PRECISE INCLUSION RELATIONS AMONG

BERGMAN-BESOV AND BLOCH-LIPSCHITZ SPACES AND H^{∞} ON THE UNIT BALL OF ℂ^{N}

H. TURGAY KAPTANOĞLU AND A. ERSİN ÜREYEN

Abstract. We describe exactly and fully which of the spaces of holomorphic functions in the title are included in which others. We provide either new results or new proofs. More importantly, we construct explicit functions in each space that show our relations are strict and the best possible. Many of our inclusions turn out to be sharper than the Sobolev imbeddings.

1. Introduction

The purpose of this paper is to provide a complete description of the inclusion relations among the spaces mentioned in the title by providing proofs of the missing cases and of simpler proofs of the known cases as well as exhibiting explicit examples in all cases that show that the inclusions are strict and the best possible.

Let 𝔹 be the unit ball in ℂ^{N} with respect to the usual hermitian inner product ⟨z, w⟩ = z_{1}\overline{w}_{1} + ⋯ + z_{N} \overline{w}_{N} and the associated norm |z| = \sqrt{⟨z, z⟩}. Let H(𝔹) and H^{∞} denote the spaces of all and bounded holomorphic functions on 𝔹, respectively.

We let ν be the Lebesgue measure on 𝔹 normalized so that ν(𝔹) = 1. For q ∈ ℝ, we also define on 𝔹 the measures

dν_{q}(z) := (1 − |z|^{2})^{q} dν(z).

For 0 < p < ∞, we denote the Lebesgue classes with respect to ν_{q} by L_{q}^{p}, using also the notation L_{0}^{p} = L^{p}. The Lebesgue class of essentially bounded functions on 𝔹 with respect to any ν_{q} is the same (see [K3, Proposition 2.3]); we denote it by L^{∞}. For α ∈ ℝ, we also define the weighted classes

L_{α}^{∞} := { φ measurable on 𝔹 : (1 − |z|^{2})^{α} φ(z) ∈ L^{∞} }

so that L_{0}^{∞} = L^{∞}, which are normed by

∥φ∥_{L_{α}^{∞}} := ess sup_{z∈𝔹} (1 − |z|^{2})^{α} |φ(z)|.

For q > −1 and 0 < p < ∞, the weighted Bergman spaces are A_{q}^{p} = L_{q}^{p} ∩ H(𝔹). To extend this family to all real q, we resort to derivatives. Given q ∈ ℝ and 0 < p < ∞, let m be a nonnegative integer such that q + pm > −1. Then the Bergman-Besov space B_{q}^{p} consists of all f ∈ H(𝔹) for which

Date: 19 June 2018.

2010 Mathematics Subject Classification. Primary 32A37, 32A36, 32A18, 30H05; Secondary 30H25, 30H20, 46E15, 32A05, 30B10, 42B25, 32W05.

Key words and phrases. Bergman, Besov, Bloch, Lipschitz space, bounded holomorphic function, inclusion, Hadamard gap series, Ryll-Wojtaszczyk polynomial, Littlewood-Paley inequality, atomic decomposition, Sobolev imbedding.

(1 − |z|^{2})^{m} \frac{∂^{m}f}{∂z_{1}^{γ_{1}} ⋯ ∂z_{N}^{γ_{N}}} ∈ L_{q}^{p}

for every multi-index γ = (γ_{1}, …, γ_{N}) with γ_{1} + ⋯ + γ_{N} = m.

The spaces B_{q}^{2} are reproducing kernel Hilbert spaces whose kernels occupy a large part in our study of all B_{q}^{p} spaces. Consequently, even to define the spaces of interest in this work, it is more advantageous to use certain radial differential operators that are compatible with the kernels. So we follow [K3] and [KT] and resort to invertible radial differential operators D_{s}^{t} of order t ∈ ℝ for any s ∈ ℝ that map H(𝔹) to itself. These are described in detail in Section 2. 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).

Definition 1.1. For q ∈ ℝ and 0 < p < ∞, we define the Bergman-Besov space B_{q}^{p} to consist of all f ∈ H(𝔹) for which I_{s}^{t}f belongs to L_{q}^{p} for some s, t satisfying

(1)  q + pt > −1.

The quantity ∥f∥_{B_{q}^{p}} := ∥I_{s}^{t}f∥_{L_{q}^{p}} for any such s, t defines a norm on B_{q}^{p} for p ≥ 1 and a quasinorm for 0 < p < 1.

Definition 1.2. For α ∈ ℝ, we define the Bloch-Lipschitz space B_{α}^{∞} to consist of all f ∈ H(𝔹) for which I_{s}^{t}f belongs to L_{α}^{∞} for some s, t satisfying

(2)  α + t > 0.

The quantity ∥f∥_{B_{α}^{∞}} := ∥I_{s}^{t}f∥_{L_{α}^{∞}} for any such s, t defines a norm on B_{α}^{∞}.

Remark 1.3. By now, it is well-known that Definitions 1.1 and 1.2 are independent of s, t under (1) and (2), respectively, and also of the particular type of the derivative. Further, the norms on a given space depending on s, t are equivalent to each other under (1) or (2). For these, see, for example, [BB1, Theorem 5.12 (i)], [K3], [KT], and [ZZ]. So given a pair s, t, I_{s}^{t} imbeds B_{q}^{p} isometrically into L_{q}^{p} if and only if (1) holds, and I_{s}^{t} imbeds B_{α}^{∞} isometrically into L_{α}^{∞} if and only if (2) holds.

If q > −1, we can take t = 0 in (1) and obtain the weighted Bergman spaces B_{q}^{p} = A_{q}^{p}. Further, B_{−1}^{2} is the Hardy space H^{2}, B_{−(1+N)}^{2} is the Dirichlet space, and B_{−N}^{2} is the Drury-Arveson space. If α > 0, we can take t = 0 in (2) and obtain the weighted Bloch spaces. If α < 0, then the corresponding spaces are the holomorphic Lipschitz spaces Λ_{−α} = B_{α}^{∞}; see, for example, [R, Section 6.4].

Our use of α follows [KT] and [DÜ], which is more logical in view of the operators I_{s}^{t} and conforms well with the notation of B_{q}^{p}, so the usual Bloch space B_{0}^{∞} = B^{∞} corresponds to α = 0. Most other authors use α + 1 while [ZZ] uses −α where we use α. There is no discussion of little Bloch spaces in this paper.

The following three theorems in increasing intricacy are our main results. Unless otherwise specified, we use the full ranges of the parameters, 0 < p < ∞ and q, α, s, t ∈ ℝ, and all our results cover the standard weighted Bergman spaces as special cases.

Notation 1.4. If X_{a} is a family of spaces indexed by a ∈ ℝ, the symbol X_{<a} denotes any one of the spaces X_{b} with b < a. For functions, h_{<a} has a similar meaning.

INCLUSIONS AMONG BESOV AND BLOCH SPACES AND H^{∞} 3

Theorem 1.5. Given B_{q}^{p}, we have the inclusions

B_{< \frac{1+q}{p}}^{∞} ⊂ B_{q}^{p} ⊂ B∞_{\frac{1+N+q}{p}} .

Theorem 1.6. Let B_{q}^{p} be given.

(i) If p ≤ P, then B_{q}^{p} ⊂ B_{Q}^{P} if and only if
(3)  \frac{1 + N + q}{p} ≤ \frac{1 + N + Q}{P}.
(ii) If P < p, then B_{q}^{p} ⊂ B_{Q}^{P} if and only if
(4)  \frac{1 + q}{p} < \frac{1 + Q}{P}.

Theorem 1.7. (i) B_{q}^{p} ⊂ H^{∞} if and only if q < −(1 + N), or q = −(1 + N) and 0 < p ≤ 1.

(ii) H^{∞} ⊂ B_{q}^{p} if and only if q > −1, or q = −1 and p ≥ 2.

Theorem 1.5 can be equivalently stated from the point of view of the Bloch-Lipschitz spaces: Given B_{α}^{∞}, the inclusions B_{αp−(1+N)}^{p} ⊂ B_{α}^{∞} ⊂ B_{>αp−1}^{p} hold.

Note that both parts of Theorem 1.6 state if-and-only-if conditions, and there is no third alternative. Thus Theorem 1.6 covers all possible inclusion relations between two members of the B_{q}^{p} family of spaces.

There is a one-to-one correspondence between the points (p, q) in the right half plane of the pq-plane and the Bergman-Besov family of spaces B_{q}^{p}. The inclusions of Theorem 1.6 are shown graphically in Figure 1. There, the space B_{q}^{p} is included in all the spaces in region I and includes all the spaces in region II. The space B_{q}^{p} does not contain nor is contained in the spaces in regions III and IV, but has nonempty intersection with them since all B_{q}^{p} spaces contain all holomorphic polynomials. A very rudimentary version of this figure is in [K2, p. 731]. In Figure 1, we call the

Figure 1. If (P, Q) ∈ I, then B_{q}^{p} ⊂ B_{Q}^{P}; if (P, Q) ∈ II, then
B_{Q}^{P} ⊂ B_{q}^{p}; if (P, Q) ∈ III ∪ IV, then neither B_{q}^{p} nor B_{Q}^{P} contains
the other, but B_{q}^{p} ∩ B_{Q}^{P} ≠ ∅.

quadrant {q > −1} the Bergman zone and its complementary quadrant {q ≤ −1} the proper Besov zone. We show in Corollary 7.2 that the spaces in the proper Besov zone require some kind of a derivative in their integral norms.

The proofs of the inclusions are often known, but we simplify them, give new ones, and complete the missing cases. The real contribution and the strength of this paper is in finding categorical examples and counterexamples of functions that lie in some spaces but not in some others, whose proofs turn out to be considerably more difficult than those of inclusions.

It turns out that whenever a space is included in H^{∞} in this paper, then it is also included in the ball algebra A(𝔹) of holomorphic functions on 𝔹 that extend continuously to \overline{𝔹}. This fact is inherent in our proofs, but we make a note of it each time. Both these spaces are normed with ∥f∥_{∞} = sup_{z∈𝔹} |f(z)|.

Each inclusion in these results is strict and the best possible. Strict means that the two spaces in an inclusion are not equal. Best possible means either a space that contains a given one is the smallest possible in the family, or the inclusion result is an if-and-only-if condition. We make sure of these by exhibiting explicit functions that lie in one space but not in the other.

Moreover, each inclusion of ours is continuous, that is, if X ⊂ Y, then the inclusion map i : X → Y is continuous. This can be checked by ∥ · ∥_{Y} ≲ ∥ · ∥_{X}. Such an inequality is inherent in the proof of every inclusion we claim.

We prove Theorem 1.5 in Section 5, Theorem 1.6 in Section 6, and Theorem 1.7 in Section 7. We prove two other elementary inclusions in Section 4. Our approach to Theorems 1.5 and 1.6 is to prove each inclusion first for one value of q covering all values of p and then apply differentiation to pass to other spaces. Our proof of Theorem 1.7 is highly nontrivial and here we supply the missing cases. It uses techniques varying from atomic decomposition to Littlewood-Paley inequalities and to Ryll-Wojtaszczyk polynomials. We also show that in essence the norm of B_{q}^{p} requires a derivative of order specified by (1) in Section 7. In Section 3, we construct the example functions we use repeatedly; they have the general property that each lies in one space but not in a “nearby” space. In the last Section 8, we make a comparison of our inclusions with the holomorphic counterparts of Sobolev imbeddings. It so happens that in most cases our imbeddings are sharper than those dictated by the Sobolev imbedding theorem.

We do not make any comparisons with the Hardy spaces, because along with the Hardy Sobolev spaces and BMOA, those should be the topic of a different work. In this respect, H^{∞} is not a Hardy space, because the correct p = ∞ version of the Hardy spaces is BMOA.

2. Preliminaries

In multi-index notation, γ = (γ_{1}, …, γ_{N}) ∈ ℕ^{N} is an N-tuple of nonnegative integers, |γ| = γ_{1} + ⋯ + γ_{N}, γ! = γ_{1}! ⋯ γ_{N}!, 0^{0} = 1, and z^{γ} = z_{1}^{γ_{1}} ⋯ z_{N}^{γ_{N}}. The number of distinct multi-indices γ with |γ| = m, that is, the dimension of the space of holomorphic homogeneous polynomials of degree m in N variables is

δ_{m} = \binom{N − 1 + m}{N − 1}.

The standard basis vectors of ℂ^{N} are e_{j} = (0, …, 0, 1, 0 …, 0) with 1 in the jth position, j = 1, …, N. An overbar \overline{( )} indicates complex conjugate for functions and closure for sets. A quasinorm is given by the inequality ∥f +g∥ ≤ C(∥f∥+∥g∥) for some constant C > 1 in place of the triangle inequality. We use the term norm even when we mean quasinorm. The inner product of a space of functions X is denoted [ · , ·]_{X}. The pth power summable sequence spaces are denoted ℓ^{p}.

Let 𝕊 be the unit sphere in ℂ^{N}. When N = 1, 𝔹 is the unit disc 𝔻 and 𝕊 is the unit circle 𝕋. We let σ be the Lebesgue measure on 𝕊 normalized so that σ(𝕊) = 1. For 0 < p ≤ ∞, we denote the Lebesgue classes with respect to σ by L^{p}(σ). The polar coordinates formula that relates σ and ν is the one in [R, § 1.4.3].

Let’s also recall the definition of the Hardy spaces on 𝔹. For 0 < p < ∞, we say an f ∈ H(𝔹) belongs to H^{p} whenever

∥f∥_{H^{p}}^{p} = sup_{0<r<1} ∫_{𝕊} |f(rζ)|^{p} dσ(ζ) < ∞.

Since σ is finite, clearly H^{∞} ⊂ H^{p}.

The Pochhammer symbol (a)_{b} is given by

(a)_{b} := \frac{Γ(a + b)}{Γ(a)}

when a and a + b are off the pole set −ℕ of the gamma function Γ. In particular, (a)_{0} = 1 and for k a positive integer, (a)_{k} = a(a + 1) ⋯ (a + k − 1). The Stirling formula yields

(5) \frac{Γ(c + a)}{Γ(c + b)} ∼ c^{a−b},  \frac{(a)_{c}}{(b)_{c}} ∼ c^{a−b},  and  \frac{(c)_{a}}{(c)_{b}} ∼ c^{a−b}  (Re c → ∞),

where x ∼ y means both x = O(y) and y = O(x) for all x, y in question. If only x = O(y), we write x ≲ y.

An f ∈ H(𝔹) can be written in terms of its homogeneous expansion and its Taylor series as

f(z) = ∑_{k=0}^{∞} f_{k}(z) = ∑_{|γ|=0}^{∞} f_{γ}z^{γ}

in which f_{k} is a holomorphic homogeneous polynomial in z_{1}, …, z_{N} of degree k.

Definition 2.1. For q ∈ ℝ and z, w ∈ 𝔹, the Bergman-Besov kernels are

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

where _{2}F_{1} ∈ H(𝔻) is the usual hypergeometric function.

These kernels for q < −(1 + N) appear in the literature first in [BB1, p. 13]. Let the coefficient of ⟨z, w⟩^{k} in the series expansion of K_{q}(z, w) be c_{k}(q). Note that c_{0}(q) = 1, c_{k}(q) > 0 for any k, and by (5),

c_{k}(q) ∼ k^{N+q} (k → ∞),

for all q. The kernel K_{q} is the reproducing kernel of the Hilbert space B_{q}^{2}. These facts, coupled with the binomial expansion of ⟨z, w⟩^{k} give us another norm

∥z^{γ}∥_{B_{q}^{2}}^{2} = \frac{1}{c_{|γ|}(q)} \frac{γ!}{|γ|!} ∼ \frac{1}{|γ|^{N+q}} \frac{γ!}{|γ|!}

for B_{q}^{2} that is equivalent to the ones given in Definition 1.1. It also follows either from here by polarization or by [R, Proposition 1.4.8] that [z^{γ_{1}}, z^{γ_{2}}]_{B_{q}^{2}} = 0 if γ_{1} ≠ γ_{2}. Thus f ∈ H(𝔹) belongs to B_{q}^{2} if and only if

(6)  ∑_{γ} \frac{|f_{γ}|^{2}}{|γ|^{N+q}} \frac{γ!}{|γ|!} < ∞.

Definition 2.2. For any s, t ∈ ℝ, we define the radial differential operator D_{s}^{t} on H(𝔹) by

D_{s}^{t}f := ∑_{k=0}^{∞} d_{k}(s, t)f_{k} := ∑_{k=0}^{∞} \frac{c_{k}(s + t)}{c_{k}(s)}f_{k}.

Note that d_{0}(s, t) = 1, d_{k}(s, t) > 0 for any k, and

(7)  d_{k}(s, t) ∼ k^{t}  (k → ∞),

for any s, t by (5). So D_{s}^{t} is a continuous operator on H(𝔹) and is of order t; for a proof of a similar continuity result, see [GKÜ, Theorem 3.2]. In particular, D_{s}^{t}z^{γ} = d_{|γ|}(s, t)z^{γ} for any multi-index γ, and hence D_{s}^{t}(1) = 1. More importantly,

(8)  D_{s}^{0} = I,  D_{s+t}^{u}D_{s}^{t} = D_{s}^{t+u},  and  (D_{s}^{t})^{−1} = D_{s+t}^{−t}

for any s, t, u. Thus any D_{s}^{t} maps H(𝔹) onto itself.

Explicit forms of the norms of B_{q}^{p} and B_{α}^{∞} given in Definitions 1.1 and 1.2 are

(9)  ∥f∥_{B_{q}^{p}} = ∫_{𝔹} |D_{s}^{t}f(z)|^{p}(1 − |z|^{2})^{q+pt} dν(z)  (q + pt > −1),
(10)  ∥f∥_{B_{α}^{∞}} = sup_{z∈𝔹} |D_{s}^{t}f(z)|(1 − |z|^{2})^{α+t}  (α + t > 0).

One of the best things about the D_{s}^{t} is that they allow us to pass easily from one kernel to the other and from one space to the other in the same family. First, it is immediate that

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

for any q, t, where differentiation is performed on the holomorphic variable z. But the more versatile result is the following.

Theorem 2.3. Let q, p, α, s, t be arbitrary. Then the maps D_{s}^{t} : B_{q}^{p} → B_{q+pt}^{p} and D_{s}^{t} : B_{α}^{∞} → B_{α+t}^{∞} are isomorphisms, and isometries when the parameters of the norms of the spaces are chosen appropriately.

Proof. See [KÜ, Proposition 3.1] and [K4, Corollary 8.5]. The proofs require no more than Remark 1.3 and (8). See also [P, Corollary 3.9] for a different proof.

Note that this theorem works both ways since t < 0 is a possibility. □

Remark 2.4. Invertibility of D_{s}^{t} implies that only the zero function has zero norm in B_{q}^{p} or B_{α}^{∞}. The other types of derivatives mentioned in Remark 1.3 that can be used in place of the D_{s}^{t} are powers of the holomorphic gradient and the usual radial derivative given by

∇f(z) = (\frac{∂f}{∂z_{1}} , …, \frac{∂f}{∂z_{N}} )  and  Rf(z) = ⟨∇f(z), \overline{z}⟩.

Integrals of these derivatives define seminorms for the spaces B_{q}^{p} or B_{α}^{∞}.

The holomorphic automorphism of 𝔹 that exchanges 0 and z is the map

(12)  φ_{z}(w) = \frac{z − P_{z}(w) − \sqrt{1 − |z|^{2}}(I − P_{z})(w)}{1 − ⟨w, z⟩}  (w ∈ 𝔹),

where P_{z}(w) = ⟨w, z⟩z/|z|^{2} is the projection on the complex line passing through 0 and z. It reduces to the well-known φ_{z}(w) = (z − w)/(1 − \overline{z}w) for w ∈ 𝔻 when N = 1. 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)|.

This metric is invariant under compositions with the automorphisms of 𝔹. We denote the balls centered at a with radius r in the Bergman metric by b(a, r). A sequence {a_{k}} in 𝔹 is called separated if there is a ρ > 0 such that d(a_{k}, a_{m}) ≥ ρ for all k ≠ m, and we call ρ the separation constant.

The following growth rate estimate turns out to be surprisingly effective for obtaining several inclusion relations.

Lemma 2.5. If f ∈ B_{q}^{p}, then for any s, t satisfying q + pt > −(1 + N), we have

|D_{s}^{t}f(z)| ≲ \frac{∥f∥_{B_{q}^{p}}}{(1 − |z|^{2})^{(1+N+q+pt)/p}} (z ∈ 𝔹).

Proof. When f belongs to the Bergman space A_{q}^{p}, q > −1, and t = 0, the result is derived from the subharmonicity of |f|^{p} using Möbius transformations and is in [BB1, Corollary 3.5 (ii)], although rediscovered later several times. If f belongs to the general Besov space B_{q}^{p}, by Definition 1.1 and Remark 1.3, D_{s}^{t}f ∈ A_{q+pt}^{p} for any s, t satisfying (1). By the same reason, ∥f∥_{B_{q}^{p}} = ∥I_{s}^{t}f∥_{L_{q}^{p}} = ∥D_{s}^{t}f∥_{A_{q+pt}^{p}}. The result for t satisfying (1) follows by applying the Bergman space case to D_{s}^{t}f.

What we have so far can be written also in the form ∥f∥_{B_{(1+N+q)/p}^{α}} ≲ ∥f∥_{B_{q}^{p}}. But by Remark 1.3, the parameter t used in the norm ∥f∥_{B_{(1+N+q)/p}^{α}} < ∞ can be as low as to satify q + pt > −(1 + N). □

Corollary 2.6. If f ∈ B_{−(1+N)}^{p}, then ∥f∥_{B^{∞}} ≲ ∥f∥_{B_{−(1+N)}^{p}}.

This corollary appears also in [HY, Proposition 3.3] as well as in [BB1, Corollary 5.5] more generally.

3. Basic Examples

We now develop and collect interesting functions that lie in certain Besov or Bloch spaces but not in certain others. We use them frequently for the strict and the best possible inclusion results.

Example 3.1. The functions in the Hilbert spaces B_{q}^{2} can be characterized by their Taylor series, so it is easy to write a function h ∈ B_{Q}^{2} \ B_{q}^{2} if q < Q. Let

h(z) := ∑_{k} k^{(2(N−1)+q+Q)/4}z_{1}^{k} (z ∈ 𝔹).

Then by (6),

∥h∥_{B_{q}^{2}}^{2} ∼ ∑_{k} \frac{1}{k^{1−(Q−q)/2}} = ∞  while  ∥h∥_{B_{Q}^{2}}^{2} ∼ ∑_{k} \frac{1}{k^{1+(Q−q)/2}} < ∞.

Example 3.2. An example that is essential for the Bloch-Lipschitz spaces is the family of functions

f_{α}(z) := ∑_{k=0}^{∞} c_{k}(α − (1 + N))z_{1}^{k} (z ∈ 𝔹)

indexed by α ∈ ℝ. For any branch of the logarithm, by Definition 2.1,

(13) f_{α}(z) = \frac{1}{(1 − z_{1})^{α}} (α > 0)  and  f_{0}(z) = \frac{1}{z_{1}} log \frac{1}{1 − z_{1}}  (z ∈ 𝔹).

Note that, by (11),

(14)  D_{α−(1+N)}^{t}f_{α} = f_{α+t}.

Now Definition 1.2 and Remark 1.3 show immediately that f_{1} ∈ B_{1}^{∞}. The same reasoning shows that also f_{1} ∉ B_{β}^{∞} if β < 1. Applying (14) and Theorem 2.3 yields

(15)  f_{α} ∈ B_{α}^{∞} \ B_{<α}^{∞}  (α ∈ ℝ).

Further, [YO, (16) and (17)] say that

(16)  f_{< \frac{1+N}{p}} ∈ A_{0}^{p}  but  f_{\frac{1+N}{p}} ∉ A_{0}^{p}.

Finally, applying (14) and Theorem 2.3 yields as before

(17)  f_{< \frac{1+N+q}{p}} ∈ B_{q}^{p}  but  f_{\frac{1+N+q}{p}} ∉ B_{q}^{p}.

We give a proof of (16) simpler than the one in [YO]. For α > 0, by (13), for w = rζ with ζ ∈ 𝕊 and r ≥ 0, we have

|f_{α}(w)|^{p} = \frac{1}{|1 − w_{1}|^{αp}} = \frac{1}{|1 − ⟨rζ, e_{1}⟩|^{αp}} .

Then by polar coordinates,

∫_{𝔹} |f|^{p} dν ∼ ∫_{0}^{1} ∫_{𝕊} \frac{dσ(ζ)}{|1 − ⟨re_{1}, ζ⟩|^{αp}} r^{2N−1} dr.

If α = (1 + N)/p, then αp = 1 + N, and [R, Proposition 1.4.10] yields

∫_{𝔹} |f|^{p} dν ∼ ∫_{0}^{1} \frac{r^{2N−1}}{1 − r^{2}} dr,

which diverges by Calculus. If α < (1+N)/p, then αp < 1+N, say αp−N = c < 1, and [R, Proposition 1.4.10] again yields

∫_{𝔹} |f|^{p} dν ≲ ∫_{0}^{1} \frac{r^{2N−1}}{(1 − r^{2})^{c}} dr,

which converges by Calculus, where we use ≲ to incorporate the cases c ≤ 0 too. Since c_{k}(α − (1 + N)) ∼ k^{α−1}, it is clear that f_{α} ∈ H^{∞} if and only if α < 0.

We say a sequence {n_{k}} in ℕ has Hadamard gaps if there is a τ > 1 such that n_{k+1} ≥ τn_{k}. For such a sequence, if the homogeneous expansion of an f ∈ H(𝔹) has the form f = ∑_{k} f_{n_{k}}, we write f ∈ HG.

Theorem 3.3. Let H = ∑_{k} H_{n_{k}} ∈ HG.

(i) H ∈ B_{q}^{p} if and only if ∑_{k} n_{k}^{−(1+q)}∥H_{n_{k}}∥_{L^{p}(σ)}^{p} < ∞.

(ii) H ∈ B_{α}^{∞} if and only if sup_{k} n_{k}^{−α}∥H_{n_{k}}∥_{L^{∞}(σ)} < ∞.

Proof. See [K4, Lemmas 9.4 and 9.2]. See also [ZZ, Propositions 61 and 63]. For unweighted Bergman spaces A_{0}^{p}, see also [YO, Proposition 3]. For the spaces B_{α}^{∞} with α > −1, see also [YO, Proposition 2]. □

Several of our examples are constructed using the Ryll-Wojtaszczyk polynomials W_{m}, m = 0, 1, 2, . . .. Each W_{m} is a homogeneous polynomial of degree m with the properties

(18)  ∥W_{m}∥_{L^{∞}(σ)} = 1  and  ∥W_{m}∥_{L^{p}(σ)} ≳ 1 (0 < p < ∞).

These polynomials are invented with the first property and the second with p = 2 in [RW, Theorem 1.2]. The second property for general p is due to [U, Corollary 1]. Clearly also

(19)  ∥W_{m}∥_{L^{p}(σ)} ≤ ∥W_{m}∥_{L^{∞}(σ)} = 1.

When N = 1, we can simply take W_{m} = z^{m}. However, taking something like z_{1}^{m} for simplicity when N > 1 would not be as useful, because it does not satisfy the second property in (18).

Example 3.4. The example that is indispensable for the Bergman-Besov spaces is the family of functions

G_{qp}(z) := ∑_{k} 2^{k(1+q)/p} W_{2^{k}}(z) (z ∈ 𝔹)

indexed by q and p. By Theorem 3.3 and (18), it is clear that G_{qp} ∉ B_{q}^{p}. But the real question is to determine those spaces B_{Q}^{P} that G_{qp} lies in. The answer is that G_{qp} lies in B_{Q}^{P} if and only if (1 + q)/p < (1 + Q)/P, that is, G_{qp} lies in those B_{Q}^{P} whose (P, Q) lies precisely in regions I and IV of Figure 1.

If (1 + q)/p < (1 + Q)/P, then by Theorem 3.3 and (19),

∑_{k} 2^{−k(1+Q)}(2^{k(1+q)/p})^{P} ∥W_{2^{k}}∥_{L^{P} (σ)}^{P} ≤ ∑_{k} \frac{1}{2^{kP ((1+Q)/P −(1+q)/p)}} < ∞,

and G_{qp} ∈ B_{Q}^{P}. On the other hand, if (1 + q)/p ≥ (1 + Q)/P, then by Theorem 3.3 and (18),

∑_{k} 2^{−k(1+Q)}(2^{k(1+q)/p})^{P} ∥W_{2^{k}}∥_{L^{P} (σ)}^{P} ≳ ∑_{k} \frac{1}{2^{kP ((1+Q)/P −(1+q)/p)}} = ∞,

and G_{qp} ∉ B_{Q}^{P}.

By Theorem 3.3 and (18), it is clear that G_{qp} ∈ B_{α}^{∞} if and only if (1 + q)/p ≤ α. Further, G_{qp} ∈ H^{∞} by (18) if q < −1. On the other hand, if G_{−1,p} ∈ H^{∞} were

true, then also G_{−1,p} ∈ H^{2}. Then we would have ∥G_{−1,p}∥_{H^{2}} = ∑_{k} ∥W_{2^{k}}∥_{L^{2}(σ)} < ∞

by [R, Proposition 1.4.8]. But this is impossible since ∑_{k} ∥W_{2^{k}}∥_{L^{2}(σ)} = ∞ by (18).

The use of Ryll-Wojtaszczyk polynomials to show exclusions between pairs of function spaces is advocated in [YO] and [O].

Example 3.5. We now construct functions in every Besov space using the atomic decomposition idea. The atomic decomposition of Besov spaces is developed in several places starting with [CR, Theorem 2], but most proofs use hypotheses that are too much for our purposes, so we construct our functions from scratch using minimal assumptions. We start with a sequence {a_{k}} in 𝔹 that is merely separated with separation constant 2ρ. Given q and p, we also take a sequence λ = {λ_{k}} in ℓ^{p}. For 0 < p ≤ 1, we take an s satisfying the inequality 1 + N + q < p(1 + N + s); for 1 < p < ∞, we take an s satisfying 1 + q < p(1 + s). We set

F_{qp}(z) := ∑_{k} λ_{k}(1 − |a_{k}|^{2})^{1+N+s−(1+N+q)/p} K_{s}(z, a_{k}) (z ∈ 𝔹).

We start by showing that the series defining F_{qp} converges uniformly on any compact set M ⊂ 𝔹 and hence F_{qp} ∈ H(𝔹). Let z ∈ M; then |K_{s}(z, a_{k})| ∼ 1 for any k, and

|F_{qp}(z)| ≲ ∑_{k} |λ_{k}|(1 − |a_{k}|^{2})^{1+N+s−(1+N+q)/p}.

First for 0 < p ≤ 1, by the first choice of s, the power on 1 − |a_{k}|^{2} is positive and hence

|F_{qp}(z)| ≲ ∑_{k} |λ_{k}| ≤ ∑_{k} |λ_{k}|^{p} < ∞ (z ∈ M).

Second for 1 < p < ∞, by the Hölder inequality and that λ ∈ ℓ^{p}, we have

|F_{qp}(z)| ≲ (∑_{k}(1 − |a_{k}|^{2})^{(p(1+N+s)−(1+N+q))/(p−1)})^{(p−1)/p} (z ∈ M).

Call the power on 1−|a_{k}|^{2} by p̃. By the second choice of s, we see that p̃ > N. Then since the balls b(a_{k}, ρ) are disjoint, using [K4, Lemmas 2.1 and 2.2], we obtain

|F_{qp}(z)|^{p/(p−1)} ≲ ∑_{k}(1 − |a_{k}|^{2})p̃ ∼ ∑_{k} ∫_{b(a_{k},ρ)}(1 − |w|^{2})^{p̃−(1+N)} dν(w)

= ∫_{⋃_{k} b(a_{k},ρ)}(1 − |w|^{2})^{p̃−(1+N)} dν(w) ≤ ∫_{𝔹}(1 − |w|^{2})^{p̃−(1+N)} dν(w)

for all z ∈ M. But the last integral is finite.

To show that F_{qp} ∈ B_{q}^{p}, we define a linear map T by T λ := F_{qp} for λ ∈ ℓ^{p}. Let t satisfy (1); then s + t > −1 for any value of p. Then using (11),

D_{s}^{t}T λ(z) = ∑_{k} λ_{k} \frac{(1 − |a_{k}|^{2})^{1+N+s−(1+N+q)/p}}{(1 − ⟨z, a_{k}⟩)^{1+N+s+t}} (z ∈ 𝔹).

First for 0 < p ≤ 1,

|I_{s}^{t}T λ(z)|^{p} ≤ (1 − |z|^{2})^{pt} ∑_{k} |λ_{k}|^{p} \frac{(1 − |a_{k}|^{2})^{p(1+N+s)−(1+N+q)}}{|1 − ⟨z, a_{k}⟩|^{p(1+N+s+t)}} .

Then by [R, Proposition 1.4.10],

∥T λ∥_{B_{q}^{p}}^{p} ≤ ∑_{k} |λ_{k}|^{p}(1 − |a_{k}|^{2})^{p(1+N+s)−(1+N+q)} ∫_{𝔹} \frac{(1 − |z|^{2})^{q+pt}}{|1 − ⟨z, a_{k}⟩|^{p(1+N+s+t)}} dν(z)

∼ ∑_{k} |λ_{k}|^{p} = ∥λ∥_{ℓ^{p}}^{p}.

Second for 1 < p < ∞, by [K4, Lemma 2.1],

|D_{s}^{t}T λ(z)| ≲ ∑_{k} |λ_{k}|\frac{(1 − |a_{k}|^{2})^{1+N−(1+N+q)/p}}{ν(b(a_{k}, ρ))} ∫_{b(a_{k},ρ)} \frac{(1 − |w|^{2})^{s}}{|1 − ⟨z, w⟩|^{1+N+s+t}} dν(w)

and |I_{s}^{t}T λ(z)| ≤ Sϕ(z), where

ϕ(w) = ∑_{k} |λ_{k}|\frac{(1 − |a_{k}|^{2})^{1+N−(1+N+q)/p}}{ν(b(a_{k}, ρ))} χ_{ν(b(a_{k},ρ))}(w),

S is as in [Z, Theorem 2.10], and χ denotes the characteristic function. Now we use that {a_{k}} is separated and [K4, Lemma 2.2] to write

∥ϕ∥_{L_{q}^{p}}^{p} = ∑_{k} |λ_{k}|^{p} \frac{(1 − |a_{k}|^{2})^{p(1+N)−(1+N+q)}}{ν(b(a_{k}, ρ))^{p}} ν_{q}(b(a_{k}, ρ))

∼ ∑_{k} |λ_{k}|^{p} \frac{(1 − |a_{k}|^{2})^{p(1+N)−(1+N+q)}}{(1 − |a_{k}|^{2})^{p(1+N)}} (1 − |a_{k}|^{2})^{1+N+q}

= ∑_{k} |λ_{k}|^{p} = ∥λ∥_{ℓ^{p}}^{p}.

Now [Z, Theorem 2.10] implies that S : L_{q}^{p} → L_{q}^{p} is bounded by the conditions imposed above on s, t. Then ∥T λ∥_{B_{q}^{p}} ≤ ∥Sϕ∥_{L_{q}^{p}} ≲ ∥ϕ∥_{L_{q}^{p}} ≲ ∥λ∥_{ℓ^{p}}.

Thus T : ℓ^{p} → B_{q}^{p} is bounded and F_{qp} = T λ ∈ B_{q}^{p} for any value of p. Note that we need the separation property of {a_{k}} only for p > 1.

4. Some Initial Inclusions

Here we take care of a few better-known and straightforward inclusions that are part of the full picture although not listed among our main results.

Theorem 4.1. (i) If q < Q, then B_{q}^{p} ⊂ B_{Q}^{p}.

(ii) If α < β, then B_{α}^{∞} ⊂ B_{β}^{∞}.

Proof. For both (i) and (ii), the inclusion follows directly from Definitions 1.1 and 1.2 and Remark 1.3.

The inclusion in (ii) is strict because of (15). This part appears earlier in [YO,

(12) and (13)] with f_{α} for α > 0 and in [KT, Example 2.2].

The inclusion in (i) is strict too, because G_{qp} ∈ B_{Q}^{p} \ B_{q}^{p} by Example 3.4, since (p, Q) lies in region I with respect to B_{q}^{p}. A similar example when N = 1 appears earlier in [KÜ, Example 4.5]. □

Theorem 4.2. B_{<0}^{∞} ⊂ H^{∞} ⊂ B^{∞}.

Proof. The left hand inclusion on 𝔹 is in [R, Theorem 6.4.10]. There is a different proof in [KT, Theorem 5.4] using Bergman projections. Both proofs show also B_{<0}^{∞} ⊂ A(𝔹).

The right hand inclusion for N = 1 is commonly proved using the Schwarz-Pick lemma, but the proof for N > 1 is nowhere to be found, so we provide one. Let f ∈ H^{∞}, and without loss of generality, assume f : 𝔹 → 𝔻 so that ∥f∥_{∞} ≤ 1. Let w = f(z), and set g = φ_{w} ∘ f ∘ φ_{z}, where φ_{w} on 𝔻 and φ_{z} on 𝔹 are as in (12). Then g ∈ H(𝔹), g(0) = 0, and |g′(0)ζ| ≤ 1 for all ζ ∈ 𝔹, where g′(0) is called the hyperbolic derivative of f at z; see [K1, p. 651]. The chain rule yields that g′(0) = (1 − |w|^{2})^{−1}∇f(z)φ′_{z}(0), where φ′_{z}(0) = −(1 − |z|^{2})P_{z} + \sqrt{1 − |z|^{2}}(I − P_{z}) by [R, Theorem 2.2.2 (ii)]. Since |w| ≤ 1, using the special value ζ = z, we obtain that |∇f(z)|(1 − |z|^{2})|z| ≤ 1 for all z ∈ 𝔹. This proves that f ∈ B^{∞}.

By Example 3.2, the function f_{0} ∈ B^{∞} \H^{∞} shows that the right hand inclusion is strict. If α < 0, choose β such that α < β < 0. Then f_{β} ∈ H^{∞} \ B_{α}^{∞} by (15), and the left hand inclusion is also strict. □

5. A Bergman-Besov and a Bloch-Lipschitz Space

In this section, we prove Theorem 1.5. This theorem already appears in [ZZ, Theorem 66] with a proof that follows the same long path as the proof of the case q = 0 of unweighted Bergman spaces in [YO, Theorem]. However, once the result is established for this case, we can use the idea in Example 3.2 to differentiate and obtain the full result in all Besov spaces. We follow a different path though. We prove the right hand inclusion for a different value of q, because it has a much more direct proof. For the left hand inclusion, we simplify the proof given in [YO].

The right hand inclusion for general q appears in [BB1, Corollary 5.5]. It is probably known for some time that the Lipschitz spaces and the usual Bloch space B^{∞} belong to all Bergman spaces, which are special cases of the left hand inclusion and are direct consequences of Remark 1.3.

Proof of Theorem 1.5. Corollary 2.6 supplies us with a sufficiently general instance of the right hand inclusion, which is

(20)  B_{−(1+N)}^{p} ⊂ B_{0}^{∞}.

We take from [YO, Theorem (a)] the case q = 0 of the left hand inclusion, which is

(21)  B_{< \frac{1}{p}}^{∞} ⊂ A_{0}^{p}.

Here’s a proof of this that depends on Remark 1.3 and is slightly simpler than the one given in [YO]. It suffices to take 0 < α < 1/p and show that B_{α}^{∞} ⊂ A_{0}^{p}; for such an α, the norm ∥ · ∥_{B_{α}^{∞}} does not require any derivative. If f ∈ B_{α}^{∞}, then |f(z)|^{p} ≲ (1 − |z|^{2})^{−αp} for all z ∈ 𝔹. Then by polar coordinates,

∫_{𝔹} |f|^{p} dν ≲ ∫_{0}^{1} \frac{r^{2N−1}}{(1 − r^{2})^{αp}} ∫_{𝕊} dσ dr ≲ ∫_{0}^{1} \frac{r}{(1 − r^{2})^{αp}} dr,

which is finite by Calculus since αp < 1.

To boost all these to B_{q}^{p} with arbitrary q, we simply apply D_{s}^{(1+N+q)/p} to the spaces in (20), and we apply D_{s}^{q/p} to the spaces in (21). These yield all the inclusions claimed in the statement of the theorem by Theorem 2.3.

We follow the proof of [YO, Theorem (b)] and exhibit functions that show the inclusions just obtained are strict and the best possible. We use (15) and (17) three times. The right hand inclusion is strict because we have f_{\frac{1+N+q}{p}} ∈ B∞_{\frac{1+N+q}{p}} \ B_{q}^{p},

and the left hand inclusion is strict because we have f_{\frac{1+q}{p}} ∈ B_{q}^{p} \ B_{< \frac{1+q}{p}}^{∞}. Next, if

β < η < (1 + N + q)/p, then f_{η} ∈ B_{q}^{p} \ B_{β}^{∞}, and in view of Theorem 4.1, this shows the right hand inclusion is the best possible. Finally, G_{qp} ∈ B∞_{\frac{1+q}{p}} \ B_{q}^{p} by Example

3.4, and this shows the left hand inclusion is the best possible. □

6. Two Bergman-Besov Spaces

This section is devoted to a new proof of Theorem 1.6. Its both parts appear also in [ZZ, Theorems 69 and 70], but their proofs rely on difficult Carleson measure results. We give totally different direct proofs of the two parts based on two representative cases, Theorem 1.5, and the differentiation idea in Example 3.2. Our proof also sheds more light on the relationships among the B_{q}^{p} spaces.

There are several earlier partial results. [H, p. 703] and [NOW, Theorem 1] have a mixture of the two cases in certain Bergman spaces. Again for Bergman spaces with 0 < p < P = 1 and (2 + q)/p = 2 + Q, the inclusion in (i) is obtained in [S, Theorem 3]. With 0 < p < P ≤ 1 and (1 + N + q)/p = (1 + N + Q)/P, the inclusion in (i) is shown in [CR, Proposition 4.2] still for Bergman spaces. The full inclusion in (i) for Besov spaces is in [BB1, Theorem 5.13]. With 1 < p < P < ∞ and q = Q = −(1 + N), the inclusion in (i) is developed in [HY, Proposition 3.3] using Möbius invariance, and [P, Theorem 2.5] adds the p = 1 case to the previous inclusions. None of these results contain the only if parts.

Proof of Theorem 1.6. (i) Assume p ≤ P and (3). We follow the method of [GKÜ, Proposition 13.3] to obtain the desired inclusion. First let s, t satisfy (1) with q = −(1 + N) and p; then s, t clearly satisfy (2) with α = 0. Let also f ∈ B_{−(1+N)}^{p}. Then by Corollary 2.6,

∥f∥_{B_{−(1+N)}^{P}}^{P} = ∫_{𝔹} |I_{s}^{t}f|^{P} dν_{−(1+N)} = ∫_{𝔹} |I_{s}^{t}f|^{P −p} |I_{s}^{t}f|^{p} dν_{−(1+N)}

≤ ∥f∥_{B^{∞}}^{P −p} ∫_{𝔹} |I_{s}^{t}f|^{p} dν_{−(1+N)} ≲ ∥f∥_{B_{−(1+N)}^{p}}^{P −p}∥f∥_{B_{−(1+N)}^{p}}^{p} = ∥f∥_{B_{−(1+N)}^{p}}^{P},

which gives us one of the two fundamental inclusions B_{−(1+N)}^{p} ⊂ B_{−(1+N)}^{P}.

To pass to the remaining Besov spaces, we use (3) and call its two fractions −t and −T; then it holds that q + pt = −(1 + N) = Q + P T. We apply D_{s}^{−T} to both sides of the fundamental inclusion in the previous paragraph. Theorem 2.3 implies B_{−(1+N)−pT}^{p} ⊂ B_{Q}^{P}, that is, B_{q+p(t−T)}^{p} ⊂ B_{Q}^{P}. Since t − T ≥ 0, we obtain B_{q}^{p} ⊂ B_{Q}^{P} by Theorem 4.1.

Conversely, suppose that p ≤ P and B_{q}^{p} ⊂ B_{Q}^{P}. Since B_{Q}^{P} ⊂ B∞_{\frac{1+N+Q}{P}} by Theorem 1.5, we also have B_{q}^{p} ⊂ B∞_{\frac{1+N+Q}{P}}. By the best possible assertion of that theorem, we conclude that (3) holds.

(ii) Assume now P < p and (4). The proof is a variant of those of [GKÜ, Proposition 13.2] and part (i). For any r > −1, the finiteness of the measure ν_{r} gives us the other fundamental inclusion A_{r}^{p} ⊂ A_{r}^{P}.

Next we pass to the remaining Besov spaces. Let

r = \frac{Qp − qP}{p − P} ,  that is,  \frac{−r + Q}{P} = \frac{−r + q}{p} .

Then r > −1 by (4). Now call the common value of the two fractions on the right t; then r + P t = Q and r + pt = q. We apply D_{s}^{t} to the fundamental inclusion in the previous paragraph. Theorem 2.3 implies that B_{q}^{p} ⊂ B_{Q}^{P}.

Conversely, suppose that P < p and B_{q}^{p} ⊂ B_{Q}^{P}. Since B_{< \frac{1+q}{p}}^{∞} ⊂ B_{q}^{p} by Theorem

1.5, also B_{< \frac{1+q}{p}}^{∞} ⊂ B_{Q}^{P}. By the best possible assertion of that theorem, we can only

conclude that (4) holds with ≤. We prove that equality cannot occur by assuming (1 + q)/p = (1 + Q)/P and showing that the claimed inclusion does not hold. We do it through a gap series of Ryll-Wojtaszczyk polynomials once again. Let

(22)  G̃_{QP} = ∑_{k} \frac{2^{k(1+Q)/P}}{k^{1/P}} W_{2^{k}}.

Figure 2. If (p, q) ∈ V, then B_{q}^{p} ⊂ H^{∞}; if (p, q) ∈ VI, then
H^{∞} ⊂ B_{q}^{p}; if (p, q) ∈ VII, then neither B_{q}^{p} nor H^{∞} contains the
other, but B_{q}^{p} ∩ H^{∞} ≠ ∅.

By Theorem 3.3, (18), and (19), we see that G̃_{QP} ∈ B_{q}^{p} \ B_{Q}^{P}. This proves (4). In the last part, G_{QP} in place of G̃_{QP} would not work, because the G_{QP} are designed for the regions in Figure 1, and a point (P, Q) satisfying P < p and (1 + q)/p = (1 + Q)/P lies not in region I but on its boundary with respect to B_{q}^{p}.

(i), (ii) The inclusions are the best possible since they are given by if and only if conditions. They are also strict because Example 3.4 says exactly that. □

7. A Bergman-Besov Space and H^{∞}

We are now ready to prove Theorem 1.7. We have nothing new to say on the inclusion part, but for certain cases in the strict and the best possible parts, the functions developed in Section 3 are not good enough, and we have to attempt even more elaborate constructions.

The inclusions of Theorem 1.7 are shown graphically in Figure 2. The space H^{∞} includes all the B_{q}^{p} spaces in region V and is included all the B_{q}^{p} in region VI. A B_{q}^{p} in region VII has nonempty intersection with H^{∞} without one including the other.

Proof of Theorem 1.7. (i) The inclusions are already proved in [ZZ, Theorems 21 and 22] in their entirety as well as B_{q}^{p} ⊂ A(𝔹). What is left is to show that they are strict and the best possible. Strictness is easy. Example 3.4 explains that G_{qp} ∉ B_{q}^{p}. But (18) shows that G_{qp} ∈ H^{∞} for q ≤ −(1 + N) and any p.

For the best possible claim, we consider the two cases

q > −(1 + N) and 0 < p ≤ 1,  or  q = −(1 + N) and 1 < p < ∞.

Example 3.5 furnishes us with F_{qp} ∈ B_{q}^{p}, and we have to pick suitable {a_{k}} and {λ_{k}} to force F_{qp} ∉ H^{∞} in these two cases. We take b_{k} = 1 − 2^{−k} and a_{k} = b_{k}e_{1}. Then P_{a_{k}}(a_{m}) = a_{m}, and for m < k, we compute that |a_{k}−a_{m}| = (2^{k}−2^{m})2^{−(k+m)} and |1 − ⟨a_{k}, a_{m}⟩| = 2^{−k} + 2^{−m} − 2^{−(k+m)} < (2^{k} + 2^{m})2^{−(k+m)}. Hence

|φ_{a_{k}}(a_{m})| ≥ \frac{2^{k} − 2^{m}}{2^{k} + 2^{m}} ≥ \frac{1}{4}  and  d(a_{k}, a_{m}) ≥ tanh^{−1} \frac{1}{4} =: 2ρ;

in other words, {a_{k}} is separated.

For c ∈ ℂ, we write sgn(c) = c/|c| if c ≠ 0 and sgn(0) = 0. Next we pick λ_{k} = k^{−1−sgn(1+N+q)/p}; so λ_{k} = 1/k for q = −(1 + N) and λ_{k} = 1/k^{1+1/p} for q ≠ −(1 + N); but for both cases under consideration, {λ_{k}} ∈ ℓ^{p}. Also by the choice of s in Example 3.5, in both cases 1 + N + s > 0, which means that the kernel used in the definition of F_{qp} is binomial and not hypergeometric. If z = re_{1}, then ⟨z, a_{k}⟩ = rb_{k}. Then

F_{qp}(re_{1}) = ∑_{k} \frac{1}{k^{1+sgn(1+N+q)/p}} \frac{(1 − b_{k}^{2})^{1+N+s−(1+N+q)/p}}{(1 − rb_{k})^{1+N+s}} .

Since 1 − b_{k}^{2} ∼ 1 − b_{k}, by the monotone convergence theorem,

lim_{r→1^{−}} F_{qp}(re_{1}) ∼ ∑_{k} \frac{(1 − b_{k})^{−(1+N+q)/p}}{k^{1+sgn(1+N+q)/p}} = ∑_{k} \frac{2^{k(1+N+q)/p}}{k^{1+sgn(1+N+q)/p}} = ∞

in both cases, and thus F_{qp} ∉ H^{∞}.

(ii) Evidently H^{∞} lies in all weighted Bergman spaces A_{q}^{p} simply by the finiteness of the measures ν_{q} for q > −1. The next question is whether the Besov spaces B_{−1}^{p} at the boundary of the Bergman zone are large enough to include H^{∞}. Here a Littlewood-Paley inequality helps us. For p ≥ 2, [CK, Theorem] states that

∫_{𝔹}(|∇f(z)|^{2} − |Rf(z)|^{2})^{p/2}(1 − |z|^{2})^{p/2−1} dν(z) ≲ ∥f∥_{H^{p}}^{p}.

But |Rf(z)| ≤ |z| |∇f(z)|. Substituting this into the Littlewood-Paley inequality, we obtain

∫_{𝔹} |∇f(z)|^{p}(1 − |z|^{2})^{p−1} dν(z) ≲ ∥f∥_{H^{p}}^{p} (2 ≤ p < ∞).

By virtue of Remark 2.4, this says nothing but H^{p} ⊂ B_{−1}^{p} for p ≥ 2. In fact, because both are Hilbert spaces, B_{−1}^{2} = H^{2}. Thus H^{∞} ⊂ B_{−1}^{p} for p ≥ 2. Note that Littlewood-Paley inequalities are in general reversed for 0 < p < 2.

With the same {a_{k}} in the proof of part (i) and λ_{k} = k^{−1−sgn(1+q)/p}, we have F_{qp} ∈ B_{q}^{p} \ H^{∞} with the same proof as above in all q, p combinations mentioned in the statement of this part since now 1 + N + q ≥ N > 0. This shows strictness.

By construction, G_{qp} ∉ B_{q}^{p}, and for q < −1 and any p, also G_{qp} ∈ H^{∞} by Example 3.4; this shows that the inclusions of this part are the best possible for p ≥ 2. Here’s another proof of this fact. If H^{∞} ⊂ B_{q}^{p} for some q < −1, let (1 + q)/p < β < 0. Then by Theorem 4.2, also B_{β}^{∞} ⊂ B_{q}^{p}. But this is impossible by the best possible conclusion of Theorem 1.5.

When q = −1 and 0 < p < 1, then from (22), G̃_{−1,p} ∈ A(𝔹) \ B_{−1}^{p} by (18), and this shows that the inclusion is the best possible for the q, p at hand.

This leaves us with proving that the inclusion is the best possible for the case q = −1 and 1 ≤ p < 2. This is the most involved part of the proof, so we isolate it as the next theorem, which is also of independent interest. □

Theorem 7.1. (i) If {c_{k}} ∈ ℓ^{2}, there is a G ∈ A(𝔹) such that [G, W_{2^{k}}]_{L^{2}(σ)} = c_{k}.

(ii) For every 1 ≤ p < 2, there is a Ğ_{p} ∈ A(𝔹) \ B_{−1}^{p}.

Proof. (i) This is just [W, Proposition] written in different words. There is a constructive proof for N = 1 in [F, Theorem]. Let’s write carefully what this part says. If G = ∑_{m} G_{m} is the homogeneous expansion of G, then we have no control

on G_{m} if m ≠ 2^{k} for some k. When m = 2^{k}, both G_{2^{k}} and W_{2^{k}} are finite sums of δ_{2^{k}} monomials possibly with different coefficients and possibly with different sets of

coefficients equal to zero; yet ∫_{𝕊} G_{2^{k}}(ζ)\overline{W_{2^{k}}(ζ)} dσ(ζ) = c_{k} by [R, Proposition 1.4.8].

(ii) We follow the sketch of the proof of [V, Lemma 1.6], which is for N = 1, and fill in all its details. Let 1 ≤ p < 2 and suppose G ∈ A(𝔹) of part (i) lies in B_{−1}^{p}. Let s, t satisfy (1) for q = −1 and such p. So D_{s}^{t}G(rζ) = ∑_{m} d_{m}(s, t)r^{m}G_{m}(ζ)

for 0 < r < 1. By [R, Proposition 1.4.8] and (7),

∫_{𝕊} D_{s}^{t}G(rζ)\overline{W_{2^{k}}(ζ)} dσ(ζ) = d_{2^{k}}r^{2^{k}} ∫_{𝕊} G_{2^{k}}(ζ)\overline{W_{2^{k}}(ζ)} dσ(ζ) = d_{2^{k}}r^{2^{k}}c_{k} ∼ 2^{tk}c_{k}r^{2^{k}}.

Then by the convexity of the pth power function and (18),

2^{ptk}|c_{k}|^{p} r^{p 2^{k}} ≲ ∫_{𝕊} |D_{s}^{t}G(rζ)|^{p} dσ(ζ) (0 < r < 1).

Now by the polar coordinates formula,

∑_{k} 2^{ptk}|c_{k}|^{p}∫_{1−2^{−k}}^{1−2^{−(1+k)}}r^{2N−1+p 2^{k}}(1−r^{2})^{−1+pt}dr ≲∫_{𝔹} |D_{s}^{t}G(z)|^{p}(1−|z|^{2})^{−1+pt}dν(z).

On the interval [1 − 2^{−k}, 1 − 2^{−(1+k)}] of length 2^{−k}, we have 1 − r^{2} ∼ 1 − r ∼ 2^{−k}, and

r^{2N−1+p 2^{k}} ≥ (1 − 2^{−k})^{2N+p 2^{k}} ≥ (e^{−2^{1−k}})^{2N+p 2^{k}} ≥ e^{−N2^{2−k}−2p} ≥ e^{−4N−2p} > 0

for large enough k since 1−x ≥ e^{−2x} for small enough x > 0. Hence the integral on this interval is ∼ 2^{−ptk}. Therefore ∑_{k} |c_{k}|^{p} ≲ ∥G∥_{B_{−1}^{p}}^{p} This implies that {c_{k}} ∈ ℓ^{p} if

G ∈ B_{−1}^{p}. If we take a {c_{k}} ∈ ℓ^{2} \ ℓ^{p}, then the function G ∈ A(𝔹) obtained in part (i) is the desired Ğ_{p} ∉ B_{−1}^{p}. □

The fact that there are functions in the ball algebra that do not lie in certain Bergman-Besov spaces has an interesting consequence: Essentially, the derivative on the function in the integral norm (9) of B_{q}^{p} cannot be dispensed with when (p, q) ∈ V ∪ VII. This fact is first noticed in [A, p. 180] for the Drury-Arveson space B_{−N}^{2}.

Corollary 7.2. Let κ : [0, ∞) → ℝ be an increasing function with κ(0) = 0, and let μ be a positive Borel measure with support in \overline{𝔹}. Define E_{κμ} as the set of all f ∈ H(𝔹) for which

lim sup_{r→1^{−}} ∫\overline{𝔹} κ(|f(rz)|) dμ(z) < ∞.

Then E_{κμ} ≠ B_{q}^{p} if q < −1, or if q = −1 and 0 < p < 2.

Proof. We imitate the proof of [BB2, Theorem 4.3] that takes care of the Hardy-Sobolev-space counterpart. Let q, p be as in the statement of the corollary, and suppose B_{q}^{p} = E_{κμ} for some κ and μ. Applying the definition of E_{κμ} to f = 1 ∈ B_{q}^{p}, we obtain κ(1)μ(\overline{𝔹}) < ∞; so μ must be finite. If f ∈ A(𝔹), then

lim sup_{r→1^{−}} ∫\overline{𝔹} κ(|f(rz)|) dμ(z) ≤ κ(∥f∥_{L^{∞}(σ)})μ(\overline{𝔹}) < ∞,

which yields that f ∈ B_{q}^{p} too. This contradicts the fact that there are functions in A(𝔹) \ B_{q}^{p} for the values of q, p considered. For q < −1, one such function is G_{qp} of Example 3.4 by (18); for q = −1 and 0 < p < 1, one such function is G̃_{−1,p} as indicated in the proof of Theorem 1.7 (ii); for q = −1 and 1 ≤ p < 2, one such function is Ğ_{p} as indicated in Theorem 7.1 (ii). □

Remark 7.3. We do not know whether or not the norm of B_{−1}^{p} with p > 2 can be written as an integral without using a derivative on the function. On the other hand, B_{−1}^{2} is the Hardy space H^{2} and its norm is the same as that of L^{2}(σ). If q > −1, then the B_{q}^{p} are the Bergman spaces and clearly have integral norms without derivatives.

The following can be considered the p = ∞ version of Corollary 7.2 and concerns the derivative in (10).

Corollary 7.4. Let κ : [0, ∞) → ℝ be an increasing function with κ(0) = 0, and let ω : 𝔹 → [0, ∞). Define E_{κω} as the set of all f ∈ H(𝔹) for which

sup_{z∈𝔹} κ(|f(z)|) ω(z) < ∞.

Then E_{κω} ≠ B_{α}^{∞} if α < 0.

Proof. Let α < 0 and suppose B_{α}^{∞} = E_{κω} for some κ and ω. Applying the definition of E_{κω} to f = 1 ∈ B_{α}^{∞}, we obtain κ(1) sup_{z∈𝔹} ω(z) < ∞; so ω is bounded. If f ∈ H^{∞},

then

sup_{z∈𝔹} κ(|f(z)|) ω(z) ≤ κ(∥f∥_{∞}) sup_{z∈𝔹} ω(z) < ∞,

which yields that f ∈ B_{α}^{∞} too. This contradicts the fact that there are functions in H^{∞} \ B_{α}^{∞} for α < 0. One such function is f_{α} of Example 3.2. □

Remark 7.5. We do not know whether or not the norm of B^{∞} can be written without using a derivative on the function. On the other hand, the B_{α}^{∞} with α > 0 have norms without derivatives by (2).

Remark 7.6. There are characterizations of Bergman-Besov and Bloch-Lipschitz spaces that do not use a derivative directly but use a difference quotient of some sort; see, for example, [Ü] and the references therein. But a difference quotient behaves very much like a derivative. When we say “without using a derivative” in Remarks 7.3 and 7.5, we exclude such characterizations too.

8. Sobolev Imbeddings

Our final intention is to compare the inclusions in Theorems 1.5 and 1.6 with those predicted by the holomorphic versions of the Sobolev imbedding theorems.

Following [AF, Chapter 3], for 1 ≤ p < ∞, we let the Sobolev space W^{m,p} be the space of all locally integrable functions on 𝔹 all of whose generalized partial derivatives of order up to and including m = 1, 2, … belong to L^{p}. The subspace of W^{m,p} consisting of holomorphic functions on 𝔹 can be regarded as the Besov space B_{−mp}^{p} in which always q = −mp ≤ −1.

The Sobolev imbedding theorem we are interested in is [AF, Theorem 4.12] and concerns the continuous inclusion of W^{m,p} in Lebesgue or Lipschitz spaces. It is for Sobolev spaces defined on some types of domains in ℝ^{n} for all of which 𝔹 is a standard example. We read this theorem by setting n = 2N and taking the intersections of all spaces with H(𝔹). We analyze our findings in five regions of the parameters.

(a) If mp < 2N (that is, −2N < q ≤ −1), the Sobolev imbedding is W^{m,p} ⊂ L^{p^{∗}},

which translates to the holomorphic setting as B_{−mp}^{p} ⊂ B_{0}^{p^{∗}}, where \frac{1}{p^{∗}} = \frac{1}{p} − \frac{m}{2N}.

The sharpest (that is, into smallest space) inclusion that Theorem 1.6 (i) gives is

B_{−mp}^{p} ⊂ B_{0}^{p_{2}}, where \frac{1}{p_{2}} = \frac{1}{p} − \frac{m}{1 + N}. Since this p_{2} > p^{∗}, our inclusion is sharper

than that of the Sobolev imbedding theorem for N > 1. The two results say the same for N = 1.

(b) If mp = 2N (that is, q = −2N), the Sobolev imbedding is W^{m,p} ⊂ L^{p^{∗}} for any p^{∗} ≥ 1, which yields B_{−2N}^{p} ⊂ B_{0}^{p^{∗}}. The inclusions that Theorem 1.6 (i) gives

are B_{−2N}^{p} ⊂ B_{0}^{p_{2}}, where \frac{1}{p_{2}} ≥ \frac{1 − N}{1 + N} \frac{1}{p}, which says 0 < p_{2} < ∞ for all N. So the

two imbeddings are equivalent for q = −2N.

(c) If 2N < mp < 2N +p (that is, −2N −p < q < −2N), the Sobolev imbedding is W^{m,p} ⊂ Λ_{β}, where β = m − 2N/p and 0 < β < 1, which yields B_{−mp}^{p} ⊂ B_{−β}^{∞}. The inclusion that Theorem 1.5 gives is B_{−mp}^{p} ⊂ B_{α}^{∞}, where α = (1 + N)/p − m. Since this α < −β, our inclusion is sharper than that of the Sobolev imbedding theorem for N > 1. The two results say the same for N = 1.

(d) If mp = 2N +p (that is, q = −2N −p), the Sobolev imbedding is W^{m,p} ⊂ Λ_{β} for any 0 < β < 1, which yields B_{−2N−p}^{p} ⊂ B_{−β}^{∞}. If also p = 1, then W^{m,1} ⊂ Λ_{1}, that is, B_{−2N−1}^{1} ⊂ B_{−1}^{∞}. The inclusion that Theorem 1.5 gives is B_{−2N−p}^{p} ⊂ B_{α}^{∞} for any 1 ≤ p < ∞, where α = (1 − N)/p − 1. Since this α < −β, our inclusion is sharper than that of the Sobolev imbedding theorem for N > 1 or p > 1. The two results say the same when both N = 1 and p = 1.

(e) If mp > 2N +p (that is, q < −2N −p), the Sobolev imbedding is W^{m,p} ⊂ Λ_{1}, which yields B_{−mp}^{p} ⊂ B_{−1}^{∞}. The inclusion that Theorem 1.5 gives is B_{−mp}^{p} ⊂ B_{α}^{∞}, where α = (1 + N)/p − m. Since now α < −1, our inclusion is sharper than that of the Sobolev imbedding theorem.

We can also compare the number of derivatives lost in the imbeddings in (c), (d), and (e), which are all in the form W^{m,p} ⊂ Λ_{β}. The loss is indicated by the difference m − β and is 2N/p. On the other hand, the derivatives needed for B_{−mp}^{p} is given by −mp + pt > −1 and is t > m − 1/p, while for Λ_{m−2N/p} it is given as t > m − 2N/p; hence the loss is m − 1/p − (m − 2N/p) = (2N − 1)/p. So in our imbeddings, we lose derivatives of order 1/p less than those lost in the Sobolev imbeddings. This difference has already been noted in [BB1, pp. 39–40].

The fact that our inclusions are stronger and our loss of derivatives is less in general than those predicted by the Sobolev imbedding theorem should come as no surprise, because our spaces consist of holomorphic functions that are very smooth on a very nice domain.

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Bi̇lkent Üni̇versi̇tesi̇, Matemati̇k Bölümü, 06800 Ankara, Türki̇ye

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

Anadolu Üni̇versi̇tesi̇, Matemati̇k Bölümü, 26470 Eski̇şehi̇r, Türki̇ye

E-mail address: aeureyen@anadolu.edu.tr
URL: https://akademik.anadolu.edu.tr/aeureyen/