Contemporary Mathematics

Analytic Properties of Besov Spaces via Bergman Projections

H. Turgay Kaptanoğlu and A. Ersi̇n Üreyen

Abstract. We consider 2-parameter Besov spaces of holomorphic functions on the unit ball of ℂ^{N}. We obtain various exclusions between Besov spaces of different parameters using gap series. We estimate the growth near the boundary and the growth of Taylor coefficients of functions in these spaces. We find the unique function with maximum value at each point of the ball in each Besov space. We base our proofs on Bergman projections and imbeddings between Lebesgue classes and Besov spaces. Special cases apply to the Hardy space H^{2}, the Arveson space, the Dirichlet space, and the Bloch space.

1. Besov Spaces

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

q ∈ ℝ,  0 < p ≤ ∞,  s ∈ ℝ,  t ∈ ℝ;

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 also consider on 𝔹 the measures

dν_{q}(z) = (1 − |z|^{2})^{q} dν(z),

which are finite only for q > −1; here |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}.

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),

where D_{s}^{t} is a radial differential operator on H(𝔹) of order t for any s and I_{s}^{0} is the identity (or inclusion) I.

2000 Mathematics Subject Classification. Primary 32A37, 32A25, 32A40, 30B10; Secondary 46E22, 46E20, 46E15, 32A36, 32A35, 32A18, 32W99.

Key words and phrases. Besov, Bergman, Bloch, Hardy, Arveson, Dirichlet space; reproducing kernel Hilbert space; radial derivative; Bergman projection; boundary growth; extremal point evaluation; Taylor coefficient.

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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).

This particular parametrization of Besov spaces was introduced in [K1]. Note that s or t are not among the parameters of the space B_{q}^{p}, and that s does not appear in (1) at all. It is one of the aims of this paper in Section 2 to emphasize that Definition 1.1 is independent of s and t as long as (1) is satisfied. It follows that the L_{q}^{p} norms of various I_{s}^{t}f under (1) are all equivalent, where the term “norm” is used even for 0 < p < 1. We call any one of them the B_{q}^{p} norm of f and denote it by ∥f∥_{B_{q}^{p}}. So any I_{s}^{t} with (1) is an isometric imbedding of B_{q}^{p} into L_{q}^{p} when the norm used on B_{q}^{p} is ∥I_{s}^{t}f∥_{L_{q}^{p}}.

It is clear from Definition 1.1 that each B_{q}^{p} space contains all constants and all polynomials, and hence is nonempty. In fact, the ball algebra is dense in any B_{q}^{p} with p < ∞ by [BB, Lemma 5.2]. From a different point of view, condition (1) is required in order that constants and polynomials have finite B_{q}^{p} norm.

It is also known that Definition 1.1 is independent of the particular form of the differential operators D_{s}^{t} used. The differential operators are often defined as coefficient multipliers on the homogeneous expansion (Taylor expansion for N = 1) of f ∈ H(𝔹). To make this more precise, let H_{k} denote the space of holomorphic homogeneous polynomials in the variables z_{1}, …, z_{N} of total degree k = 0, 1, 2, . . .. An f ∈ H(𝔹) determines a unique expansion f = ∑_{k=0}^{∞} f_{k} with f_{k} ∈ H_{k}, where f_{k}(z) = c_{k}z^{k} for N = 1. Then D_{s}^{t}f = ∑_{k=0}^{∞} d_{k}f_{k}, where the dependence of d_{k} on s, t is not explicitly shown.

In this work, we use the D_{s}^{t} and d_{k} given in [K3, Definition 3.1], which we do not repeat here. The coefficients satisfy

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

for any s, t, where x ∼ y means that |x/y| is bounded above and below by two positive constants that are independent of the parameters in question (k here). It turns out that each D_{s}^{t} is a continuous differential operator of order t on H(𝔹), which is actually integral for t < 0. Further, D_{s}^{0} = I for any s, D_{−N}^{1} = I + R where R is the usual radial derivative, D_{s}^{t}(1) = d_{0} ≠ 0, and D_{s+t}^{u}D_{s}^{t} = D_{s}^{u+t}. Thus each D_{s}^{t} is invertible on H(𝔹) with two-sided inverse

(D_{s}^{t})^{−1} = D_{s+t}^{−t}.

After a short explanation on the role of the parameters q and p in Section 2, we relate Besov spaces to well-known spaces in Section 3. There it becomes clear that Besov spaces are natural continuation of Bergman spaces, and that Hardy spaces are just at the edge. We obtain some inclusion and exclusion relations between Besov spaces with different parameters in Section 4. In Section 5, we introduce the extended Bergman projections and investigate their orthogonality. Then come the applications of Bergman projections. We estimate the growth of Besov functions near the boundary of the ball in Section 6, and the growth of their Taylor coefficients in Section 7. The determination of the unique function yielding the maximality of point evaluations is also in Section 6.

We often use the Pochhammer symbol

(a)_{b} = \frac{Γ(a + b)}{Γ(a)}

when a and a + b lie off the pole set −ℕ of the gamma function Γ. For fixed a, b, Stirling formula gives

\frac{Γ(c + a)}{Γ(c + b)} ∼ c^{a−b}  and  \frac{(a)_{c}}{(b)_{c}} ∼ c^{a−b}  (Re c → ∞). (3)

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

An unadorned C represents a constant whose value varies from one formula to another but never depends on the functions involved or the parameters being investigated.

2. Two Parameters Suffice

The purpose of this section is to explain that the order t of the derivative D_{s}^{t} used in defining B_{q}^{p} is not a property of the space as long as (1) holds.

Required reading on holomorphic Besov spaces on 𝔹 is [BB]. The space B_{q}^{p} is called A_{Q,t}^{p} there ([BB, p. 36]), where Q = 1+q +pt, and Q > 0 is always assumed, which corresponds to our (1). Also 0 < p ≤ ∞ and t ∈ ℝ. (As a matter of fact, Q = 0 is a possibility there too, but this case pertains to Hardy Sobolev spaces, on which [BB] is also required reading.) We caution that the measure v_{q} has power q − 1 on 1 − |z|^{2} there ([BB, §0.3]), while our ν_{q} has q. Conversely, we call the space A_{Q,t}^{p} there the space B_{Q−1−pt}^{p} here.

As far as D_{s}^{t} or Definition 1.1 are concerned, the value of s makes no difference; see [BB, p. 41]. We need it in such exact formulas as Theorem 5.2 (9). Its value is irrelevant in most other results in this paper.

The following result is the content of [BB, Theorem 5.12 (i)].

Theorem 2.1. Suppose

\frac{Q_{1} − Q_{2}}{p} = t_{1} − t_{2}. (4)

Then A_{Q_{1},t_{1}}^{p} = A_{Q_{2},t_{2}}^{p}.

If we substitute 1 + q_{j} + pt_{j} > 0 for Q_{j} in (4), j = 1, 2, then (4) takes the form q_{1} = q_{2} =: q, and the conclusion of Theorem 2.1 becomes A_{Q_{1},t_{1}}^{p} = A_{Q_{2},t_{2}}^{p} = B_{q}^{p}. In other words, the space B_{q}^{p} remains unchanged by switching from t_{1} to t_{2} the order of the derivative in Definition 1.1 as long as both satisfy (1) with the same q and p.

This result is proved in a different way in [K3, Corollary 4.2]. Yet another proof for 1 ≤ p ≤ ∞ is given in Remark 5.4. Thus B_{q}^{p} spaces are all the Besov spaces that are defined by a norm of pure (Bergman) type. The modifier “diagonal” is used for these spaces in [AFJP, Remark 5.2] to tell them apart among the Besov spaces defined by a norm of mixed (Hardy and Bergman) type.

The B_{q}^{p} spaces are sometimes called Sobolev spaces, and generalize Bergman spaces to q ≤ −1. However, the name “Besov space” suits them better, because for q ≤ −1, their definition requires differentiation, and this differentitation is of fractional order; moreover, there is the connection with Besov spaces with mixed-type norms. Passing from Theorem 2.1 to Definition 1.1 brings out another interesting point. In the usual definition of Sobolev spaces, the maximum order of derivatives is specified; in B_{q}^{p} spaces, (1) specifies a “minimum” order of derivatives.

3. Special and Extremal Cases

Besov spaces include many known spaces as special cases. Here we fix the exact location of the special cases within the Besov-space family and the location of Besov spaces in reference to yet other known spaces. To begin with, each point in the right half plane of the pq-plane can be matched to exactly one B_{q}^{p} with p ≠ ∞. This point of view is illustrated in [K1, (2)].

Clearly, (1) is satisfied with t = 0 if q > −1. Then for q > −1, Besov spaces B_{q}^{p} are precisely the weighted Bergman spaces with the same parameters. So that part of the half plane above the line q = −1 is the Bergman region, and the part on or below it is the proper Besov region. Another critical line is q = −(N + 1). The spaces below it consist only of bounded functions; see Section 6. Several of our results below exhibit a change of behavior either at q = −1 or at q = −(N + 1).

Continuing, when p = ∞, (1) takes the form

t > 0

independently of q. Combined with the fact that L_{q}^{∞} = L^{∞} for any q (see [K3, Proposition 2.3]), we see that all B_{q}^{∞} spaces are one and the same, and taking t = 1 shows that they coincide with the Bloch space B. This is the only Besov space not in the right half plane. The little Bloch space B_{0} consists of those f ∈ B for which some I_{s}^{t}f with t > 0 vanishes on ∂𝔹.

We can pass between Besov spaces with the same p by differentiation or integration. The following result, inherent in the definition of the spaces A_{Q,t}^{p} in [BB], was rediscovered in [P, Corollary 3.9] for s > N and s + t > N.

Proposition 3.1. For any s, t, q and 0 < p < ∞, D_{s}^{t}(B_{q}^{p}) = B_{q+pt}^{p} is an isometric isomorphism when appropriate norms are used in the two spaces.

Proof. Let f ∈ B_{q}^{p} and put g = D_{s}^{t}f. Take u so large that q + p(t + u) > −1. Then D_{s+t}^{u}g = D_{s+t}^{u}D_{s}^{t}f = D_{s}^{t+u}f and I_{s}^{t+u}f lies in L_{q}^{p}. This is equivalent to saying that I_{s+t}^{u}g lies in L_{q+pt}^{p}. Hence g ∈ B_{q+pt}^{p} and the norms ∥f∥_{B_{q}^{p}} = ∥I_{s}^{t+u}f∥_{L_{q}^{p}} and ∥g∥_{B_{q+pt}^{p}} = ∥I_{s+t}^{u}g∥_{L_{q+pt}^{p}} are equal. We are done by showing D_{s}^{−t}(B_{q+pt}^{p}) = B_{q}^{p} the same way.

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]. Let b_{k}^{q} be the coefficient of ⟨z, w⟩^{k} in the series expansion of K_{q}. To see that _{2}F_{1} is the right choice to use in K_{q} for lower values of q, we can check by (3) that

b_{k}^{q} ∼ k^{N+q} (k → ∞)

irrespectively of the value of q. Thus f(z) = ∑_{α} c_{α}z^{α} ∈ B_{q}^{2} if and only if

∑_{α≠0} \frac{1}{|α|^{N+q}} \frac{α!}{|α|!} |c_{α}|^{2} < ∞. (5)

This also shows that K_{q} is bounded if and only if q < −(N + 1). We further have

D_{q}^{t}K_{q}(z, w) = K_{q+t}(z, w) (6)

similar to Proposition 3.1, where differentiation is performed on the holomorphic variable z.

The form of K_{q} clearly shows that B_{−(N+1)}^{2} is the Dirichlet space of the ball since K_{−(N+1)}(z, w) = −⟨z, w⟩^{−1} log(1 − ⟨z, w⟩), B_{−N}^{2} is the Arveson space, and B_{−1}^{2} is the Hardy space H^{2}.

For p ≠ 2 and p ≠ ∞, we can compare the norms of a monomial z^{α} in the two spaces to see how close the space B_{−1}^{p} comes to the space H^{p}. With the aid of [K3, Proposition 2.1], one easily sees that

∥z^{α}∥_{B_{q}^{p}}^{p} = d_{|α|}^{p} ∫_{𝔹} |z^{α}|^{p} (1 − |z|^{2})^{q+pt} dν(z)

= d_{|α|}^{p} \frac{N! Γ(1 + q + pt) ∏_{j=1}^{N} Γ(1 + pα_{j}/2)}{Γ(N + 1 + q + pt + p|α|/2)},

where (1) holds. On the other hand, by [R, Definition 5.6.1] and [K3, Proposition 2.1],

∥z^{α}∥_{H^{p}}^{p} = ∫_{∂𝔹} |ζ^{α}|^{p} dΣ(ζ) = \frac{(N − 1)! ∏_{j=1}^{N} Γ(1 + pα_{j}/2)}{Γ(N + p|α|/2)}, (7)

where Σ is the surface measure on the boundary ∂𝔹 of the ball normalized with Σ(∂𝔹) = 1. Now applying (2) and (3), we have

\frac{∥z^{α}∥_{B_{−1}^{p}}^{p}}{∥z^{α}∥_{H^{p}}^{p}} ∼ |α|^{tp} \frac{Γ(N + p|α|/2)}{Γ(N + pt + p|α|/2)} ∼ |α|^{0} = 1 (|α| → ∞).

Remark 3.2. Yet according to [BB, Theorem 5.12 (ii) and (iii)], B_{−1}^{p} ⊂ H^{p} if 0 < p ≤ 2 and H^{p} ⊂ B_{−1}^{p} if 2 ≤ p ≤ ∞ with continuous inclusions. In [BGP, p. 840], it is shown, using lacunary series in N = 1, that these inclusions are proper except for p = 2. This reference makes further comparison of the two families.

The Hardy spaces H^{p} are recovered as a limiting case of B_{−1}^{p} as the order t of the derivative D_{s}^{t} used in the definition of B_{−1}^{p} tends to 0. This is seen easily by considering the equivalent norm

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

for B_{−1}^{p}, where t satisfies (1) with q = −1. If f ∈ B_{q}^{p}, then D_{s}^{t}f belongs to the Bergman space B_{−1+pt}^{p} and

|∥f|∥_{B_{−1}^{p}}^{p} = \frac{(pt)_{N}}{N!} ∫_{𝔹} |D_{s}^{t}f(z)|^{p} dν_{−1+pt}(z),

which is the pth power of the norm of D_{s}^{t}f in B_{−1+pt}^{p} with respect to the normalized version of the measure ν_{−1+pt}. It is known that these normalized weighted volume measures converge weak-∗ to Σ as t → 0^{+} as noted in [BB, §0.3]. Here is a proof. Switching to polar coordinates and letting

dμ_{t}(ρ) = \frac{2N (pt)_{N}}{N!} ρ^{2N−1} (1 − ρ^{2})^{−1+pt} dρ

for 0 ≤ ρ ≤ 1, we have

|∥f|∥_{B_{−1}^{p}}^{p} = ∫_{0}^{1} dμ_{t}(ρ) ∫_{∂𝔹} |D_{s}^{t}f(ρζ)|^{p} dΣ(ζ).

To compute the distribution function F_{t}(x) = μ_{t}([0, x]) of μ_{t}, we let y = 1 − ρ^{2} and obtain

F_{t}(x) = \frac{2N (pt)_{N}}{N!} ∫_{0}^{x} ρ^{2N−1} (1 − ρ^{2})^{−1+pt} dρ

= \frac{(pt)_{N}}{(N − 1)!} ∫_{1−x^{2}}^{1} y^{−1+pt} (1 − y)^{N−1} dy

= (pt)_{N} ∑_{j=0}^{N−1} \frac{(−1)^{j}}{j! (N − 1 − j)!} ∫_{1−x^{2}}^{1} y^{−1+pt+j} dy

= (pt)_{N} ∑_{j=0}^{N−1} \frac{(−1)^{j}}{j! (N − 1 − j)! (pt + j)}[1 − (1 − x^{2})^{pt+j}]

for 0 ≤ x < 1. A computation with the beta function shows that F_{t}(1) = 1. But lim_{t→0^{+}} F_{t}(x) = 0 if 0 ≤ x < 1 and lim_{t→0^{+}} F_{t}(1) = 1. This means that weak-∗ limit of μ_{t} as t → 0^{+} is the unit point mass at ρ = 1. Consequently,

lim_{t→0^{+}} |∥f|∥_{B_{−1}^{p}}^{p} = ∫_{∂𝔹} |f(ζ)|^{p} dΣ(ζ) = ∥f∥_{H^{2}}^{p} (f ∈ B_{−1}^{p})

since D_{s}^{0} = I.

4. Inclusions and Exclusions

Various inclusions among Besov spaces follow directly from Definition 1.1. For example, B_{q_{1}}^{p} ⊂ B_{q_{2}}^{p} if q_{1} < q_{2} for fixed p. For fixed q > −1 (in the Bergman region), B_{q}^{p_{1}} ⊃ B_{q}^{p_{2}} if p_{1} < p_{2}. These are shown graphically in [K1, (2)]. Many others can be derived from [BB, Theorem 5.13]. This result is very general and several of its special cases have been rediscovered later in other papers. For example, the inclusion B_{−(N+1)}^{p_{1}} ⊂ B_{−(N+1)}^{p_{2}} for p_{1} < p_{2} follows immediately from it, but has been obtained also later for p_{1} > 1 using the Möbius invariance of these particular Besov spaces. For connections between Besov, Bergman, Hardy, and Hardy Sobolev spaces, one should first look at again [BB, Theorem 5.13] and [BB, Theorem 5.12 (ii) and (iii)]. Let’s note that Hardy spaces are labeled A_{0,0}^{p} and Hardy Sobolev spaces A_{0,t}^{p} in this source.

It is also interesting to show that Besov spaces with different parameters are different from each other. For the subfamily of spaces B_{q}^{2}, we can use the orthogonality of the monomials to construct explicit series that lie in B_{q_{2}}^{2} but not in B_{q_{1}}^{2} for q_{1} < q_{2}. This was done in [AK, Example 2.4] for q ≥ −(N + 1); as mentioned in the note added in proof at the end of the paper, it works for all q.

To construct further examples for exclusions, we set N = 1 in the remaining part of this section and consider lacunary series, mentioned earlier in Remark 3.2.

Definition 4.1. A power series f(z) = ∑_{k} c_{k}z^{n_{k}} ∈ H(𝔻) is said to have Hadamard gaps (f ∈ HG) if n_{k+1}/n_{k} ≥ λ for all k for some λ > 1.

The following result appears for all q in [K2]; for particular values of q, it has been noted earlier by various authors.

Theorem 4.2. A gap series f(z) = ∑_{k} c_{k}z^{n_{k}} ∈ HG belongs to B_{q}^{p} if and only if ∑_{k} n_{k}^{−(1+q)} |c_{k}|^{p} converges.

Proof. This is done by adapting [M, Theorem 1] to our case.

Similar to the inclusion relation for the spaces B_{−(N+1)}^{p} noted above, we have the following, where we concentrate on the proper Besov region for N = 1.

Corollary 4.3. Let q ≤ −1 and p_{1} < p_{2}. Then B_{q}^{p_{1}} ∩ HG ⊂ B_{q}^{p_{2}} ∩ HG.

Example 4.4. Let q ≤ −1 and p_{1} < p_{2}. Choose c_{k} = k^{−1/p_{1}} 2^{k(1+q)/p_{2}} and n_{k} = 2^{k}, and consider the power series f formed with them. Then f ∈ B_{q}^{p_{2}}\B_{q}^{p_{1}} by Theorem 4.2.

Example 4.5. Let q_{1} < q_{2}. Choose c_{k} = k^{−(1+1/p)} 2^{k(1+q_{2})/p} and n_{k} = 2^{k}, and consider the power series f formed with them. Then f ∈ B_{q_{2}}^{p}\B_{q_{1}}^{p} by Theorem 4.2 again. With appropriate modifications, this example works also for N > 1.

5. Bergman Projections and Orthogonality

In the remaining part of the paper, we consider only 1 ≤ p ≤ ∞, as our proofs depend on Definition 5.1 and Theorem 5.2, the latter of which is not available for smaller p.

Definition 5.1. (Extended) Bergman projections are the linear transformations P_{s} defined by

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

for suitable f.

Theorem 5.2. [K3] The operator P_{s} : L_{q}^{p} → B_{q}^{p} is bounded if and only if

q + 1 < p (s + 1) (8)

Given an s satisfying (8), if t satisfies (1), then

P_{s}I_{s}^{t}f = \frac{N!}{(1 + s + t)_{N}} f =: \frac{1}{C_{s+t}} f (f ∈ B_{q}^{p}). (9)

Thus P_{s} : L_{q}^{p} → B_{q}^{p} is onto and C_{s+t}I_{s}^{t} : B_{q}^{p} → L_{q}^{p} is a right inverse for it. A rudimentary case of (9) for p = ∞ is in [C, Corollary 13]. In the Bergman region, it is possible to take s = q except when p = 1. In the proper Besov region, s must be strictly greater than q for any value of p; however, s = −q works for any p, and s = 0 works for any p and q in this region.

When written explicitly, (9) is an integral representation for f ∈ B_{q}^{p}. In a sense, this paper is about showing how powerful such a formula can be.

When p = ∞, Theorem 5.2 is about operators from L^{∞} onto B, and (8) has the form

s > −1.

There is a similar result for the little Bloch space, where C denotes continuous functions on 𝔹 and C_{0} the subspace of functions in C whose restriction to ∂𝔹 is 0.

Theorem 5.3. The operator P_{s} 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 there is a constant C such that P_{s}I_{s}^{t}f = Cf for f ∈ B_{0}.

Proof. The if part and the case t = 1 are proved in [C, Theorem 2]. The equality for other t follow from the p = ∞ case of Theorem 5.2. Now consider f(w) = −w_{1}/ log(1 − |w|^{2}) ∈ C_{0}. It is easily shown that P_{s}f fails to exist if s ≤ −1, and this proves the only-if part.

Remark 5.4. Let’s show once again that Besov spaces are fully described by the parameters q and p, for 1 ≤ p ≤ ∞ now. Suppose t_{1}, t_{2} satisfy (1) with the same q, p, and that g = I_{s}^{t_{1}}f ∈ L_{q}^{p}, where s satisfies (8). By Theorem 5.2, P_{s}I_{s}^{t_{1}}f = Cf. Apply I_{s}^{t_{2}} to both sides to get V_{s}^{t_{2}}g = CI_{s}^{t_{2}}f. The operator V_{s}^{t_{2}} := I_{s}^{t_{2}}P_{s} is bounded on L_{q}^{p} by [K3, Remark 5.2 and Theorem 2.4] because of (1) and (8). Then also I_{s}^{t_{2}}f ∈ L_{q}^{p}.

The operator P_{s} is not a projection on a subspace in the true sense of the word, because B_{q}^{p} need not be subspace of L_{q}^{p}. However, for any t satisfying (1), M_{qs}^{pt} := C_{s+t}I_{s}^{t}(B_{q}^{p}) is an isometric copy (up to a constant multiple) of B_{q}^{p} in L_{q}^{p} and thus a closed subspace of L_{q}^{p} by Definition 1.1 and the discussion on norms following it. Then (9) shows that V_{s}^{t} = C_{s+t}I_{s}^{t}P_{s} is a projection indeed on M_{qs}^{pt} for any s satisfying (8). To determine whether or not this projection is orthogonal when p = 2, we proceed by computing the exact operator norm of P_{s}.

Proposition 5.5. If P_{s} : L_{q}^{2} → B_{q}^{2} is bounded and the norm on B_{q}^{2} is ∥I_{s}^{t}(·)∥_{L_{q}^{2}}, then

∥P_{s}∥ = \frac{N! \sqrt{Γ(1 − q + 2s) Γ(1 + q + 2t)}}{Γ(N + 1 + s + t)}.

Proof. Let f ∈ L_{q}^{2}. First,

P_{s}f(z) = ∑_{k=0}^{∞} b_{k}^{s} ∫_{𝔹} f(w) (1 − |w|^{2})^{−q+s} ⟨z, w⟩^{k} dν_{q}(w) (10)

The spaces Y_{k} = (1 − |z|^{2})^{−q+s}H_{k} lie in L_{q}^{2} by (8); and they, as well as H_{k}, are pairwise orthogonal by [FR, Proposition 2.4 (23)]. Let Y be the closure of the span of Y_{k}; P_{s} annihilates the orthogonal complement of Y in L_{q}^{2} by (10). Denote by P the orthogonal projection from L_{q}^{2} to Y. Then P f(z) = ∑_{k=0}^{∞}(1 − |z|^{2})^{−q+s}f_{k}(z) for some f_{k} ∈ H_{k}, and P_{s}f = P_{s}P f. By replacing f by P f in the integral in (10), we obtain

P_{s}f(z) = ∑_{k=0}^{∞} b_{k}^{s} ∫_{𝔹} f_{k}(w) (1 − |w|^{2})^{−q+2s} ⟨z, w⟩^{k} dν(w)

by the orthogonality of Y_{k}. Hence, by [FR, Proposition 2.4 (26)],

P_{s}f(z) = \frac{N!}{(1 − q + 2s)_{N}} ∑_{k=0}^{∞} b_{k}^{s} \frac{k!}{(N + 1 − q + 2s)_{k}} f_{k}(z).

Note that (8) now has the form −q+2s > −1. Pick also t so that (1) is satisfied, that is, q + 2t > −1. Together s + t > −1. Then, using the orthogonality of H_{k} and [FR, Proposition 2.4 (25)] twice, we compute that

∥P_{s}f∥_{B_{q}^{2}}^{2} = ∥I_{s}^{t}P_{s}f∥_{L_{q}^{2}}^{2}

= \frac{(N!)^{2}}{(1 − q + 2s)_{N}^{2}} ∑_{k=0}^{∞} \frac{d_{k}^{2} (b_{k}^{s})^{2} (k!)^{2}}{(N + 1 − q + 2s)_{k}^{2}} ∫_{𝔹}(1 − |z|^{2})^{q+2t} |f_{k}(z)|^{2} dν(z)

= \frac{(N!)^{2}}{(1 − q + 2s)_{N} (1 + q + 2t)_{N}} ∑_{k=0}^{∞} \frac{d_{k}^{2} (b_{k}^{s})^{2} (k!)^{2}}{(N + 1 − q + 2s)_{k} (N + 1 + q + 2t)_{k}}

· ∫_{𝔹}(1 − |z|^{2})^{−q+2s} |f_{k}(z)|^{2} dν(z).

The values of the coefficients d_{k} and b_{k}^{s} both depend on whether s > −(N + 1) or not. Yet it turns out that the end result is the same in either case. We show the details for s > −(N + 1) only. Then d_{k}b_{k}^{s} = (N + 1 + s + t)_{k}/k!, and

∥P_{s}f∥_{B_{q}^{2}}^{2} = \frac{(N!)^{2} Γ(1 − q + 2s) Γ(1 + q + 2t)}{Γ(N + 1 + s + t)^{2}} ∑_{k=0}^{∞} \frac{(N + 1 − q + 2s + k)_{q−s+t}}{(N + 1 + s + t + k)_{q−s+t}}

· ∫_{𝔹}(1 − |z|^{2})^{−q+2s} |f_{k}(z)|^{2} dν(z) (11)

≤ \frac{(N!)^{2} Γ(1 − q + 2s) Γ(1 + q + 2t)}{Γ(N + 1 + s + t)^{2}} ∑_{k=0}^{∞} ∫_{𝔹}(1 − |z|^{2})^{−q+2s} |f_{k}(z)|^{2} dν(z)

= \frac{(N!)^{2} Γ(1 − q + 2s) Γ(1 + q + 2t)}{Γ(N + 1 + s + t)^{2}} ∥f∥_{L_{q}^{2}}^{2},

where the inequality follows from [FR, Proposition 2.6 (33)], which is also valid for x = 0 when y = 0.

To show that equality holds, we take f(z) = (1 − |z|^{2})^{−q+s} f_{k}(z) with f_{k} ∈ H_{k}. By (11), we have

∥P_{s}f∥_{B_{q}^{2}}^{2} = \frac{(N!)^{2} Γ(1 − q + 2s) Γ(1 + q + 2t)}{Γ(N + 1 + s + t)^{2}} \frac{(N + 1 − q + 2s + k)_{q−s+t}}{(N + 1 + s + t + k)_{q−s+t}} ∥f∥_{L_{q}^{2}}^{2}.

The second fraction has limit 1 as k → ∞ by [FR, Proposition 2.6]. This yields the desired result.

Corollary 5.6. If V_{s}^{t} : L_{q}^{2} → M_{qs}^{2t} is bounded, then

∥V_{s}^{t}∥ = \frac{\sqrt{Γ(1 − q + 2s) Γ(1 + q + 2t)}}{Γ(1 + s + t)}.

Given an s satisfying (8), taking t = −q + s always satisfies (1), and this makes ∥V_{s}^{−q+s}∥ = 1. Therefore V_{s}^{−q+s} : L_{q}^{2} → M_{qs}^{2(−q+s)} is an orthogonal projection. For the classical Bergman space B_{0}^{2}, we need s > −1/2 and t = 0 is customary. Then Corollary 5.6 reduces to [FR, Theorem (9)]. However, we can also take t = s with B_{0}^{2}, say s = t = 1. Then V_{1}^{1} : L^{2} ⟶ \frac{1}{2}(N + 1)(N + 2)(1 − |z|^{2})D_{1}^{1}(B_{0}^{2}) is also an orthogonal projection although P_{1} : L^{2} → B_{0}^{2} is not orthogonal. So by adjusting the range space by I_{s}^{t}, we can generate orthogonal projections from previously nonorthogonal ones.

Adjusting by I_{s}^{t} is useful in other formulas, too, when s, t satisfy (8) and (1). For example, we have P_{s}C_{s+t}I_{s}^{t}\overline{f(z)} = \overline{f(0)}, where D_{s}^{t}\overline{f} is defined as \overline{D_{s}^{t}f}. Also, ∥I_{s}^{t}f∥_{L_{q}^{p}} ≤ C ∥I_{s}^{t}(Re f)∥_{L_{q}^{p}} for some C.

6. Boundary Growth and Maximal Point Evaluations

We now deduce several properties of Besov spaces as consequences of Bergman projections. Recall that our standing hypothesis is 1 ≤ p ≤ ∞.

Theorem 6.1. Given q, 1 ≤ p < ∞, and s, t to be used in ∥ · ∥_{B_{q}^{p}}, there is a constant C such that for all f ∈ B_{q}^{p} and z ∈ 𝔹,

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

This theorem as well as the next two corollaries can be found in a form that covers also the case 0 < p < 1 in [BB, Lemma 5.6]. The novelty here is that we have quick proofs based on Theorem 5.2.

Proof. First let 1 < p < ∞. We begin by applying Theorem 5.2 (9) to f ∈ B_{q}^{p} with s, t satisfying (8), (1), and also s > −(N + 1) for convenience. This gives

f(z) = C ∫_{𝔹} \frac{I_{s}^{t}f(w) (1 − |w|^{2})^{−q+s}}{(1 − ⟨z, w⟩)^{N+1+s}} dν_{q}(w) (z ∈ 𝔹).

Applying Hölder inequality with p′ = p/(p − 1), we obtain

|f(z)| ≤ C ∥f∥_{B_{q}^{p}}(∫_{𝔹} \frac{(1 − |w|^{2})^{(−q+s)p′}}{|1 − ⟨z, w⟩|^{(N+1+s)p′}} dν_{q}(w))^{1/p′}

= C ∥f∥_{B_{q}^{p}}(∫_{𝔹} \frac{(1 − |w|^{2})^{a}}{|1 − ⟨z, w⟩|^{N+1+a+c}} dν(w))^{1/p′},

where a = (−q + ps)/(p − 1) > −1 by (8) and c/p′ = (N + 1 + q)/p. A glance at [R, Proposition 1.4.10] yields all three inequalities for p ≠ 1. Since the Lebesgue norms are continuous functions of p, the proof is completed by letting p → 1^{+}.

When q = −1, we see that the spaces B_{−1}^{p} have the same growth rate near ∂𝔹 as those of Hardy spaces H^{p} given in [R, Theorem 7.2.5 (a)]. Another similarity between the two families has already been noted in Section 3.

Corollary 6.2. Given q, r, u, 1 ≤ p < ∞, and s, t to be used in ∥ · ∥_{B_{q}^{p}}, there is a constant C such that for all f ∈ B_{q}^{p} and z ∈ 𝔹,

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

Proof. Just combine Theorem 6.1 with Proposition 3.1. Note that u < 0 is a possibility.

When p = ∞, these results take the following form. The proof is similar to and easier than that above, or we can just set p = ∞.

Corollary 6.3. Given r, u and s, t to be used in ∥ · ∥_{B}, there is a constant C such that for all f ∈ B and z ∈ 𝔹,

|D_{r}^{u}f(z)| ≤ C ∥f∥_{B} \begin{cases}(1 − |z|^{2})^{−u}, & if u > 0; \\ log(1 − |z|^{2})^{−1}, & if u = 0; \\ 1, & if u < 0. \end{cases}

That the above results do not depend on the particular s, t used in their proofs confirms the work of Section 2.

Corollary 6.4. If q < −(N + 1) and 1 ≤ p < ∞, then B_{q}^{p} ⊂ H^{∞}, the space of bounded holomorphic functions on 𝔹, and the inclusion is continuous.

Corollary 6.5. Let q > −(N + 1) and 1 ≤ p < ∞. If f ∈ B_{q}^{p}, then

lim_{|z|→1^{−}}(1 − |z|^{2})^{(N+1+q)/p} |f(z)| = 0.

Further, the exponent (N + 1 + q)/p on (1 − |z|^{2})^{−1} in Theorem 6.1 cannot be replaced by a smaller one.

Proof. The first claim is proved the same way as [R, Theorem 7.2.5] by the properties of the dilates of f given in the proof of [BB, Lemma 5.2]. For the second claim, we modify the example given following [R, Theorem 7.2.5] that contains the case q = 0. Let c < (N + 1 + q)/p and g(z) = (1 − z_{1})^{−c} = (1 − ⟨z, e_{1}⟩)^{−c}, where e_{1} = (1, 0, …, 0). We take a t satisfying (1) and compute ∥g∥_{B_{q}^{p}} using (6). By [R, Proposition 1.4.10], g ∈ B_{q}^{p}. So B_{q}^{p} has an element whose growth rate has exponent c for any c lower than the one claimed.

Here is another application. A general result containing also the case 0 < p < 1 is in [BB, Theorem 5.7], but the proof now is trivial given the above results.

Theorem 6.6. Given a compact set E in 𝔹, q, p, r, u, and s, t to be used in ∥ · ∥_{B_{q}^{p}}, there is a constant C such that

sup_{z∈E} |D_{r}^{u}f(z)| ≤ C ∥f∥_{B_{q}^{p}}.

Therefore evaluation of f and of any D_{r}^{u}f at any point of 𝔹 is a bounded linear functional. Consequently, the Besov space B_{q}^{p} is a complete space.

Proof. The first and second statements follow immediately from Theorem 6.1. It is a standard matter to obtain the last statement from these.

Associated to point evaluations is the extremal problem of finding

sup { f(a) > 0 : a ∈ 𝔹, f ∈ B_{q}^{p}, ∥f∥_{B_{q}^{p}} = ∥I_{s}^{t}f∥_{L_{q}^{p}} = 1 }, (12)

which has been solved for Bergman spaces in [V]. More generally, we have the following result.

Theorem 6.7. For 0 < p < ∞, the extremal function solving (12) exists and is unique.

Proof. Take s, t satisfying (1) and put Q = q + pt > −1. Then D_{s}^{t}f ∈ B_{Q}^{p}, which is in the Bergman region; we use ∥f∥_{B_{q}^{p}} = ∥D_{s}^{t}f∥_{B_{Q}^{p}}. First take a = 0. By the proof of [V, Theorem] applied to D_{s}^{t}f, we have

D_{s}^{t}f(0) ≤ exp(C_{Q} ∫_{𝔹} log |D_{s}^{t}f(z)| dν_{Q}(z)) ≤ C_{Q}^{1/p} ∥f∥_{B_{q}^{p}} = C_{Q}^{1/p}.

The great advantage of using the radial differential operators D_{s}^{t} now becomes apparent. We have D_{s}^{t}f(0) = d_{0}f(0). Hence f(0) ≤ C_{Q}^{1/p}/d_{0}, and equality holds if

and only if f is constant. Thus the unique extremal function at 0 is f_{0}(z) = C_{Q}^{1/p}/d_{0}.

To pass to other a ∈ 𝔹, we use an idea in [RS, Theorem 5.2]. By Proposition 3.1, the map U_{a} = D_{s+t}^{−t}T_{a}D_{s}^{t} is a surjective involutive isometry on B_{q}^{p}, because the T_{a} in [V, Lemma] has the same properties on B_{Q}^{p}. Therefore the unique extremal function at a is

f_{a}(z) = U_{a}f_{0}(z) = D_{s+t}^{−t}T_{a}C_{Q}^{1/p}(z) = C_{q+pt}^{1/p} D_{s+t}^{−t}(\frac{1 − |a|^{2}}{(1 − ⟨z, a⟩)^{2}} )^{(N+1+q+pt)/p}.

When s = (N + 1 + q + pt)2/p − (N + 1) − t, we can make f_{a} a little more explicit by using (6). Then

f_{a}(z) = (\frac{(1 + q + pt)_{N}}{N!} )^{1/p} (1 − |a|^{2})^{(N+1+q+pt)/p} K_{(N+1+q+pt)2/p−(N+1)−t}(z, a).

When q > −1, we can take t = 0, and f_{a} reduces to the function given in [V, Theorem] when we take into account the difference in the normalization of the measures.

We can also consider the problem of maximizing the values of derivatives of functions in B_{q}^{p}, but by Theorem 3.1, this reduces to problem (12) in another Besov space, so we omit the details. A particular case is in [RS, Theorem 5.3].

Another topic closely related to boundary growth is the growth of the integral means M_{r}(f, R) of functions in a B_{q}^{p} space; see [BB, p. 23] for a definition of these means. We are content with summarizing the situation in this matter.

For Besov spaces in the Bergman region and for the Bloch space, the problem has been dealt with in [BB, Theorem 3.2] in considerable detail. A Besov space B_{q}^{p} in the region q < −1 lies in a Hardy space H^{p_{2}} with p_{2} > p by [BB, Theorem 5.13] and hence lies in H^{p}. This is true also for the spaces B_{−1}^{p} for 0 < p ≤ 2 by Remark 3.2. More strongly, by Corollary 6.4, any B_{q}^{p} space with q < −(N + 1) lies in H^{∞}. Then one can consult the numerous results in [D, Chapter 5] on the growth of the integral means of Hardy spaces.

These remarks leave out the case of B_{−1}^{p} with p > 2. Let N = 1 for now and f ∈ B_{−1}^{p}. Then f′ ∈ B_{p−1}^{p}, which is a Bergman space. By [BB, Theorem 3.2], we have M_{p}(f′, R) = O((1 − R^{2})^{−1}). Then [GP, Theorem 1] yields that M_{p}(f, R) = O((log(1 − R^{2})^{−1})^{b}) for all b > 1/2 and that this result is sharp.

7. Taylor Coefficients

In this section, our standing hypotheses are N = 1 and 1 ≤ p ≤ ∞.

Theorem 7.1. Given q, p and s, t to be used in ∥ · ∥_{B_{q}^{p}}, there is a constant C such that for all f ∈ B_{q}^{p}, we have

|c_{k}| ≤ C ∥f∥_{B_{q}^{p}} k^{(1+q)/p},

where c_{k} = f′(0)/k!.

Notice the natural similarity with Theorem 4.2.

Proof. As in the proof of Theorem 6.1, we begin by applying Theorem 5.2 (9) to f ∈ B_{q}^{p} with s, t satisfying (8), (1), and also s > −2 for convenience. This gives

f(z) = C ∫_{𝔻} \frac{I_{s}^{t}f(w)}{(1 − z\overline{w})^{2+s}} dA_{s}(w) (z ∈ 𝔻),

where A is the normalized area measure. Then

f^{(k)}(z) = C (2 + s)_{k} ∫_{𝔻} \frac{I_{s}^{t}f(w) \overline{w}^{k}}{(1 − z\overline{w})^{2+s+k}} dA_{s}(w)

and

c_{k} = \frac{f^{(k)}(0)}{k!} = C \frac{(2 + s)_{k}}{k!} ∫_{𝔻} I_{s}^{t}f(w) \overline{w}^{k} (1 − |w|^{2})^{−q+s} dA_{q}(w).

Consider first 1 < p < ∞. Applying Hölder inequality with p′ = p/(p−1), switching to polar coordinates, and letting y = |w|^{2} yield

|c_{k}| ≤ C \frac{(2 + s)_{k}}{k!} ∥f∥_{B_{q}^{p}}(∫_{𝔻} |w|^{p′k} (1 − |w|^{2})^{(−q+s)p′+q} dA(w))^{1/p′}

= C ∥f∥_{B_{q}^{p}} \frac{Γ(2 + s + k)}{Γ(1 + k)}(∫_{0}^{1} y^{1+p′k/2} (1 − y)^{−q(p′−1)+p′s} dy)^{1/p′}

= C ∥f∥_{B_{q}^{p}} \frac{Γ(2 + s + k)}{Γ(1 + k)}(\frac{Γ(p′k/2 + 2)}{Γ(p′k/2 − q(p′ − 1) + p′s + 3)})^{1/p′},

where −q(p′−1)+p′s > −1 by (8), and factors not involving k are gathered together in C. Now we estimate using (3). Thus

|c_{k}| ≤ C ∥f∥_{B_{q}^{p}} k^{1+s} (k^{(q−ps)/(p−1)−1})^{(p−1)/p} ≤ C ∥f∥_{B_{q}^{p}} k^{(1+q)/p}.

The cases p = 1 and p = ∞ follow as in Section 6, or the case p = ∞ can be directly computed. Likewise, the final result does not depend on the particular s, t used in the proof, once again confirming the work of Section 2.

The case q = −2 of Theorem 7.1 has been taken care of in [Zh, Theorem 8]. There is a stronger result for 0 < p < 1 in [BK, Theorem 2.8].

The next step in Taylor coefficients is Hardy-Littlewood-type results, and these have already been derived in [BK] for q > −1. They extend to all q readily as follows.

Theorem 7.2. Let f(z) = ∑_{k} c_{k}z^{k}. For 0 < p ≤ 2, there is a constant C such that

∑_{k} \frac{k^{p−2}}{k^{1+q}} |c_{k}|^{p} < C ∥f∥_{B_{q}^{p}}.

For 2 ≤ p < ∞, there is a constant C such that

∥f∥_{B_{q}^{p}} < C ∑_{k} \frac{k^{p−2}}{k^{1+q}} |c_{k}|^{p}.

The exponents on k are sharp.

Proof. First, ∥f∥_{B_{q}^{p}} = ∥g∥_{B_{Q}^{p}}, where g = D_{s}^{t}f and Q = q + pt > −1 for some s, t satisfying (1). Second, g(z) = D_{s}^{t}f(z) ∼ ∑_{k} k^{t}c_{k}z^{k}. We are done by using [BK, Theorems 2.1 and 2.2] on g.

When p = 2, the exponent on k is the same as that in (5) for N = 1, which also shows that this exponent is sharp. It is impossible to distinguish between B_{−1}^{p} and H^{p} in this respect. See [D, Chapter 6] for the results on Hardy spaces. The case q = −2 for p > 1 first appears in [HW, Theorem 2]. The very special case q = −2 and p = 1 is also in [Zh, Theorem 8 (2)].

All the results in this section generalize to N > 1, although they become less pleasing in appearance. We omit them, but refer the reader to [BK, Theorem 3.4] for more attractive-looking results lacking q in the exponents and thus more reminiscent of the classical Hardy-Littlewood inequalities.

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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/

Department of Mathematics, Anadolu University, Eski̇şehi̇r 26470, Turkey

    E-mail address: ureyen@gmail.com