HARMONIC AND SUPERHARMONIC MAJORANTS ON THE DISK

ALEXANDER BORICHEV ARTUR NICOLAU PASCAL J. THOMAS

Abstract. It is proved that a positive function on the unit disk admits a harmonic majorant if and only if a certain explicit upper envelope of it admits a superharmonic majorant. The (logarithmic) Lipschitz regularity of this superharmonic majorant is discussed.

1. Definitions and statements

Let 𝔻 denote the open unit disk in the complex plane, and H^{+}(𝔻) the cone of positive harmonic functions on 𝔻. We would like to describe the nonnegative functions φ : 𝔻 ⟼ ℝ_{+} which admit a harmonic majorant, that is, h ∈ H^{+}(𝔻) such that h ≥ φ. This question arises in problems about the decrease of bounded holomorphic functions in the unit disk, as well as in the description of free interpolating sequences for the Nevanlinna class. See [4], where an answer is given in terms of duality with the measures that act on positive harmonic functions. The aim of this note is to reduce this problem first to the finiteness of a certain best Lipschitz majorant function, and then to the existence of a merely superharmonic (nontrivial) majorant.

Let the hyperbolic (or Poincaré) distance ρ on the disk be defined by dρ(z) := (1 − |z|^{2})^{−1}|dz|. This is invariant under biholomorphic maps from the disk to itself. Explicitly, if we first define the pseudohyperbolic (or Gleason) distance by

d(z, w) := |\frac{z − w}{1 − z\overline{w}}| ,

then

ρ(z, w) = \frac{1}{2} log \frac{1 + d(z, w)}{1 − d(z, w)}. (1.1)

For h ∈ H^{+}(𝔻), the classical Harnack inequality reads, for 0 < r < 1, θ ∈ ℝ,

\frac{1 − r}{1 + r}h(0) ≤ h(re^{iθ}) ≤ \frac{1 + r}{1 − r}h(0).

This implies that the function log h is Lipschitz with constant 2 with respect to the hyperbolic distance. We will say that a positive valued function F is Log-Lipschitz (with constant C) if and only if | log F (z) − log F (w)| ≤ Cρ(z, w) for all z, w ∈ 𝔻.

Key words and phrases. harmonic majorants, positive harmonic functions, superharmonic functions, Harnack inequality ; 2000 Mathematics Subject Classification 31A05.

All authors supported by the PICS program no 1019 of Generalitat de Catalunya and CNRS. First and second author also supported by European Commission Research Training Network HPRN-CT-2000-00116. Second author supported by DGICYT grant BFM2002-00571 and the CIRIT grant 2001-SGR00431.

Theorem 1. A nonnegative function φ admits a superharmonic and Log-Lipschitz majorant with constant C ≥ 2, if and only if φ admits a harmonic majorant.

Since the infimum of two harmonic functions is not in general harmonic, there is no smallest harmonic majorant for a given function. On the other hand, the cone Sp(𝔻) of superharmonic functions is stable under finite infima. Denote by R(φ) the reduced function of φ, i.e.

R(φ)(z) := inf{u(z) : u ∈ Sp(𝔻), u ≥ φ on 𝔻}

(see e.g. [2]). We use the convention that R(φ) ≡ ∞ if there is no (non identically infinite) superharmonic majorant. The reduced function could also be called “superharmonic envelope” as in [8]. The reduced function is not in general superharmonic, because it can fail to be lower semicontinuous. J. W. Green [3, Theorem 2] proved that when φ is continuous, then R(φ), if finite, is also continuous, therefore superharmonic, so the infimum in its definition is really a minimum; furthermore, R(φ) is harmonic in the open set {z : R(φ)(z) > φ(z)}.

If instead of taking the infimum of all superharmonic functions above φ, we restrict ourselves to those which are Log-Lipschitz with a given constant C, the corresponding infimum R_{C}(φ) (if finite) will again be Log-Lipschitz with constant C, hence will be the smallest Log-Lipschitz superharmonic majorant of φ with constant C. Theorem 1 means that φ admits a harmonic majorant if and only if R_{C}(φ) is finite for any given C ≥ 2.

In order to study the finiteness of R_{C}(φ), it would be nice to be able to proceed in two steps, first dealing with the Lipschitz property, then with superharmonicity. The smallest Log-Lipschitz majorant with constant C of a given nonnegative function φ is (see [7], or [5] for a more recent survey)

L_{C}(φ)(z) = exp{sup_{w∈𝔻}(φ(w) − Cρ(w, z))}.

Clearly, L_{C}(φ) ≤ R_{C}(φ), so R(L_{C}(φ)) ≤ R_{C}(φ). That inequality can be strict when C < 2.

Proposition 2. For every 0 < γ ≤ β < 2 there exists a bounded positive function H on the disc such that

R(L_{γ}(H)) ≠ R_{β}(H).

But we do not know whether R(L_{C}(φ)) = R_{C}(φ) holds for C ≥ 2. Nevertheless, if a Log-Lipschitz function admits a superharmonic majorant s, then the invariant averages of s will provide us with a superharmonic majorant with a weak Log-Lipschitz property, and so we obtain the following result.

Theorem 3. If there exists C > 0 such that L_{C}(φ) admits a superharmonic majorant, then φ admits a harmonic majorant.

Note that it would not help to perform our two steps in the reverse order. Typical data for many problems of harmonic majorants are functions φ which vanish everywhere except on a discrete subset [4]. For such φ, R(φ) = φ.

Of course, Theorem 3 raises the question of criteria to ensure the finiteness of the reduced function of Log-Lipschitz data. In this general direction, one should note results of Koosis [6, p. 77] and Cole and Ransford [1, Theorem 1.3] which imply that when φ is merely continuous,

R(φ)(x) = sup{∫ φdμ : μ ∈ I_{x}} = sup{∫ φdμ : μ ∈ H_{x}},

where I_{x} denotes the set of Jensen measures for x, and H_{x} denotes the set of harmonic measures for x with respect to a domain ω ⊂⊂ 𝔻. A perhaps more computable characterization of R(φ) is given by [6, Théorème, p. 80] : let D_{H}(z, r) stand for the disc of center z and radius r with respect to the hyperbolic distance ρ, and dβ(z) := (1 − |z|^{2})^{−2}dm(z) be the invariant measure on the disk, where dm denotes the 2-dimensional Lebesgue measure. Given a real-valued continuous function F on 𝔻, let

MF (z) := sup_{r>0} \frac{1}{β(D_{H}(z, r))} ∫_{D_{H}(z,r)} F (w)dβ(w),

and define F^{(0)} := F, F^{(k+1)} := MF^{(k)}. Then, arguing as in [6], one can check that

Rφ(z) = lim_{n→∞} φ^{(n)}(z).

So, Theorem 3 says that φ admits a harmonic majorant if and only if the sequence L_{C}(φ)^{(n)}(0) remains bounded.

The paper is organized as follows. The next section is devoted to studying the dyadic analogue of the problem of harmonic majorants. Theorems 1 and 3 are proved in sections 3 and 5 respectively. Proposition 2 is proved in section 4.

We would like to thank the referee for his very careful reading of our paper.

2. A discrete model

Recall that any positive harmonic function h is the Poisson integral of a finite positive measure μ on the boundary of the disk, h(z) = ∫_{0}^{2π} P_{z}(e^{iθ}) dμ(θ), where

P_{z}(e^{iθ}) := \frac{1}{2π} \frac{(1 − |z|^{2})}{|z − e^{iθ}|^{2}}.

The following considerations concern the simpler case of functions which are generated by the “square” kernel

K_{z}(e^{iθ}) := \frac{1}{|I_{z}|}χ_{I_{z}}(e^{iθ}), where I_{z} := \{e^{iθ} : z ∈ Γ_{α}(e^{iθ})\}.

Here χ_{E} stands for the characteristic function of the set E, and Γ_{α}(e^{iθ}) for the Stolz angle of aperture α and vertex e^{iθ}. The “square” integral of a finite measure μ is defined by ∫_{0}^{2π} K_{z}(e^{iθ}) dμ(θ) = μ(I_{z})/|I_{z}|.

Consider the usual partition of ∂𝔻 in dyadic arcs, for any n in ℤ_{+}:

I_{n,k} := {e^{iθ} : θ ∈ [2πk2^{−n}, 2π(k + 1)2^{−n})}, 0 ≤ k < 2^{n}. (2.1)

Note that |I_{n,k}| = 2π2^{−n}. To this subdivision we associate the Whitney partition in “dyadic squares” of the unit disk :

Q_{n,k} := {re^{iθ} : e^{iθ} ∈ I_{n,k}, 1 − 2^{−n} ≤ r < 1 − 2^{−n−1}}.

It is well known and easy to see that there exists a constant c_{α} such that for any z ∈ Q_{n,k},P_{z} ≥ c_{α}K_{I_{n,k}}. This implies that sufficient conditions for majorization by “Kharmonic functions” yield sufficient conditions for majorization by (true) harmonic functions.

Theorem 4. Given a collection of nonnegative data {p_{n,k}} ⊂ ℝ_{+}, there exists a finite positive measure μ on ∂𝔻 such that

\frac{μ(I_{n,k})}{|I_{n,k}|} ≥ p_{n,k} (2.2)

if and only if there exists a constant S such that

∑_{n,k} p_{n,k}|I_{n,k}| ≤ S (2.3)

where the sum is taken over any disjoint subfamily of the whole family of dyadic arcs {I_{n,k}} defined in (2.1).

Proof. Condition (2.3) is clearly necessary with S = μ(∂𝔻). To prove the converse direction, let us consider the following modified data:

p̃_{n,k} := \frac{1}{|I_{n,k}|} sup \{∑ p_{q,j}|I_{q,j}|\} ,

where the supremum is taken over any disjoint subfamily {I_{q,j}} of the family of all dyadic subarcs of the given I_{n,k}. Observe that p̃_{n,k} ≥ p_{n,k}, and that p̃_{n,k} verifies the following discrete superharmonicity property :

p̃_{n,k} ≥ \frac{1}{2}(p̃_{n+1,2k} + p̃_{n+1,2k+1}).

Assuming (2.3) we will construct a sequence of positive measures μ_{n} of bounded total mass, uniformly distributed on each arc I_{n,j}, such that for all m ≤ n,

μ_{n}(I_{m,k}) ≥ |I_{m,k}|p̃_{m,k}. (2.4)

Let μ_{0} be the uniform measure of total mass S on the arc I_{0,0} = ∂𝔻. The hypothesis (2.3) coincides with (2.4) in this case. Assuming that μ_{m}, m ≤ n have already been constructed satisfying (2.4), we will choose μ_{n+1}. Fix j, 0 ≤ j < 2^{n}. Then I_{n,j} = I_{n+1,2j} ∪ I_{n+1,2j+1}, so (2.4) implies in particular that

μ_{n}(I_{n,j}) ≥ |I_{n+1,2j}|p̃_{n+1,2j} + |I_{n+1,2j+1}|p̃_{n+1,2j+1}.

Now choose α, β ≥ 0 such that

μ_{n}(I_{n,j}) = α + β, α ≥ |I_{n+1,2j}|p̃_{n+1,2j}, β ≥ |I_{n+1,2j+1}|p̃_{n+1,2j+1},

and set μ_{n+1}(I_{n+1,2j}) = α, μ_{n+1}(I_{n+1,2j+1}) = β. This defines a measure μ_{n+1} which verifies (2.4) at rank n + 1 for m = n + 1. It also satisfies

μ_{n+1}(I_{m,j}) = μ_{n}(I_{m,j}), ∀m ≤ n,

so (2.4) is verified by μ_{n+1} for all m ≤ n + 1. This bounded sequence of measures contains a weakly convergent subsequence, whose limit μ will satisfy (2.2).

3. Proof of Theorem 1

We will prove the following slightly more general fact, which will be useful in Section 5.

Proposition 5. Let u be a positive superharmonic function on 𝔻 such that, for any z, w ∈ 𝔻,

|log u(z) − log u(w)| ≤ C_{1}(1 + ρ(z, w)). (3.1)

Then there exists h ∈ H^{+}(𝔻) such that h ≥ u.

Proof. The Riesz representation theorem tells us that there exists a positive measure ν = −Δu in the sense of distributions, and a positive harmonic function h_{0} such that

u(z) = h_{0}(z) + ∫_{𝔻} log \frac{1}{d(z, w)}dν(w) =: h_{0}(z) + u_{0}(z).

Let δ ∈ (0, 1), to be chosen later. Then define

u_{0}(z) = ∫_{\{w:d(z,w)≤δ\}} log \frac{1}{d(z, w)}dν(w) + ∫_{\{w:d(z,w)>δ\}} log \frac{1}{d(z, w)}dν(w)

=: u_{1}(z) + u_{2}(z).

For d(z, w) > δ, we have

log \frac{1}{d(z, w)} ≤ C_{δ} \frac{1 − |zw|^{2}}{|1 − z\overline{w}|^{2}} (1 − |w|^{2}),

which is harmonic in z. Note that

∞ > u(0) ≥ ∫_{𝔻} log \frac{1}{|w|}dν(w) ≥ ∫_{𝔻}(1 − |w|)dν(w), (3.2)

so that the harmonic fuction

h_{2}(z) := C_{δ} ∫_{𝔻} \frac{1 − |zw|^{2}}{|1 − z\overline{w}|^{2}} (1 − |w|^{2})dν(w)

is bigger than u_{2}. Now we only need to find a harmonic majorant for the remaining term u_{1}.

A sequence {z_{k}} ⊂ 𝔻 is called uniformly dense if there exists 0 < r_{1} < r_{2} satisfying 1. D_{H}(z_{j}, r_{1}) ∩ D_{H}(z_{k}, r_{1}) = ∅ for any j ≠ k,

2. 𝔻 ⊂ ∪_{k}D_{H}(z_{k}, r_{2}). Here D_{H}(z, r) denotes the hyperbolic disk with center z and radius r.

Lemma 6. For δ ∈ (0, 1) a properly chosen absolute constant, there exist a uniformly dense sequence {z_{k}} and a positive harmonic function h_{1} such that for any k, u_{1}(z_{k}) ≤ h_{1}(z_{k}).

Conclusion of the Proof of Proposition 5.

Accepting this Lemma, if we write h_{3} := h_{0} + h_{1} + h_{2}, we see that we have u(z_{k}) ≤ h_{3}(z_{k}), for any k. Now by Harnack’s inequality, for any z ∈ D_{H}(z_{k}, r_{2}), h_{3}(z) ≥ e^{−2r_{2}}h_{3}(z_{k}); while by (3.1) —note that this is the only step in this argument where this hypothesis is used—

u(z) ≤ exp[C_{1}(1 + r_{2})]u(z_{k}) ≤ exp[C_{1}(1 + r_{2})]h_{3}(z_{k}) ≤ exp[C_{1}(1 + r_{2})]e^{2r_{2}}h_{3}(z).

So we have found a harmonic majorant of u.

Proof of Lemma 6 First let T_{n,j} := {z ∈ Q_{n,j} : ρ(z, ∂Q_{n,j}) > δ}. For δ small enough, we have

m(Q_{n,j}) ≥ m(T_{n,j}) ≥ C(δ)m(Q_{n,j}),

where m is the 2-dimensional Lebesgue measure, and 0 < C(δ) < 1.

Choose in each T_{n,j} a point z_{n,j} such that

u_{1}(z_{n,j}) ≤ \frac{1}{m(T_{n,j})} ∫_{T_{n,j}} u_{1}(ζ)dm(ζ).

It is enough to estimate this last average. We apply Fubini’s theorem:

\frac{1}{m(T_{n,j})} ∫_{T_{n,j}} u_{1}(z)dm(z) = \frac{1}{m(T_{n,j})} ∫_{T_{n,j}} ∫_{\{w:d(z,w)≤δ\}} log \frac{1}{d(z, w)}dν(w)dm(ζ)

≤ \frac{1}{m(T_{n,j})} ∫_{Q_{n,j}} (∫_{T_{n,j}} log \frac{1}{d(z, w)}dm(z)) dν(w) ≤ C ∫_{Q_{n,j}} dν(w),

where the last inequality is due to the following explicit estimate : for w ∈ Q_{n,j},

∫_{T_{n,j}} log \frac{1}{d(z, w)}dm(z) ≤ ∑_{k=0}^{∞} log \frac{1}{2^{−k−1}} m (\{z : 2^{−k−1} < d(w, z) ≤ 2^{−k}\})

≤ C ∑_{k=0}^{∞} k2^{−2n−2k} ≤ C2^{−2n} ≤ Cm(T_{n,j}).

Now we set

p_{n,j} := \frac{1}{|I_{n,j}|} ∫_{Q_{n,j}}(1 − |w|)dν(w) ≈ ∫_{Q_{n,j}} dν(w).

Since the “squares” Q_{n,j} are disjoint, the condition (3.2) implies that

∑_{n,j} |I_{n,j}|p_{n,j} < ∞. (3.3)

Hence, {p_{n,j}} satisfies (2.3) and Theorem 4 provides a positive measure μ on the unit circle such that for any dyadic arc I_{n,j},

μ(I_{n,j}) ≥ |I_{n,j}|p_{n,j}. (3.4)

However, whenever condition (3.3) is verified, a more direct construction can be applied. Namely, one may take dμ = f(e^{iθ})dθ where

f(e^{iθ}) := ∑_{I_{n,k}∋e^{iθ}} p_{n,k}.

Observe that ∫_{0}^{2π} f(e^{iθ})dθ = ∑ |I_{n,k}|p_{n,k}, and since f(e^{iθ}) ≥ p_{n,k} whenever e^{iθ} ∈ I_{n,k}, the measure μ satisfies (3.4).

Then, by the remarks before Theorem 4, there exists a harmonic function h_{1} such that

h_{1}(z_{n,j}) ≥ Cp_{n,j} ≥ C ∫_{Q_{n,j}} dν(w) ≥ u_{1}(z_{n,j}).

□

4. Proof of Proposition 2

The gist of this proof is that the Poisson kernel itself cannot be log-Lipschitz with a constant better than 2.

Denote f(z) = Re\frac{1+z}{1−z} = P_{z}(1). Fix γ > 0. For small δ > 0 consider the function

g(z) = f((1 − δ)z),

positive and harmonic in \frac{1}{1−δ}𝔻. For 0 < ε < δ/2 consider

g_{ε}(z) = g(\frac{z}{1 − ε}), z ∈ 𝔻.

Denote

M_{δ} = sup_{0<ε<δ/2, z∈𝔻}∥∇(log g_{ε})(z)∥.

Put

h(z) = g_{ε}(z), z ∈ (1 − ε)∂𝔻.

Lemma 7. For sufficiently small ε < δ/2, for any w ∈ (1 − ε)∂𝔻, z ∈ 𝔻, we have

|log g_{ε}(w) − log g_{ε}(z)| ≤ M_{δ}|z − w| ≤ γρ(z, w),

where ρ is defined as in (1.1).

Applying this to the special case where z ∈ (1 − ε)∂𝔻, we see that for sufficiently small ε, log h satisfies the Lipschitz condition with respect to ρ with constant γ.

Consider the γ-log-Lipschitz (with respect to ρ) extension of h to 𝔻:

H(z) = sup_{w∈(1−ε)∂𝔻}h(w)e^{−γρ(w,z)}.

It also follows from the Lemma that for sufficiently small ε < δ/2,

H ≤ g_{ε} on 𝔻.

Proof of Lemma 7.

To prove the second inequality, note that if |\frac{z−w}{1−zw̄} | ≤ \frac{1}{2}, then |1 − zw̄| ≤ u(ε) → 0 as ε → 0, and then (since log \frac{1+s}{1−s} ≥ 2s, 0 ≤ s < 1)

2ρ(w, z) ≥ 2|\frac{z − w}{1 − zw̄}| ≥ \frac{2}{u(ε)}|z − w|.

If |\frac{z−w}{1−zw̄} | > \frac{1}{2}, and |z − w| ≤ \frac{γ}{2M_{δ}}, then

2ρ(w, z) ≥ log 3 > 1 ≥ \frac{2M_{δ}}{γ} |z − w|.

Finally, if |z − w| > \frac{γ}{2M_{δ}}, then

|\frac{z − w}{1 − zw̄}| ≥ 1 − u_{1}(ε, \frac{γ}{2M_{δ}} ),

where u_{1}(ε, \frac{γ}{2M_{δ}} ) → 0 as ε → 0, and

2ρ(w, z) ≥ log \frac{1}{u_{1}(ε, \frac{γ}{2M_{δ}} )} ≥ \frac{2M_{δ}}{γ} ,

as ε → 0. This completes the proof of Lemma 7. □

By definition of H, we have L_{γ}(H) = H.

Since R(H) is superharmonic, and H coincides with g_{ε} on (1 − ε)∂𝔻, with g_{ε} harmonic in 𝔻, we obtain

R(H) ≥ g_{ε} on (1 − ε)𝔻.

Furthermore, since H ≤ g_{ε} on 𝔻,

R(H) ≤ g_{ε} on 𝔻.

As a result,

R(H) = g_{ε} on (1 − ε)𝔻,

and, writing z = x + iy,

\frac{∂}{∂x}(log R(H))(0) = \frac{∂}{∂x}(log g_{ε})(0) = \frac{1 − δ}{1 − ε} \frac{∂}{∂x}(log f)(0) = 2\frac{1 − δ}{1 − ε}.

Therefore, the function R(H) is no better than 2\frac{1−δ}{1−ε}-log-Lipschitz, and

R(L_{γ}(H)) = R(H) ≠ R_{c}(H), c < 2\frac{1 − δ}{1 − ε}.

This completes the proof of Proposition 2.

5. Comparison of upper envelopes

The aim of this section is to prove Theorem 3. Proposition 5 shows that it will follow from the following result.

Lemma 8. Given C_{0} > 0, there is a C_{1} ≥ C_{0} such that if φ is Log-Lipschitz with constant C_{0}, and admits a non-trivial superharmonic majorant, then there exists a superharmonic function v such that v ≥ φ and v verifies (3.1) with constant C_{1}.

Proof. Let u ∈ Sp(𝔻) such that u ≥ φ. We will notice (Lemma 9) that an averaged version of u always satisfies (3.1), and that (up to a multiplicative constant) it provides the regular superharmonic majorant we are looking for.

For any δ ∈ (0, \frac{1}{2}], let D_{H}(z, δ) stand, as above, for a hyperbolic disk of radius δ centered at z. Let dβ(z) := (1 − |z|^{2})^{−2}dm(z) be the invariant measure on the disk. For any Möbius automorphism ϕ of the disk, any measurable function f and any measurable set E, we have (see e.g. [9, (2.19), p. 19])

∫_{E} f ∘ ϕ dβ = ∫_{ϕ(E)} f dβ.

For any measurable function g on the unit disk, let

g_{δ}(z) := \frac{1}{β(D_{H}(z, δ))} ∫_{D_{H}(z,δ)} g dβ.

Since φ is Log-Lipschitz with constant C_{0}, we have φ(w) ≥ e^{−C_{0}δ}φ(z) for any w ∈ D_{H}(z, δ), so that, being an average of such values, φ_{δ}(z) ≥ e^{−C_{0}δ}φ(z). Now φ ≤ u implies that

φ(z) ≤ e^{C_{0}δ}φ_{δ}(z) ≤ e^{C_{0}δ}u_{δ}(z).

The proof of Lemma 8 will conclude with the next two Lemmas.

Lemma 9. There exists an absolute constant κ such that for any positive valued superharmonic function u, and for any z, w in 𝔻 such that d(z, w) ≤ δ/4, one has u_{δ}(w) ≤ κu_{δ}(z), and therefore u_{δ} verifies (3.1).

Proof. Recall that since u is superharmonic, for any z ∈ 𝔻 and r_{1} < r_{2},

\frac{1}{β(D_{H}(z, r_{1}))} ∫_{D_{H}(z,r_{1})} u dβ ≥ \frac{1}{β(D_{H}(z, r_{2}))} ∫_{D_{H}(z,r_{2})} u dβ. (5.1)

This fact is clear when z = 0, because dβ has radial density, and we can reduce ourselves to this case by composing u with an appropriate Möbius automorphism of 𝔻.

Pick a constant K < 1 such that ρ(z, w) ≤ δ/4 implies that D_{H}(w, Kδ) ⊂ D_{H}(z, δ). Then (5.1) implies

u_{δ}(w) ≤ u_{Kδ}(w) ≤ \frac{1}{β(D_{H}(w, Kδ))} ∫_{D_{H}(z,δ)} u dβ,

by the inclusion of discs and positivity of u. But

\frac{1}{β(D_{H}(w, Kδ))} ∫_{D_{H}(z,δ)} u dβ ≤ κ\frac{1}{β(D_{H}(z, δ))} ∫_{D_{H}(z,δ)} u dβ = κu_{δ}(z),

since β(D_{H}(w, Kδ)) and β(D_{H}(z, δ)) are comparable.

Lemma 10. Let u be a positive superharmonic function on 𝔻. Then u_{δ} is also superharmonic.

Proof. If we were dealing with superharmonic functions on the plane, we could simply use invariance of harmonicity under translations to define suitable averaged functions. In 𝔻, we need invariance under Möbius automorphisms. The adequate machinery happens to have been developed in the more general case of the unit ball of ℂ^{n}.

We will follow the notations and use the results of [9, Chapter 4, pp. 34–39], itself inspired by [10]. Let ϕ_{z} denote the unique involutive Möbius automorphism of the disk which exchanges z and 0. Define the invariant convolution of two measurable functions by

(f ∗ g) (z) := ∫_{𝔻} f(w) (g ∘ ϕ_{z}) (w) dβ(w),

whenever the integral makes sense. This operation is commutative [9, bottom of page 34]. If we set

Ω_{δ}(z) := \frac{1}{β(D_{H}(0, δ))}χ_{D_{H}(0,δ)}(z),

then f_{δ} = f ∗ Ω_{δ}. Note that D_{H}(0, δ) = D(0, tanh δ) so that Ω_{δ} is a radial function.

The M-subharmonic functions defined in [9, Chapter 4, (4.1)] reduce for n = 1 to ordinary subharmonic functions. The invariant Laplacian (Laplace-Beltrami operator for the Bergman metric of the ball) reduces in the case n = 1 to

Δ̃ = 2(1 − |z|^{2})^{2} \frac{∂^{2}}{∂z∂z̄},

so that C^{2} superharmonic functions g can be characterized as those such that Δ̃g ≤ 0. Since our function Ω_{δ} is not smooth, we need to perform an approximation argument. It will be enough to show that u∗Ω_{δ} can be approximated from below by an increasing sequence of C^{2} superharmonic functions. Pick an increasing sequence of smooth, nonnegative, radial functions Ω_{δ,n} so that lim_{n→∞} Ω_{δ,n} = Ω_{δ} almost everywhere. Then the monotone convergence theorem tells us that u ∗ Ω_{δ,n} converges to u ∗ Ω_{δ}, and the sequence is clearly increasing. For f ∈ C^{2}(𝔻), by [9, (4.11), p. 36],

(Δ̃f)(a) = lim_{r→0} \frac{4}{r^{2}} [(f ∗ Ω_{r})(a) − f(a)].

Now, twice applying Ulrich’s lemma about associativity of the invariant convolution when the middle element is radial [9, Lemma 4.5, p. 36], we have

Δ̃(u ∗ Ω_{δ,n}) = lim_{r→0} \frac{4}{r^{2}} [(u ∗ Ω_{δ,n}) ∗ Ω_{r} − u ∗ Ω_{δ,n}] = lim_{r→0} \frac{4}{r^{2}} [u ∗ (Ω_{δ,n} ∗ Ω_{r}) − u ∗ Ω_{δ,n}]

= lim_{r→0} \frac{4}{r^{2}} [u ∗ (Ω_{r} ∗ Ω_{δ,n}) − u ∗ Ω_{δ,n}] = lim_{r→0} \frac{4}{r^{2}} [(u ∗ Ω_{r}) ∗ Ω_{δ,n} − u ∗ Ω_{δ,n}]

= lim_{r→0} \frac{4}{r^{2}} [(u ∗ Ω_{r} − u) ∗ Ω_{δ,n}] .

Since u is superharmonic, u ∗ Ω_{r} − u ≤ 0, and since Ω_{δ,n} ≥ 0, we finally have Δ̃(u ∗ Ω_{δ,n}) ≤ 0.

References

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[8] Ransford, T. J. “Jensen measures”, in : Approximation, Complex Analysis and Potential Theory (Montréal, Qc, 2000), 221–237, NATO Sci. Sér. II, Math-Phys-Chem 37, Kluwer, Dordrecht, 2001.

[9] Stoll, M. Invariant Potential Theory in the Unit Ball of ℂ^{n}, London Mathematical Society Lecture Note Series 199, Cambridge University Press, London, 1994.

[10] Ulrich, D. “Radial limits of M-subharmonic functions”, Trans. Amer. Math. Soc. 292 (1985), 501–518.

Alexander Borichev, LaBAG, Université Bordeaux 1, 351 cours de la Libération 33405 Talence CEDEX, France.

Alexander.Borichev@math.u-bordeaux.fr

Artur Nicolau, Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain.

artur@mat.uab.es

Pascal J. Thomas, Laboratoire de Mathématiques Emile Picard, UMR CNRS 5580, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse CEDEX, France.

pthomas@cict.fr