MÖBIUS-INVARIANT HARMONIC FUNCTION SPACES ON THE UNIT DISC

H. TURGAY KAPTANOĞLU AND A. ERSİN ÜREYEN

Abstract. We investigate and identify Möbius-invariant harmonic function spaces on the unit disc. We derive their fundamental properties and establish connections among various topologies on them. We show that the harmonic Bloch space b^{∞} is the “maximal” and the Besov space b_{−2}^{1} is the minimal invariant complete semi-normed space. There is only one invariant semi-Hilbert space and it is the harmonic Dirichlet space b_{−2}^{2}.

1. Introduction

Let 𝔻 ⊂ ℂ be the unit disc and let Aut(𝔻), the automorphism group, be the group of holomorphic, bijective maps of 𝔻. Let (E, ρ_{E}) be a complete semi-normed linear space of holomorphic functions on 𝔻. Roughly speaking, (E, ρ_{E}) is called Möbius invariant if for all f ∈ E and φ ∈ Aut(𝔻), the composition f ∘ φ is in E and ρ_{E}(f ∘ φ) ∼ ρ_{E}(f). Möbius-invariant holomorphic function spaces have been studied intensively in the past. It is shown in [11] that the Bloch space B^{∞} is “maximal”, and in [2] that the Besov space B^{1} is the minimal invariant space. In [1] invariant semi-Hilbert spaces are considered and it is shown that the Dirichlet space is the unique Möbius-invariant semi-Hilbert space. In higher dimensions analogous results are obtained for the unit ball in ℂ^{n} in [10, 15], and for the polydisc in [8].

Consider now the harmonic analogue of the above problem. A Möbius transformation of ℝ̂^{n} := ℝ^{n} ∪ {∞} is a finite composition of reflections (inversions) in planes and spheres. Let 𝔹_{n} ⊂ ℝ^{n} be the real unit ball and let M(𝔹_{n}) be the group of Möbius transformations that map 𝔹_{n} to itself. If u is harmonic on 𝔹_{n} and φ ∈ M(𝔹_{n}), the composition u ∘ φ need not be harmonic. Nevertheless, if we multiply u ∘ φ with a correction factor, the function

K_{φ}(u)(x) = (\frac{1 − |φ(x)|^{2}}{1 − |x|^{2}} )^{(n−2)/2} u(φ(x)) (x ∈ 𝔹_{n})

is harmonic on 𝔹_{n}. The first factor above stems from applying the Kelvin transform and because of this K_{φ}(u) is called the Kelvin-Möbius transform of u; for details see [9].

Note that when n = 2, the correction factor in the Kelvin-Möbius transform disappears and K_{φ}(u) is just u ∘ φ. Because of this there are differences between the cases n = 2 and n ≥ 3. For example, when n ≥ 3, there are normed spaces containing unbounded functions that are Kelvin-Möbius invariant, but this is not true when n = 2 and to obtain interesting invariant spaces it is necessary to allow semi-norms. Due to this fact even the definitions of invariant spaces are different in the cases n = 2 and n ≥ 3. The case n ≥ 3 is worked out in [9], where the minimal and the maximal invariant spaces are determined and the unique invariant Hilbert space is identified. The purpose of this work is to settle the remaining case n = 2.

Date: 21 July 2021.

2010 Mathematics Subject Classification. Primary 31A05; Secondary 46E15, 46E20, 47A15.

Key words and phrases. Möbius transformation, Möbius-invariant harmonic function space, harmonic Bergman-Besov space, harmonic Bloch space.

Our results are similar to those in the holomorphic setting given, for example, in [1, 2], which is to be expected since harmonic and holomorphic functions on the unit disc are intimately related. A close look at works on holomorphic spaces reveals that connections between convergence in a semi-norm and pointwise convergence play essential roles in the proofs, although not always explicitly stated; for a careful treatment of these issues, see the recent monograph [6, Chapter 6]. We will start with slightly different but basic assumptions when defining Möbius-invariant harmonic spaces. More importantly, we will be meticulous in establishing the relations among various kinds of topologies, semi-norm, norm, pointwise convergence, and make clear how and where the conditions on each are used in the proofs.

We endow ℝ^{2} with the usual inner product x · y = x_{1}y_{1} + x_{2}y_{2} for x = (x_{1}, x_{2}) and y = (y_{1}, y_{2}) in ℝ^{2} and with the induced norm |x| = \sqrt{x · x}. We identify ℝ^{2} with ℂ via the correspondence z = x_{1} +ix_{2} and freely pass between the real and complex notation. In the rest of this work we always have n = 2, and in real notation we denote the unit disc by 𝔹 = 𝔹_{2} and identify it with 𝔻.

Let E be a linear space. A map ρ_{E} : E → [0, ∞) is called a semi-norm on E if it satisfies the axioms of a norm except that ρ_{E}(u) can be zero for non-zero u ∈ E. A map ⟨·, ·⟩_{E} : E × E → ℂ is called a semi-inner product on E if it satisfies the axioms of an inner product except that ⟨u, u⟩_{E} can be zero for non-zero u ∈ E. The natural topology of a semi-inner product space is with respect to the induced seminorm ρ_{E}(u) = \sqrt{⟨u, u⟩_{E}}. The semi-normed space (E, ρ_{E}) is called complete if every Cauchy sequence in ρ_{E} is convergent in ρ_{E}. The pair (E, ⟨·, ·⟩_{E}) is called a semi-Hilbert space if it is complete with respect to its induced semi-norm. Equivalently, if (E, ρ_{E}) is a semi-normed space, we can let Z = { u ∈ E : ρ_{E}(u) = 0 } and consider the quotient space Ĕ := E/Z and the corresponding ρ̆Ĕ on it. Then (Ĕ, ρ̆Ĕ) is a genuine normed space and is a Banach space if and only if (E, ρ_{E}) is complete. Similarly, (E, ⟨·, ·⟩_{E}) is a semi-Hilbert space if and only if (Ĕ, ⟨·, ·⟩Ĕ) is a Hilbert space. For more information, see [14, Section 1.2].

We now define a Möbius-invariant harmonic function space following [11] and [2]. Denote by h(𝔹) the space of all complex-valued harmonic functions on 𝔹 endowed with the topology of uniform convergence on compact subsets. Let (E, ρ_{E}) be a semi-normed space of harmonic functions on 𝔹. A non-zero continuous linear functional L on h(𝔹) is called decent on E if L|_{E} is also continuous in the seminorm topology. We denote by 1 the constant function whose value is 1. If f : 𝔻 → ℂ and θ ∈ [0, 2π], we define f_{θ} by f_{θ}(z) := f(e^{iθ}z) for z ∈ 𝔻.

Definition 1.1. Let E be a linear space of harmonic functions on 𝔹 and ρ_{E} be a complete semi-norm on E. We call (E, ρ_{E}) Möbius invariant if the following properties hold:

(i) 1 ∈ E and ρ_{E}(1) = 0. We also assume E is non-trivial in the sense that it contains a non-constant function.

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

(iii) For every u ∈ E and φ ∈ M(𝔹), u ∘ φ belongs to E and

ρ_{E}(u ∘ φ) ≤ Cρ_{E}(u),

for some constant C independent of u and φ. (iv) For each u ∈ E, the map θ ↦ u_{θ} is continuous from [0, 2π] to (E, ρ_{E}). If the constant in (iii) is C = 1, then we have ρ_{E}(u ∘ φ) = ρ_{E}(u) for every u ∈ E and φ ∈ M(𝔹) and in this case we call (E, ρ_{E}) strictly Möbius invariant.

If (E, ρ_{E}) is Möbius invariant, then ρ_{E}(u ∘ φ) ∼ ρ_{E}(u) for all u ∈ E. Here, the notation A ∼ B means that |A|/|B| is bounded above and below by positive constants that are independent of the parameters or functions involved. If |A|/|B| is bounded above, we write A ≲ B.

If (E, ρ_{E}) is Möbius invariant, then it is possible to define a new, equivalent semi-norm on E with which E becomes strictly Möbius invariant; see [2, p. 111].

The condition (i) in Definition 1.1 does not appear in [1, 2]. We use it in Lemma 5.3 and Lemma 6.2 below where we connect pointwise convergence and convergence in (E, ρ_{E}). Condition (iv) is needed in Proposition 5.5 which is the basis of the properties obtained at the end of Section 5.

The harmonic function spaces we are interested in are the Bergman-Besov spaces b_{q}^{p} (q ∈ ℝ) and the Bloch space b^{∞}. The harmonic Bloch space b^{∞} consists of all u ∈ h(𝔹) such that the semi-norm

(1)  ρ_{b^{∞}}(u) := sup_{x∈𝔹} (1 − |x|^{2})|∇u(x)| < ∞,

where ∇u(x) = (u_{x_{1}}, u_{x_{2}}) is the gradient of u and u_{x_{i}} = ∂u/∂x_{i}, i = 1, 2. The space b^{∞} satisfies the conditions (i)–(iii) of Definition 1.1, but not the condition (iv), which can be seen by taking u(z) = log(1 − z). Therefore b^{∞} is not Möbius invariant in the sense of Definition 1.1. Nevertheless, it has the following maximality property which is the harmonic analogue of the main result of [11].

Theorem 1.2. Let (E, ρ_{E}) be a Möbius-invariant harmonic function space. Then E ⊂ b^{∞} and there is a constant C > 0 such that ρ_{b^{∞}}(u) ≤ Cρ_{E}(u) for every u ∈ E.

Remark 1.3. The conditions (i) and (iv) of Definition 1.1 are actually not needed in the above theorem. That is, if (E, ρ_{E}) satisfies conditions (ii) and (iii) of Definition 1.1, then E is continuously contained in b^{∞}.

We next define harmonic Bergman-Besov spaces for the whole range q ∈ ℝ. For this we need some preparation. Let dA be the normalized area measure on 𝔹 and for q ∈ ℝ, let

dA_{q}(x) := c_{q}(1 − |x|^{2})^{q} dA(x).

The measure dA_{q} is finite only when q > −1 and in this case we set c_{q} so that A_{q}(𝔹) = 1. When q ≤ −1, we set c_{q} = 1. The case q = −2 is especially important for us since dA_{−2}(x) = (1 − |x|^{2})^{−2} dA(x) is the Möbius-invariant measure on 𝔹. For 1 ≤ p < ∞, we denote the Lebesgue classes with respect to dA_{q} by L^{p}(dA_{q}). To denote higher-order partial derivatives we use multi-indices, where a multi-index α = (α_{1}, α_{2}) is a pair of non-negative integers and |α| := α_{1} + α_{2}. For a smooth function f we write

∂^{α}f := \frac{∂^{|α|}f}{∂x_{1}^{α_{1}}∂x_{2}^{α_{2}}} .

So by definition, a function u is harmonic if (∂^{(2,0)} + ∂^{(0,2)})u = 0.

Definition 1.4. Let 1 ≤ p < ∞ and q ∈ ℝ. Pick a non-negative integer N so that

(2)  q + pN > −1.

The harmonic Bergman-Besov space b_{q}^{p} consists of all u ∈ h(𝔹) such that for every multi-index α with |α| = N,

(1 − |x|^{2})^{N}∂^{α}u ∈ L^{p}(dA_{q}).

A norm on b_{q}^{p} is

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

The space b_{q}^{p} does not depend on the choice of N as long as (2) is satisfied. The norm ∥ · ∥_{b_{q}^{p}} in (3) depends on N but we do not indicate this since different choices of N give rise to equivalent norms; see [7] for details. In the region q > −1, choosing N = 0 shows b_{q}^{p} = h(𝔹) ∩ L^{p}(dA_{q}), the weighted harmonic Bergman space. The region q ≤ −1 is the Besov region, where we need to consider derivatives of u. We also endow b_{q}^{p} with the semi-norm

ρ_{b_{q}^{p}}(u) := ∥u − u(0)∥_{b_{q}^{p}}

which vanishes on constants. The next theorem determines all Möbius-invariant harmonic Bergman-Besov spaces.

Theorem 1.5. Let 1 ≤ p < ∞ and q ∈ ℝ. Then (b_{q}^{p}, ρ_{b_{q}^{p}}) is Möbius invariant if and only if q = −2.

In particular, there are no Möbius-invariant standard weighted Bergman spaces. In all papers dealing with holomorphic spaces, no such only-if-type result is proved.

Taking p = 2 in Theorem 1.5 shows that the space b_{−2}^{2} is a Möbius-invariant semi-Hilbert space. It is the harmonic Dirichlet space endowed with the semi-norm (choosing N = 1)

(4)  ρ_{b_{−2}^{2}}^{2}(u) = ∫_{𝔹} |∇u(x)|^{2} dA(x)

which is induced by the semi-inner product

⟨u, v⟩_{b_{−2}^{2}} = ∫_{𝔹} ∇u(x) · \overline{∇v(x)} dA(x).

The Dirichlet space (b_{−2}^{2}, ⟨·, ·⟩_{b_{−2}^{2}}) is actually strictly Möbius invariant and it is the only strictly Möbius-invariant semi-Hilbert space. This is our next theorem.

Theorem 1.6. (i) The semi-Hilbert space (b_{−2}^{2}, ⟨·, ·⟩_{b_{−2}^{2}}) is strictly Möbius in-

variant. (ii) If (H, ⟨·, ·⟩_{H}) is a strictly Möbius-invariant semi-Hilbert space, then H = b_{−2}^{2}

and ⟨·, ·⟩_{H} = C⟨·, ·⟩_{b_{−2}^{2}} for some C > 0.

The Möbius-invariant spaces b_{−2}^{p} (1 ≤ p < ∞) grow larger as p increases; see [7, Proposition 13.3]. This suggests that b_{−2}^{1} might be the smallest Möbius-invariant space. The next theorem shows that this is true.

Theorem 1.7. The space b_{−2}^{1} is the smallest Möbius-invariant space. More precisely, if (E, ρ_{E}) is Möbius invariant, then b_{−2}^{1} ⊂ E and there exists a C > 0 such that ρ_{E}(u) ≤ Cρ_{b_{−2}^{1}}(u) for every u ∈ b_{−2}^{1}.

Finally, we consider subspaces of the Fréchet space h(𝔹) and show that the only Möbius-invariant closed subspaces are the trivial ones.

Theorem 1.8. Let A be a closed subspace of h(𝔹). If u ∘ φ is in A for every u ∈ A and φ ∈ M(𝔹), then A = {0}, A = {constants} or A = h(𝔹).

The proofs of Theorems 1.2 and 1.8 are in Section 3. The proof of Theorem 1.5 is in Section 4, of Theorem 1.6 in Section 6, and of Theorem 1.7 in Section 7. In Section 5, we collect together general properties of Möbius-invariant harmonic function spaces not covered by the above theorems. In Section 2, we provide background on Möbius transformations, harmonic functions, and Bergman-Besov spaces.

2. Preliminaries

In this section we first recall some elementary facts about Möbius transformations and harmonic functions on the unit disc. We then review some properties of harmonic and holomorphic Bergman-Besov spaces that we use in the sequel.

2.1. Möbius Transformations. For a ∈ 𝔻, let

(5)  φ_{a}(z) := \frac{a − z}{1 − āz}  (z ∈ 𝔻).

The group Aut(𝔻) is generated by φ_{a}’s and rotations; if φ ∈ Aut(𝔻), then φ = e^{iθ}φ_{a} for some θ ∈ [0, 2π] and a ∈ 𝔻. Elements of Aut(𝔻) are holomorphic, so they are orientation preserving. On the other hand, M(𝔹) consists of compositions of reflections in lines and circles and its elements can be orientation preserving or reversing. Throughout this work we reserve the letter γ for the conjugation function

γ(z) = z̄.

The group M(𝔹) is generated by φ_{a}’s, rotations and γ. If φ ∈ M(𝔹) is a composition of an even number of reflections, then φ is holomorphic and φ ∈ Aut(𝔻). If φ ∈ M(𝔹) is a composition of an odd number of reflections, then φ̄ is holomorphic and φ = γ ∘ ψ for some ψ ∈ Aut(𝔻). If we write \overline{Aut}(𝔻) := \{γ ∘ φ : φ ∈ Aut(𝔻)\}, then

(6)  M(𝔹) = Aut(𝔻) ⋃ \overline{Aut}(𝔻).

We record the following elementary lemma for future reference.

Lemma 2.1. Let φ ∈ Aut(𝔻). Then there exists φ̂ ∈ Aut(𝔻) such that

γ ∘ φ = φ̂ ∘ γ.

Proof. If φ = e^{iθ}φ_{a}, take φ̂ = e^{−iθ} ∘ φā. □

2.2. Harmonic Functions on the Unit Disc. A harmonic function u on 𝔻 is the sum of a holomorphic and a conjugate holomorphic function in the form

\begin{aligned}u(z) & = ∑_{m=0}^{∞} a_{m}z^{m} + ∑_{m=1}^{∞} b_{m}z̄^{m}, \\ & =: f(z) + g(z̄) = f(z) + g ∘ γ(z), \end{aligned} (7)

where the series converge absolutely and uniformly on compact subsets of 𝔻, and f and g are holomorphic on 𝔻. We use this form many times and state it as a lemma.

Lemma 2.2. If u is harmonic on 𝔻, then there exist functions f and g holomorphic on 𝔻 such that u = f + g ∘ γ. The functions f and g are unique up to a constant summand.

As stated in Definition 1.1, for Möbius-invariant spaces we use semi-norms that vanish on constants. So when we refer to Lemma 2.2, the constant summand will have no effect and it will not be important which f or g we use.

Let

(8)  \frac{∂}{∂z} := \frac{1}{2} (\frac{∂}{∂x_{1}} − i\frac{∂}{∂x_{2}} )  and  \frac{∂}{∂z̄} := \frac{1}{2} (\frac{∂}{∂x_{1}} + i\frac{∂}{∂x_{2}} )

be the Wirtinger operators. Then

(9)  \frac{∂}{∂x_{1}} = \frac{∂}{∂z} + \frac{∂}{∂z̄}  and  \frac{∂}{∂x_{2}} = i (\frac{∂}{∂z} − \frac{∂}{∂z̄}) .

If u is harmonic on 𝔻 and f and g are as in Lemma 2.2, then

(10)  \frac{∂u}{∂z} = f′(z)  and  \frac{∂u}{∂z̄} = g′(z̄).

Also a harmonic function composed on the right with a holomorphic or conjugate holomorphic function remains harmonic.

A continuous linear functional L on the Fréchet space h(𝔹) is continuous if and only if there exists a compact F ⊂ 𝔹 and a constant C > 0 such that

(11)  |L(u)| ≤ C sup{|u(x)| : x ∈ F }  (u ∈ h(𝔹)).

If (E, ρ_{E}) is a semi-normed space of harmonic functions and L is a linear functional on E, then L is continuous if and only if there exists a constant C > 0 such that

(12)  |L(u)| ≤ Cρ_{E}(u)  (u ∈ E).

2.3. Holomorphic and Harmonic Bergman-Besov Spaces. Let H(𝔻) be the set of all holomorphic functions on 𝔻. Definition of holomorphic Bergman-Besov spaces B_{q}^{p} (q ∈ ℝ) is similar to that of harmonic ones.

Definition 2.3. Let 1 ≤ p < ∞ and q ∈ ℝ. Pick a non-negative integer N so that q + pN > −1. The holomorphic Bergman-Besov space B_{q}^{p} consists of all f ∈ H(𝔻) such that (1 − |z|^{2})^{N}f^{(N)} ∈ L^{p}(dA_{q}). A norm on B_{q}^{p} is

∥f∥_{B_{q}^{p}}^{p} = ∑_{k=0}^{N−1} |f^{(k)}(0)|^{p} + ∫_{𝔻} |(1 − |z|^{2})^{N}f^{(N)}(z)|^{p} dA_{q}(z).

Again, the space B_{q}^{p} does not depend on the choice of N as long as q + pN > −1 and different choices of N give rise to equivalent norms. For q > −1, choosing N = 0 shows B_{q}^{p} = H(𝔻) ∩ L^{p}(dA_{q}) is the holomorphic weighted Bergman space. We endow B_{q}^{p} also with the semi-norm ρ_{B_{q}^{p}}(f) := ∥f − f(0)∥_{B_{q}^{p}}.

The following lemma shows how taking derivatives moves a function from one Bergman-Besov space to another in both holomorphic and harmonic cases.

Lemma 2.4. (i) Suppose f ∈ H(𝔻) and m ≥ 1. Then f ∈ B_{q}^{p} if and only if f^{(m)} ∈ B_{q+pm}^{p} and

∥f∥_{B_{q}^{p}} ∼ ∑_{k=0}^{m−1} |f^{(k)}(0)| + ∥f^{(m)}∥_{B_{q+pm}^{p}}.

(ii) Suppose u ∈ h(𝔹) and m ≥ 1. Then u ∈ b_{q}^{p} if and only if ∂^{α}u ∈ b_{q+pm}^{p} for every multi-index α with |α| = m and

∥u∥_{b_{q}^{p}} ∼ ∑_{|α|<m} |∂^{α}u(0)| + ∑_{|α|=m} ∥∂^{α}u∥_{b_{q+pm}^{p}}.

In Definition 1.4 (and similarly in Definition 2.3), partial derivatives can be replaced with the more convenient radial derivatives. If f ∈ H(𝔻), then the radial derivative Rf of f is defined as Rf(z) := zf′(z). So, if f(z) = ∑_{m=0}^{∞} c_{m}z^{m}, then

Rf(z) = ∑_{m=1}^{∞} mc_{m}z^{m}.

For k ≥ 2, we define R^{k}f := R^{k−1}Rf = ∑_{m=1}^{∞} m^{k}c_{m}z^{m}. If u ∈ h(𝔹), then the radial derivative R_{h}u of u is defined as R_{h}u(x) := x · ∇u(x). If u is written as in (7), then

R_{h}u(z) = ∑_{m=1}^{∞} m (a_{m}z^{m} + b_{m}z̄^{m}) .

For k ≥ 2, we define R_{h}^{k}u := R_{h}^{k−1}R_{h}u = ∑_{m=1}^{∞} m^{k}(a_{m}z^{m} + b_{m}z̄^{m}).

Lemma 2.5. Let p ≥ 1 and q ∈ ℝ. Pick N ≥ 1 so that q + pN > −1.

(i) f ∈ B_{q}^{p} if and only if (1 − |z|^{2})^{N}R^{N}f ∈ L^{p}(dA_{q}) and

∥f∥_{B_{q}^{p}}^{p} ∼ |f(0)|^{p} + ∫_{𝔻} |(1 − |z|^{2})^{N}R^{N}f(z)|^{p} dA_{q}(z).

(ii) u ∈ b_{q}^{p} if and only if (1 − |x|^{2})^{N}R_{h}^{N}u ∈ L^{p}(dA_{q}) and

∥u∥_{b_{q}^{p}}^{p} ∼ |u(0)|^{p} + ∫_{𝔹} |(1 − |x|^{2})^{N}R_{h}^{N}u(x)|^{p} dA_{q}(x).

Parts (i) of Lemmas 2.4 and 2.5 are well known. For their parts (ii), see [7]. When q > −1, it is clear that B_{q}^{p} ⊂ b_{q}^{p}. This continues to hold when q ≤ −1.

Lemma 2.6. Let 1 ≤ p < ∞, q ∈ ℝ and f be holomorphic on 𝔻. Then

∥f∥_{B_{q}^{p}} ∼ ∥f∥_{b_{q}^{p}} ∼ ∥f ∘ γ∥_{b_{q}^{p}}

and

ρ_{B_{q}^{p}}(f) ∼ ρ_{b_{q}^{p}}(f) ∼ ρ_{b_{q}^{p}}(f ∘ γ).

Proof. If q > −1, since the measure dA_{q} is invariant under conjugation,

∥f ∘ γ∥_{b_{q}^{p}} = ∫_{𝔻} |f(z̄)|^{p} dA_{q}(z) = ∫_{𝔻} |f(z)|^{p} dA_{q}(z) = ∥f∥_{b_{q}^{p}} = ∥f∥_{B_{q}^{p}}.

If q ≤ −1, pick N ≥ 1 so that q + pN > −1. Note that since f is holomorphic, Rf = R_{h}f. Thus, using both parts of Lemma 2.5,

∥f∥_{b_{q}^{p}} ∼ |f(0)|^{p} + ∫_{𝔹} |(1 − |x|^{2})^{N}R_{h}^{N}f(x)|^{p} dA_{q}(x)
= |f(0)|^{p} + ∫_{𝔻} |(1 − |z|^{2})^{N}R^{N}f(z)|^{p} dA_{q}(z) ∼ ∥f∥_{B_{q}^{p}}.

The equivalence ∥f ∘ γ∥_{b_{q}^{p}} ∼ ∥f∥_{b_{q}^{p}} follows from the fact that R_{h}^{N}(f ∘ γ) = R_{h}^{N}f ∘ γ and the conjugation invariance of dA_{q}. For the assertion about semi-norms just replace f with f − f(0). □

For a proof of the following estimate see [12, Proposition 1.4.10].

Lemma 2.7. For z ∈ 𝔻, t > −1 and c ∈ ℝ, we have

∫_{𝔻} \frac{(1 − |w|^{2})^{t}}{|1 − zw̄|^{2+t+c}} dA(w) ∼ \begin{cases}\frac{1}{(1 − |z|^{2})^{c}} & if c > 0, \\ 1 + log \frac{1}{1 − |z|^{2}} & if c = 0, \\ 1 & if c < 0. \end{cases}

3. The Maximal Space

In this section we prove Theorems 1.2 and 1.8 which are the harmonic counterparts of the Theorem and Corollary 2 of [11]. In the holomorphic case the proof of the Rubel-Timoney theorem is based on the lemma below. We denote the holomorphic Bloch space by B^{∞} which consists of all f ∈ H(𝔻) such that the semi-norm

ρ_{B^{∞}}(f) := sup_{z∈𝔻} (1 − |z|^{2})|f′(z)| < ∞.

Lemma 3.1 ([11], p. 46). Let f ∈ H(𝔻) and m ≥ 1. If

sup_{φ∈Aut(D)} |(f ∘ φ)^{(m)}(0)| < ∞,

then f ∈ B^{∞} and

ρ_{B^{∞}}(f) ≲ sup_{φ∈Aut(D)} |(f ∘ φ)^{(m)}(0)|.

Our first aim is to obtain an analogue of the above lemma for harmonic functions. As a preliminary result, we first show the following simple connection between the harmonic Bloch space b^{∞} and the holomorphic Bloch space B^{∞}.

Lemma 3.2. Let u be harmonic on 𝔻 and f and g be as in Lemma 2.2. Then u ∈ b^{∞} if and only if f, g ∈ B^{∞} and

ρ_{b^{∞}}(u) ∼ ρ_{B^{∞}}(f) + ρ_{B^{∞}}(g).

Proof. It is clear that ρ_{b^{∞}}(u) defined in (1) is equivalent to

sup_{z∈𝔻} (1 − |z|^{2}) (|u_{x_{1}}(z)| + |u_{x_{2}}(z)|)

which, by (8) and (9), is equivalent to

sup_{z∈𝔻} (1 − |z|^{2}) (|\frac{∂u}{∂z}(z)| + |\frac{∂u}{∂z̄}(z)|) .

Finally, note that by (10), the above quantity is equivalent to ρ_{B^{∞}}(f)+ρ_{B^{∞}}(g). □

Lemma 3.3. Let u ∈ h(𝔹) and m ≥ 1. If

sup_{φ∈M(𝔹)} |\frac{∂^{m}(u ∘ φ)}{∂z^{m}}(0)| < ∞,

then u ∈ b^{∞} and

ρ_{b^{∞}}(u) ≲ sup_{φ∈M(𝔹)} |\frac{∂^{m}(u ∘ φ)}{∂z^{m}}(0)| .

Proof. Write u = f + g ∘ γ as in Lemma 2.2. We first deal with f. Let φ ∈ Aut(𝔻). By Lemma 2.1, there exists φ̂ ∈ Aut(𝔻) such that

u ∘ φ = f ∘ φ + g ∘ γ ∘ φ = f ∘ φ + g ∘ φ̂ ∘ γ.

Since the second function g ∘ φ̂ ∘ γ is conjugate holomorphic we have

\frac{∂^{m}(u ∘ φ)}{∂z^{m}} = \frac{d^{m}(f ∘ φ)}{dz^{m}}.

Then by hypothesis and the fact that Aut(𝔻) ⊂ M(𝔹),

sup_{φ∈Aut(𝔻)} |(f ∘ φ)^{(m)}(0)| ≤ sup_{φ∈M(𝔹)} |\frac{∂^{m}(u ∘ φ)}{∂z^{m}}(0)| < ∞.

Lemma 3.1 shows that f ∈ B^{∞} and

ρ_{B^{∞}}(f) ≲ sup_{φ∈M(𝔹)} |\frac{∂^{m}(u ∘ φ)}{∂z^{m}}(0)| .

To deal with g, let φ ∈ Aut(𝔻). Then ψ = γ ∘ φ ∈ M(𝔹). By Lemma 2.1, γ ∘ φ = φ̂ ∘ γ for some φ̂ ∈ Aut(𝔻) and u ∘ ψ = f ∘ φ̂ ∘ γ + g ∘ φ. Since the first function f ∘ φ̂ ∘ γ is conjugate holomorphic,

\frac{∂^{m}(u ∘ ψ)}{∂z^{m}} = \frac{d^{m}(g ∘ φ)}{dz^{m}}

and

sup_{φ∈Aut(𝔻)} |(g ∘ φ)^{(m)}(0)| ≤ sup_{ψ∈M(𝔹)} |\frac{∂^{m}(u ∘ ψ)}{∂z^{m}}(0)| < ∞.

Lemma 3.1 shows that g ∈ B^{∞} and

ρ_{B^{∞}}(g) ≲ sup_{ψ∈M(𝔹)} |\frac{∂^{m}(u ∘ ψ)}{∂z^{m}}(0)| .

The desired result now follows from Lemma 3.2. □

We follow the approach of [13] and obtain Theorems 1.2 and 1.8 as consequences of the following theorem.

Theorem 3.4. Let L be a non-zero continuous linear functional on h(𝔹). If

K = sup{|L(u ∘ φ)| : φ ∈ M(𝔹)} < ∞

for some u ∈ h(𝔹), then u ∈ b^{∞} and ρ_{b^{∞}}(u) ≤ C_{L}K, where the constant C_{L} depends on L but is independent of u.

Proof. Since L ≠ 0 on h(𝔻), it cannot vanish on the whole set {1, z^{m}, z̄^{m} : m ≥ 1}. We can assume that there exists m_{0} ≥ 0 such that L(z^{m_{0}}) ≠ 0 since, if necessary, we can replace L with L̃, where L̃(v) = L(v ∘ γ).

Define a new linear functional L on h(𝔻) by

L(v) := \frac{1}{2π} ∫_{0}^{2π} L(v_{θ})e^{−im_{0}θ} dθ.

Using condition (11) it is straightforward to show that L is continuous on h(𝔻). It is also elementary to verify that the following properties hold

\begin{aligned}L(z^{m_{0}}) = L(z^{m_{0}}) & ≠ 0; \\ L(z^{m}) & = 0,  m ≥ 0, m ≠ m_{0}; \\ L(z̄^{m}) & = 0,  m ≥ 1. \end{aligned} (13)

Suppose the hypothesis of the theorem holds for u. Then

(14)  |L(u ∘ φ)| ≤ K  (φ ∈ M(𝔹)).

Write u as a series as in (7). By the uniform convergence of this series on compact subsets of 𝔻, the continuity of L on h(𝔻), and (13),

L(u) = ∑_{m=0}^{∞} a_{m}L(z^{m}) + ∑_{m=1}^{∞} b_{m}L(z̄^{m}) = a_{m_{0}}L(z^{m_{0}}) = a_{m_{0}}L(z^{m_{0}})

and since

a_{m_{0}} = \frac{1}{m_{0}!} \frac{∂^{m_{0}}u}{∂z^{m_{0}}}(0),

we deduce

|\frac{∂^{m_{0}}u}{∂z^{m_{0}}}(0)| ≤ \frac{m_{0}!}{|L(z^{m_{0}})|}|L(u)|.

Replacing u by u ∘ φ and using (14) we obtain

|\frac{∂^{m_{0}}(u ∘ φ)}{∂z^{m_{0}}}(0)| ≤ \frac{m_{0}!}{|L(z^{m_{0}})|}K

for all φ ∈ M(𝔹). If m_{0} ≥ 1, then by Lemma 3.3, u ∈ b^{∞} and ρ_{b^{∞}}(u) ≤ C_{L}K. If m_{0} = 0, then u is a bounded harmonic function and an application of Cauchy’s estimate for harmonic functions ([3, p. 33]) shows u ∈ b^{∞} and ρ_{b^{∞}}(u) ≤ C_{L}K. □

Proof of Theorem 1.2. Let L be a decent linear functional on E. So L ≠ 0 is continuous on h(𝔹) and (12) holds. Pick u ∈ E. For every φ ∈ M(𝔹), by (12) and the Möbius invariance of E,

|L(u ∘ φ)| ≲ ρ_{E}(u ∘ φ) ≲ ρ_{E}(u).

Theorem 3.4 implies u ∈ b^{∞} and ρ_{b^{∞}}(u) ≲ ρ_{E}(u). □

Proof of Theorem 1.8. Suppose A ≠ h(𝔹). By the Hahn-Banach theorem, there exists a non-zero continuous linear functional L on h(𝔹) which vanishes on A. If u ∈ A, then

sup{|L(u ∘ φ)| : φ ∈ M(𝔹)} = 0,

since u ∘ φ ∈ A for every φ ∈ M(𝔹). Theorem 3.4 shows that ρ_{b^{∞}}(u) = 0 and so u must be constant. □

4. Möbius-Invariant Harmonic Bergman-Besov Spaces

In this section we prove Theorem 1.5 and thereby determine all Möbius-invariant harmonic Bergman-Besov spaces. In the holomorphic case, when 1 ≤ p < ∞ and q = −2, the spaces B_{−2}^{p} are Möbius invariant. This is clear when 1 < p < ∞, since if we choose N = 1 in Definition 2.3,

ρ_{B_{−2}^{p}}(f) = ∫_{𝔻} |f′(z)|^{p} (1 − |z|^{2})^{p−2} dA(z)

and for φ ∈ Aut(𝔻),

(15)  ρ_{B_{−2}^{p}}(f ∘ φ) = ρ_{B_{−2}^{p}}(f)

by the identity

|φ′(z)| = \frac{1 − |φ(z)|^{2}}{1 − |z|^{2}} .

If p = 1, choosing N = 2 in Definition 2.3 shows

ρ_{B_{−2}^{1}}(f) = |f′(0)| + ∫_{𝔻} |f′′(z)| dA(z),

and it is shown in [2, Theorem 8] that ρ_{B_{−2}^{1}}(f ∘ φ) ∼ ρ_{B_{−2}^{1}}(f) for φ ∈ Aut(𝔻).

In the proof of Theorem 1.5 we will use the Möbius invariance of B_{−2}^{p} and to do this we first establish a connection between holomorphic Bergman-Besov spaces B_{q}^{p} and harmonic Bergman-Besov spaces b_{q}^{p}.

Lemma 4.1. Let 1 ≤ p < ∞ and q ∈ ℝ. Let u ∈ h(𝔹) be written in the form u = f + g ∘ γ as in Lemma 2.2. Then u ∈ b_{q}^{p} if and only if f, g ∈ B_{q}^{p} and

ρ_{b_{q}^{p}}(u) ∼ ρ_{B_{q}^{p}}(f) + ρ_{B_{q}^{p}}(g).

Proof. Suppose f, g ∈ B_{q}^{p}. Then

ρ_{b_{q}^{p}}(u) = ρ_{b_{q}^{p}}(f + g ∘ γ) ≤ ρ_{b_{q}^{p}}(f) + ρ_{b_{q}^{p}}(g ∘ γ) ∼ ρ_{B_{q}^{p}}(f) + ρ_{B_{q}^{p}}(g),

where the last equivalence follows from Lemma 2.6. Thus u ∈ b_{q}^{p} and the estimate ρ_{b_{q}^{p}}(u) ≲ ρ_{B_{q}^{p}}(f) + ρ_{B_{q}^{p}}(g) holds.

To see the other direction, let u ∈ b_{q}^{p}. By Lemma 2.4 (ii), u_{x_{i}} ∈ b_{q+p}^{p} and

∥u_{x_{i}}∥_{b_{q+p}^{p}} ≲ ∥u − u(0)∥_{b_{q}^{p}} (i = 1, 2).

Then by (8) and (10), ∂u/∂z = f′ and ∂u/∂z̄ = g′ ∘ γ are in b_{q+p}^{p} and

∥f′∥_{b_{q+p}^{p}} ≲ ∥u − u(0)∥_{b_{q}^{p}}  and  ∥g′ ∘ γ∥_{b_{q+p}^{p}} ≲ ∥u − u(0)∥_{b_{q}^{p}}.

Applying Lemma 2.6 shows that f′ and g′ are in the holomorphic space B_{q+p}^{p} and

∥f′∥_{B_{q+p}^{p}} ≲ ∥u − u(0)∥_{b_{q}^{p}}  and  ∥g′∥_{B_{q+p}^{p}} ≲ ∥u − u(0)∥_{b_{q}^{p}}.

Finally, by Lemma 2.4 (i), f and g are in B_{q}^{p} and

∥f − f(0)∥_{B_{q}^{p}} ∼ ∥f′∥_{B_{q+p}^{p}} ≲ ∥u − u(0)∥_{b_{q}^{p}} and ∥g − g(0)∥_{B_{q}^{p}} ≲ ∥u − u(0)∥_{b_{q}^{p}}.

Thus ρ_{B_{q}^{p}}(f) + ρ_{B_{q}^{p}}(g) ≲ ρ_{b_{q}^{p}}(u). □

We are now ready to prove Theorem 1.5.

Proof of Theorem 1.5. To see the if part we check the conditions stated in Definition 1.1. The conditions (i), (ii) and (iv) actually hold for every (b_{q}^{p}, ρ_{b_{q}^{p}}), q ∈ ℝ. It is clear that 1 ∈ b_{q}^{p} and for the decent linear functional one can take, for example, u ↦ u_{x_{1}}(0). Verifying condition (iv) for q > −1 is straightforward. When q ≤ −1, take N so that q + pN > −1. Then R_{h}^{N}u ∈ b_{q+pN}^{p} and condition (iv) follows from the fact that R_{h}^{N}u_{θ} = (R_{h}^{N}u)_{θ}. It remains to verify that condition (iii) holds when q = −2.

Let u ∈ b_{−2}^{p} and write it in the form u = f + g ∘ γ as in Lemma 2.2. By Lemma 4.1, f and g are in B_{−2}^{p}. Let φ ∈ M(𝔹). Then by (6), either φ ∈ Aut(𝔻) or φ ∈ \overline{Aut}(𝔻). If φ ∈ Aut(𝔻), by Lemma 2.1, there exists φ̂ ∈ Aut(𝔻) such that

u ∘ φ = f ∘ φ + g ∘ γ ∘ φ = f ∘ φ + g ∘ φ̂ ∘ γ.

The right-hand side is a representation of u ∘ φ in the form stated in Lemma 2.2 and we have

ρ_{b_{−2}^{p}}(u ∘ φ) ∼ ρ_{B_{−2}^{p}}(f ∘ φ) + ρ_{B_{−2}^{p}}(g ∘ φ̂) ∼ ρ_{B_{−2}^{p}}(f) + ρ_{B_{−2}^{p}}(g) ∼ ρ_{b_{−2}^{p}}(u),

where the first and the last equivalences follow from Lemma 4.1 and the second from Möbius invariance of B_{−2}^{p}. If φ ∈ \overline{Aut}(𝔻), then φ = γ ∘ ψ for some ψ ∈ Aut(𝔻) and exchanging the roles of f and g, we again obtain ρ_{b_{−2}^{p}}(u ∘ φ) ∼ ρ_{b_{−2}^{p}}(u). Thus condition (iii) of Definition 1.1 is satisfied and b_{−2}^{p} is Möbius invariant.

We next show the only if part, that is, if q ≠ −2, then condition (iii) in Definition 1.1 does not hold. For 1/2 < r < 1, let f_{r}(z) := 1 − rz. Then

(16)  ρ_{b_{q}^{p}}(f_{r}) = ρ_{b_{q}^{p}}(1 − rz) = rρ_{b_{q}^{p}}(z) ∼ 1.

With φ_{r} as in (5), we have f_{r} ∘ φ_{r}(z) = (1 − r^{2})/(1 − rz) and by Lemma 2.6,

ρ_{b_{q}^{p}}(f_{r} ∘ φ_{r}) ∼ ρ_{B_{q}^{p}}(f_{r} ∘ φ_{r}) = (1 − r^{2})ρ_{B_{q}^{p}} (\frac{1}{1 − rz} ) .

To compute the right-hand side, pick N ≥ 1 so that q + pN > −1. Since the k^{th} derivative of 1/(1 − rz) at z = 0 is k!r^{k}, by Definition 2.3 we have

ρ_{b_{q}^{p}}^{p}(f_{r} ∘ φ_{r}) ∼ (1 − r^{2})^{p} ( ∑_{k=1}^{N−1}(k!r^{k})^{p} + ∫_{𝔻} \frac{(1 − |z|^{2})^{q+pN}}{|1 − rz|^{p(N+1)}} dA(z)) .

Set M_{r} = ∑_{k=1}^{N−1}(k!r^{k})^{p} and note that it stays bounded as r → 1^{−}. We estimate the above integral using Lemma 2.7 with c = p(N + 1) − (q + pN) − 2 = p − q − 2.

If c > 0, then

ρ_{b_{q}^{p}}^{p}(f_{r} ∘ φ_{r}) ∼ (1 − r^{2})^{p}M_{r} + (1 − r^{2})^{q+2}

and the right-hand side approaches 0 or ∞ as r → 1^{−} when q ≠ −2. If c = 0, then ρ_{b_{q}^{p}}^{p}(f_{r} ∘ φ_{r}) approaches 0 as r → 1^{−} since (1 − r^{2})^{p} dominates log 1/(1 − r^{2}). If c < 0, then ρ_{b_{q}^{p}}^{p}(f_{r} ∘ φ_{r}) approaches 0 as r → 1^{−}.

Thus, by (16), the estimate ρ_{b_{q}^{p}}(f_{r} ∘φ_{r}) ∼ ρ_{b_{q}^{p}}(f_{r}) can not hold when q ≠ −2. □

5. Properties of Möbius-Invariant Spaces

We begin this section by explaining why we allow semi-norms in Definition 1.1. Suppose (E, ∥ · ∥) is a normed space of harmonic functions and point evaluations are bounded on E. If E contains an unbounded function, then E cannot be Möbius invariant. To see this, let u ∈ E and a_{m} ∈ 𝔻 be such that |u(a_{m})| → ∞. Because point evaluation at 0 is bounded we have

|u(a_{m})| = |u ∘ φ_{a_{m}}(0)| ≤ C∥u ∘ φ_{a_{m}}∥.

This shows that ∥u∘φ_{a_{m}}∥ → ∞ and we cannot have ∥u∘φ_{a_{m}}∥ ∼ ∥u∥. Note that the Besov spaces b_{−2}^{p}, except when p = 1, and the Bloch space b^{∞} contain unbounded functions. Therefore to call these spaces Möbius invariant we need to give up requiring norms and allow semi-norms vanishing on constants. The downside of this approach is point evaluations are not bounded on these semi-normed spaces and additional care is needed in dealing with convergence issues.

To establish a connection between pointwise convergence and convergence in a Möbius-invariant space we first introduce a norm on the semi-normed space (E, ρ_{E}) that will make point evaluations bounded.

Lemma 5.1. Let (E, ρ_{E}) be a Möbius-invariant space of harmonic functions. Then (i) ρ_{E}(u) = 0 if and only if u is constant, (ii) for any u ∈ E and any constant C ∈ ℂ, we have ρ_{E}(u + C) = ρ_{E}(u).

Proof. Semi-norm of a constant function is 0 by assumption (i) in Definition 1.1. If u is non-constant, then ρ_{b^{∞}}(u) ≠ 0 and so ρ_{E}(u) ≠ 0 by Theorem 1.2. □

For a Möbius-invariant space (E, ρ_{E}), define

(17)  ∥u∥_{E} := |u(0)| + ρ_{E}(u).

Lemma 5.2. (E, ∥ · ∥_{E}) is a complete normed space.

Proof. That ∥ · ∥_{E} is a norm follows from Lemma 5.1 (i). To see the completeness, let (u_{m}) be a Cauchy sequence in (E, ∥ · ∥_{E}). Then (u_{m}(0)) is Cauchy and (u_{m}) is Cauchy in (E, ρ_{E}). So, there exists a ∈ ℂ such that u_{m}(0) → a and there exists u ∈ E such that ρ_{E}(u_{m} − u) → 0. Let ũ := u − u(0) + a. Then ũ ∈ E since E contains constants and u_{m} → ũ in (E, ∥ · ∥_{E}) because (u_{m} − ũ)(0) = u_{m}(0) − a → 0 and ρ_{E}(u_{m} − ũ) = ρ_{E}(u_{m} − u + u(0) − a) = ρ_{E}(u_{m} − u) → 0 by Lemma 5.1 (ii). □

For x ∈ 𝔹, let Λ_{x} : E → ℂ, Λ_{x}(u) := u(x) be the point-evaluation functional.

Lemma 5.3. Let (E, ρ_{E}) be a Möbius-invariant space of harmonic functions and let ∥ · ∥_{E} be as in (17). Then Λ_{x} is bounded on (E, ∥ · ∥_{E}).

Proof. Fix x ∈ 𝔹 and let u ∈ E. By Theorem 1.2, u is in b^{∞} and ρ_{b^{∞}}(u) ≲ ρ_{E}(u). Then

|u(x) − u(0)| = |∫_{0}^{1} ∇u(tx) · x dt| ≤ ρ_{b^{∞}}(u) ∫_{0}^{1} \frac{dt}{1 − t^{2}|x|^{2}} ≲ ρ_{b^{∞}}(u) ≲ ρ_{E}(u).

Thus |u(x)| ≲ |u(0)| + ρ_{E}(u) = ∥u∥_{E}. □

Note that although Λ_{x} is continuous on (E, ∥ · ∥_{E}), it is not continuous on (E, ρ_{E}) since that would violate (12) for a constant u. On the other hand, whereas ρ_{E}(u ∘ φ) ∼ ρ_{E}(u), it is not true that ∥u ∘ φ∥_{E} ∼ ∥u∥_{E} for all φ ∈ M(𝔹). However, if we consider only the rotations and the conjugation, this equivalence is true.

Lemma 5.4. Let (E, ρ_{E}) be a Möbius-invariant space of harmonic functions and let ∥ · ∥_{E} be as in (17). Then

(i) ∥u_{θ}∥_{E} ∼ ∥u∥_{E} and ∥u ∘ γ∥_{E} ∼ ∥u∥_{E}, (ii) for fixed u ∈ E, the map θ ↦ u_{θ} is continuous from [0, 2π] to (E, ∥ · ∥_{E}).

Proof. Part (i) is true because u_{θ}(0) = u(0) and u ∘ γ(0) = u(0). Part (ii) follows from Definition 1.1 (iv) and the fact that ∥u_{θ_{1}} − u_{θ_{2}}∥_{E} = ρ_{E}(u_{θ_{1}} − u_{θ_{2}}). □

We next show the harmonic analogues of Propositions 1 and 2 of [2]. The ideas are taken from [2], but more details are provided and we will be cautious about convergence issues.

Proposition 5.5. Let (E, ρ_{E}) be a Möbius-invariant space of harmonic functions, u ∈ E and m ∈ ℤ. Define the function U pointwise by

(18)  U(z) = \frac{1}{2π} ∫_{0}^{2π} u(e^{iθ}z)e^{−imθ} dθ  (z ∈ 𝔻).

Then U ∈ E, ∥U∥_{E} ≲ ∥u∥_{E} and ρ_{E}(U) ≲ ρ_{E}(u).

Proof. We work in the Banach space (E, ∥ · ∥_{E}) and consider the Bochner integral

V = \frac{1}{2π} ∫_{0}^{2π} u_{θ} e^{−imθ} dθ.

For Bochner integrals see, for example, [4, Appendix E]. The above integral exists by the completeness of (E, ∥ · ∥_{E}) (Lemma 5.2) and the continuity in Lemma 5.4 (ii). Hence V ∈ E and by [4, Proposition E.5] and Lemma 5.4 (i),

∥V ∥_{E} ≤ \frac{1}{2π} ∫_{0}^{2π} ∥u_{θ} e^{−imθ}∥_{E} dθ = \frac{1}{2π} ∫_{0}^{2π} ∥u_{θ}∥_{E} dθ ≲ \frac{1}{2π} ∫_{0}^{2π} ∥u∥_{E} dθ = ∥u∥_{E}.

By Lemma 5.3, for every z ∈ 𝔻, the point evaluation functional Λ_{z} is bounded on (E, ∥ · ∥_{E}). Applying [4, Proposition E.11] we see that

V (z) = Λ_{z}(V) = Λ_{z} (\frac{1}{2π} ∫_{0}^{2π} u_{θ} e^{−imθ} dθ) = \frac{1}{2π} ∫_{0}^{2π} u_{θ}(z) e^{−imθ} dθ = U(z),

for every z ∈ 𝔻. Thus U = V and we conclude that U ∈ E and ∥U∥_{E} ≲ ∥u∥_{E}. To see ρ_{E}(U) ≲ ρ_{E}(u), replace u with u − u(0) and apply the same argument. □

Lemma 5.6. Let (E, ρ_{E}) be a Möbius-invariant space of harmonic functions. Let u ∈ E and write u as a series as in (7). If a_{m_{0}} ≠ 0 for some m_{0} ≥ 1, then z^{m_{0}} ∈ E and if b_{m_{0}} ≠ 0 for some m_{0} ≥ 1, then z̄^{m_{0}} ∈ E.

Proof. If a_{m_{0}} ≠ 0 for some m_{0} ≥ 1, define U as in (18). A calculation shows U(z) = a_{m_{0}}z^{m_{0}} and U ∈ E by Proposition 5.5. The argument for z̄^{m_{0}} is similar. □

Now we prove the harmonic counterpart of [2, Proposition 2].

Theorem 5.7. A Möbius-invariant space of harmonic functions (E, ρ_{E}) contains all z^{m} and z̄^{m}, m ≥ 1. Moreover, ρ_{E}(z^{m}) ≲ mρ_{E}(z) and ρ_{E}(z̄^{m}) ≲ mρ_{E}(z).

Proof. By assumption E contains a non-constant function. Let u ∈ E be such that in the series representation (7) at least one of a_{m_{0}} ≠ 0 or b_{m_{0}} ≠ 0 with m_{0} ≥ 1. If a_{m_{0}} ≠ 0, then by Lemma 5.6, z^{m_{0}} ∈ E. If b_{m_{0}} ≠ 0, then by the same lemma, z̄^{m_{0}} ∈ E. But then z^{m_{0}} = z̄^{m_{0}} ∘ γ is in E. We conclude that z^{m_{0}} ∈ E for some m_{0} ≥ 1.

Next, by Möbius invariance, z^{m_{0}} ∘ φ_{r} = φ_{r}^{m_{0}} ∈ E for every −1 < r < 1. In the series expansion of the holomorphic function φ_{r}^{m_{0}}, the coefficient of z is non-zero for r ≠ 0 because (φ_{r}^{m_{0}})′(0) = −m_{0}r^{m_{0}−1}(1 − r^{2}) ≠ 0. Thus z ∈ E by Lemma 5.6, and φ_{r} ∈ E (−1 < r < 1) by Möbius invariance. Since

(19)  φ_{r}(z) = \frac{r − z}{1 − rz} = (r − z) ∑_{m=0}^{∞} r^{m}z^{m} = r − (1 − r^{2}) ∑_{m=1}^{∞} r^{m−1}z^{m},

z^{m} ∈ E for every m ≥ 1 by Lemma 5.6.

We next estimate ρ_{E}(z^{m}). By (19),

\frac{1}{2π} ∫_{0}^{2π} φ_{r}(e^{iθ}z)e^{−imθ} dθ = −(1 − r^{2})r^{m−1}z^{m} (m ≥ 1),

and so by Proposition 5.5,

(1 − r^{2})r^{m−1}ρ_{E}(z^{m}) ≲ ρ_{E}(φ_{r}) ∼ ρ_{E}(z).

Set r^{2} = (m − 1)/m. Then 1 − r^{2} = 1/m and

\frac{1}{r^{m−1}} = (\frac{m}{m − 1})^{(m−1)/2} = ((1 + \frac{1}{m − 1})^{m−1})^{1/2} ≤ e^{1/2}.

We conclude that ρ_{E}(z^{m}) ≲ e^{1/2}mρ_{E}(z). The estimate of ρ_{E}(z̄^{m}) follows from Möbius invariance. □

Corollary 5.8. A Möbius-invariant space of harmonic functions contains all harmonic polynomials.

Finally we show that harmonic polynomials are dense.

Lemma 5.9. Let (E, ρ_{E}) be a Möbius-invariant space of harmonic functions. Then harmonic polynomials are dense in (E, ∥ · ∥_{E}) and in (E, ρ_{E}).

Proof. We first work in the normed space (E, ∥ · ∥_{E}). Pick u ∈ E and represent u as a series as in (7). Let

K_{m}(θ) := ∑_{k=−m}^{m} (1 − \frac{|k|}{m + 1}) e^{−ikθ} (0 ≤ θ ≤ 2π),

be the Fejer kernel. The Bochner integral

U_{m} = \frac{1}{2π} ∫_{0}^{2π} u_{θ}K_{m}(θ) dθ

belongs to E, and since point evaluations are bounded on (E, ∥·∥_{E}) by Lemma 5.3, applying [4, Proposition E.11] shows that for all z ∈ 𝔻,

U_{m}(z) = \frac{1}{2π} ∫_{0}^{2π} u(e^{iθ}z) ( ∑_{k=−m}^{m} (1 − \frac{|k|}{m + 1}) e^{−ikθ}) dθ

= ∑_{k=0}^{m} (1 − \frac{k}{m + 1}) a_{k}z^{k} + ∑_{k=1}^{m} (1 − \frac{k}{m + 1}) b_{k}z̄^{k}.

Thus U_{m} is a harmonic polynomial. Using the fact that {K_{m}} is a positive approximate identity, it is straightforward to show U_{m} → u in (E, ∥ · ∥_{E}) as m → ∞. To see that harmonic polynomials are dense in (E, ρ_{E}), note that ρ_{E}(U_{m} − u) ≤ ∥U_{m} − u∥_{E} → 0. □

6. Uniqueness of the Semi-Hilbert Space

In this section we show that the harmonic Dirichlet space b_{−2}^{2} is the unique strictly Möbius-invariant semi-Hilbert space. By Theorem 1.5, (b_{−2}^{2}, ρ_{b_{−2}^{2}}) is Möbius invariant. Let us first show that it is strictly Möbius invariant.

Lemma 6.1. For all u ∈ b_{−2}^{2} and φ ∈ M(𝔹), the composition u ∘ φ is in b_{−2}^{2} and ρ_{b_{−2}^{2}}(u ∘ φ) = ρ_{b_{−2}^{2}}(u).

Proof. Note that by (4), (8) and (9), we have

ρ_{b_{−2}^{2}}^{2}(u) = 2 ∫_{𝔻} (|\frac{∂u}{∂z}(z)|^{2} + |\frac{∂u}{∂z̄}(z)|^{2}) dA(z).

Pick u ∈ b_{−2}^{2} and let u = f + g ∘ γ as in Lemma 2.2. By (10) and conjugation invariance of dA,

ρ_{b_{−2}^{2}}^{2}(u) = 2 ∫_{𝔻} (|f′(z)|^{2} + |g′(z̄)|^{2}) dA(z) = 2 ∫_{𝔻} (|f′(z)|^{2} + |g′(z)|^{2}) dA(z).

Let φ ∈ M(𝔹). Then either φ ∈ Aut(𝔻) or φ ∈ \overline{Aut}(𝔻). If φ ∈ Aut(𝔻), as in the proof of Lemma 3.3, there exists φ̂ ∈ Aut(𝔻) such that

u ∘ φ = f ∘ φ + g ∘ γ ∘ φ = f ∘ φ + g ∘ φ̂ ∘ γ

and

\frac{∂(u ∘ φ)}{∂z} = (f ∘ φ)′(z)  and  \frac{∂(u ∘ φ)}{∂z̄} = (g ∘ φ̂)′(z̄).

Then

ρ_{b_{−2}^{2}}^{2}(u ∘ φ) = 2 ∫_{𝔻} (|(f ∘ φ)′(z)|^{2} + |(g ∘ φ̂)′(z̄)|^{2}) dA(z)
= 2 ∫_{𝔻} (|(f ∘ φ)′(z)|^{2} + |(g ∘ φ̂)′(z)|^{2}) dA(z)
= 2 ∫_{𝔻} (|f′(z)|^{2} + |g′(z)|^{2}) dA(z) = ρ_{b_{−2}^{2}}^{2}(u),

where in the second equality we use conjugation invariance of dA and in the third equality we use (15) with p = 2. The case φ ∈ \overline{Aut}(𝔻) is dealt with similarly as done a few times above. □

An elementary calculation shows that if u is written as a series as in (7), then

(20)  ρ_{b_{−2}^{2}}^{2}(u) = 2 ∑_{m=1}^{∞} m (|a_{m}|^{2} + |b_{m}|^{2}) .

We next show that b_{−2}^{2} is the only strictly Möbius-invariant semi-Hilbert space. In the proof, as in [1], we use the functions

(21)  1 − rφ_{r}(z) = \frac{1 − r^{2}}{1 − rz} = (1 − r^{2}) ∑_{m=0}^{∞} r^{m}z^{m},  (−1 < r < 1, z ∈ 𝔻).

The Taylor series above converges pointwise. Let us verify that it also converges in a Möbius-invariant space. As a preliminary we first show the following lemma.

Lemma 6.2. Let (E, ρ_{E}) be a Möbius-invariant space of harmonic functions. Suppose (u_{m}) is convergent in the space (E, ρ_{E}) and u_{m} → u pointwise. Then the limit of (u_{m}) in (E, ρ_{E}) is u.

Proof. With ∥ · ∥_{E} as in (17), we pass to the normed space (E, ∥ · ∥_{E}) where point evaluations are bounded by Lemma 5.3. The sequence (u_{m}) is Cauchy in (E, ∥·∥_{E}) because (u_{m}) is Cauchy in (E, ρ_{E}) and (u_{m}(0)) is Cauchy. Then, by Lemma 5.2, there exists v ∈ E such that ∥u_{m} − v∥_{E} → 0. By Lemma 5.3, u_{m} → v pointwise and thus v = u. This shows u ∈ E and ρ_{E}(u_{m} − u) ≤ ∥u_{m} − u∥_{E} → 0. □

Proposition 6.3. Let (E, ρ_{E}) be a Möbius-invariant space of harmonic functions. The series (1 − r^{2}) ∑_{m=0}^{∞} r^{m}z^{m} in (21) converges to 1 − rφ_{r} in the space (E, ρ_{E}).

Proof. By the estimate ρ_{E}(z^{m}) ≲ m in Theorem 5.7, the series ∑_{m=0}^{∞} r^{m}z^{m} is Cauchy and therefore converges in the complete space (E, ρ_{E}). The proposition now follows from Lemma 6.2. □

Proof of Theorem 1.6 (ii). As ρ_{H}(u ∘ φ) = ρ_{H}(u), by polarization, we have

(22)  ⟨u ∘ φ, v ∘ φ⟩_{H} = ⟨u, v⟩_{H}  (u, v ∈ H, φ ∈ M(𝔹)).

Taking φ(z) = e^{iθ}z, a rotation, we see that

⟨z^{m}, z^{k}⟩_{H} = ⟨e^{imθ}z^{m}, e^{ikθ}z^{k}⟩_{H} = e^{i(m−k)θ}⟨z^{m}, z^{k}⟩_{H}.

This shows ⟨z^{m}, z^{k}⟩_{H} = 0 when m ≠ k. Replacing one or both z with z̄ we obtain a similar equality. We list all these below which include ⟨1, 1⟩_{H} = 0.

\begin{aligned}⟨z^{m}, z^{k}⟩_{H} & = 0,  m ≠ k (m, k ≥ 0), \\ ⟨z̄^{m}, z̄^{k}⟩_{H} & = 0,  m ≠ k (m, k ≥ 0), \\ ⟨z^{m}, z̄^{k}⟩_{H} & = 0,  (m, k ≥ 0). \end{aligned} (23)

Also taking φ(z) = γ(z) = z̄ shows

(24)  ⟨z̄^{m}, z̄^{m}⟩_{H} = ⟨z^{m}, z^{m}⟩_{H}  (m ≥ 1).

As done in [1], for −1 < r < 1, we compute ⟨1−rφ_{r}, 1−rφ_{r}⟩_{H} in two ways. One using Möbius invariance, the other by using the series expansion in Proposition 6.3. First,

⟨1 − rφ_{r}, 1 − rφ_{r}⟩_{H} = ⟨1 − rz, 1 − rz⟩_{H} = r^{2}⟨z, z⟩_{H}

by (22) and (23). Second, by Proposition 6.3 and (23),

⟨1 − rφ_{r}, 1 − φ_{r}⟩_{H} = (1 − r^{2})^{2}⟨ ∑_{m=0}^{∞} r^{m}z^{m}, ∑_{m=0}^{∞} r^{m}z^{m}⟩_{H}

= (1 − r^{2})^{2} ∑_{m=0}^{∞} r^{2m}⟨z^{m}, z^{m}⟩_{H}.

Equating the above and using ⟨1, 1⟩_{H} = 0, we see that for all −1 < r < 1,

r^{2}⟨z, z⟩_{H} = (1 − r^{2})^{2} ∑_{m=0}^{∞} r^{2m}⟨z^{m}, z^{m}⟩_{H}

= r^{2}⟨z, z⟩_{H} + ∑_{m=2}^{∞} r^{2m} (⟨z^{m}, z^{m}⟩_{H} − 2⟨z^{m−1}, z^{m−1}⟩_{H} + ⟨z^{m−2}, z^{m−2}⟩_{H}) .

Thus for m ≥ 2, we must have

⟨z^{m}, z^{m}⟩_{H} − 2⟨z^{m−1}, z^{m−1}⟩_{H} + ⟨z^{m−2}, z^{m−2}⟩_{H} = 0.

Solving this equation recursively we obtain

⟨z^{m}, z^{m}⟩_{H} = m⟨z, z⟩_{H} (m ≥ 2).

Let λ := \sqrt{⟨z, z⟩_{H}/2}. Using also (24) we conclude that

(25)  ⟨z̄^{m}, z̄^{m}⟩_{H} = ⟨z^{m}, z^{m}⟩_{H} = λ^{2}2m  (m ≥ 0).

We now show H = b_{−2}^{2} and ρ_{H}(u) = λρ_{b_{−2}^{2}}(u) for every u ∈ H. By polarization this implies ⟨u, v⟩_{H} = λ^{2}⟨u, v⟩_{b_{−2}^{2}} for u, v ∈ H.

To establish a connection between convergence in the space H and pointwise convergence, we pass from semi-inner product to inner product. Define

(26)  ⟨⟨u, v⟩⟩ := λ^{2}u(0)\overline{v(0)} + ⟨u, v⟩_{H}  (u, v ∈ H).

By Lemma 5.1 (i), ⟨⟨·, ·⟩⟩ is a true inner product with the corresponding norm

⫴u⫴^{2} = λ^{2}|u(0)|^{2} + ρ_{H}^{2}(u).

The above norm is slightly different from the one defined in (17), but is equivalent to it and therefore the properties obtained in Section 5 for (E, ∥ · ∥_{E}) hold also for (H, ⫴ · ⫴). Same is true for b_{−2}^{2} when endowed with the norm

∥u∥_{b_{−2}^{2}}^{2} = |u(0)|^{2} + ρ_{b_{−2}^{2}}^{2}(u).

If p(z) = ∑_{m=0}^{M} a_{m}z^{m} + ∑_{m=1}^{M} b_{m}z̄^{m} is a harmonic polynomial, then by (20), (23) and (25),

(27)  ⫴p⫴ = λ∥p∥_{b_{−2}^{2}}.

To see that H ⊂ b_{−2}^{2}, pick u ∈ H. By Lemma 5.9, there is a sequence (p_{m}) of harmonic polynomials such that ⫴p_{m} − u⫴ → 0 and by Lemma 5.3, p_{m}(z) → u(z) pointwise for every z ∈ 𝔻. By (27), (p_{m}) is Cauchy in the complete space (b_{−2}^{2}, ∥ · ∥_{b_{−2}^{2}}) and so there exists v ∈ b_{−2}^{2} such that ∥p_{m} − v∥_{b_{−2}^{2}} → 0. But then by Lemma 5.3, (p_{m}) converges pointwise to v and we conclude that u = v ∈ b_{−2}^{2}. In addition, by (27),

⫴u⫴ = lim_{m→∞} ⫴p_{m}⫴ = λ lim_{m→∞} ∥p_{m}∥_{b_{−2}^{2}} = λ∥u∥_{b_{−2}^{2}},

which implies ρ_{H}(u) = λρ_{b_{−2}^{2}}(u). The inclusion b_{−2}^{2} ⊂ H can be verified in the same way. □

7. The Minimal Space

To show the minimality of b_{−2}^{1} we use atomic decomposition. Atomic decomposition of harmonic Bergman spaces b_{q}^{p} (q > −1) is obtained in [5, Theorem 3] and this is extended to all q ∈ ℝ in [7]. The following lemma follows from [7, Theorem 10.1] by taking p = 1, q = −2, s = −1, and using the formula in the last paragraph of [7, Section 14].

Lemma 7.1. There exists a sequence (a_{m}) ∈ 𝔻 such that the following holds. For every u ∈ b_{−2}^{1}, there exists a sequence (c_{m}) ∈ ℓ^{1} such that ∥(c_{m})∥_{ℓ^{1}} ∼ ∥u∥_{b_{−2}^{1}} and

(28)  u(z) = ∑_{m=1}^{∞} c_{m}(1 − |a_{m}|^{2}) (\frac{1}{1 − ā_{m}z} + \frac{1}{1 − a_{m}z̄} − 1) .

The above series converges absolutely and uniformly on compact subsets of 𝔻 and also in the norm ∥ · ∥_{b_{−2}^{1}}.

Proof of Theorem 1.7. Let (a_{m}) ∈ 𝔻 be as asserted in Lemma 7.1. Pick arbitrary u ∈ b_{−2}^{1} and let (c_{m}) be such that

(29)  ∥(c_{m})∥_{ℓ^{1}} ∼ ∥u∥_{b_{−2}^{1}}

and (28) holds. By the identities

\frac{1 − |a_{m}|^{2}}{1 − ā_{m}z} = 1 − ā_{m}φ_{a_{m}}(z)  and  \frac{1 − |a_{m}|^{2}}{1 − a_{m}z̄} = 1 − a_{m}φā_{m}(z̄),

we have

(30)  u(z) = ∑_{m=1}^{∞} c_{m} (1 + |a_{m}|^{2} − ā_{m}φ_{a_{m}}(z) − a_{m}φā_{m}(z̄)) .

The series on the right converges pointwise to u. Let us show that it also converges to u in the space (E, ρ_{E}). Writing φā_{m}(z̄) = (φā_{m} ∘ γ)(z) and using first Lemma 5.1 (ii), then |a_{m}| < 1, and finally Möbius invariance, we obtain

\begin{aligned}ρ_{E} (1 + |a_{m}|^{2} − ā_{m}φ_{a_{m}} − a_{m}φā_{m} ∘ γ) & = ρ_{E} (−ā_{m}φ_{a_{m}} − a_{m}φā_{m} ∘ γ) \\ & ≤ ρ_{E}(φ_{a_{m}}) + ρ_{E}(φā_{m} ∘ γ) \\ & ≲ ρ_{E}(z). \end{aligned} (31)

Because (c_{m}) ∈ ℓ^{1}, the series in (30) is Cauchy (convergent) in E and applying Lemma 6.2 shows that u is in E. In addition, by (31) and (29),

ρ_{E}(u) = ρ_{E} ( ∑_{m=1}^{∞} c_{m} (1 + |a_{m}|^{2} − ā_{m}φ_{a_{m}} − a_{m}φā_{m} ∘ γ)) ≲ ∑_{m=1}^{∞} |c_{m}| ≲ ∥u∥_{b_{−2}^{1}}.

To finish the proof all we need is to replace ∥u∥_{b_{−2}^{1}} above with ρ_{b_{−2}^{1}}(u). This is easily done by replacing u with u − u(0) and using Lemma 5.1 (ii). □

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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, 26470 Eski̇şehi̇r, Türki̇ye

Email address: aeureyen@eskisehir.edu.tr