1 − |z|^{2})^{α} |f′(z)| is bounded on 𝔻. It is also known that (1 − |z|^{2}) f′(z) can be replaced by (1 − |z|^{2})^{k} f^{(k)}(z). In the ball, we replace f′ by its radial derivative, better yet, by D_{s}^{1}f. Then it is easy to see that the weighted Bloch space is the limiting case as p → ∞ of the space B_{p(α−1)}^{p}. Since the lower index of a Besov space can be any real number, now it is clear that the positivity restriction on α is superfluous. For uniformity of notation with Besov spaces, we change the parameter α to α + 1 since D_{s}^{u}f(z) goes hand in hand with (1 − |z|^{2})^{u}.

Definition 8.2. For any α ∈ ℝ, we define the weighted Bloch space B^{α} to consist of all f ∈ H(𝔹) for which

(1 − |z|^{2})^{α} I_{s}^{u}f(z) (21)

is bounded on 𝔹 for some s, u satisfying

α + u > 0. (22)

The weighted little Bloch space B_{0}^{α} is the subspace of B^{α} consisting of those f for which the quantity in (21) vanishes on ∂𝔹 for some s, u satisfying (22).

With this labeling of parameters, the usual Bloch and little Bloch spaces are B^{0} and B_{0}^{0}. We show below the independence of Definition 8.2 of s, u satisfying (22). Note the similarity of (22) to the first inequality in (14). By [Zh4, Theorems 7.17 and 7.18], the spaces B^{α} and B_{0}^{α} for α < 0 are the holomorphic Lipschitz spaces Λ_{−α} and Λ_{−α,0} of the ball. By (22), when α > 0, no derivative is required to define these spaces. Thus the spaces B^{α} for α > 0 are the so-called growth spaces A^{−α}; see [HKZ, Definition 4.13]. However, for the usual Bloch space and all Lipschitz spaces, as |α| increases, increasingly higher order derivatives are required.

The following characterization of these spaces extends Theorem 8.1 and is proved along the same lines. The α here must not be confused with the α of Theorem 1.5.

Theorem 8.3. A function h ∈ H(𝔹) lies in B^{α} if and only if dμ = |I_{s}^{u}h|^{p} dν_{α+q} is a q-Carleson measure for some q, p, s, and u satisfying (22). Such an h lies in B_{0}^{α} if and only if μ is a vanishing q-Carleson measure.

Corollary 8.4. Definition 8.2 is independent of s, u satisfying (22).

Proof. Theorem 8.3 does not depend on s or u as long as (22) is fulfilled.

Thus the L^{∞} norms of the quantities (1 − |z|^{2})^{α} I_{s}^{u}f(z) for different values of s, u satisfying (22) are equivalent, and we put

∥f∥_{B^{α}} = ∥f∥_{B_{0}^{α}} = sup_{z∈𝔹} (1 − |z|^{2})^{α} |I_{s}^{u}f(z)|.

This fact allows us to pass between different weighted Bloch spaces by taking derivatives or integrals as in Proposition 3.2.

Corollary 8.5. For any α, s, t, D_{s}^{t}(B^{α}) = B^{α+t} and D_{s}^{t}(B_{0}^{α}) = B_{0}^{α+t} are isometric isomorphisms under the equivalence of norms.

Corollary 8.6. Suppose α, q, p are related by q − αp > −1. Then B^{α} ⊂ B_{q}^{p}, and the inclusion is continuous.

Proof. We recall that the Bloch space B lies in every Bergman space A_{Q}^{p} for which Q > −1, take derivatives of order α, and use Corollary 8.5 and Proposition 3.2.

Corollary 8.7. Suppose α, q, p are related by q − αp = −(N + 1). Then B_{q}^{p} ⊂ B^{α}, and the inclusion is continuous.

Proof. This time, we recall that each Besov space B_{−(N+1)}^{p} lies in the Bloch space B and proceed as in the previous corollary.

In particular, when α > 0, the last two corollaries give inclusion relations between Besov spaces and growth spaces A^{−α}, and between Besov spaces and Lipschitz spaces Λ_{−α} when α < 0. In fact, precisely the case α < 0 of Corollary 8.7 is proved in [BB, Theorem 5.14 (i)] by other means.

9. Connections with Gap Series

In this section, gap series are the common thread. We first decide which Borel measures on the ball are finite in terms of the same imbedding we have used in characterizing Carleson measures. We go ahead with constructing Carleson measures from holomorphic functions under some conditions easily entailed by gap series in Besov spaces. We then see that Carleson measures made with holomorphic functions give rise to Besov functions.

Definition 9.1. Let f = ∑_{k} F_{k} ∈ H(𝔹), where F_{k} is a homogeneous polynomial of degree n_{k}. The series f is said to have Hadamard gaps and we write f ∈ HG if there is an ω > 1 such that n_{k+1}/n_{k} ≥ ω for all k.

When N = 1, of course F_{k}(z) = c_{k}z^{n_{k}} and ∥F_{k}∥_{H^{∞}} = ∥F_{k}∥_{H^{p}} = c_{k}.

Lemma 9.2. If f = ∑_{k} F_{k} ∈ HG and ∥F_{k}∥_{H^{∞}} ≤ C n_{k}^{α}, where n_{k} is the degree of F_{k}, then f ∈ B^{α}.

Proof. This is really a combination of the techniques of [Yam, Theorem 1] and [Ti, Proposition 4.16]. Although the former reference considers only α > −1, its proof is valid word by word for the case α ≤ −1 too.

Our first theorem helps identify those Carleson measures that are finite.

Theorem 9.3. A positive Borel measure μ on 𝔹 is finite if and only if some I_{s}^{u} with (22) satisfied maps the weighted Bloch space B^{α} into L^{p}(μ_{αp}) for some p continuously.

Proof. The idea of the proof is in [AFP, Theorem 16]. If μ is a finite measure, then trivially ∫_{𝔹} |I_{s}^{u}f|^{p} dμ_{αp} ≤ ∥f∥_{B^{α}}^{p} ∫_{𝔹} dμ < ∞. Conversely, assume

∫_{𝔹} |I_{s}^{u}f|^{p} dμ_{αp} ≤ C ∥f∥_{B^{α}}^{p} (f ∈ B^{α}) (23)

for some p, s, u satisfying (22). Pick multi-indices λ_{k} with |λ_{k}| = n_{k}, where n_{k} is as in Definition 9.1, such that the components λ_{k_{j}}, j = 1, …, N, are all nonzero, and let F_{k}(z) = n_{k}^{α} z^{λ_{k}}. Then f = ∑_{k} F_{k} ∈ HG ∩ B^{α} by Lemma 9.2. The nonzero condition assures that we have |F_{k}(z)| ≥ C n_{k}^{α} |z|^{n_{k}} for some C and all z ∈ 𝔹. By homogeneity and (7),

D_{s}^{u}f(e^{iθ}z) = ∑_{k} _{s}^{u}d_{n_{k}}F_{k}(e^{iθ}z) ∼ ∑_{k} n_{k}^{u} e^{in_{k}θ} F_{k}(z).

If we replace f(z) by f(e^{iθ}z) on the left of (23), integrate with respect to θ, then apply Fubini theorem and [Zy, Theorem V.8.20], we obtain

∥f∥_{B^{α}}^{p} ≥ C ∫_{𝔹}(1 − |z|^{2})^{αp+pu} \frac{1}{2π} ∫_{0}^{2π}|∑_{k} n_{k}^{u} F_{k}(z) e^{in_{k}θ}|^{p} dθ dμ(z)

∼ ∫_{𝔹}(1 − |z|^{2})^{αp+pu} (∑_{k} n_{k}^{2u} |F_{k}(z)|^{2})^{p/2} dμ(z)

≥ C ∫_{𝔹}(1 − |z|^{2})^{αp+pu} (∑_{k} n_{k}^{2α+2u} |z|^{2n_{k}})^{p/2} dμ(z)

≥ C ∫_{𝔹}(1 − |z|^{2})^{αp+pu} (∑_{k} k^{2α+2u−1} |z|^{2k})^{p/2} dμ(z)

∼ ∫_{𝔹}(1 − |z|^{2})^{αp+pu} \frac{1}{(1 − |z|^{2})^{αp+pu}} dμ(z) = μ(𝔹)

after some obvious inequalities and a binomial expansion.

If α = 0 and q > −1, then B ⊂ B_{q}^{p} = A_{q}^{p} and ∥f∥_{A_{q}^{p}} ≤ C ∥f∥_{B}. Then (23) is satisfied automatically for any q-Carleson measure μ and any f ∈ B by Theorem 1.3 (iii). Thus we recapture the fact that Bergman-space Carleson measures are finite.

An equivalent statement to Theorem 9.3 is this: Given a positive Borel measure μ on 𝔹, the measure μ_{−αp} is finite if and only if some I_{s}^{u} with (22) satisfied maps B^{α} into L^{p}(μ) continuously. In this form, the particular case N = 1, α > −1, and positive integer t of Theorem 9.3 is in [PR, Theorem 3.2].

We next give some conditions on functions in Besov spaces that generate q-Carleson measures. Let J(k) = { j ∈ ℕ : 2^{k} ≤ j < 2^{k+1} } = { j ∈ ℕ : j/2 < 2^{k} ≤ j } for k = 0, 1, 2, . . .. The following result is in [KÜ, Theorem 4.2].

Lemma 9.4. A power series f(z) = ∑_{k} F_{k} ∈ HG belongs to B_{q}^{p} if and only if ∑_{k} n_{k}^{−(1+q)} ∥F_{k}∥_{H^{p}}^{p} < ∞, where n_{k} is the degree of F_{k}.

Lemma 9.5. Let q, p, t be related by (1). Let f(z) = ∑_{k} F_{k} ∈ H(𝔹) be given by its homogeneous expansion. Suppose that

∑_{k=0}^{∞} 2^{−(1+q)k}( ∑_{j∈J(k)} ∥F_{j}∥_{H^{∞}})^{p} < ∞.

Then dμ = |I_{s}^{t}f|^{p} dν_{1+2q} is a q-Carleson measure.

Proof. Let z = Rζ ∈ 𝔹, where ζ ∈ ∂𝔹, and put M_{k} = ∥F_{k}∥_{H^{∞}} for simplicity. Then |F_{k}(Rζ)| = R^{k}|F_{k}(ζ)| ≤ R^{k}M_{k}, and |D_{s}^{t}f(z)| ≤ ∑_{k} _{s}^{t}d_{k}M_{k}R^{k} ≤ C ∑_{k} k^{t}M_{k}R^{k} by (7). Take a u satisfying (1) when substituted for t. Then the left side of Theorem 1.3 (iv) is

= sup_{w∈𝔹} (1 − |w|^{2})^{N+1+q+pu} ∫_{𝔹} \frac{(1 − R^{2})^{1+2q+pt+pu}}{|1 − ⟨Rζ, w⟩|^{(N+1+q+pu)2}} |D_{s}^{t}f(Rζ)|^{p} dν(z)

≤ C sup_{w∈𝔹} (1 − |w|^{2})^{N+1+q+pu} ∫_{0}^{1}(1 − R^{2})^{1+2q+pt+pu} R^{2N−1} (∑_{k} k^{t}M_{k}R^{k})^{p}

· ∫_{∂𝔹} \frac{1}{|1 − ⟨ζ, Rw⟩|^{(N+1+q+pu)2}} dσ(ζ) dR

≤ C sup_{w∈𝔹} (1 − |w|^{2})^{N+1+q+pu} ∫_{0}^{1} \frac{(1 − R^{2})^{1+2q+pt+pu}}{(1 − R^{2} |w|^{2})^{N+(1+q+pu)2}} (∑_{k} k^{t}M_{k}R^{k})^{p} dR

≤ C ∫_{0}^{1}(1 − R^{2})^{q+pt} (∑_{k} k^{t}M_{k}R^{k})^{p} dR ≤ C ∑_{k=0}^{∞} 2^{−(1+q+pt)k}( ∑_{j∈J(k)} j^{t}M_{j})^{p}

≤ C ∑_{k=0}^{∞} 2^{−(1+q+pt)k} 2^{ptk}( ∑_{j∈J(k)} M_{j})^{p} < ∞,

where we have used [R, Proposition 1.4.10], [MP, Theorem 1], and the definition of J(k).

Corollary 9.6. If f = ∑_{k} F_{k} ∈ B_{q}^{p} ∩ HG and the sequence {∥F_{k}∥_{H^{∞}}/∥F_{k}∥_{H^{p}}} is bounded, then dμ = |I_{s}^{t}f|^{p} dν_{1+2q} is a q-Carleson measure.

Note that the boundedness condition is automatic for N = 1 since then the sequence constantly is 1. It is also satisfied if each F_{k} is a function of a single variable that may vary with k.

Proof. Let f have parameter ω of Definition 9.1. It is well-known that for this kind of f the number of n_{k} in J(k) is at most 1 + ⌊log_{ω} 2⌋ independently of k. Coupled with Lemma 9.4 and the way J(k) is defined, this yields

∞ > ∑_{k=0}^{∞} n_{k}^{−(1+q)}∥F_{k}∥_{H^{p}}^{p} ∼ ∑_{k=0}^{∞} 2^{−(1+q)k}( ∑_{j∈J(k)} ∥F_{j}∥_{H^{p}})^{p}

> C ∑_{k=0}^{∞} 2^{−(1+q)k}( ∑_{j∈J(k)} ∥F_{j}∥_{H^{∞}})^{p}

We are done by Lemma 9.5.

We have a partial converse.

Proposition 9.7. Let f ∈ H(𝔹), q, p, t be related by (1), and dμ = |I_{s}^{t}f|^{p} dν_{1+2q}. If μ is a q-Carleson measure, then f ∈ B_{Q}^{p} for any Q > q.

Proof. Pick u such that Q = 1 + 2q + pu. Then (1) is satisfied with u in place of t. By Corollary 1.4, μ_{pu} is a finite measure. This just means f ∈ B_{Q}^{p}.

There are similar results in [GP, §3] using Hardy-space Carleson measures and functions in B_{−1}^{p} when N = 1.

10. Fejér-Riesz and Hardy-Littlewood Inequalities

We obtain two kinds of inequalities on the behavior of the derivatives of functions in Besov spaces on lower-dimensional subspaces and spheres. They are interesting and new in their own right, even for Bergman spaces. They also reduce to two classical inequalities of Hardy spaces in the limit of a special case, thereby providing simple and direct proofs of them without resorting to interpolation.

First some notation. Write ℂ^{N} = ℂ_{1} × ⋯ × ℂ_{N} and ℂ_{m} = ℝ_{m} × iℝ_{m}. Put

L_{j,k} = ℝ_{1} × ⋯ × ℝ_{j} × ℂ_{j+1} × ⋯ × ℂ_{j+k} × {0} × ⋯ × {0}

for j = 1, …, N and k = 0, …, N −j. Put also L_{0,k} = ℂ_{1}×⋯×ℂ_{k}×{0}×⋯×{0} for k = 1, …, N. Denote the normalized Lebesgue measure on L_{j,k} by ν^{j,k}, and the one on L_{0,k} by ν^{0,k}. This notation should not be confused with our basic notation ν_{q} of (2).

Theorem 10.1. Given q and j, k as above, there is a constant C such that for all p, s and t satisfying (1), we have

∫_{L_{j,k}∩𝔹}(1 − |z|^{2})^{N+1+q−(1+j+2k)/2} |I_{s}^{t}f(z)|^{p} dν^{j,k}(z) ≤ C ∥f∥_{B_{q}^{p}}^{p}  (f ∈ B_{q}^{p})

and

∫_{L_{0,k}∩𝔹}(1 − |z|^{2})^{N+1+q−(1+k)} |I_{s}^{t}f(z)|^{p} dν^{0,k}(z) ≤ C ∥f∥_{B_{q}^{p}}^{p}  (f ∈ B_{q}^{p}).

Proof. First let dμ(z) = (1 − |z|^{2})^{N+1+q−(1+j+2k)/2} dν^{j,k}(z). By Lemma 2.1 and (6), we have

μ(b(w, r)) = ∫_{b(w,r)}(1 − |z|^{2})^{N+1+q−(1+j+2k)/2} dν^{j,k}(z)
∼ (1 − |w|^{2})^{N+1+q−(1+j+2k)/2} ν^{j,k}(b_{ψ}(w, tanh r))

for fixed r. It is explained in [R, §2.2.7] that b_{ψ}(w, tanh r) is an ellipsoid whose intersection with the complex line [w] is a disc of radius ∼ (1 − |w|^{2}) and whose intersection with the real (2N − 2)-dimensional space perpendicular to [w] is a ball of radius ∼ \sqrt{1 − |w|^{2}}. Then ν^{j,k}(b_{ψ}(w, tanh r)) ≤ C (1 − |w|^{2}) (\sqrt{1 − |w|^{2}})^{j−1+2k} and μ(b(w, r)) ≤ C (1 − |w|^{2})^{N+1+q}. Therefore μ is a q-Carleson measure and the first inequality follows by the equivalence of Theorem 1.3 (i) and (iii).

Analogously, we obtain ν^{0,k}(b_{ψ}(w, tanh r)) ≤ C (1 − |w|^{2})^{2} (\sqrt{1 − |w|^{2}})^{2k−2}. Then letting dμ(z) = (1 − |z|^{2})^{N+1+q−(1+k)} dν^{0,k}(z) and arguing as above yield the second inequality.

For k = 1, …, N, let 𝔹_{k} denote the unit ball of ℂ^{k} imbedded in ℂ^{N} in the same way as L_{0,k} is imbedded in ℂ^{N}. Clearly L_{0,k} ∩ 𝔹 = 𝔹_{k}. Let also σ^{k} denote the normalized Lebesgue measure on ∂𝔹_{k}.

Corollary 10.2. The restriction to 𝔹_{k} imbeds B_{q}^{p} continuously into B_{N−k+q}^{p}(𝔹_{k}).

Before stating the next result, for k = 1, …, N, let’s define the generalized integral means of an f ∈ H(𝔹) by

M_{p}(f, k, R) = (∫_{∂𝔹_{k}} |f(Rζ, 0′)|^{p} dσ^{k}(ζ))^{1/p} (0 < R < 1, 0 < p < ∞),

where 0′ is an (N − k)-tuple of zeros.

Theorem 10.3. Let q, 0 < p ≤ p_{1} < ∞, and k = 1, …, N be given. Put

q_{1} = p_{1}(\frac{N + 1 + q}{p} − \frac{k + 1}{p_{1}}).

Then there is a constant C such that for all s, t with q + pt > −1, we have

∫_{0}^{1}(1 − R^{2})^{q_{1}+p_{1}t} M_{p_{1}}^{p_{1}}(D_{s}^{t}f, k, R) dR ≤ C ∥f∥_{B_{q}^{p}}^{p} (f ∈ B_{q}^{p}). (24)

Proof. We have q_{1} ≥ q and q_{1} +p_{1}t > −1 too. Clearly ν_{q_{1}}^{0,k} is a q_{1}-Carleson measure on 𝔹_{k}, and Theorem 1.3 (iii) implies (24) with C ∥f∥_{B_{q_{1}}^{p_{1}} (𝔹_{k})}^{p_{1}} instead on the right

side. By Corollary 10.2, this quantity is ≤ C ∥f∥_{B_{−N+k+q_{1}}^{p_{1}}}^{p_{1}} .

But (N + 1 + q)/p = (N + 1 + (−N + k + q_{1}))/p_{1}, which is precisely the condition to be satisfied for [BB, Theorem 5.13] translated to our notation by the correspondence A_{1+q+pt,t}^{p} = B_{q}^{p} between their labeling of spaces and ours. Hence B_{q}^{p} is continuously included in B_{−N+k+q_{1}}^{p_{1}}, and this completes the proof.

Remark 10.4. Now we take q = −1, replace the norm ∥ · ∥_{B_{−1}^{2}} by (20), and then let t → 0^{+} in Theorem 10.1 as we did in Remark 6.2. This results in the case c = 1 of the two inequalities for Hardy spaces of the ball in [M, Theorem 1]. Further letting N = 1 in the resulting first inequality, which forces j = 1 and k = 0, we arrive at the classical Fejér-Riesz inequality for Hardy spaces. When k = N in the second inequality, we cannot let t → 0^{+} without an extra requirement such as c > 1 in [M, Theorem 1 (2)].

Doing the same norm replacement and taking the same limit for q = −1 in Theorem 10.3 give [M, Theorem 4] in the case λ = q, which is our p_{1}. Further letting N = 1, which forces k = 1, we find a classical inequality of Hardy and Littlewood for Hardy spaces; see [D, p. 146].

Although we cannot obtain sharp constants in the classical inequalities, it is worth noting that our proofs follow very easily from Theorem 1.3.

Let Q = q + pt > −1 and g = D_{s}^{t}f in the last two theorems; then g ∈ A_{Q}^{p}, a Bergman space, and ∥f∥_{B_{q}^{p}} = ∥g∥_{A_{Q}^{p}}. From this point of view, Theorems 10.1 and 10.3 are results for Bergman spaces. In this form, the case k = N of Theorem 10.3 can be found in [BB, Theorem 3.2 (iii)] more generally with one extra free parameter, yet less generally only for 1 < p < ∞. All other cases of these two theorems, including the cases without derivatives, seem to be new for Bergman spaces too.

11. Cesàro Operators

In [AS], the authors consider two operators defined through differentiation, multiplication, and then integration. One of them is a generalization of the Cesàro operator, and they analyze it on weighted Bergman spaces on 𝔻. This operator is later generalized to mixed norm spaces on 𝔹 using first-order radial derivatives; see [H]. The other (companion) operator is investigated on weighted Bloch and weighted BMOA spaces in [Yo2] and [Yo1].

Here we generalize this companion operator to arbitrary-order radial differentiation and integration, and analyze it on Besov spaces, hence also on weighted Bergman spaces. We also prove that the exponential of a holomorphic function lies in Bergman spaces if the function gives rise to a Carleson measure.

Let s, t be arbitrary, g ∈ H(𝔹), and define

T_{g}f = D_{s+t}^{−t}M_{g}D_{s}^{t}f (f ∈ H(𝔹)),

where M_{g} represents the operator of multiplication by g. The two radial differential operators are inverses of each other by (8).

This operator, for certain values of s, t, shows up as a generalized Toeplitz operator with a holomorphic symbol on Arveson and general Dirichlet spaces in [AK2, §4]. Ordinarily, a Toeplitz operator with a holomorphic symbol is just a multiplication operator, but the generalization to all Dirichlet spaces requires the introduction of derivatives, and when g is holomorphic, the new Toeplitz operators reduce to T_{g}. This connection has already been predicted in [AS, p. 338], where they call their S_{g} a “distant cousin of Toeplitz operators”, which is essentially our T_{g} with t = 1, and not distant at all.

More interesting is the fact that T_{g} is the link between the weakly convergent families of functions in B_{q}^{2} spaces in Remark 4.8 and the Berezin transforms of functions in B_{q}^{2}. So given q and p = 2, let s satisfy (13), set t = −q + s, and put

H_{q}(z, w) = \frac{K_{s}(z, w)}{\sqrt{K_{−q+2s}(w, w)}} ∈ B_{q}^{2}.

Recall that H_{q} is some kind of a normalized kernel. Using (9) and (8) we have

[T_{g}H_{q}(·, w), H_{q}(·, w)]_{q}

= (1 − |w|^{2})^{N+1−q+2s}[I_{s}^{−q+s}D_{−q+2s}^{q−s}M_{g}D_{s}^{−q+s}K_{s}(·, w), I_{s}^{−q+s}K_{s}(·, w)]_{L_{q}^{2}}

= (1 − |w|^{2})^{N+1−q+2s}[(1 − |z|^{2})^{(−q+s)2} g(z) K_{−q+2s}(z, w), K_{−q+2s}(z, w)]_{L_{q}^{2}}

= B_{q}^{−q+2s}g(w) (g ∈ H(𝔹))

since −q + 2s > −1 and t = −q + s. This identity is in accord with the classical definition of the Berezin symbol of an operator. For further applications of this idea, see [AK2, §5].

Theorem 11.1. Given q, p, pick s, t satisfying (1), and let dμ = |g|^{p} dν_{q}, where g ∈ H(𝔹). The operator T_{g} : B_{q}^{p} → B_{q}^{p} defined using the selected s, t is bounded (respectively, compact) if and only if μ is a q-Carleson (respectively, vanishing q-Carleson) measure. For 1 ≤ c < ∞, the operator T_{g} : B_{q}^{2} → B_{q}^{2} belongs to the Schatten ideal S^{2c} if and only if the Berezin transform B_{q}^{u}μ belongs to L_{−(N+1)}^{c} for u > −1.

Proof. Consider I_{s}^{t} : B_{q}^{p} → L^{p}(μ), which is the main operator in Section 5. Let f ∈ B_{q}^{p}. All three claims follow at once from the equality ∥I_{s}^{t}f∥_{L^{p}(μ)} = ∥T_{g}f∥_{B_{q}^{p}} and Theorems 5.8 and 5.12. It also follows that ∥T_{g}∥ ≤ ∥g∥_{H^{∞}}.

It is important that the proof would not go through if we did not have our characterization of Carleson measures using I_{s}^{t} as an imbedding.

Corollary 11.2. Given q and g ∈ H(𝔹), suppose that |g|^{p} dν_{q} is a q-Carleson measure. Then e^{g} ∈ ⋂_{Q>−1} A_{Q}^{2}.

Proof. We follow the proof of [AS, Corollary 4] and show little detail. By assumption, T_{g} with q + 2t > 0 is bounded on B_{q}^{2} by Theorem 11.1. It is easily computed that T_{g}(1) = _{s}^{t}d_{0} D_{s+t}^{−t}(g) and T_{g}^{k}(1) = _{s}^{t}d_{0} D_{s+t}^{−t}(g^{k}) for k = 1, 2, …, where 1 is the constant function. Put also T_{g}^{0} = I. Then the series

\frac{1}{_{s}^{t}d_{0}} ∑_{k=0}^{∞} \frac{T_{g}^{k}(1)}{k!} = D_{s+t}^{−t} ∑_{k=0}^{∞} \frac{g^{k}}{k!}

converges in the operator norm topology and has sum D_{s+t}^{−t}(e^{g}) ∈ B_{q}^{2} by Proposition 3.2. Therefore e^{g} ∈ D_{s}^{t}(B_{q}^{2}) = A_{q+2t}^{2}. But t can be chosen so that Q = q + 2t > −1 is arbitrary.

One way that the measure μ in Theorems 8.3 and 11.1 is the same is by having g = D_{s}^{u}h and α + u = 0. But the second equality contradicts (22). In view of Corollary 8.5, we conclude that the g in Corollary 11.2 need not belong to any of the weighted Bloch spaces.

Acknowledgments

Parts of this work were done while the author’s host institution was Middle East Technical University and during his sabbatical visit to the University of Virginia. He thanks Middle East Technical University for granting the sabbatical leave. He also thanks the Department of Mathematics of the University of Virginia, the operator theory group there, and especially Thomas Kriete and Barbara MacCluer, for their hospitality.

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Department of Mathematics, Bi̇lkent University, Ankara 06800, Turkey

   E-mail address: kaptan@fen.bilkent.edu.tr
   URL: http://www.fen.bilkent.edu.tr/∼kaptan/