REPRODUCING KERNELS AND RADIAL DIFFERENTIAL OPERATORS FOR HOLOMORPHIC AND HARMONIC BESOV

SPACES ON UNIT BALLS: A UNIFIED VIEW

H. TURGAY KAPTANOĞLU

Abstract. We investigate some relations between the reproducing kernels of Hilbert spaces of holomorphic and harmonic functions on the unit balls and the radial differential operators acting on the spaces that allow their characterization via integrals of their derivatives on the balls. We compare and contrast the holomorphic and harmonic cases.

1. Introduction

The purpose of this paper is to make a short survey of the most basic characteristics of the two-parameter Besov spaces of holomorphic and harmonic functions on the unit balls with standard weights by stressing the similarities between the two categories and by paying particular attention to the reproducing kernels of those that are Hilbert spaces and to the radial differential operators that are so convenient in defining the spaces via integral norms.

We restrict our attention strictly to spaces with Bergman-type norms, and do not mention any spaces with Bloch-type norms. We also omit from consideration the full three-parameter Besov family. So we do not mention any Hardy-Sobolev spaces or spaces with Hardy-Bergman mixed-type norms.

The exposition is aligned with the interests and the research of the author. Nor an attempt is made to give exact references for most classical results.

2. Notation and Preliminaries

We start by introducing the notation used throughout the paper. The Pochhammer symbol (a)_{b} for a, b ∈ ℂ is defined 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. This is a shifted factorial since (a)_{k} = a(a + 1) ⋯ (a + k − 1) for positive integer k. In particular, (1)_{k} = k! and (·)_{0} = 1. The Pochhammer symbol seems more useful than the gamma function itself because of its more controlled polynomial growth as one of its parameters increases without bound; see [TE]. The Stirling formula gives

Date: 9 July 2010.

2000 Mathematics Subject Classification. Primary 46E22, 46E20, 46E15, 26A33, 32A37, 32A36, 31B05, 31C25, 33C55, 47B34.

Key words and phrases. Besov, Dirichlet, Drury-Arveson, Hardy, Bergman space, reproducing kernel Hilbert space, radial differential operator, spherical harmonic.

This research is partially supported by TÜBİTAK under Research Project Grant 108T329.

The author thanks A. E. Üreyen of Anadolu University and S. Gergün of Çankaya University for their input on harmonic spaces.

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

where A ∼ B means that |A/B| is bounded above and below by two positive constants. Such constants that are independent of the parameters in the equation are all denoted by the generic unadorned upper case C.

We alternately work in ℂ^{N} and ℝ^{n}, where N ≥ 1 and n ≥ 2. Equating n = 2N allows for a comparison in even dimensions. The Hermitian inner product and the norm of ℂ^{N} are ⟨z, w⟩ = z_{1}\overline{w}_{1} + ⋯ + z_{N} \overline{w}_{N} and |z| = \sqrt{⟨z, z⟩}. The inner (dot) product and the norm of ℝ^{n} are x · y = x_{1}y_{1} + ⋯ + x_{n}y_{n} and |x| = \sqrt{x · x}.

We denote the unit balls and the unit spheres with respect to | · | in ℂ^{N} or ℝ^{n} by 𝔹 and 𝕊. We often write z = Rζ, w = rω ∈ ℂ^{N}, x = Rξ, y = rη ∈ ℝ^{n} with R = |z| or R = |x|, r = |w| or r = |y|, and ζ, ω, ξ, η ∈ 𝕊. When N = 1 or n = 2, we talk about the unit disc 𝔻 and the unit circle 𝕋 instead.

We denote by H(𝔹) the space of holomorphic functions f on 𝔹 ∈ ℂ^{N}, and by h(𝔹) the space of harmonic functions u on 𝔹 ∈ ℝ^{n}, both with the Fréchet topology of uniform convergence on compact subsets. Members of h(𝔹) are the complexvalued real-analytic functions annihilated on 𝔹 by the usual Laplacian.

Let ν be the volume measure on 𝔹 normalized as ν(𝔹) = 1, and let σ be the surface measure on 𝕊 normalized as σ(𝕊) = 1. For q ∈ ℝ, define on 𝔹 also the measures

dν_{q}(·) = C_{q} (1 − | · |^{2})^{q} dν(·).

These measures are finite only for q > −1 and then we choose C_{q} so that ν_{q}(𝔹) = 1. For q ≤ −1, we set C_{q} = 1. We denote the Lebesgue classes with respect to ν_{q} by L_{q}^{p}. The exact value of C_{q} is

(2)  C_{q} = \frac{(1 + q)_{n/2}}{(1)_{n/2}} = \frac{(1 + n/2)_{q}}{(1)_{q}}  (q > −1).

Thus C_{q} ∼ n^{q} for n large, and C_{q} ∼ q^{n/2} for q large. To use this in ℂ^{N}, we just replace n/2 by N. Also C_{q} = 1 + q when n = 2 (N = 1) and naturally C_{0} = 1.

We 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}} for z = (z_{1}, …, z_{N}) ∈ ℂ^{N}, and 0^{0} = 1. To use them in ℝ^{n}, we just change N to n and z to x.

Proposition 2.1. We have the orthogonality relation

∫_{𝕊} ζ^{α} \overline{ζ}^{β} dσ(ζ) = 0 (α ≠ β)

and the exact value

∫_{𝕊} |ζ^{α}|^{2} dσ(ζ) = \frac{α!}{(N)_{|α|}} .

The proof of Proposition 2.1 is in [R, Propositions 1.4.8 and 1.4.9] .

A complex-valued polynomial p of the real or complex variable v = (v_{1}, v_{2}, …) is called homogeneous of degree l if p(τv) = τ^{l} p(v) for τ ∈ ℝ. Its gradient is ∇p = (∂p/∂v_{1}, ∂p/∂v_{2}, …), and its usual radial derivative is

Rp(v) = v · ∇p(v) = \frac{d}{dτ} [p(τv)]_{τ=1}.

By homogeneity,

(3)  Rp = lp.

Using the Pochhammer symbol, the binomial expansion is written as

(4)  \frac{1}{(1 − z)^{Q}} = ∑_{k=0}^{∞} \frac{(Q)_{k}}{(1)_{k}} z^{k}  (Q > 0, z ∈ 𝔻).

It is a special case of the classical hypergeometric function

2F_{1}(a, b; c; z) = ∑_{k=0}^{∞} \frac{(a)_{k} (b)_{k}}{(c)_{k} (1)_{k}} z^{k} (a, b ∈ ℝ, c > 0, z ∈ 𝔻),

which we use with suitable values of a, b, c depending on Q to extend the region of validity of the binomial expansion to Q ≤ 0. Both expansions converge absolutely and uniformly on compact subsets of 𝔻, and thus both functions belong to H(𝔻). We also have their variants belonging to h(𝔻) below.

We denote an integral inner product on a space X of functions on a set E by [·, ·]_{X} and the corresponding norm on it by ∥ · ∥_{X}.

Definition 2.2. We call a function K(·, ·) on E × E a reproducing kernel for a Hilbert space X of functions defined on E if K(λ, ·) ∈ X for each λ ∈ E and

g(λ) = [g(·), K(λ, ·)]_{X} (g ∈ X, λ ∈ E).

For more on reproducing kernel Hilbert spaces including the following results, see [AM, Chapter 2]. The first one is just the Riesz lemma together with its converse.

Theorem 2.3. A Hilbert space X of functions on E admits a reproducing kernel if and only if the point evaluation functional g ↦ g(λ) at every λ ∈ E is a nonzero continuous functional on X.

Theorem 2.4. A sesquilinear function K(·, ·) is a reproducing kernel on E if and only if it is positive semidefinite there and nonzero on the diagonal of E × E. Such a function generates a unique Hilbert space of functions on E whose members are of the same class as K(λ, ·).

That K is positive semidefinite means that

(5)  ∑_{j=1}^{l} ∑_{k=1}^{l} \overline{a}_{j} a_{k} K(λ_{j}, λ_{k}) ≥ 0

for a_{1}, …, a_{l} ∈ ℂ and distinct λ_{1}, …, λ_{l} ∈ E. Sesquilinearity reduces to bilinearity for real-valued functions.

Corollary 2.5. An infinite sum of reproducing kernels on 𝔹 for subspaces of H(𝔹) or h(𝔹) converging uniformly on compact subsets of 𝔹 × 𝔹 is also a reproducing kernel on 𝔹 for a Hilbert subspace of H(𝔹) or h(𝔹).

Theorem 2.6. Suppose X is a Hilbert space with reproducing kernel K and that

\{e_{1}, e_{2}, …\} is an orthonormal basis for X. Then K(λ, μ) = ∑_{j=1}^{∞} e_{j}(λ) \overline{e}_{j}(μ).

3. Spherical Harmonics

For any N ≥ 1, members of H(𝔹) are decribed using power series in one or more variables. The same is true for members of h(𝔻), where the extensions of the Fourier series on 𝕋 to the disc suffice. However, for members of h(𝔹) with n > 2, there is something more elaborate which extends what is known for n = 2.

For m = 0, 1, 2, …, let H_{m} be the space of complex-valued polynomials of the form ∑_{|α|=m} b_{α}x^{α} on ℝ^{n} that are also harmonic; such polynomials are homogeneous

of degree m. Functions in H_{m} are determined by their restrictions to 𝕊 by homogeneity; these restrictions are called spherical harmonics; we identify restricted and nonrestricted spaces. Clearly, each H_{m} is finite dimensional with dimension

(6)  h_{m} = dim H_{m} = \frac{n − 2 + 2m}{m} \frac{(n − 1)_{m−1}}{(1)_{m−1}} ∼ m^{n−2}  (m → ∞).

Hence each H_{m} is a closed subspace of L^{2}(σ) and thus a Hilbert space with respect to the inner product of L^{2}(σ).

Similar to the theory of Fourier series, we have

(7)  L^{2}(σ) = ⨁_{m=0}^{∞} H_{m},

which is an orthogonal direct sum, meaning that H_{m} is orthogonal to H_{k} for m ≠ k with respect to the inner product of L^{2}(σ), and every function in L^{2}(σ) can be expanded in a unique series of polynomials in the H_{m} converging in L^{2}(σ).

The point evaluation functional at each point η ∈ 𝕊 trivially is a bounded linear functional on H_{m}. Thus, by Theorem 2.3, there is a unique function Z_{m}(·, η) ∈ H_{m} such that

(8)  u(η) = ∫_{𝕊} f(·) Z_{m}(·, η) dσ  (u ∈ H_{m}).

In other words, Z_{m} is the reproducing kernel of H_{m}; it is called the zonal harmonic of degree m. It is real-valued and symmetric in its two variables. When m = 0, H_{0} is just the constants, h_{0} = 1, and Z_{0} ≡ 1.

Other properties of zonal harmonics are that

(9)  |Z_{m}(ξ, η)| ≤ Z_{m}(ξ, ξ) = h_{m}  (ξ, η ∈ 𝕊);

that Z_{m}(T (ξ), T (η)) = Z_{m}(ξ, η) for any orthogonal transformation T; and that Z_{m}(ξ, ·) is constant on each parallel of 𝕊 orthogonal to ξ; all for any ξ, η ∈ 𝕊.

Let {Y_{m1}, …, Y_{mh_{m}}} be a basis for H_{m} that is orthonormal with respect to the inner product of L^{2}(σ). Then by Theorem 2.6,

(10)  Z_{m}(ξ, η) = ∑_{j=1}^{h_{m}} Y_{mj}(ξ) \overline{Y}_{mj}(η).

The spaces H_{m}, the functions in them, the Z_{m}, and the Y_{mj} are extended to 𝔹 by homogeneity; for example, Z_{m}(x, y) = R^{m} r^{m} Z_{m}(ξ, η). Such extensions are sometimes called solid spherical harmonics. It is well-known that the Poisson kernel P for 𝔹 is the sum of all zonal harmonics. This idea also extends P to 𝔹 in both variables. Thus we have

(11)  P(x, y) = ∑_{m=0}^{∞} Z_{m}(x, y) = ∑_{m=0}^{∞} ∑_{j=1}^{h_{m}} Y_{mj}(x) \overline{Y}_{mj}(y)

= \frac{1 − R^{2} r^{2}}{(1 − 2 x · y + R^{2} r^{2})^{n/2}} (x, y ∈ 𝔹). (12)

The picture equivalent to spherical and zonal harmonics for H(𝔹) is much simpler and more explicit. Spherical harmonics correspond to the space of all holomorphic polynomials of the form

(13)  ∑_{|α|=k} b_{α}z^{α}

whose dimension is

θ_{k} = \frac{(N)_{k}}{(1)_{k}} ∼ k^{N−1} (k → ∞).

Further, an integration similar to (8) with f(ζ) = ζ^{α} using Proposition 2.1 and the uniqueness of kernels show that the counterparts of zonal harmonics are the sesquilinear functions

W_{k}(ζ, ω) = \frac{(N)_{k}}{(1)_{k}} ⟨ζ, ω⟩^{k}.

This also implies that the W_{k} are positive kernels on 𝕊. The equivalent of (7) is obvious by the same proposition.

Other properties of zonal harmonics carry over here too, and in fact trivially due to the explicit form of the W_{k}. We have that |W_{k}(ζ, ω)| ≤ W_{k}(ζ, ζ) = θ_{k}; that W_{k}(U(ζ), U(ω)) = W_{k}(ζ, ω) for any unitary transformation U; and that W_{k}(ζ, ·) is constant on each parallel of 𝕊 orthogonal to ζ with respect to ⟨·, ·⟩; all for any ζ, ω ∈ 𝕊.

Extending to 𝔹 by homogeneity as above and summing as in (11) using (4), we obtain

(14)  ∑_{k=0}^{∞} W_{k}(z, w) = \frac{1}{(1 − ⟨z, w⟩)^{N}} = C(z, w)  (z, w ∈ 𝔹),

which is the Cauchy-Szegő kernel for 𝔹.

Going back to the zonal harmonics, it is well-known that the Z_{m}(ξ, η) are multiples of the Gegenbauer (ultraspherical) polynomials G_{n}^{λ}, which are defined by the generating function

\frac{1}{(1 − 2vτ + τ^{2})^{κ}} = ∑_{n=0}^{∞} G_{n}^{κ}(v) τ^{n},

whose denominator is a power of the denominator of the Poisson kernel. Precisely,

Z_{m}(ξ, η) = h_{m} \frac{(1)_{m}}{(n − 2)_{m}} G_{n}^{n/2−1}(ξ · η) = \frac{n − 2 + 2m}{n − 2} G_{n}^{n/2−1}(ξ · η)

for n > 2; see [F, p. 403]. This shows another similarity between the holomorphic and harmonic cases. The kernels W_{k} and Z_{m} depend on their variables via the associated inner product of the underlying space.

Remark 3.1. Yet there is an inherent difference between the harmonic and holomorphic cases due to the difference between the H_{m} and their holomorphic counterparts, and their dimensions h_{m} and θ_{k}.

The kernels W_{k} and Z_{m} remain reproducing kernels when their domains are extended from 𝕊 to 𝔹, because (5) is satisfied with the r_{j} a_{j} replacing a_{j}, where r_{j} = |λ_{j}| for λ_{j} ∈ 𝔹.

4. Bergman Spaces

Definition 4.1. Let q > −1 and p ≥ 1. The weighted Bergman space A_{q}^{p} is the space of all f ∈ H(𝔹) that lie also in L_{q}^{p}. Similarly, the weighted harmonic Bergman space b_{q}^{p} is the space of all u ∈ h(𝔹) that lie also in L_{q}^{p}. The norms of A_{q}^{p} and b_{q}^{p} are those of L_{q}^{p}, and the same is true of the inner products for p = 2.

Polynomials and functions bounded on 𝔹 belong to L_{q}^{p} whenever ν_{q} is finite; this explains the condition q > −1. In Section 8, we have the variant (25) of this inequality for Besov spaces when q ≤ −1.

The restriction to p ≥ 1 is for having a unified presentation of both holomorphic and harmonic spaces. The reason is that |f|^{p} is subharmonic for all p > 0 when f ∈ H(𝔹), but only for p ≥ 1 when f ∈ h(𝔹). The proof of [DS, Theorem 1.1] shows where exactly this property is used.

Bergman spaces of either kind are Banach spaces, and when p = 2, reproducing kernel Hilbert spaces. Using the orthonormal basis

\{ \sqrt{\frac{(1 + N + q)_{|α|}}{α!}} z^{α} : α ∈ ℕ^{N} \}

and Theorem 2.6, the reproducing kernel of A_{q}^{2} for q > −1 is computed as

K_{q}(z, w) = \frac{1}{(1 − ⟨z, w⟩)^{1+N+q}} = ∑_{k=0}^{∞} \frac{(1 + N + q)_{k}}{(1)_{k}} ⟨z, w⟩^{k}

= ∑_{k=0}^{∞} \frac{(1 + N + q)_{k}}{(N)_{k}} W_{k}(z, w) = ∑_{k=0}^{∞} c_{k}(q) W_{k}(z, w), (15)

where z, w ∈ 𝔹, and the c_{k}(q) are defined. We call the K_{q} introduced so far binomial kernels.

Because of the lack of a simple summable basis, the computation of the reproducing kernels of harmonic Bergman spaces has to follow a different path. The reproducing kernel of b_{q}^{2} for q > −1 is computed to be

(16)  R_{q}(z, w) = ∑_{m=0}^{∞} \frac{(1 + n/2 + q)_{m}}{(n/2)_{m}} Z_{m}(x, y) = ∑_{m=0}^{∞} γ_{m}(q) Z_{m}(x, y),

where x, y ∈ 𝔹 and the γ_{m}(q) are defined; see [M, Proposition 3]. Additionally see [CR, (3.1)] or [BP, (1)] for integer or real q > −1; see also [Li1, p. 25], [Li2], and [ABR, Chapter 8] for q = 0.

When interpreted as above, the forms of K_{q} and R_{q} turn out to be exactly the same in even dimensions with the identification n = 2N. It would be interesting to know if the R_{q} could be expressed in a form that resembles the binomial form of the K_{q}, at least for even n.

There are other similarities. We have c_{0}(q) = 1 = γ_{0}(q) for all q. More importantly,

(17)  c_{k}(q) ∼ k^{1+q}  (k → ∞)  and  γ_{m}(q) ∼ m^{1+q}  (m → ∞).

An important property we utilize most is that K_{q} and R_{q} are both weighted infinite sums zonal harmonics or their equivalents, which are reproducing kernels on their own. A careful look at (15) shows that the c_{k}(q) and the K_{q} make sense for q > −(1 + N) although Bergman spaces are defined only for q > −1. For all such extended q, (17) still holds and yields that

(18)  |K_{q}(z, w)| ≤ C ∑_{k=0}^{∞} k^{N+q} (Rr)^{k}  (z, w ∈ 𝔹, q > −(1 + N)).

This shows that the series giving K_{q} converges absolutely and uniformly on compact subsets of 𝔹 × 𝔹. Thus by Corollary 2.5, additionally for each −(1 + N) < q ≤ −1, K_{q} is the reproducing kernel for a unique Hilbert space of holomorphic functions on 𝔹; let’s call these spaces D_{q}. It is easy to know what this space is when q = −1; since c_{k}(−1) = 1 for all k, by (14), D_{−1} is the Hardy space H^{2}(𝔹).

Similarly, the γ_{m}(q) and the R_{q} also make sense in a larger interval of q, this time for q > −(1 + n/2). For all such extended q, (17) still holds, and with the help of (9) and (6), yields that

|R_{q}(x, y)| ≤ C ∑_{m=0}^{∞} m^{1+q} (Rr)^{m} Z_{m}(ξ, ξ) = C ∑_{m=0}^{∞} m^{1+q} (Rr)^{m} h_{m}

≤ C ∑_{m=0}^{∞} m^{n−1+q} (Rr)^{m} (x, y ∈ 𝔹, q > −(1 + n/2)). (19)

As above, the convergence is absolute and uniform on compact subsets which allows us to conclude that R_{q} is the reproducing kernel of a unique Hilbert space of harmonic functions on 𝔹 for each −(1 + n/2) < q ≤ −1 as well; let’s call these spaces Δ_{q}. The space for q = −1 is known again since γ_{m}(−1) = 1 for all m; by (11), Δ_{−1} is the harmonic Hardy space h^{2}(𝔹); see [ABR, Chapter 6]. The closed form (12) of its kernel, the Poisson kernel, is a binomial-type expression. We do not know whether or not the remaining R_{q} for q > −(1 + n/2) have such closed-form or binomial-type expressions.

Remark 4.2. We see here a small difference between the holomorphic and harmonic cases. Due to the different growth rates of θ_{k} and h_{m} for large k and m, the growth rates of the coefficients of (Rr)^{k} and (Rr)^{m} are different.

Two main points of interest are to extend this construction to all real values of q, and to describe the nonBergman Hilbert spaces in a way similar to Bergman spaces, that is, through integral norms.

5. Homogeneous Expansions and Radial Derivatives

An f ∈ H(𝔹) and a u ∈ h(𝔹) has a homogeneous expansion in the form

(20)  f = ∑_{k=0}^{∞} f_{k},  u = ∑_{m=0}^{∞} u_{m},

where f_{k} is a polynomial of the form (13) and u_{m} ∈ H_{m}, which converge absolutely and uniformly on compact subsets of 𝔹; see [R, Theorem 1.5.6] and [ABR, Corollary 5.23]. Their radial derivatives take the form

Rf(z) = ∑_{k=1}^{∞} k f_{k}(z), Ru(x) = ∑_{m=1}^{∞} m u_{m}(x)

by (3), which also clearly converge absolutely and uniformly on compact subsets of 𝔹. Thus Rf ∈ H(𝔹) and Ru ∈ h(𝔹) as well.

Remark 5.1. For j = 1, 2, …, R^{j} has the index of summation raised to the jth power; so this power determines the order of the radial derivative. By raising the index to a power t ∈ ℝ, t > 0, one often considers radial derivatives of fractional order.

Further, R(⟨z, w⟩) = ⟨z, w⟩, where R acts on the holomorphic variable z. Then

(21)  RK_{q}(z, w) = (1 + N + q) ⟨z, w⟩ K_{1+q}(z, w)  (q > −(1 + N)),

and in particular, K_{0} = C + \frac{1}{N} RC. A similar statement for R_{q} is

(22)  R_{q}(x, y) = \frac{2}{n (1)_{q}} (\frac{d}{dt})^{1+q}[t^{n/2+q} P(tx, y)]_{t=1}  (q ∈ ℕ);

see [M, p. 29]. In particular, R_{0} = P + \frac{1}{n/2} RP, where differentiation is applied

only on one of the variables x, y, and it does not matter which by symmetry; see [ABR, p. 157].

So the radial derivative allows us to pass from a Bergman Hilbert space to another one whose lower index is greater by a positive integer. It would be nice if the same passage could be done if the difference between the indices is not an integer, and also if it is negative. Easy modifications would also eliminate the unnecessary multiplicative constants above. The particular relations for K_{0} and R_{0} also suggest that combinations that do not annihilate constants are more useful than pure radial derivatives.

All these are possible if we have radial derivatives of order any real number t that are suited to the particular forms of the reproducing kernels.

6. Besov Kernels

We are now ready to present the extensions of the reproducing kernels K_{q} and R_{q} to all real q. The key idea is to keep unchanged the growth rates expressed in (17) of the coefficients of W_{k} and Z_{m} in the series expansions of K_{q} and R_{q}. We replace c_{k}(q) and γ_{m}(q) by new ones for q ≤ −(1 + N) or q ≤ −(1 + n/2) that still satisfy (17).

The holomorphic case has already been dealt with in [BB1, p. 13]. Here we modify it only slightly to have c_{0}(q) = 1.

Definition 6.1. We set

c_{k}(q) = \frac{(1)_{k}^{2}}{(1 − (N + q))_{k} (N)_{k}} (q ≤ −(1 + N)).

The harmonic case is done only recently in [GKÜ1, Definition 3.1]; it is fashioned after the holomorphic case.

Definition 6.2. We set

γ_{m}(q) = \frac{(1)_{m}^{2}}{(1 − (n/2 + q))_{m} (n/2)_{m}} (q ≤ −(1 + n/2)).

Now c_{0}(q) = 1 = γ_{0}(q) for all q ∈ ℝ. More importantly, for all q ∈ ℝ and k, m ∈ ℕ, c_{k}(q) > 0 and γ_{m}(q) > 0. Using (1) and considering (15) and (16) in conjunction with Definitions 6.1 and 6.2, we easily see that (17) now holds for all q ∈ ℝ. Then for every q ∈ ℝ,

K_{q}(z, w) = ∑_{k=0}^{∞} c_{k}(q) W_{k}(z, w)  and  R_{q}(x, y) = ∑_{m=0}^{∞} γ_{m}(q) Z_{m}(x, y)

converge uniformly and absolutely on compact subsets of 𝔹 × 𝔹, are reproducing kernels on 𝔹, and define unique Hilbert spaces of holomorphic or harmonic functions on 𝔹. Let’s again call these Hilbert spaces D_{q} and Δ_{q}. Also for uniformity, let’s set D_{q} as the Bergman space A_{q}^{2} for q > −1, and let’s set Δ_{q} as the Bergman space b_{q}^{2} for q > −1. Evidently, all the R_{q} are real-valued and symmetric in their two variables, and all the K_{q} are sesquilinear.

It is easy to see that

K_{q}(z, w) = _{2}F_{1}(1, 1; 1 − (N + q); ⟨z, w⟩) (q ≤ −(1 + N)).

It is also classical that the Gegenbauer polynomials, and hence the zonal harmonics, are finite hypergeometric series, but we do not know whether or not weighted infinite sums of them, the R_{q} for q ≤ −(1 + n/2), have representations in terms of some special functions.

Now for all q ∈ ℝ, we can write

K_{q}(z, w) = ∑_{k=0}^{∞} c_{k}(q) \frac{(N)_{k}}{(1)_{k}} ∑_{|α|=k} \frac{k!}{α!} z^{α} \overline{w}^{α} = ∑_{α} \frac{c_{k}(q) (N)_{k}}{α!} z^{α} \overline{w}^{α},

which shows that ∥1∥_{D_{q}} = 1 and

∥z^{α}∥_{D_{q}}^{2} = \frac{α!}{c_{k}(q) (N)_{k}} (q ∈ ℝ).

Consequently, an f(z) = ∑_{α} f_{α} z^{α} ∈ H(𝔹) belongs to D_{q}, q ∈ ℝ, if and only if

∥f∥_{D_{q}}^{2} = ∑_{α} |f_{α}|^{2} \frac{α!}{c_{|α|}(q) (N)_{|α|}} < ∞.

For an equivalent formulation in Δ_{q}, we consider the breaking up (10) of Z_{m}(x, y) into smaller pieces. Then

R_{q}(x, y) = ∑_{m=0}^{∞} γ_{m}(q) ∑_{j=1}^{h_{m}} Y_{mj}(x) \overline{Y}_{mj}(y),

which shows that

∥Y_{mj}∥_{Δ_{q}}^{2} = \frac{1}{γ_{m}(q)} (q ∈ ℝ, j = 1, …, h_{m}),

and ∥1∥_{Δ_{q}} = 1/γ_{0}(q) = 1. Moreover, each term u_{m} in the homogeneous expansion of a u ∈ h(𝔹) has an expansion in terms of the basis {Y_{m1}, …, Y_{mh_{m}}}; that is,

u(x) = ∑_{m=0}^{∞} R^{m} u_{m}(ξ) = ∑_{m=0}^{∞} R^{m} ∑_{j=1}^{h_{m}} u_{mj} Y_{mk}(ξ) = ∑_{m=0}^{∞} ∑_{j=1}^{h_{m}} u_{mj} Y_{mk}(x),

where

u_{mj} = \frac{1}{R^{m}} ∫_{𝕊} u_{m}(x) \overline{Y_{mj}(ξ)} dσ(ξ)

independently of any 0 < R < 1.

Theorem 6.3. Let q ∈ ℝ. A function u ∈ h(𝔹) belongs to Δ_{q} if and only if

(23)  ∥u∥_{Δ_{q}}^{2} = ∑_{m=0}^{∞} ∑_{j=1}^{h_{m}} \frac{|u_{mj}|^{2}}{γ_{m}(q)} < ∞.

This results appears in [GKÜ1, Theorem 3.8].

Going back to the holomorphic case, the counterpart of the Y_{mj}(x) are the functions

V_{α}(z) = \begin{matrix}√ & \overline{(N)_{|α|}} \\ √ & \overline{α!} \end{matrix} R^{|α|} ζ^{α}

that are orthonormal with respect to the inner product of L^{2}(σ) by Proposition 2.1. For each k = |α|, there are θ_{k} such functions, and

∥V_{α}∥_{D_{q}}^{2} = \frac{1}{c_{|α|}(q)} (q ∈ ℝ, |α| = k).

An f ∈ H(𝔹) can be written as f(z) = ∑_{α} f̃_{α} V_{α}(z), and it belongs to D_{q}, q ∈ ℝ, if

and only if

(24)  ∥f∥_{D_{q}}^{2} = ∑_{α} \frac{|f̃_{α}|^{2}}{c_{|α|}(q)} < ∞.

The equivalent norms (23) and (24) are useful for another purpose too. Using them, it is easy to see that if q_{1} < q_{2}, then D_{q_{1}} ⊂ D_{q_{2}} and Δ_{q_{1}} ⊂ Δ_{q_{2}}. Further, examples are constructed in [AK, Example 2.4] and [GKÜ1, Example 3.10] that show that these inclusions are proper.

7. Radial Differential Operators

We now take one step further the fractional-order radial derivatives that we have already noted.

Definition 7.1. Let s, t ∈ ℝ be unrestricted. We define positive numbers d_{k}(s, t) and operators 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},

and other positive numbers δ_{m}(s, t) and operators D_{s}^{t} on h(𝔹) by

D_{s}^{t}u = ∑_{m=0}^{∞} δ_{m}(s, t) u_{m} = ∑_{m=0}^{∞} \frac{γ_{m}(s + t)}{γ_{m}(s)} u_{m},

where f ∈ H(𝔹) and u ∈ h(𝔹) are given by their homogeneous expansions (20), and c_{k} and γ_{m} are from Definitions 6.1 and 6.2 and equations (15) and (16).

Definition 7.1 is the essence of the connection between the Bergman-Besov kernels and the radial differential operators that help characterize the spaces they generate.

For any s ∈ ℝ, D_{s}^{0} = I, the identity. If a function is homogeneous of order l, then any D_{s}^{t} applied on it is also homogeneous of degree l; in particular D_{s}^{t}(H_{m}) = H_{m}.

Further, d_{k}(s, t) ∼ k^{t} as k → ∞ and δ_{m}(s, t) ∼ m^{t} as m → ∞ by (1). Then by [Ara, Theorem 5], an appropriate D_{s}^{t} maps H(𝔹) or h(𝔹) into itself continuously, and hence D_{s}^{t}f is also holomorphic and D_{s}^{t}u is also harmonic on 𝔹.

We select d_{k}(s, t) and δ_{m}(s, t) in order to have

D_{q}^{t}K_{q}(z, w) = K_{q+t}(z, w)  and  D_{q}^{t}R_{q}(x, y) = R_{q+t}(x, y)  (q, t ∈ ℝ),

where D_{s}^{t} are applied only on the first variable just like R. These immediately generalize (21) and (22) in two (t > 0 or t < 0) directions. Remark 5.1 hints that many choices are possible for the d_{k} and the δ_{k}. All authors have their own particular coefficients, but for many formulas our choices seem to be the most natural and exact. Thus the reproducing kernels of Hilbert spaces determine the particular natural forms of the radial differential operators acting on them.

The last two paragraphs show that every D_{s}^{t} is a radial differential operator of fractional order t ∈ ℝ. In particular, D_{−N}^{1} = I + R for H(𝔹) and D_{−n/2}^{1} = I + R for h(𝔹). The D_{s}^{t} are actually integral operators when t < 0.

Moreover, d_{k}(s, t) ≠ 0 for all k including k = 0, and δ_{m}(s, t) ≠ 0 for all m including m = 0. It is also clear that D_{s+t}^{v}D_{s}^{t} = D_{s}^{v+t} for all choices of s, t, v ∈ ℝ. Then every D_{s}^{t} is invertible on H(𝔹) or h(𝔹) with two-sided inverse (D_{s}^{t})^{−1} = D_{s+t}^{−t}. The usual radial derivative R obviously is not invertible.

8. Besov Spaces

For s, t ∈ ℝ, let’s define further operators I_{s}^{t} by

I_{s}^{t}f(z) = (1 − |z|^{2})^{t} D_{s}^{t}f(z)  (f ∈ H(𝔹)),
I_{s}^{t}u(x) = (1 − |x|^{2})^{t} D_{s}^{t}u(x)  (u ∈ h(𝔹)).

Thus I_{s}^{0} = I as well for any s ∈ ℝ. It turns out that the I_{s}^{t} are more useful than the D_{s}^{t}.

Definition 8.1. Let q ∈ ℝ, 1 ≤ p < ∞, and pick s, t such that

(25)  q + pt > −1.

The Besov space B_{q}^{p} is the space of all f ∈ H(𝔹) for which I_{s}^{t}f lies in L_{q}^{p}. The harmonic Besov space b_{q}^{p} is the space of all u ∈ h(𝔹) for which I_{s}^{t}u lies in L_{q}^{p}. We also define ∥f∥_{B_{q}^{p}} = ∥I_{s}^{t}f∥_{L_{q}^{p}} and ∥u∥_{b_{q}^{p}} = ∥I_{s}^{t}u∥_{L_{q}^{p}}, and similarly for the inner products for p = 2.

That this definition is independent of s, t as long as (25) is satisfied and that different s, t satisying (25) give equivalent norms have been shown for the holomorphic case in [BB1, Theorem 5.12 (i)], which is an excellent source on the spaces B_{q}^{p}; see also [KÜ, Remark 5.4] for another proof. For the harmonic case, it is in [Li1, Theorem 3]; see also [GKÜ1, Proposition 3.6].

A norm of the harmonic Besov space b_{q}^{p} written explicitly takes the form

∥f∥_{b_{q}^{p}} = (C_{q} ∫_{𝔹} |D_{s}^{t}u(x)|^{p} (1 − |x|^{2})^{q+pt} dν(x))^{1/p}.

For B_{q}^{p}, we just change u to f and x to z. This formula explains why (25) is essential and that it replaces the condition q > −1 of Bergman spaces. The invertibility of D_{s}^{t} yields that only the zero function has norm 0, and hence the Besov norms are genuine norms.

The condition (25) ensures that Besov spaces contain all polynomials of the right kind (holomorphic or harmonic). In fact, such polynomials are dense in every Besov space; this follows from the fact that they are dense in Bergman spaces and the fact that the D_{s}^{t} acting on homogeneous polynomials of an order produce polynomials of the same order.

Note that if q > −1, (25) is satisfied with t = 0, and then the Besov space is nothing but the Bergman space with the same parameters. Here the use of the parameter q for Besov spaces is nonstandard; we follow [K1] and [GKÜ1]. But our q for Besov spaces is selected in such a way that it extends the q of the Bergman spaces to values q ≤ −1 in the most natural manner.

The equivalence of norms extends farther on Besov Hilbert spaces.

Theorem 8.2. Let q ∈ ℝ be unrestricted. The space B_{q}^{2} and the the space D_{q} coincide with equivalent norms. The space b_{q}^{2} and the the space Δ_{q} coincide with equivalent norms.

This result is in [BB1, p. 35] for the holomorphic case; it is in [GKÜ1, Theorem 3.7] for the harmonic case.

Remark 8.3. We have noted above that B_{−1}^{2} is the Hardy space H^{2}(𝔹). However, this identification does not extend to other p. It is known that B_{−1}^{p} ⊂ H^{p}(𝔹) for 1 ≤ p < 2 and H^{p}(𝔹) ⊂ B_{−1}^{2} for 2 < p < ∞, where the inclusions are proper; see [GP, p. 336].

When q ≤ −1, a positive-order D_{s}^{t} is required by (25). The question whether or not B_{q}^{2} can be characterized by an integral using f itself has been raised and answered negatively for q = −N in [Arv, p. 180]. This has been extended negatively to all q < −1 in [BB2, Theorem 4.3]. The answer clearly is positive for q = −1 using the surface measure σ on 𝕊 since B_{−1}^{2} = H^{2}.

It is inherent in Definition 8.1 that every I_{s}^{t} satisfying (25) imbeds B_{q}^{p} or b_{q}^{p} into the Lebesgue class L_{q}^{p}. Imbedding spaces via the I_{s}^{t} or their cousins is just as important as imbedding them via the inclusion; see [Lu, Theorem 2] and [K2, Theorem 1.3]. The boundedness of maps in the reverse direction is of considerable interest for many properties of Besov spaces can be derived from them.

Definition 8.4. For s ∈ ℝ, the integral operators

P_{s}φ(z) = ∫_{𝔹} K_{s}(z, w) φ(w) dν_{s}(w)  and  Q_{s}φ(x) = ∫_{𝔹} R_{s}(x, y) φ(y) dν_{s}(y)

acting on φ in suitable Lebesgue classes are called (extended) Bergman projections.

Theorem 8.5. For 1 ≤ p < ∞, P_{s} is a bounded operator from L_{q}^{p} onto B_{q}^{p} if and only if

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

Given an s satisfying (26), if t satisfies (25), then

(27)  P_{s}I_{s}^{t}f = \frac{1}{C_{s+t}} f  (f ∈ B_{q}^{p}),

where C_{s+t} is as in (2) with the identification n/2 = N.

Note that (26) and (25) together imply that s + t > −1.

Theorem 8.5 says that the imbeddings I_{s}^{t} for which (25) holds are right inverses for the Bergman projections P_{s} for which (26) holds. In this most general form for the holomorphic Besov spaces, this result is in [K1, Theorem 1.2]. For the harmonic Besov spaces, such a general theorem will appear in [GKÜ2]. For the boundedness of Q_{s} on the harmonic Bergman spaces on the upper half space, see [KNY, Theorem 4.3]. It is surprising that the equivalent condition for the boundedness of Bergman projections is the same for holomorphic or harmonic Bergman-Besov spaces on the ball or on the upper half space; it is (26).

The maps P_{s} and Q_{s} are true projections on subspaces B_{q}^{p} or b_{q}^{p} of L_{q}^{p} for q > −1 and s = q. For other values of q and s, I_{s}^{t}P_{s} is a true projection onto the subspace I_{s}^{t}(B_{q}^{p}) of L_{q}^{p}, which is an isomorphic copy of B_{q}^{p} inside L_{q}^{p}; see [KÜ, Corollary 5.6].

The relations in (27), when written explicitly, are integral representations for functions in B_{q}^{p} specific to the space. They are

f(z) = \frac{(1 + s + t)_{N}}{(1)_{N}} ∫_{𝔹} D_{s}^{t}f(w) K_{s}(z, w) (1 − |w|^{2})^{s+t} dν(w) (f ∈ B_{q}^{p}, z ∈ 𝔹),

where 1 ≤ p < ∞, and s and t satisfy (26) and (25). It is the existence of formulas of this kind that make the Bergman projections so useful.

9. Special Cases

By definition, for q > −1, Besov spaces are actually Bergman spaces. We call the range q ≤ −1 the proper Besov range. There are other values of q and p for which the Besov spaces have special names and properties, especially for p = 2.

We have already noted that B_{−1}^{2} = H^{2} and b_{−1}^{2} = h^{2}, and for N = 1, the reproducing kernel K_{−1} takes the form

K_{−1}(z, w) = \frac{1}{1 − z \overline{w}} (z, w ∈ 𝔻).

A kernel that has the same form for N > 1 is

K_{−N}(z, w) = \frac{1}{1 − ⟨z, w⟩} (z, w ∈ 𝔹).

The corresponding space B_{−N}^{2} is called the Drury-Arveson space. It is the space that is considered the right generalization of the Hardy space H^{2}(𝔻) of the disc to higher dimensions from the point of view of operator theory and Nevanlinna-Pick interpolation; see [Arv]. Note also that

K_{−(1+N)}(z, w) = \frac{1}{⟨z, w⟩} log \frac{1}{1 − ⟨z, w⟩};

thus B_{−(1+N)}^{2} is the Dirichlet space; that’s why we have initially named the family B_{q}^{2} as D_{q}.

In the plane where N = 1, n = 2, and ℂ = ℝ^{2}, there are closer relations between the holomorphic and the harmonic cases. Now θ_{k} = 1 and h_{m} = 2 for all k and m ≥ 1; z = x and w = y; also ⟨z, w⟩ = z \overline{w} and Z_{m}(x, y) = x^{m} \overline{y}^{m} +\overline{x}^{m} y^{m}. Further

c_{m}(q) = γ_{m}(q) = \begin{cases}\frac{(2 + q)_{m}}{m!} , & if q > −2; \\ \frac{m!}{(−q)_{m}} , & if q ≤ −2; \end{cases}

K_{q}(z, w) = \begin{cases}\frac{1}{(1 − z \overline{w})^{2+q}} , & if q > −2; \\ _{2}F_{1}(1, 1; −q; z \overline{w}), & if q ≤ −2; \end{cases}

R_{q}(x, y) = K_{q}(x, y) + \overline{K_{q}(x, y)} − 1.

At some critical values of q, certain crucial changes occur at the reproducing kernels K_{q} and R_{q}. There are 2 critical values of q in the holomorphic category. At q = −1, we pass between Bergman and proper Besov spaces. At q = −(1 + N), we pass between binomial (Bergman-type) kernels and hypergeometric kernels. This value q = −(1 + N) is also the passage point between unbounded kernels and bounded kernels. By (18), the series giving K_{q} converges absolutely and uniformly on \overline{𝔹} × \overline{𝔹} for q < −(1 + N), which yields that the spaces B_{q}^{2} for such q consist of bounded holomorphic functions on 𝔹 that extend continuously to 𝕊.

By contrast, in the harmonic category, there are 3 critical values of q. This difference arises from the dissimilarities expressed in Remarks 3.1 and 4.2. The behavior at q = −1 is as expected; we pass between Bergman and proper Besov spaces, that is, between spaces that require derivatives in their integral norms and those that do not. At q = −(1 + n/2), we pass between Bergman-type kernels and kernels that are not. At q = −n, we pass between unbounded kernels and bounded ones. Now by (19), the series giving R_{q} converges absolutely and uniformly on \overline{𝔹} × \overline{𝔹} for q < −n, which yields that the spaces b_{q}^{2} for such q consist of bounded harmonic functions on 𝔹 that extend continuously to 𝕊. When n = 2, the latter two critical values are merged and is the same as the critical value for the holomorphic category with N = 1.

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Department of Mathematics, Bi̇lkent University, 06800 Ankara, Turkey

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