KELVIN-MÖBIUS-INVARIANT HARMONIC FUNCTION SPACES ON THE REAL UNIT BALL

H. TURGAY KAPTANOĞLU AND A. ERSİN ÜREYEN

Abstract. We define the Kelvin-Möbius transform of a function harmonic on the unit ball of ℝ^{n} and determine harmonic function spaces that are invariant under this transform. When n ≥ 3, in the category of Banach spaces, the minimal Kelvin-Möbius-invariant space is the Bergman-Besov space b_{−(1+n/2)}^{1} and the maximal invariant space is the Bloch space b_{(n−2)/2}^{∞}. There exists a unique strictly Kelvin-Möbius-invariant Hilbert space, and it is the Bergman-Besov space b_{−2}^{2}. There is a unique Kelvin-Möbius-invariant Hardy space.

1. Introduction

Let 𝔻 ⊂ ℂ be the open unit disc and Aut(𝔻) be its automorphism group, that is, the group of holomorphic, bijective maps of 𝔻. The Bloch space B consists of all holomorphic functions f on 𝔻 such that the seminorm

ρ_{B}(f) := sup{(1 − |z|^{2})|f′(z)| : z ∈ 𝔻}

is finite. The Bloch space is Möbius invariant in the sense that if f ∈ B, then for every φ ∈ Aut(𝔻), f ∘ φ is in B and ρ_{B}(f ∘ φ) = ρ_{B}(f). The quantity (1 − |z|^{2})f′(z) is sometimes called the invariant derivative of f at z since its modulus is Möbius invariant in this particular sense.

More generally, let E be a linear space of holomorphic functions on 𝔻 that is complete with respect to a seminorm ρ_{E}. Roughly speaking, E is called Möbius invariant if for every f ∈ E and φ ∈ Aut(𝔻), f ∘ φ belongs to E and ρ_{E}(f ∘ φ) = ρ_{E}(f) (or the weaker condition ρ_{E}(f ∘ φ) ∼ ρ_{E}(f) holds). We say “roughly speaking” because there should be a few technical restrictions on E; for details see [3], [5] and [25]. Among all Möbius-invariant spaces, the Bloch space B is the largest [25] and the Besov space B^{1} is the smallest [5]. If one considers Hilbert spaces, the Dirichlet space is the unique Möbius-invariant Hilbert space [3]. Similar results with domains other than 𝔻 are obtained in various sources, in [24] and [32] the domain is the unit ball of ℂ^{n}, in [21] the domain is the polydisc and [4] considers bounded symmetric domains.

The purpose of this paper is to study the harmonic analogue of this problem on the unit ball 𝔹 of ℝ^{n} with respect to the standard norm |x|^{2} = x · x = x_{1}^{2} + ⋯ + x_{n}^{2}. Let h(𝔹) be the Fréchet space of all complex-valued harmonic functions on 𝔹 endowed with the topology of uniform convergence on compact subsets, and M(𝔹) be the group of Möbius transformations of 𝔹; see subsection 2.2. In the harmonic case the first problem one encounters is that if f ∈ h(𝔹) and φ ∈ M(𝔹), then f ∘ φ need not be harmonic. To remedy this problem note that any φ ∈ M(𝔹) can be written as a composition of an orthogonal transformation and an inversion. Composing a harmonic function with an orthogonal transformation preserves harmonicity. So, harmonicity is lost in f ∘ φ because of composition with an inversion. On the other hand, one can compose with an inversion and still preserve harmonicity provided one multiplies by a correction factor and this is called the Kelvin transform. Therefore we need to combine composition with φ with the Kelvin transform and this leads us to the following definition.

Date: 9 April 2021.

2010 Mathematics Subject Classification. Primary 31B05; Secondary 46B70, 46E15, 46E20, 46E22, 47A15.

Key words and phrases. Möbius transformation, Kelvin-Möbius transform, Kelvin-Möbius-invariant space, harmonic Bergman-Besov space, weighted harmonic Bloch space, atomic decomposition, complex interpolation.

Definition 1.1. Let f ∈ h(𝔹) and φ ∈ M(𝔹). The Kelvin-Möbius transform K_{φ}(f) of f is defined as

K_{φ}(f)(x) := (\frac{1 − |φ(x)|^{2}}{1 − |x|^{2}} )^{(n−2)/2} f(φ(x)).

The factor multiplying f ∘φ stems from the Kelvin transform; see subsection 2.3. If f is harmonic on 𝔹, then so is K_{φ}(f). This is verified later, but in the literature there are also different proofs. See, for example, [2, Corollary 2.3], [9, §2.3], [19, Proposition 3.1] or [23, Theorem 2].

If the dimension is n = 2, the first factor disappears and we just have K_{φ}(f) = f ∘ φ. Because of this there are differences between the cases n = 2 and n ≥ 3. When n = 2, the harmonic case is very similar to the holomorphic case studied in [3, 5, 25]; nevertheless there are details that require attention. We deal with that in a different work [22] and throughout this paper we consider the case n ≥ 3.

Remark 1.2. We show in subsection 2.3 that K_{φ} : h(𝔹) → h(𝔹) is invertible with K_{φ}^{−1} = K_{φ^{−1}}.

We now define a Kelvin-Möbius-invariant harmonic function space following [25] and [31]. Let (E, ∥ · ∥_{E}) be a Banach space of harmonic functions on 𝔹. A non-zero continuous linear functional L on h(𝔹) is called decent on E if L is also continuous on E with respect to the norm ∥ · ∥_{E}.

We denote by 1 the constant function whose value is 1.

Definition 1.3. Let n ≥ 3. A Banach space (E, ∥ · ∥_{E}) of harmonic functions on 𝔹 is called Kelvin-Möbius invariant if the following properties hold:

(i) E contains 1.

(ii) There exists a decent linear functional on E.

(iii) For every f ∈ E and φ ∈ M(𝔹), the Kelvin-Möbius transform K_{φ}(f) belongs to E and

(1)  ∥K_{φ}(f)∥_{E} ≤ C∥f∥_{E},

for some constant C independent of f and φ.

Since E is linear we can write the first condition as (i) E contains constant functions.

If the constant in (1) is C = 1, then by Remark 1.2, ∥K_{φ}(f)∥_{E} = ∥f∥_{E} for every f ∈ E and φ ∈ M(𝔹) and in this case we call E strictly Kelvin-Möbius invariant as done in [5, Definition 1]. In the general case we only have ∥K_{φ}(f)∥_{E} ∼ ∥f∥_{E}, where we use the notation A ∼ B to mean that A/B is bounded from above and below by some constants that are independent of the parameters involved. If A/B is just bounded above, we write A ≲ B.

Remark 1.4. If E is Kelvin-Möbius invariant, then we can define a new, equivalent norm on E with which E becomes strictly Kelvin-Möbius invariant (see [5, p. 111]). Thus weakening strictly Kelvin-Möbius invariance and using Kelvin-Möbius invariance with “∼” is not very important.

Remark 1.5. We note that in the holomorphic case the invariant space is taken as a complete semi-normed space. In the harmonic case the same is true when n = 2 but when n ≥ 3 we do not need semi-norms because of the extra multiplying factor in K_{φ}, and E in Definition 1.3 is a Banach space.

The harmonic function spaces we are mainly interested in this work are Bergman-Besov spaces b_{q}^{p} and weighted Bloch spaces b_{s}^{∞} with q, s ∈ ℝ in both. Let ν be the normalized Lebesgue measure on 𝔹 and for q ∈ ℝ, let

dν_{q}(x) := c_{q}(1 − |x|^{2})^{q} dν(x).

The measure ν_{q} is finite only when q > −1 and in this case we pick the constant c_{q} so that ν_{q}(𝔹) = 1. When q ≤ −1, we just set c_{q} = 1. For 0 < p < ∞, we denote the Lebesgue classes with respect to ν_{q} by L_{q}^{p}.

For q > −1, the well-known harmonic Bergman space b_{q}^{p} is defined as b_{q}^{p} := h(𝔹) ∩ L_{q}^{p} with norm ∥f∥_{b_{q}^{p}} := ∥f∥_{L_{q}^{p}}. The next definition extends this class to all q ∈ ℝ. To denote partial derivatives we use multi-indices and write

∂^{α}f := \frac{∂^{|α|}f}{∂x_{1}^{α_{1}} ⋯ ∂x_{n}^{α_{n}}} ,

where the multi-index α = (α_{1}, …, α_{n}) is an n-tuple of nonnegative integers and |α| = α_{1} + ⋯ + α_{n}.

Definition 1.6. Let 0 < p < ∞ and q ∈ ℝ. Pick a nonnegative integer N such that

(2)  q + pN > −1.

The harmonic Bergman-Besov space b_{q}^{p} consists of all f ∈ h(𝔹) such that

(1 − |x|^{2})^{N}∂^{α}f ∈ L_{q}^{p}

for every multi-index α with |α| = N. A norm (quasinorm when 0 < p < 1) on b_{q}^{p} is

(3)  ∥f∥_{b_{q}^{p}}^{p} := ∑_{|α|<N} |∂^{α}f(0)|^{p} + ∑_{|α|=N} ∫_{𝔹} |(1 − |x|^{2})^{N}∂^{α}f(x)|^{p} dν_{q}(x).

When 0 < p < 1, while we still use the notation ∥ · ∥_{b_{q}^{p}}, b_{q}^{p} is not a normed space, but it is a complete metric space with respect to the metric d(f, g) = ∥f − g∥_{b_{q}^{p}}^{p}.

The space b_{q}^{p} does not depend on the choice of N. Different choices of N satisfying (2) give rise to equivalent norms and in the notation ∥ · ∥_{b_{q}^{p}} we do not indicate the dependence on N. The partial derivatives in the above definition can be replaced with radial derivatives or various other suitable differential operators.

The region q > −1 is the Bergman region in which we take N = 0, and q ≤ −1 is the proper Besov region. When p = 2, b_{q}^{2} is a Hilbert space endowed with the inner product

(4) ⟨f, g⟩_{b_{q}^{2}} := ∑_{|α|<N} ∂^{α}f(0)\overline{∂^{α}g(0)} + ∑_{|α|=N} ∫_{𝔹} ∂^{α}f(x)\overline{∂^{α}g(x)}(1 − |x|^{2})^{2N} dν_{q}(x),

with N satisfying (2). In the special case q = −n, the spaces b_{−n}^{p} are studied in [20, 29]. For the whole family b_{q}^{p} with q ∈ ℝ, see [17] (when p ≥ 1) and [13] (when 0 < p < 1).

We first determine which harmonic Bergman-Besov spaces are Kelvin-Möbius invariant. In the Bergman region q > −1, this is very easy.

Lemma 1.7. Let n ≥ 3. Let 1 ≤ p < ∞ and q > −1. Then b_{q}^{p} is Kelvin-Möbius invariant if and only if q = p(n − 2)/2 − n and in this case b_{q}^{p} is strictly Kelvin-Möbius invariant.

Figure 1. Equation of the ray l is q = p(n − 2)/2 − n, p ≥ 1. By Lemma 1.7, if (p, q) is on the solid part of the ray l, then b_{q}^{p} is Kelvin-Möbius invariant.

It is clear that 1 ∈ b_{q}^{p}, and it is well known that point-evaluation functionals are bounded on Bergman spaces. To check condition (iii) of Definition 1.3, let f ∈ b_{q}^{p} and φ ∈ M(𝔹). Then

∥K_{φ}(f)∥_{b_{q}^{p}}^{p} = c_{q} ∫_{𝔹} |f(φ(x))|^{p} (\frac{1 − |φ(x)|^{2}}{1 − |x|^{2}} )^{p(n−2)/2} (1 − |x|^{2})^{q} dν(x).

Using p(n − 2)/2 = q + n, changing variables as x̃ = φ(x), and using (12) below, we immediately obtain ∥K_{φ}(f)∥_{b_{q}^{p}} = ∥f∥_{b_{q}^{p}}. We show the only-if part of Lemma 1.7 in Section 4 within the proof of Theorem 1.8.

Lemma 1.7 tells that if (p, q) is on the solid part of the ray l in Figure 1, then the space b_{q}^{p} is Kelvin-Möbius invariant. This suggests that the Banach spaces corresponding to the dashed part of the ray l may also be Kelvin-Möbius invariant. This is true and it is our first theorem.

Theorem 1.8. Let n ≥ 3. Let p ≥ 1 and q ∈ ℝ. Then b_{q}^{p} is Kelvin-Möbius invariant if and only if q = p(n − 2)/2 − n.

Of course the interesting part of Theorem 1.8 is the case q ≤ −1. We prove this part in Section 4 using complex interpolation.

By Remark 1.4, when q = p(n − 2)/2 − n, the space b_{q}^{p} is strictly Kelvin-Möbius invariant when endowed with a suitable equivalent norm.

Remark 1.9. Theorem 1.8 is true also when n = 2. In this case q = p(n−2)/2−n = −2 is less than −1 and there is no Kelvin-Möbius-invariant harmonic Bergman space. When n = 2, as in the holomorphic case, all Kelvin-Möbius-invariant spaces belong to the proper Besov region and lie on the horizontal line q = −2.

Observe that when p = 2, the corresponding q in Theorem 1.8 is q = −2 and the Hilbert space (b_{−2}^{2}, ⟨·, ·⟩_{b_{−2}^{2}}) is Kelvin-Möbius invariant. The space b_{−2}^{2} is strictly Kelvin-Möbius invariant when endowed with a suitable inner product ⟨·, ·⟩_{I} which we now describe.

For m ≥ 0, let H_{m}(𝕊) be the vector space of spherical harmonics of degree m and δ_{m} be its dimension (see subsection 3.1 for more details). Let {Y_{m}^{j} : j = 1, …, δ_{m}} be an orthonormal basis of H_{m}(𝕊). If f is harmonic on 𝔹, then f has the expansion

f(x) = ∑_{m=0}^{∞} ∑_{j=1}^{δ_{m}} f_{m}^{j}Y_{m}^{j}(x) (x ∈ 𝔹),

where f_{m}^{j}’s are complex numbers. The above series converges absolutely and uniformly on compact subsets of 𝔹. By [16, Theorem 3.8] or [17, Theorem 5.1], the space b_{−2}^{2} can also be described as

b_{−2}^{2} = \{f = ∑_{m=0}^{∞} ∑_{j=1}^{δ_{m}} f_{m}^{j}Y_{m}^{j} ∈ h(𝔹) : ∥f∥_{I}^{2} = ∑_{m=0}^{∞} ∑_{j=1}^{δ_{m}} \frac{m + n/2 − 1}{n/2 − 1}|f_{m}^{j}|^{2} < ∞\}.

The above norm is equivalent to the norm(s) given in Definition 1.6 and is induced by the inner product

(5)  ⟨f, g⟩_{I} := ∑_{m=0}^{∞} ∑_{j=1}^{δ_{m}} \frac{m + n/2 − 1}{n/2 − 1}f_{m}^{j}\overline{g_{m}^{j}}  (f, g ∈ b_{−2}^{2}).

When endowed with the above inner product ⟨·, ·⟩_{I}, the space b_{−2}^{2} is strictly Kelvin-Möbius-invariant Hilbert space and it is the only strictly Kelvin-Möbius-invariant Hilbert space. This is our second theorem.

Theorem 1.10. (i) The Hilbert space (b_{−2}^{2}, ⟨·, ·⟩_{I}) is strictly Kelvin-Möbius invariant. (ii) If (H, ⟨·, ·⟩_{H}) is a strictly Kelvin-Möbius-invariant Hilbert space, then H = b_{−2}^{2} and ⟨·, ·⟩_{H} = C⟨·, ·⟩_{I} for some C > 0.

In Proposition 6.1, we give an integral description of the inner product ⟨·, ·⟩_{I} similar to (4), but in terms of radial derivatives.

When (p, q) is on the ray q = p(n − 2)/2 − n, p ≥ 1, the Kelvin-Möbius-invariant spaces b_{q}^{p} increase as p increases. This is a consequence of [15, Theorem 1.2]. If p = 1, the corresponding q is −(1 + n/2), and this suggests that b_{−(1+n/2)}^{1} might be the smallest Kelvin-Möbius-invariant space. This is true and it is our next theorem.

Theorem 1.11. Let n ≥ 3. The space b_{−(1+n/2)}^{1} is the smallest Kelvin-Möbius-invariant space. More precisely, if E is Kelvin-Möbius invariant, then b_{−(1+n/2)}^{1} ⊂ E and there exists a C > 0 such that ∥f∥_{E} ≤ C∥f∥_{b_{−(1+n/2)}^{1}} for every f ∈ b_{−(1+n/2)}^{1}.

We next determine the largest Kelvin-Möbius-invariant space. For this we need to consider weighted harmonic Bloch spaces.

Definition 1.12. Let s ∈ ℝ. Pick a non-negative integer N so that s + N > 0. The weighted harmonic Bloch space b_{s}^{∞} consists of all f ∈ h(𝔹) such that for every multi-index α with |α| = N,

sup_{x∈𝔹} (1 − |x|^{2})^{s+N} |∂^{α}f(x)| < ∞.

A norm on b_{s}^{∞} is

∥f∥_{b_{s}^{∞}} := ∑_{|α|<N} |∂^{α}f(0)| + ∑_{|α|=N} sup_{x∈𝔹} (1 − |x|^{2})^{s+N} |∂^{α}f(x)| .

As before, the space b_{s}^{∞} does not depend on the choice of N as long as s+N > 0 and different choices of N give rise to equivalent norms (see [14] for more details). When n ≥ 3 we are interested in the space b_{(n−2)/2}^{∞}. Choosing N = 0 shows

(6) b_{(n−2)/2}^{∞} = {f ∈ h(𝔹) : ∥f∥_{b_{(n−2)/2}^{∞}} = sup_{x∈𝔹} (1 − |x|^{2})^{(n−2)/2}|f(x)| < ∞}.

It is immediate that the space b_{(n−2)/2}^{∞} is strictly Kelvin-Möbius invariant. Clearly 1 ∈ b_{(n−2)/2}^{∞} and for a decent linear functional, we can take any point-evaluation functional. That ∥K_{φ}(f)∥_{b_{(n−2)/2}^{∞}} = ∥f∥_{b_{(n−2)/2}^{∞}} is obvious.

By [15, Theorem 1.3], the increasing family of Kelvin-Möbius-invariant spaces b_{q}^{p}, p ≥ 1, q = p(n − 2)/2 − n are all included in the weighted Bloch space b_{(n−2))/2}^{∞}. This suggests that b_{(n−2)/2}^{∞} might be the largest Kelvin-Möbius-invariant space. This is true and it is our next theorem.

Theorem 1.13. Let n ≥ 3. The space b_{(n−2)/2}^{∞} is the largest Kelvin-Möbius-invariant space. More precisely, if E is Kelvin-Möbius invariant, then E ⊂ b_{(n−2)/2}^{∞} and there exists a C > 0 such that ∥f∥_{b_{(n−2)/2}^{∞}} ≤ C∥f∥_{E} for every f ∈ E.

For completeness we also look at the spaces b_{q}^{p}, 0 < p < 1. Because these are not normed spaces, they cannot be Kelvin-Möbius invariant in the sense of Definition 1.3. Nevertheless if q = p(n − 2)/2 − n, the properties (i)-(iii) in Definition 1.3 do hold even when 0 < p < 1, where we understand ∥ · ∥ as a quasinorm.

Theorem 1.14. Let n ≥ 3. Let 0 < p < 1 and q ∈ ℝ. Then the properties (i)-(iii) in Definition 1.3 hold for b_{q}^{p} if and only if q = p(n − 2)/2 − n.

We note that the spaces b_{q}^{p}, 0 < p < 1, q = p(n − 2)/2 − n are smaller than b_{−(1+n/2)}^{1} by [15, Theorem 1.2]. This, however, does not contradict Theorem 1.11 since these spaces are not Banach spaces. An analogous result concerning Möbius invariant holomorphic function spaces on the unit ball of ℂ^{n} is obtained in [33].

Another well-studied space of harmonic functions is the class of harmonic Hardy spaces. For each n ≥ 3, one Hardy space can be shown to be Kelvin-Möbius invariant by using [19, Theorem 1.2]. We discuss this in Remark 4.4.

Lastly, we consider subspaces of h(𝔹) and show that there is no non-trivial closed subspace of h(𝔹) that is invariant under taking Kelvin-Möbius transform.

Theorem 1.15. Let n ≥ 3 and A ⊂ h(𝔹) be a closed subspace. If K_{φ}(f) ∈ A for every f ∈ A and φ ∈ M(𝔹), then A = {0} or A = h(𝔹).

Corollary 1.16. Let n ≥ 3 and f ∈ h(𝔹) be non-zero. Then span{K_{φ}(f) : φ ∈ M(𝔹)} is dense in h(𝔹).

The paper is organized as follows. In Section 2 we justify Definition 1.1 and present some elementary properties of the Kelvin-Möbius transform. In Section 3 we review reproducing kernels and atomic decompositions of harmonic Bergman-Besov spaces b_{q}^{p}. We consider the Kelvin-Möbius invariance of the b_{q}^{p} spaces and prove Theorems 1.8 and 1.14 in Section 4. In Section 5 we first show that point-evaluation functionals are bounded on Kelvin-Möbius-invariant spaces and as a consequence determine the minimal and maximal invariant spaces. We prove Theorem 1.15 also in this section. Section 6 is devoted to the Hilbert-space case. We repeat that in this paper we take n ≥ 3.

2. Kelvin-Möbius Transform

In this section we first recall some known facts about Möbius transformations. For more detail about these transformations see [1, 7, 30]. We then justify the definition of Kelvin-Möbius transform and prove some of its elementary properties.

2.1. Orthogonal Transformations. We denote the group of all orthogonal transformations of ℝ^{n} by O(n). Fixing an orthonormal basis of ℝ^{n} we can represent each element of O(n) with an orthogonal matrix of size n × n. Identifying matrices of size n × n with elements of ℝ^{n^{2}} induces a natural topology on such matrices which makes O(n) a compact topological group. The elements of O(n) whose corresponding matrices have determinant 1 form a subgroup of O(n) denoted by SO(n). We denote the normalized Haar measure on SO(n) by μ_{0}, where normalized means μ_{0}(SO(n)) = 1.

For a proof of the lemma below see [11, Theorem 3.1] or [26, Lemma 1.4.7 (3)].

Lemma 2.1. Let f be continuous on 𝕊 and η ∈ 𝕊. Then

∫_{SO(n)} f(U(η)) dμ_{0}(U) = ∫_{𝕊} f(ζ) dσ(ζ).

2.2. Möbius Transformations. A Möbius transformation of ℝ̂^{n} := ℝ^{n} ∪ {∞} is a finite composition of inversions in spheres and reflections in planes. We denote the group of all Möbius transformations that map the unit ball 𝔹 to itself by M(𝔹).

For a ∈ 𝔹, the involutive Möbius transformation φ_{a} that exchanges a and 0 is defined by

φ_{a}(x) := \frac{(1 − |a|^{2})(a − x) + |x − a|^{2}a}{[x, a]^{2}},

where

[x, a] := \sqrt{1 − 2x · a + |x|^{2}|a|^{2}}.

The map φ_{a} can be decomposed into simple maps. Let J be the inversion with respect to the unit sphere 𝕊,

J(x) := x^{∗} := \frac{x}{|x|^{2}} .

Note that when a ≠ 0,

(7)  [x, a] = |a| |x − a^{∗}|.

More generally, for c ∈ ℝ^{n} and r > 0, let 𝕊(c, r) be the sphere with center c and radius r, and let J_{c,r} denote the inversion with respect to 𝕊(c, r),

J_{c,r}(x) := c + r^{2}J(x − c).

For 0 ≠ a ∈ 𝔹, φ_{a} can be decomposed as

φ_{a} = −P_{a} ∘ J_{a^{∗},ρ},

where

(8)  ρ := \frac{\sqrt{1 − |a|^{2}}}{|a|}

and

P_{a}(x) := x − 2\frac{x · a}{|a|^{2}}a

is the reflection about the plane passing through the origin and perpendicular to the vector a. For details see [1, Section 2.6] which uses T_{a} = −φ_{a}. The sphere 𝕊(a^{∗}, ρ) is orthogonal to 𝕊 and thus J_{a^{∗},ρ}(𝔹) = 𝔹.

Let φ ∈ M(𝔹) be arbitrary and a = φ^{−1}(0). If a = 0, then φ is an orthogonal transformation. If a ≠ 0, then by [30, Theorem 2.1.2] there exists an orthogonal transformation U ∈ O(n) such that φ = U ∘ φ_{a} and φ can be decomposed as

(9)  φ = U ∘ (−P_{a}) ∘ J_{a^{∗},ρ}.

Thus, since P_{a} is also orthogonal, we can write φ as composition of an orthogonal transformation and an inversion.

We list some identities involving φ. The most useful identity is

(10)  1 − |φ(x)|^{2} = \frac{(1 − |a|^{2})(1 − |x|^{2})}{[x, a]^{2}}  (a = φ^{−1}(0)).

The ratio [x, y]^{2}/((1 − |x|^{2})(1 − |y|^{2})) is Möbius invariant, that is, for every x, y ∈ 𝔹 and φ ∈ M(𝔹),

(11)  \frac{[φ(x), φ(y)]^{2}}{(1 − |φ(x)|^{2})(1 − |φ(y)|^{2})} = \frac{[x, y]^{2}}{(1 − |x|^{2})(1 − |y|^{2})}.

The derivative φ′(x) of φ = (φ_{1}, …, φ_{n}) : 𝔹 → 𝔹, is the n × n matrix

φ′(x) = [\frac{∂φ_{i}}{∂x_{j}} ]_{i,j=1}^{n} .

The absolute value of the Jacobian determinant of φ is ([30, Theorem 3.3.1])

(12)  | det(φ′(x))| = (\frac{1 − |φ(x)|^{2}}{1 − |x|^{2}} )^{n} .

2.3. Kelvin-Möbius Transform. The Kelvin transform of f with respect to the unit sphere 𝕊 is defined by

K_{𝕊}(f)(x) := \frac{1}{|x|^{n−2}} f(J(x)).

More generally, Kelvin transform of f with respect to the sphere 𝕊(c, r) is defined by (see [18, p. 39])

K_{𝕊(c,r)}(f)(x) := \frac{r^{n−2}}{|x − c|^{n−2}} f(J_{c,r}(x)).

If f is harmonic on a domain Ω ⊂ ℝ^{n}, then K_{𝕊(c,r)}(f) is harmonic on J_{c,r}(Ω).

Let φ ∈ M(𝔹), φ^{−1}(0) = a ≠ 0 and φ have the decomposition (9). By (7) and (8),

K_{𝕊(a^{∗},ρ)}(f)(x) = \frac{(1 − |a|^{2})^{(n−2)/2}}{[x, a]^{n−2}} f(J_{a^{∗},ρ}(x)).

Because J_{a^{∗},ρ}(𝔹) = 𝔹, if f is harmonic on 𝔹, then K_{𝕊(a^{∗},ρ)}(f) is harmonic on 𝔹. Replacing f with f ∘ U ∘ (−P_{a}) we deduce that if f ∈ h(𝔹), then so is

\frac{(1 − |a|^{2})^{(n−2)/2}}{[x, a]^{n−2}} f(φ(x)) = (\frac{1 − |φ(x)|^{2}}{1 − |x|^{2}} )^{(n−2)/2} f(φ(x)),

where we also use the formula (10). The function above is the Kelvin-Möbius transform of f.

If φ = U ∈ O(n), then we just have

(13)  K_{U}(f) = f ∘ U.

For future reference let us also record that if f = 1, then

(14)  K_{φ_{a}}(1)(x) = \frac{(1 − |a|^{2})^{(n−2)/2}}{[x, a]^{n−2}} .

Some basic properties of the transform K_{φ} are listed in the following lemma.

Lemma 2.2. Let φ, ψ ∈ M(𝔹). The Kelvin-Möbius transform K_{φ} : h(𝔹) → h(𝔹) satisfies the following properties:

(i) K_{φ} is linear. (ii) K_{ψ} ∘ K_{φ} = K_{φ∘ψ}. (iii) K_{φ} is one-to-one and onto and K_{φ}^{−1} = K_{φ^{−1}}. (iv) K_{φ_{a}}^{−1} = K_{φ_{a}}.

(v) K_{φ} is continuous.

Proof. Part (i) is clear. Part (ii) is pure computation. Part (iii) follows from part (ii) and part (iv) is true because φ_{a} is an involution. To see part (v) suppose f_{m} → f in h(𝔹), that is, f_{m} converges to f uniformly on compact subsets of 𝔹. We have

K_{φ}(f_{m})(x) = (\frac{1 − |φ(x)|^{2}}{1 − |x|^{2}} )^{(n−2)/2} f_{m}(φ(x))

and if x lies in the compact set |x| ≤ r < 1, then there exists s < 1 such that |φ(x)| ≤ s. The first factor on the right is bounded and second factor converges uniformly to f(φ(x)) since f_{m} converges uniformly to f on |x| ≤ s. Hence K_{φ}(f_{m}) converges uniformly to K_{φ}(f) on |x| ≤ r. □

3. Harmonic Bergman-Besov spaces

This section is for review purposes. We recall some known facts about zonal harmonics and harmonic Bergman-Besov spaces b_{q}^{p} that will be used in the sequel. For more detail about zonal harmonics see [6, Chapter 5], and about the spaces b_{q}^{p} see [17] and [13].

3.1. Spherical and Zonal Harmonics. Let L^{2}(𝕊) be the Hilbert space of square integrable functions on 𝕊 with respect to the inner product ⟨f, g⟩ = ∫_{𝕊} f\overline{g} dσ, where σ is the normalized surface area measure on 𝕊. Let H_{m}(ℝ^{n}) denote the complex vector space of all homogeneous harmonic polynomials of degree m in n real variables. Restriction of an element of H_{m}(ℝ^{n}) to 𝕊 is called a (surface) spherical harmonic of degree m. The collection H_{m}(𝕊) of all spherical harmonics of degree m is a finite-dimensional subspace of L^{2}(𝕊) with dimension δ_{m}.

For m ≥ 0, let {Y_{m}^{j}, j = 1, …, δ_{m}} be an orthonormal basis of H_{m}(𝕊). If m ≠ k, then H_{m}(𝕊) ⊥ H_{k}(𝕊) in L^{2}(𝕊), so

(15)  ∫_{𝕊} Y_{m}^{j}(ξ)\overline{Y_{k}^{i}(ξ)} dσ(ξ) = 0

unless m = k and j = i. For η ∈ 𝕊, the point-evaluation functional f ↦ f(η) is bounded on H_{m}(𝕊) and therefore there exists a unique Z_{m}(·, η) ∈ H_{m}(𝕊) such that for all f ∈ H_{m}(𝕊)

f(η) = ∫_{𝕊} f(ξ)\overline{Z_{m}(ξ, η)} dσ(ξ).

The zonal harmonic Z_{m}(ξ, η) is real-valued, it is symmetric in its variables, and in terms of Y_{m}^{j} it equals

Z_{m}(ξ, η) = ∑_{j=1}^{δ_{m}} Y_{m}^{j}(ξ)\overline{Y_{m}^{j}(η)} (ξ, η ∈ 𝕊).

The above formula extends to ℝ^{n} × ℝ^{n} by homogeneity

(16)  Z_{m}(x, y) = ∑_{j=1}^{δ_{m}} Y_{m}^{j}(x)\overline{Y_{m}^{j}(y)}  (x, y ∈ ℝ^{n}),

where Y_{m}^{j}(x) = |x|^{m}Y_{m}^{j}(ξ) for x = |x|ξ.

When n ≥ 3, as a function of x, 1/[x, a]^{n−2} is harmonic on \overline{𝔹} by (7). Its homogeneous expansion is given in the following lemma.

Lemma 3.1. Let n ≥ 3 and a ∈ 𝔹. Then

\frac{1}{[x, a]^{n−2}} = ∑_{m=0}^{∞} \frac{n/2 − 1}{m + n/2 − 1}Z_{m}(x, a),

where the series converges uniformly for x ∈ \overline{𝔹}.

Proof. Lemma is clear when a = 0, so we assume a ≠ 0. For d > 0, the Gegenbauer polynomial G_{m}^{d} of degree m is defined by the generating function

\frac{1}{(1 − 2rt + t^{2})^{d}} = ∑_{m=0}^{∞} G_{m}^{d}(r) t^{m}.

Writing x = |x|ξ, a = |a|η, we see that

\frac{1}{[x, a]^{n−2}} = \frac{1}{(1 − 2ξ · η |x||a| + |x|^{2}|a|^{2})^{n/2−1}} = ∑_{m=0}^{∞} G_{m}^{n/2−1}(ξ · η)|x|^{m}|a|^{m}.

It is known that (see, for example, [17, Equation (14.8)])

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

Thus

\frac{1}{[x, a]^{n−2}} = ∑_{m=0}^{∞} \frac{n/2 − 1}{m + n/2 − 1}Z_{m}(ξ, η)|x|^{m}|a|^{m} = ∑_{m=0}^{∞} \frac{n/2 − 1}{m + n/2 − 1}Z_{m}(x, a),

where the last equality follows from the homogeneity of Z_{m}. For fixed a ∈ 𝔹, the series converges uniformly for x ∈ \overline{𝔹} since |Z_{m}(ξ, η)| ≲ m^{n−2} (see [6, Proposition 5.27 (e) and Exercise 10, p. 107]). □

3.2. Reproducing Kernels of Harmonic Bergman-Besov Spaces. For all q ∈ ℝ, the space b_{q}^{2} is a reproducing kernel Hilbert space. We denote the reproducing kernel by R_{q}(x, ·). When q > −1, the natural inner product on b_{q}^{2} is ⟨f, g⟩_{b_{q}^{2}} := ∫_{𝔹} f\overline{g} dν_{q} and with respect to this inner product (see [12, p. 164])

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

where the Pochhammer symbol (a)_{b} 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.

When q ≤ −1, it is necessary to consider derivatives of the functions in the inner product and there are various choices. Different choices of the inner product would give rise to different reproducing kernels and vice versa. We follow the approach of [16] and extend the reproducing kernels to whole q ∈ ℝ in the following way. Define

(17)  γ_{m}(q) := \begin{cases}\frac{(n/2 + q + 1)_{m}}{(n/2)_{m}} , & if q > −(1 + n/2); \\ \frac{(m!)^{2}}{(1 − (n/2 + q))_{m}(n/2)_{m}} , & if q ≤ −(1 + n/2); \end{cases}

and

R_{q}(x, y) := ∑_{m=0}^{∞} γ_{m}(q)Z_{m}(x, y).

For an (integral) inner product that makes this R_{q}(x, y) the reproducing kernel of b_{q}^{2} (q ∈ ℝ) see [17, Theorem 5.2]. Every R_{q}(x, y) is symmetric in its variables since every Z_{m}(x, y) is.

Using the reproducing kernels above we can define radial differential operators D_{s}^{t} of order t for every t, s ∈ ℝ. If f ∈ h(𝔹) with homogeneous expansion f = ∑_{m=0}^{∞} f_{m}, then we define

D_{s}^{t}f := ∑_{m=0}^{∞} \frac{γ_{m}(s + t)}{γ_{m}(s)}f_{m}.

The operators D_{s}^{t} are compatible with reproducing kernels in the sense that for every t, s ∈ ℝ

(18)  D_{s}^{t}R_{s}(x, y) = R_{s+t}(x, y),

where the operator acts on the first variable. Because of this in the study of the properties of harmonic Bergman-Besov spaces it is more convenient to use the operators D_{s}^{t} rather than the partial derivatives. Similar to Definition 1.6 the spaces b_{q}^{p} can be described in terms of the operators D_{s}^{t}. For 0 < p < ∞ and q ∈ ℝ, pick t ∈ ℝ such that q + pt > −1 and any s ∈ ℝ. Then f ∈ b_{q}^{p} if and only if

(19)  _{(t,s)}∥f∥_{b_{q}^{p}}^{p} := ∫_{𝔹} |(1 − |x|^{2})^{t}D_{s}^{t}f(x)|^{p} dν_{q}(x) < ∞

and the norm (quasinorm when 0 < p < 1) _{(t,s)}∥ · ∥_{b_{q}^{p}}^{p} is equivalent to the norm ∥ · ∥_{b_{q}^{p}} in (3); see [17, Theorems 1.1 and 1.2] and [13, Theorem 1.1]. We need the D_{s}^{t} in the proof of the only-if part of Theorems 1.8 and 1.14.

As suggested by Theorem 1.10, for our purposes the most important kernel is R_{−2}. In this case, by (17) and Lemma 3.1, the following closed formula holds.

Lemma 3.2. Let n ≥ 3. Then

R_{−2}(x, y) = \frac{1}{[x, y]^{n−2}} (x, y ∈ 𝔹).

Combining the above lemma with (14) we see that

(20)  K_{φ_{a}}(1)(x) = \frac{(1 − |a|^{2})^{(n−2)/2}}{[x, a]^{n−2}} = (1 − |a|^{2})^{(n−2)/2}R_{−2}(x, a).

For a proof of the following estimate of the weighted integrals of powers of R_{q}(x, y), see [17, Theorem 1.5].

Lemma 3.3. Let q ∈ ℝ, a > 0, and b > −1. Set c = a (n + q) − (n + b). Then

∫_{𝔹} |R_{q}(x, y)|^{a}(1 − |y|^{2})^{b} dν(y) ∼ \begin{cases}\frac{1}{(1 − |x|^{2})^{c}} , & if c > 0; \\ 1 + log \frac{1}{1 − |x|^{2}} , & if c = 0; \\ 1, & if c < 0. \end{cases}

3.3. Atomic Decomposition of Harmonic Bergman-Besov Spaces. Every function in the space b_{q}^{p} can be written as an infinite sum of reproducing kernels (atoms). For harmonic Bergman spaces (q > −1) this is proved in [10] and is extended to all q ∈ ℝ in [17, Theorem 10.1] when p ≥ 1 and in [13, Theorem 1.4] when 0 < p < 1. To describe this atomic decomposition we need some definitions.

For a, b ∈ 𝔹, the pseudo-hyperbolic metric d_{H}(a, b) is defined by d_{H}(a, b) := |φ_{a}(b)|. It is well known that d_{H} is Möbius invariant,

(21)  d_{H}(φ(a), φ(b)) = d_{H}(a, b)  (φ ∈ M(𝔹), a, b ∈ 𝔹).

For a ∈ 𝔹 and 0 < r < 1, let D(a, r) := {x ∈ 𝔹 : d_{H}(x, a) < r} be the pseudohyperbolic ball with center a and radius r. A sequence (a_{m}) in 𝔹 is called r-separated if the balls D(a_{m}, r) are pairwise disjoint. The sequence (a_{m}) is called an r-lattice if 𝔹 = ∪_{m=1}^{∞}D(a_{m}, r) and (a_{m}) is r/2-separated.

Theorem 3.4. Let 0 < p < ∞ and q ∈ ℝ. Choose s such that

q + 1 < p(s + 1), if p ≥ 1;
q + n < p(s + n), if 0 < p < 1.

There exists an r_{0} > 0 such that if (a_{m}) is an r-lattice with r < r_{0}, then the following hold:

(i) For every (c_{m}) ∈ ℓ^{p}, the function

(22)  f(x) = ∑_{m=1}^{∞} c_{m}(1 − |a_{m}|^{2})^{s+n−(q+n)/p} R_{s}(x, a_{m})

is in b_{q}^{p} and ∥f∥_{b_{q}^{p}} ≲ ∥(c_{m})∥_{ℓ^{p}}. The series in (22) converges to f absolutely and uniformly on compact subsets of 𝔹 and also in ∥ · ∥_{b_{q}^{p}}.

(ii) For every f ∈ b_{q}^{p}, there exists (c_{m}) ∈ ℓ^{p} with ∥(c_{m})∥_{ℓ^{p}} ≲ ∥f∥_{b_{q}^{p}} such that the representation (22) holds.

4. Kelvin-Möbius-Invariant Harmonic Bergman-Besov Spaces

In this section we consider harmonic Bergman-Besov spaces b_{q}^{p} for the whole range 0 < p < ∞, q ∈ ℝ and prove Theorems 1.8 and 1.14. That is, we show that the properties (i)-(iii) in Definition 1.3 hold for b_{q}^{p} if and only if q = p(n − 2)/2 − n. Note that clearly 1 ∈ b_{q}^{p} for every p and q and point-evaluation functionals are bounded on all b_{q}^{p} by [17, Theorem 13.1] and [13, Theorem 5.1]. Therefore all we need to check is condition (iii) of Definition 1.3.

We first prove the if parts of Theorems 1.8 and 1.14 and defer the only-if parts to the end of the section. We begin with the case 0 < p ≤ 1 which we handle by using atomic decomposition.

Proposition 4.1. Let n ≥ 3, 0 < p ≤ 1 and q = p(n − 2)/2 − n. Then there exists a constant C > 0 such that ∥K_{φ}(f)∥_{b_{q}^{p}} ≤ C∥f∥_{b_{q}^{p}} for every f ∈ b_{q}^{p} and φ ∈ M(𝔹).

Proof. We apply Theorem 3.4 with s = −2 which is possible since n ≥ 3. Let r_{0} be as asserted in that theorem and pick an r-lattice (a_{m}) with r < r_{0}. Let f ∈ b_{q}^{p} and φ ∈ M(𝔹) be arbitrary. By Theorem 3.4 (ii), there exists (c_{m}) ∈ ℓ^{p} with ∥(c_{m})∥_{ℓ^{p}} ≲ ∥f∥_{b_{q}^{p}} such that

f(x) = ∑_{m=1}^{∞} c_{m}(1 − |a_{m}|^{2})^{(n−2)/2}R_{−2}(x, a_{m}) = ∑_{m=1}^{∞} c_{m}K_{φ_{am}} (1)(x),

where the second equality follows from (20). Apply K_{φ} to f and pass it through the sum to each term of the series. This is possible by Lemma 2.2 (v) and the uniform convergence of the series on compact subsets of 𝔹. Applying also Lemma 2.2 (ii) we obtain

K_{φ}(f) = ∑_{m=1}^{∞} c_{m}K_{φ} ∘ K_{φ_{am}} (1) = ∑_{m=1}^{∞} c_{m}K_{φ_{am} ∘φ}(1).

Let b_{m} = φ^{−1}(a_{m}). By (21), (b_{m}) is an r-lattice too. Because (φ_{a_{m}} ∘ φ)(b_{m}) = 0, there exist U_{m} ∈ O(n) such that φ_{a_{m}} ∘ φ = U_{m} ∘ φ_{b_{m}} and

K_{φ}(f) = ∑_{m=1}^{∞} c_{m}K_{φ_{am} ∘φ}(1) = ∑_{m=1}^{∞} c_{m}K_{U_{m}∘φ_{bm}} (1) = ∑_{m=1}^{∞} c_{m}K_{φ_{bm}} (1),

where the last equality holds because K_{U_{m}∘φ_{bm}} (1) = K_{φ_{bm}} ∘K_{U_{m}}(1) and K_{U_{m}}(1) = 1 by (13). Using (20) again we obtain

K_{φ}(f)(x) = ∑_{m=1}^{∞} c_{m}(1 − |b_{m}|^{2})^{(n−2)/2}R_{−2}(x, b_{m}).

Finally, Theorem 3.4 (i) implies ∥K_{φ}(f)∥_{b_{q}^{p}} ≲ ∥(c_{m})∥_{ℓ^{p}} and we conclude that ∥K_{φ}(f)∥_{b_{q}^{p}} ≲ ∥f∥_{b_{q}^{p}}. □

Proposition 4.1 shows that property (iii) in Definition 1.3 holds for b_{q}^{p} when 0 < p ≤ 1, q = p(n − 2)/2 − n and this proves the if part of Theorem 1.14. We separate the Banach space case p = 1.

Corollary 4.2. The Bergman-Besov space b_{−(1+n/2)}^{1} is Kelvin-Möbius invariant.

We next prove the if part of Theorem 1.8. In view of Lemma 1.7 we only need to deal with the q ≤ −1 case, that is, we need to establish that the spaces corresponding to the dashed part of the ray l in Figure 1 are Kelvin-Möbius invariant.

Note that on the ray l in Figure 1, when q = 0, the corresponding p is 2n/(n−2) and by Lemma 1.7, the unweighted Bergman space b_{0}^{2n/(n−2)} is Kelvin-Möbius invariant. Next, observe that the coordinates of the left end point of the ray l are p = 1, q = −(1 + n/2), and the corresponding space b_{−(1+n/2)}^{1} is Kelvin-Möbius invariant by Corollary 4.2. To finish the proof it suffices to show that the spaces corresponding to the line segment joining the points (1, −(1 + n/2)) and (2n/(n − 2), 0) in Figure 1 are Kelvin-Möbius invariant. We do this by using complex interpolation.

The complex interpolation space between two Bergman-Besov spaces is determined in [17, Theorem 13.5] which we repeat below.

Theorem 4.3. Let 1 ≤ p_{0} < p_{1} < ∞ and q_{0}, q_{1} ∈ ℝ. If

(23)  \frac{1}{p} = \frac{1 − θ}{p_{0}} + \frac{θ}{p_{1}}

for some 0 < θ < 1 and

(24)  \frac{q}{p} = \frac{(1 − θ)q_{0}}{p_{0}} + \frac{θq_{1}}{p_{1}} ,

then [b_{q_{0}}^{p_{0}}, b_{q_{1}}^{p_{1}}]_{θ}, the complex interpolation space between b_{q_{0}}^{p_{0}} and b_{q_{1}}^{p_{1}}, is b_{q}^{p}.

Proof of the if part of Theorem 1.8. With p_{0} = 1, q_{0} = −(1+n/2), p_{1} = 2n/(n−2), q_{1} = 0, and p_{0} < p < p_{1}, if θ satisfies (23) and q satisfies (24), then the complex interpolation space b_{q}^{p} given by Theorem 4.3 lies on the ray l in Figure 1.

Now consider the linear transformation K_{φ}(f) acting on f. Corollary 4.2 says that K_{φ} is bounded on b_{−(1+n/2)}^{1}, and Lemma 1.7 says that K_{φ} is bounded on

b_{0}^{2n/(n−2)}, both uniformly for φ ∈ M(𝔹). On the other hand, [8, Theorem 4.1.2] says that the complex interpolation method is an interpolation functor, which means that K_{φ} is also bounded on b_{q}^{p} with 1 < p < 2n/(n − 2) and q = p(n − 2)/2 − n uniformly for φ ∈ M(𝔹). Consequently the b_{q}^{p} on the dashed part of the ray are all Kelvin-Möbius invariant. □

We now deal with the only-if parts of Theorems 1.8 and 1.14 and show that if q ≠ p(n−2)/2−n, then property (iii) in Definition 1.3 does not hold. This includes the Bergman (q > −1) case asserted in Lemma 1.7 too.

Proof of the only-if parts of Theorems 1.8 and 1.14. Consider the function f = 1. We have ∥1∥_{b_{q}^{p}} = 1 and by (20)

K_{φ_{a}}(1)(x) = (1 − |a|^{2})^{(n−2)/2}R_{−2}(x, a),

for a ∈ 𝔹. Pick t ∈ ℝ such that q + pt > −1. Applying the operator D_{−2}^{t} and using (18) we obtain

D_{−2}^{t}K_{φ_{a}}(1) = (1 − |a|^{2})^{(n−2)/2}D_{−2}^{t}R_{−2}(x, a) = (1 − |a|^{2})^{(n−2)/2}R_{t−2}(x, a).

Thus by (19)

∥K_{φ_{a}}(1)∥_{b_{q}^{p}}^{p} ∼ (1 − |a|^{2})^{p(n−2)/2} ∫_{𝔹} |R_{t−2}(x, a)|^{p} (1 − |x|^{2})^{q+pt} dν(x).

We estimate the above integral using Lemma 3.3. We check three cases depending on the sign of c = p(n − 2) − q − n.

If p(n − 2) − q − n > 0, then

∥K_{φ_{a}}(1)∥_{b_{q}^{p}}^{p} ∼ \frac{1}{(1 − |a|^{2})^{p(n−2)/2−q−n}} ,

and right-hand side tends to ∞ or 0 as |a| → 1^{−} unless q = p(n − 2)/2 − n.

If p(n − 2) − q − n = 0, then ∥K_{φ_{a}}(1)∥_{b_{q}^{p}}^{p} → 0 as |a| → 1^{−} since (1 − |a|^{2})^{p(n−2)/2} dominates log(1/(1 − |a|^{2})).

If p(n − 2) − q − n < 0, then ∥K_{φ_{a}}(f)∥_{b_{q}^{p}}^{p} → 0 as |a| → 1^{−}.

Thus ∥K_{φ_{a}}(1)∥_{b_{q}^{p}} ∼ ∥1∥_{b_{q}^{p}} = 1 as |a| → 1^{−} only if q = p(n − 2)/2 − n. □

Remark 4.4. We finish this section by looking at another class of harmonic function spaces, harmonic Hardy spaces. For 1 ≤ p < ∞, the space h^{p} consists of all f ∈ h(𝔹) such that

∥f∥_{h^{p}}^{p} := sup_{0≤r<1} ∫_{𝕊} |f(rξ)|^{p} dσ(ξ) < ∞.

For φ ∈ M(𝔹), the norm of K_{φ} : h^{p} → h^{p} is computed in [19, Theorem 1.2]. It follows from this theorem that when n ≥ 3, h^{p} is Kelvin-Möbius invariant if and only if p = 2(n − 1)/(n − 2). We note that in the Bergman-Besov case if we take q = −1 in Theorem 1.8, we find the same p. In view of Theorems 1.11 and 1.13, the inclusions b_{−(1+n/2)}^{1} ⊂ h^{2(n−1)/(n−2)} ⊂ b_{(n−2)/2}^{∞} follow from this invariance. We also note that although there is no Möbius-invariant holomorphic Hardy space on the unit ball of ℂ^{n}, there is one Kelvin-Möbius-invariant harmonic Hardy space.

5. Minimal and Maximal Spaces

We show in this section that of all the Kelvin-Möbius-invariant spaces, the Bergman-Besov space b_{−(1+n/2)}^{1} is minimal and the weighted Bloch space b_{(n−2)/2}^{∞} is maximal. A major part of the proof is to verify that point-evaluation functionals are bounded on Kelvin-Möbius-invariant spaces. We begin with some lemmas. Recall that we always take n ≥ 3.

Lemma 5.1. span{1/[x, a]^{n−2} : a ∈ 𝔹} is dense in h(𝔹).

Proof. It is well known that harmonic polynomials are dense in h(𝔹). Let g be a harmonic polynomial. It is clear that g belongs to every b_{q}^{p}, in particular it belongs to b_{−(1+n/2)}^{1}. We apply Theorem 3.4 to b_{−(1+n/2)}^{1} with s = −2 which shows that there exist suitable a_{m} ∈ 𝔹 and (c_{m}) ∈ ℓ^{1} such that

g(x) = ∑_{m=1}^{∞} c_{m}(1 − |a_{m}|^{2})^{(n−2)/2}R_{−2}(x, a_{m}) = ∑_{m=1}^{∞} c_{m} \frac{(1 − |a_{m}|^{2})^{(n−2)/2}}{[x, a_{m}]^{n−2}} ,

where in the second equality we use Lemma 3.2. Since the above series converges uniformly on compact subsets of 𝔹, in h(𝔹), we can approximate g with elements of span{1/[x, a]^{n−2} : a ∈ 𝔹}. □

Lemma 5.2. Let E be a Kelvin-Möbius-invariant space. Then there exists a decent linear functional L on E with L(1) ≠ 0.

Proof. As E is Kelvin-Möbius invariant, there is a continuous linear functional L ≠ 0 on h(𝔹) which is also continuous on (E, ∥ · ∥_{E}). For a ∈ 𝔹, define L_{a} by L_{a}(f) = L(K_{φ_{a}}(f)). By Lemma 2.2 (v), L_{a} is continuous on h(𝔹) and L_{a} is also continuous on (E, ∥ · ∥_{E}) since

|L_{a}(f)| = |L(K_{φ_{a}}(f))| ≤ ∥L∥ ∥K_{φ_{a}}(f)∥_{E} ≲ ∥L∥ ∥f∥_{E},

by Kelvin-Möbius invariance of E, where ∥L∥ is the norm of L on (E, ∥ · ∥_{E}). If L_{a}(1) = 0 for every a ∈ 𝔹, then by (14) we would have L(1/[x, a]^{n−2}) = 0 for every a ∈ 𝔹. But by Lemma 5.1 this would imply L = 0. Thus there exists a ∈ 𝔹 such that L_{a}(1) ≠ 0 and L_{a} is the asserted decent linear functional. □

Theorem 5.3. Let E be a Kelvin-Möbius-invariant space. Then the point evaluation functional f ↦ f(0) is bounded on E.

Proof. Let L be a decent linear functional on E with L(1) ≠ 0 which exists by Lemma 5.2. Define a new linear functional L on h(𝔹) by

(25)  L(f) = ∫_{SO(n)} L(f ∘ U) dμ_{0}(U)  (f ∈ h(𝔹)),

where μ_{0} is as in subsection 2.1. The above integral is well defined since the map U ↦ f ∘ U is continuous from SO(n) to h(𝔹) and SO(n) is compact.

We first verify that L is continuous on h(𝔹). A linear functional Λ on h(𝔹) is continuous if and only if there exists a compact set K ⊂ 𝔹 and a constant C > 0 such that |Λ(f)| ≤ C sup{|f(x)| : x ∈ K}, for every f ∈ h(𝔹) [25, p. 1]. Hence there exists a K_{r} = {x ∈ 𝔹 : |x| ≤ r < 1} and a C > 0 such that

|L(f)| ≤ C sup{|f(x)| : x ∈ K_{r}} (f ∈ h(𝔹)).

The same inequality also holds when |L(f)| is replaced by |L(f ∘ U)| since K_{r} is O(n)-invariant. Thus

|L(f)| ≤ ∫_{SO(n)} |L(f ∘ U)| dμ_{0}(U) ≤ C sup{|f(x)| : x ∈ K_{r}},

because μ_{0} is normalized. This shows that L is continuous on h(𝔹).

It is clear that L(1) = L(1) ≠ 0. We proceed to show that L(Y_{m}^{j}) = 0 for every j = 1, 2, …, δ_{m}, m = 1, 2, …, where Y_{m}^{j} is as in subsection 3.1. For this, for fixed f ∈ h(𝔹), we consider the vector-valued integral

(26)  ∫_{SO(n)} f ∘ U dμ_{0}(U)

as defined in [27, Definition 3.26]. Noting that the map U ↦ f ∘ U is continuous from SO(n) to h(𝔹), by [27, Theorem 3.27] and the remark preceding it, the integral in (26) exists in the sense that there exists a unique g ∈ h(𝔹) such that

Λ(g) = ∫_{SO(n)} Λ(f ∘ U) dμ_{0}(U)

for every continuous linear functional Λ on h(𝔹). The value of the integral in (26) is then defined to be equal to g.

Since L is continuous on h(𝔹), taking Λ = L we obtain

(27) L(Y_{m}^{j}) = ∫_{SO(n)} L(Y_{m}^{j} ∘ U) dμ_{0}(U) = L (∫_{SO(n)} Y_{m}^{j} ∘ U dμ_{0}(U)) = L(V_{m}^{j}),

where V_{m}^{j} = ∫_{SO(n)} Y_{m}^{j} ∘ U dμ_{0}(U). To find V_{m}^{j} note that for every x ∈ 𝔹, the point-evaluation functional Λ_{x} given by Λ_{x}(f) = f(x) is continuous on h(𝔹). Thus taking Λ = Λ_{x} and writing ξ = x/|x| for x ≠ 0, we see that

V_{m}^{j}(x) = Λ_{x}(V_{m}^{j}) = Λ_{x} (∫_{SO(n)} Y_{m}^{j} ∘ U dμ_{0}(U)) = ∫_{SO(n)} Λ_{x} (Y_{m}^{j} ∘ U) dμ_{0}(U)

= ∫_{SO(n)} Y_{m}^{j}(U(x)) dμ_{0}(U).

The last integral can be computed to be

∫_{SO(n)} Y_{m}^{j}(U(x)) dμ_{0}(U) = |x|^{m} ∫_{SO(n)} Y_{m}^{j}(U(ξ)) dμ_{0}(U) = |x|^{m} ∫_{𝕊} Y_{m}^{j}(ζ) dσ(ζ)
= |x|^{m}Y_{m}^{j}(0) = 0,

where in the first equality we use the homogeneity of Y_{m}^{j}, in the second equality we use Lemma 2.1, in the third equality we use the mean-value property, and in the last that m ≥ 1. Hence V_{m}^{j}(x) = 0 for x ≠ 0 and by continuity V_{m}^{j} ≡ 0. By (27) we conclude that L(Y_{m}^{j}) = 0 for m ≥ 1.

We can now find L(f) for any f ∈ h(𝔹) easily. If

f(x) = f(0) + ∑_{m=1}^{∞} ∑_{j=1}^{δ_{m}} f_{m}^{j}Y_{m}^{j}(x) (f_{m}^{j} ∈ ℂ, x ∈ 𝔹)

is the expansion of f, then, since the above series converges in h(𝔹), we have

(28)  L(f) = L(f(0)1) + ∑_{m=1}^{∞} ∑_{j=1}^{δ_{m}} f_{m}^{j}L(Y_{m}^{j}) = f(0)L(1).

We now show that L is decent on E, that is, L is continuous also on (E, ∥ · ∥_{E}). To see this, first note that since L is decent, it is continuous on (E, ∥·∥_{E}). Next, by the Kelvin-Möbius invariance of E and (13), we have ∥f ∘U∥_{E} = ∥K_{U}(f)∥_{E} ∼ ∥f∥_{E} for every f ∈ E and U ∈ SO(n). Therefore

(29) |L(f)| ≤ ∫_{SO(n)} |L(f ∘ U)| dμ_{0}(U) ≤ ∥L∥ ∫_{SO(n)} ∥f ∘ U∥_{E} dμ_{0}(U) ≲ ∥f∥_{E}.

To finish the proof, let f ∈ E be arbitrary. Then by (28) and (29),

|f(0)L(1)| = |L(f)| ≲ ∥f∥_{E},

and since L(1) ≠ 0, we conclude that |f(0)| ≲ ∥f∥_{E}. □

Corollary 5.4. Let E be a Kelvin-Möbius-invariant space. Then for every a ∈ 𝔹, the point-evaluation functional Λ_{a} : E → ℂ, Λ_{a}(f) = f(a) is bounded on E.

Proof. Since K_{φ_{a}}(f)(0) = (1 − |a|^{2})^{(n−2)/2}f(a), using that ∥K_{φ_{a}}(f)∥_{E} ∼ ∥f∥_{E} and Theorem 5.3, we obtain

{□ \atop (30)} (1 − |a|^{2})^{(n−2)/2}|f(a)| = |K_{φ_{a}}(f)(0)| ≲ ∥K_{φ_{a}}(f)∥_{E} ≲ ∥f∥_{E}.

The maximality of b_{(n−2)/2}^{∞} among Kelvin-Möbius-invariant spaces is now immediate.

Proof of Theorem 1.13. If E is Kelvin-Möbius invariant and f ∈ E, then (30) and (6) shows f ∈ b_{(n−2)/2}^{∞} and ∥f∥_{b_{(n−2)/2}^{∞}} ≲ ∥f∥_{E}. □

We next show the minimality of b_{−(1+n/2)}^{1}.

Proof of Theorem 1.11. Suppose E is Kelvin-Möbius invariant. Pick f ∈ b_{−(1+n/2)}^{1}. Applying Theorem 3.4 with p = 1, q = −(1 + n/2) and s = −2 shows that there exist a_{m} ∈ 𝔹 and (c_{m}) ∈ ℓ^{1} with ∥(c_{m})∥_{ℓ^{1}} ≲ ∥f∥_{b_{−(1+n/2)}^{1}} such that

(31)  f(x) = ∑_{m=1}^{∞} c_{m}(1 − |a_{m}|^{2})^{(n−2)/2}R_{−2}(x, a_{m}) = ∑_{m=1}^{∞} c_{m}K_{φ_{am}} (1)(x),

where the second equality follows from (20). The above series converges to f absolutely and uniformly on compact subsets of 𝔹. We claim that it converges to f also in (E, ∥ · ∥_{E}). To verify this, first note that since 1 ∈ E, for each m, K_{φ_{am}} (1) ∈ E and ∥K_{φ_{am}} (1)∥_{E} ∼ ∥1∥_{E}. Because (c_{m}) ∈ ℓ^{1},

∑_{m=1}^{∞} ∥c_{m}K_{φ_{am}} (1)∥_{E} ≲ ∥1∥_{E} ∑_{m=1}^{∞} |c_{m}| < ∞,

and the series ∑_{m=1}^{∞} c_{m}K_{φ_{am}} (1) converges in the Banach space (E, ∥ · ∥_{E}) to some g ∈ E. Since point-evaluation functionals are bounded on E by Corollary 5.4, we must have

g(x) = ∑_{m=1}^{∞} c_{m}K_{φ_{am}} (1)(x) (x ∈ 𝔹).

Comparing with (31) we conclude that f = g ∈ E and

∥f∥_{E} = ∥ ∑_{m=1}^{∞} c_{m}K_{φ_{am}} (1)∥_{E} ≤ ∑_{m=1}^{∞} |c_{m}| ∥K_{φ_{am}} (1)∥_{E}
≲ ∥1∥_{E}∥(c_{m})∥_{ℓ^{1}} ≲ ∥1∥_{E}∥f∥_{b_{−(1+n/2)}^{1}}.

This completes the proof. □

The following corollary follows from the fact that every harmonic polynomial belongs to b_{−(1+n/2)}^{1}.

Corollary 5.5. A Kelvin-Möbius-invariant space contains all harmonic polynomials.

We finish this section by proving Theorem 1.15 and its corollary. The proof utilizes same ideas we used before.

Proof of Theorem 1.15. First observe that if 1 ∈ A, then A = h(𝔹). This is because if 1 ∈ A, then by the invariance of A under the Kelvin-Möbius transform and (14), 1/[x, a]^{n−2} ∈ A for every a ∈ 𝔹. Since A is closed, A = h(𝔹) by Lemma 5.1.

Suppose that A ≠ h(𝔹). Then 1 ∉ A and since h(𝔹) is locally convex, by [27, Theorem 3.5], there exists a continuous linear functional L on h(𝔹) with L(1) = 1 and L(f) = 0 for every f ∈ A. Define L as in (25). Then, as shown in the proof of Theorem 5.3, L is continuous on h(𝔹) and for every f ∈ h(𝔹),

(32)  L(f) = f(0)L(1) = f(0)

by (28).

Now let f ∈ A. Since K_{U}(f) = f ∘ U ∈ A for every U ∈ SO(n) and L vanishes on A, we have L(f) = 0 by the definition (25). Thus, by (32), f(0) = 0 for every f ∈ A. Next, since K_{φ_{a}}(f) ∈ A for every a ∈ 𝔹, we have K_{φ_{a}}(f)(0) = 0, and because K_{φ_{a}}(f)(0) = (1 − |a|^{2})^{(n−2)/2}f(a), we conclude that f(a) = 0 for every a ∈ 𝔹. Thus f vanishes on 𝔹 and A = {0}. □

Proof of Corollary 1.16. Let A be the closure of span{K_{φ}(f) : φ ∈ M(𝔹)}. Then A is a closed subspace of h(𝔹) and A ≠ {0}. By Lemma 2.2 (ii) and (v), A is invariant under the Kelvin-Möbius transform and thus by Theorem 1.15, A = h(𝔹). □

6. The Unique Hilbert Space

We first show that b_{−2}^{2}, endowed with the inner product ⟨·, ·⟩_{I} given in (5), is strictly Kelvin-Möbius invariant.

Proof of Theorem 1.10 (i). By Theorem 1.8, (b_{−2}^{2}, ⟨·, ·⟩_{b_{−2}^{2}}) is Kelvin-Möbius invariant and since ∥ · ∥_{I} is equivalent to ∥ · ∥_{b_{−2}^{2}}, it follows that for all φ ∈ M(𝔹) and f ∈ b_{−2}^{2}

(33)  ∥K_{φ}(f)∥_{I} ∼ ∥f∥_{I}.

We need to show that exact equality holds in (33). For this we show

(34)  ⟨K_{φ}(f), K_{φ}(g)⟩_{I} = ⟨f, g⟩_{I},  for every φ ∈ M(𝔹) and f, g ∈ b_{−2}^{2}.

By Corollary 5.4, point-evaluation functionals are bounded on (b_{−2}^{2}, ⟨·, ·⟩_{I}). We first determine the corresponding reproducing kernels. For a ∈ 𝔹, let

R_{a}(x) := \frac{1}{[x, a]^{n−2}} = ∑_{m=0}^{∞} ∑_{j=1}^{δ_{m}} \frac{n/2 − 1}{m + n/2 − 1}\overline{Y_{m}^{j}(a)}Y_{m}^{j}(x),

where second equality follows from Lemma 3.1 and (16). As R_{a} is harmonic on \overline{𝔹}, it belongs to b_{−2}^{2}, and by (5), ⟨f, R_{a}⟩_{I} = f(a) for all f ∈ b_{−2}^{2}. Thus R_{a}(x) is the reproducing kernel of b_{−2}^{2} with respect to the inner product ⟨·, ·⟩_{I}.

We next compute K_{φ}(R_{a}). Because

K_{φ}(R_{a})(x) = (\frac{1 − |φ(x)|^{2}}{1 − |x|^{2}} )^{(n−2)/2} \frac{1}{[φ(x), a]^{n−2}} ,

using (11) with y replaced by φ^{−1}(a) we obtain

K_{φ}(R_{a})(x) = (\frac{1 − |φ^{−1}(a)|^{2}}{1 − |a|^{2}} )^{(n−2)/2} \frac{1}{[x, φ^{−1}(a)]^{n−2}}

= (\frac{1 − |φ^{−1}(a)|^{2}}{1 − |a|^{2}} )^{(n−2)/2} R_{φ^{−1}(a)}(x). (35)

We verify (34) first when g = R_{a}. Let a ∈ 𝔹, φ ∈ M(𝔹) and f ∈ b_{−2}^{2}. Then

⟨K_{φ}(f), K_{φ}(R_{a})⟩_{I} = (\frac{1 − |φ^{−1}(a)|^{2}}{1 − |a|^{2}} )^{(n−2)/2} ⟨K_{φ}(f), R_{φ^{−1}(a)}⟩_{I}

= (\frac{1 − |φ^{−1}(a)|^{2}}{1 − |a|^{2}} )^{(n−2)/2} K_{φ}(f)(φ^{−1}(a))

= f(a) = ⟨f, R_{a}⟩_{I},

where the first equality follows from (35), the second equality from the reproducing property, and the third equality from the definition of K_{φ}. Thus (34) holds when g ∈ span{R_{a}}_{a∈𝔹} by the linearity of K_{φ}.

Finally, let f, g ∈ b_{−2}^{2} be arbitrary. By the general properties of reproducing kernels, span{R_{a}}_{a∈𝔹} is dense in (b_{2}^{2}, ⟨·, ·⟩_{I}). So there exists a sequence (g_{m}) with g_{m} ∈ span{R_{a}}_{a∈𝔹} such that

g_{m} → g in (b_{2}^{2}, ⟨·, ·⟩_{I}).

By (33) it also holds that

K_{φ}(g_{m}) → K_{φ}(g) in (b_{2}^{2}, ⟨·, ·⟩_{I}).

Hence

⟨K_{φ}(f), K_{φ}(g)⟩_{I} = lim_{m→∞}⟨K_{φ}(f), K_{φ}(g_{m})⟩_{I} = lim_{m→∞}⟨f, g_{m}⟩_{I} = ⟨f, g⟩_{I}.

This shows (34) and the proof is complete.

Note that (33) and therefore Theorem 1.8 plays an essential role here. That is, we first verified the Kelvin-Möbius invariance of (b_{−2}^{2}, ⟨·, ·⟩_{I}) and then using this we showed its strictness. □

We next show that (b_{−2}^{2}, ⟨·, ·⟩_{I}) is essentially the only strictly Kelvin-Möbius-invariant Hilbert space.

Proof of Theorem 1.10 (ii). Let (H, ⟨·, ·⟩_{H}) be a strictly Kelvin-Möbius-invariant Hilbert space. That is, ∥K_{φ}(f)∥_{H} = ∥f∥_{H} for every φ ∈ M(𝔹) and f ∈ H. By polarization we also have

(36)  ⟨K_{φ}(f), K_{φ}(g)⟩_{H} = ⟨f, g⟩_{H}  (φ ∈ M(𝔹), f, g ∈ H).

By Corollary 5.4, the point-evaluation functional f ↦ f(a) is bounded on H for every a ∈ 𝔹. Therefore for each a ∈ 𝔹, there exists S_{a} ∈ H such that for all f ∈ H,

f(a) = ⟨f, S_{a}⟩_{H}.

We show that S_{a} is a positive scalar multiple of the function 1/[x, a]^{n−2}. For this let us first compute K_{φ}(S_{a}). By (36) and Lemma 2.2 (iii),

⟨f, K_{φ}(S_{a})⟩_{H} = ⟨K_{φ^{−1}}(f), S_{a}⟩_{H} = K_{φ^{−1}}(f)(a)

= (\frac{1 − |φ^{−1}(a)|^{2}}{1 − |a|^{2}} )^{(n−2)/2} f(φ^{−1}(a))

= (\frac{1 − |φ^{−1}(a)|^{2}}{1 − |a|^{2}} )^{(n−2)/2} ⟨f, S_{φ^{−1}(a)}⟩_{H}.

Since this is true for all f ∈ H, we conclude that

(37)  K_{φ}(S_{a}) = (\frac{1 − |φ^{−1}(a)|^{2}}{1 − |a|^{2}} )^{(n−2)/2} S_{φ^{−1}(a)},

which is the same as (35) with S_{a} in place of R_{a}.

We first determine S_{0}. Taking φ = U ∈ O(n) and a = 0 in (37) and using (13) we obtain

S_{0} ∘ U = K_{U}(S_{0}) = S_{0}.

This implies that S_{0} is constant on the spheres |x| = r, 0 < r < 1. Because S_{0} is also harmonic on 𝔹, by the mean value property, S_{0} is constant on 𝔹 and S_{0} = λ1, for some λ ∈ ℂ. Since

1 = ⟨1, S_{0}⟩_{H} = ⟨1, λ1⟩_{H} = \overline{λ}∥1∥_{H}^{2},

we see that λ = 1/∥1∥_{H}^{2} is positive.

We now find S_{a}. For any a ∈ 𝔹, taking φ = φ_{a} in (37) shows

K_{φ_{a}}(S_{a}) = \frac{1}{(1 − |a|^{2})^{(n−2)/2}} S_{0}.

Applying K_{φ_{a}}^{−1} = K_{φ_{a}^{−1}} = K_{φ_{a}} to both sides and using that S_{0} = λ1, we obtain

S_{a}(x) = \frac{1}{(1 − |a|^{2})^{(n−2)/2}} K_{φ_{a}}(S_{0})(x) = \frac{λ}{(1 − |a|^{2})^{(n−2)/2}} K_{φ_{a}}(1)(x) = \frac{λ}{[x, a]^{n−2}} ,

where in the last equality we use (14). It follows that the reproducing kernel of (H, ⟨·, ·⟩_{H}) is λ/[x, a]^{n−2} and therefore the reproducing kernel of (H, λ⟨·, ·⟩_{H}) is 1/[x, a]^{n−2}. On the other hand, by the proof of part (i), 1/[x, a]^{n−2} is the reproducing kernel of (b_{−2}^{2}, ⟨·, ·⟩_{I}). By the uniqueness of reproducing kernel Hilbert spaces ([28, Theorem 1.4]), we conclude that H = b_{−2}^{2} and λ⟨·, ·⟩_{H} = ⟨·, ·⟩_{I}. □

We finish this section by giving an integral description of the invariant inner product ⟨·, ·⟩_{I}. This is similar to [24, Definition 5.4], but since in our case the invariant Hilbert space corresponds to q = −2, first-order derivatives are sufficient. On the other hand, in the holomorphic case the Möbius-invariant Hilbert space is B_{−(n+1)}^{2} and as n gets larger higher order derivatives are needed in the integral description of the inner product.

Let f ∈ h(𝔹) and f = ∑_{m=0}^{∞} f_{m} be its homogeneous expansion. The radial derivative Rf of f is defined by

Rf := ∑_{m=1}^{∞} m f_{m}.

For n ≥ 3, let R̃ := (n/2 − 1)^{−1}R + I, where I is the identity. It is clear that

R̃f = ∑_{m=0}^{∞} \frac{m + n/2 − 1}{n/2 − 1} f_{m}.

Theorem 6.1. Let n ≥ 3. For f, g ∈ b_{−2}^{2},

⟨f, g⟩_{I} = \frac{n/2 − 1}{n/2} ∫_{𝔹} \frac{1 − |x|^{2}}{|x|} R̃f(x)\frac{1 − |x|^{2}}{|x|} \overline{R̃g(x)} dν_{−2}(x)

= \frac{n/2 − 1}{n/2} ∫_{𝔹} \frac{1}{|x|^{2}} R̃f(x)\overline{R̃g(x)} dν(x).

Note that the factor 1/|x|^{2} causes no integrability problem since n ≥ 3.

Proof. Since f ∈ b_{−2}^{2}, the first-order derivative Rf is in b_{0}^{2} by [17, Theorem 1.2]. Also since b_{−2}^{2} ⊂ b_{0}^{2}, f is in b_{0}^{2}. Thus R̃f is in b_{0}^{2} ⊂ L^{2}(dν). Similarly R̃g is in L^{2}(dν) and we can pass to polar coordinates to write

∫_{𝔹} \frac{1}{|x|^{2}} R̃f(x)\overline{R̃g(x)} dν(x) = ∫_{0}^{1} nr^{n−1} \frac{1}{r^{2}} ∫_{𝕊} R̃f(rξ)\overline{R̃g(rξ)} dσ(ξ) dr.

Let f = ∑_{m=0}^{∞} ∑_{j=1}^{δ_{m}} f_{m}^{j}Y_{m}^{j} and g = ∑_{m=0}^{∞} ∑_{j=1}^{δ_{m}} g_{m}^{j}Y_{m}^{j}, where Y_{m}^{j} is as in subsection 3.1. Then

R̃f(x) = ∑_{m=0}^{∞} ∑_{j=1}^{δ_{m}} \frac{m + n/2 − 1}{n/2 − 1}f_{m}^{j}Y_{m}^{j}(x), R̃g(x) = ∑_{m=0}^{∞} ∑_{j=1}^{δ_{m}} \frac{m + n/2 − 1}{n/2 − 1}g_{m}^{j}Y_{m}^{j}(x),

where the series converge uniformly on r𝕊. Using also the orthogonality (15), and the homogeneity of Y_{m}^{j}, we deduce that

∫_{𝔹} \frac{1}{|x|^{2}} R̃f(x)\overline{R̃g(x)} dν(x) = ∑_{m=0}^{∞} ∑_{j=1}^{δ_{m}} (\frac{m + n/2 − 1}{n/2 − 1})^{2} f_{m}^{j}\overline{g_{m}^{j}} ∫_{0}^{1} n r^{2m+n−3} dr

= \frac{n/2}{n/2 − 1} ⟨f, g⟩_{I}.  □

The integral inner product given in Proposition 6.1 is in terms of radial derivatives which are well known. If we want to use the differential operators D_{s}^{t}, then we have other choices. Note that the reproducing kernel of (b_{−2}^{2}, ⟨·, ·⟩_{I}) is R_{−2}(x, y). So [17, Theorem 5.2] provides us with an integral inner product such that R_{−2} is the reproducing kernel. For example, if we follow the proof of [17, Theorem 5.2] and choose t = 1, s_{1} = −1 and s_{2} = −2, then we see that

⟨f, g⟩_{I} = ∫_{𝔹} D_{−1}^{1}f(x)\overline{D_{−2}^{1}g(x)} dν(x).

To verify the above formula, just integrate in polar coordinates and use the homogeneous expansions of the functions. The integral on the right does not look like (conjugate) symmetric in f and g, but it actually is.

References

[1] L. A. Ahlfors, Möbius Transformations in Several Dimensions, Univ. Minnesota, Minneapolis, 1981.

[2] Ö. Akın and H. Leutwiler, On the invariance of the solutions of the Weinstein equation under Möbius transformations, in Classical and Modern Potential Theory and Applications (K. Gowrisankaran et al., eds.) Kluwer, Dordrecht, 1994, pp. 19–29.

[3] J. Arazy and S. D. Fisher, The uniqueness of the Dirichlet space among Möbius-invariant Hilbert spaces, Illinois J. Math. 29 (1985), 449–462.

[4] J. Arazy and S. D. Fisher, Invariant Hilbert spaces of analytic functions on bounded symmetric domains, in Topics in Operator Theory (L. de Branges, I. Gohberg and J. Rovnyak, eds.), Oper. Theory Adv. Appl., vol.48, Birkhäuser, Basel, 1990, pp. 67–91.

[5] J. Arazy, S. D. Fisher and J. Peetre, Möbius invariant function spaces, J. Reine Angew. Math. 363 (1985), 110–145.

[6] S. Axler, P. Bourdon and W. Ramey, Harmonic Function Theory, 2nd ed., Grad. Texts in Math., vol. 137, Springer, New York, 2001.

[7] A. F. Beardon, The Geometry of Discrete Groups, Grad. Texts in Math., vol. 91, Springer, New York, 1983.

[8] J. Bergh and J. Löfström, Interpolation Spaces: An Introduction, Grundlehren Math. Wiss., vol. 223, Springer, Berlin, 1976.

[9] B. R. Choe and K. Nam, Double integral characterizations of harmonic Bergman spaces, J. Math. Anal. Appl. 379 (2011), 889–909.

[10] R. R. Coifman and R. Rochberg, Representation theorems for holomorphic and harmonic functions in L^{p}, Astérisque 77 (1980), 12–66.

[11] R. R. Coifman and G. Weiss, Representations of compact groups and spherical harmonics, Enseign. Math. 14 (1968), 121–173.

[12] A. E. Djrbashian and F. A. Shamoian, Topics in the Theory of A_{α}^{p} Spaces, Teubner Texts Math., vol. 105, Teubner, Leipzig, 1988.

[13] Ö. F. Doğan, Harmonic Besov spaces with small exponents, Complex Var. Elliptic Equ. 65 (2020), 1051–1075.

[14] Ö. F. Doğan and A. E. Üreyen, Weighted harmonic Bloch spaces on the ball, Complex Anal. Oper. Theory 12 (2018), 1143–1177.

[15] Ö. F. Doğan and A. E. Üreyen, Inclusion relations between harmonic Bergman-Besov and weighted Bloch spaces on the unit ball, Czechoslovak Math. J. 69 (2019), 503–523.

[16] S. Gergün, H. T. Kaptanoğlu and A. E. Üreyen, Reproducing kernels for harmonic Besov spaces on the ball, C. R. Math. Acad. Sci. Paris 347 (2009), 735–738.

[17] S. Gergün, H. T. Kaptanoğlu and A. E. Üreyen, Harmonic Besov spaces on the ball, Internat. J. Math. 27 (2016), 1650070, 59 pp.

[18] L. L. Helms, Potential Theory, 2nd ed., Universitext, Springer, London, 2014.

[19] P. Hu, C. Liu, T. Liu and L. Zhou, Weighted composition operators acting on harmonic Hardy spaces, preprint, available at https://arxiv.org/abs/1708.05225

[20] M. Jevtić and M. Pavlović, Harmonic Besov spaces on the unit ball in ℝ^{n}, Rocky Mountain J. Math. 31 (2001), 1305–1316.

[21] H. T. Kaptanoğlu, Möbius-invariant Hilbert spaces in polydics, Pacific J. Math. 163 (1994), 337–360.

[22] H. T. Kaptanoğlu amd A. E. Üreyen, Möbius-invariant harmonic function spaces on the unit disc, preprint.

[23] M. Pavlović, Kelvin-Möbius transform of polyharmonic functions, preprint.

[24] M. M. Peloso, Möbius invariant spaces on the unit ball, Michigan Math. J. 39 (1992), 509– 536.

[25] L. A. Rubel and R. M. Timoney, An extremal property of the Bloch space, Proc. Amer. Math. Soc. 75 (1979), 45–49.

[26] W. Rudin, Function Theory in the Unit Ball of ℂ^{n}, Grundlehren Math. Wiss., vol. 241, Springer, Berlin, 1980.

[27] W. Rudin, Functional Analysis, 2nd ed., McGraw-Hill, New York, 1991.

[28] S. Saitoh, Theory of Reproducing Kernels and its Applications, Pitman Res. Notes Math. Series, Longman, Harlow, 1988.

[29] S. Stević, Harmonic Bloch and Besov spaces on the unit ball, Ars Combin. 91 (2009), 3–9.

[30] M. Stoll, Harmonic and Subharmonic Function Theory on the Hyperbolic Ball, London Math. Soc. Lect. Note Series, vol. 431, Cambridge University, Cambridge, 2016.

[31] R. M. Timoney, Maximal invariant spaces of analytic functions, Indiana Univ. Math. J. 31 (1982), 651–663.

[32] K. H. Zhu, Möbius invariant Hilbert spaces of holomorphic functions in the unit ball of ℂ^{n}, Trans. Amer. Math. Soc. 323 (1991), 823–842.

[33] K. H. Zhu, The Möbius invariance of Besov spaces on the unit ball of ℂ^{n}, in Complex Analysis and Applications (Y. Wang et al., eds.), World Scientific, Singapore, 2006, pp. 328-337.

Bi̇lkent Üni̇versi̇tesi̇, Matemati̇k Bölümü, 06800 Ankara, Türki̇ye

   Email address: kaptan@fen.bilkent.edu.tr
   URL: http://www.fen.bilkent.edu.tr/∼kaptan/

Eski̇şehi̇r Tekni̇k Üni̇versi̇tesi̇, Fen Fakültesi̇, Matemati̇k Bölümü, Yunusemre Kampüsü 26470 Eski̇şehi̇r, Türki̇ye

   Email address: aeureyen@eskisehir.edu.tr