CONVOLUTION EQUATIONS IN SPACES OF ENTIRE FUNCTIONS

AND RELATED PROPERTIES OF CONFORMAL MAPPINGS.

F. Haslinger

Abstract. In this paper we consider convolution equations in Fréchet and dual Fréchet algebras of entire functions. The emphasis lies on nonradial weight functions defining the algebras of entire functions. With the help of various methods from functional analysis a necessary conditon is derived for a convolution operator to be surjective. Further we construct examples of nonsurjective convolution operators. Then sufficient conditions for surjectivity of convolution operators are discussed. For one complex variable, certain properties of level sets of conformal mappings appear to be of some importance in this context. On the other hand the examples of nonsurjective convolution operators give some insight into the behavior of related conformal mappings and yield new results in the variational theory of conformal mappings.

Keywords: Convolution equations, entire functions, conformal mappings.

AMS Classification (1980): 30 C 20, 30 D 20, 32 A 15, 46 E 25.

0. Introduction.

This paper is devoted to the study of several problems in the theory of convolution equations of entire functions with nonradial growth behavior. The results can be applied, in particular, to finite or infinite order differential equations with constant coefficients. Convolution equations are interesting for both theoretical and applied problems. There is a wide literature on this subject, for excellent treatments of the most important results see Lelong and Gruman [13] and Hörmander [8] . In a certain sense our setting lies between the case of entire functions of exponential type and the case of distributions with compact support. In the first case each nontrivial convolution operator is surjective (see [14] and [15] ), whereas in the second case the concept of a slowly decreasing function yields a characterization of surjective convolution operators (see Ehrenpreis [3] or Hörmander [8] ,for more recent generalizations see also [2] and [16] ). The necessary condition established in this paper is derived from several facts from functional analysis. Apart from the open mapping theorem, one of the most important tools is an appropriate representation of the dual spaces of the Fréchet or dual Fréchet algebras of entire functions, on which the convolution operators are defined (see [6] ).The weight functions considered here are convex and their conjugate functions (see for instance [6] or [10] ) play an important role in the whole setting. The necessary condition is in a certain sense a perturbation of the procedure to pass from the original weight function to its conjugate.

In our cases surjectivity of convolution operators is, up to duality, equivalent to the solution of a corresponding division problem for entire functions (see [1] , [4] , [8] , [14] , [18] , [22]). We use this fact together with some properties of the function z ↦ exp(z^{2}) to find examples of nonsurjective convolution operators for a wide class of weighted Fréchet algebras. J.J. Metzger [17] has earlier obtained counterexamples to the corresponding division problem in a special case and with much more involved techniques from complex analysis.

We will also discuss several sufficient conditions for surjectivity. For one complex variable, we will point out that a certain growth condition for the divisor function, formulated with the help of the first derivatives of related conformal mappings, yields a new sufficient condition. It is easy to see that this condition is always satisfied in the case of radial weight functions. In the nonradial case the examples of nonsurjective convolution operators lead to some new insight into difficult properties of level sets of conformal mappings and yield new results for the variational principles of conformal mappings (see Lavrent’ev [11]).

1. Fréchet and dual Fréchet algebras of entire functions.

Let P = (p_{m})_{m=1}^{∞} be a family of convex functions

p_{m} : ℂ^{n} ⟶ ℝ

such that

p_{m}(z) ≥ p_{m+1}(z)

for each z ∈ ℂ^{n} and for each m ∈ ℕ. Let F_{P} denote the space of all entire functions

f : ℂ^{n} ⟶ ℂ

such that

∥f∥_{m} = sup{|f(z)| exp(−p_{m}(z)) : z ∈ ℂ^{n}} < ∞

for each m ∈ ℕ. F_{P} is endowed with the topology determined by the system of norms (∥.∥_{m})_{m=1}^{∞}. In this way F_{P} becomes a Fréchet space the topology of which is in general stronger than the topology of uniform convergence on the compact subsets of ℂ^{n} .

In [6] it is shown that under suitable conditions on the weights, the dual space F′_{P} of F_{P} is , via Fourier–Borel transform, topologically isomorphic to a dual Fréchet space G_{P^{∗}} of entire functions. We have to suppose that the family P = (p_{m})_{m=1}^{∞} satisfies the following conditions:

(i)

p_{m}^{∗}(w) = sup{ℜ(z.w) − p_{m}(z) : z = (z_{1}, ..., z_{n}) ∈ ℂ^{n}} < ∞

for each w = (w_{1}, ..., w_{n}) ∈ ℂ^{n} and for each m ∈ ℕ , where

z.w = ∑_{k=1}^{n} z_{k}w_{k};

(ii) for each m ∈ ℕ there exist a number l ∈ ℕ and a constant N = N(l, m) > 0 such that

p_{l}(z) ≤ p_{m}(z) − |z| ,

for |z| ≥ N;

(iii) for each m ∈ ℕ there exist a number l ∈ ℕ and a constant N = N(l, m) > 0 such that

sup{p_{l}(z + ζ) : |ζ| ≤ 1} ≤ p_{m}(z),

for |z| ≥ N .

For technical reasons it is also convenient to assume that the weights p_{m} are functions of |x_{1}|, |y_{1}|, ..., |x_{n}|, |y_{n}|, where z_{j} = x_{j} + iy_{j} , 1 ≤ j ≤ n.

By definition, the space F_{P} can be written in the form

F_{P} = ⋂_{m∈ℕ} F_{p_{m}},

where F_{p_{m}} is the space of all entire functions F such that

∥F ∥_{m}^{∗} = sup{|F (w)| exp(−p_{m}^{∗}(w)) : w ∈ ℂ^{n}} < ∞,

for some m ∈ ℕ. Since the family P is decreasing, the family P^{∗} of conjugate functions (p_{m}^{∗})_{m=1}^{∞} is increasing and we can write

G_{P^{∗}} = ⋃_{m∈ℕ} G_{p_{m}^{∗}},

where G_{p_{m}^{∗}} is the space of all entire functions F such that ∥F ∥_{m}^{∗} < ∞ for a fixed m ∈ ℕ.

Our assumptions on the weights imply that the functions

E_{w} : z ⟼ exp(z.w)

are contained in F_{P} for each w ∈ ℂ^{n} . Hence, if T is a continuous linear functional on F_{P} , we can define the Fourier–Borel transform T̂ of T by

T̂(w) = T (E_{w}) , w ∈ ℂ^{n}.

The mapping T ↦ T̂ constitutes the indicated topological isomorphism between F′_{P} with the strong topology and G_{P^{∗}}, further , we have that for each m ∈ ℕ there exist a number l ∈ ℕ and a constant C = C(l, m) > 0 such that

C∥T ∥′_{l} ≤ ∥T̂∥_{m}^{∗} ≤ ∥T ∥′_{m} , T ∈ F′_{P} ,

where

∥T ∥′_{m} = sup{|T (f)| : ∥f∥_{m} ≤ 1}

(see [6] ).

The proof of this result uses several geometric properties of the convex weight functions and, in the main step, Hörmander’s L^{2}–estimates for the solution of the inhomogeneous Cauchy–Riemann differential equations.

1.1.Definition. Let T ∈ F′_{P} be a continuous linear functional. The operator T_{T} defined by

T_{T} (f)(z) = T (f(z + .))

for f ∈ F_{P} and z ∈ ℂ^{n}, is called a convolution operator, we sometimes also write

T_{T} (f) = T ∗ f.

(See [7] ,[8] ,[13] , [14] ).

1.2.Proposition. Suppose that the family P of the weight functions satisfies the following additional condition:

(iv) for each pair (l, m) ∈ ℕ × ℕ there exists a number k ∈ ℕ such that

sup{p_{k}(z + w) − p_{m}(z) − p_{l}(w) : z, w ∈ ℂ^{n}} < ∞.

Then any convolution operator T_{T} , T ∈ F′_{P} , is a continuous linear operator mapping F_{P} into itself.

This follows from the fact that for each T ∈ F′_{P} there exist a continuous function

ψ : ℂ^{n} ⟶ ℂ

and a number l ∈ ℕ such that

T (f) = ∫_{ℂ^{n}} f(w)ψ(w) dλ(w),

for each f ∈ F_{P} ; here λ denotes the Lebesgue measure in ℂ^{n} , and

∫_{ℂ^{n}} |ψ(w)| exp(p_{l}(w)) dλ(w) < ∞

(see [6] and [7] ).

Examples.

(a) Let

p_{m}(z) = a_{1,m}|x_{1}|^{α_{1}} + b_{1,m}|y_{1}|^{β_{1}} + ⋯ + a_{n,m}|x_{n}|^{α_{n}} + b_{n,m}|y_{n}|^{β_{n}},

where (a_{j,m})_{m=1}^{∞} and (b_{j,m})_{m=1}^{∞}, 1 ≤ j ≤ n, are decreasing sequences of positive real numbers, α_{j}, β_{j} > 1, 1 ≤ j ≤ n. These functions satisfy the conditions (i),(ii) and (iii). Condition (iv) is also fulfilled if

lim_{m→∞} a_{j,m} = lim_{m→∞} b_{j,m} = 0 , 1 ≤ j ≤ n.

(b) Also functions growing faster than the functions in example (a) satisfy the conditions (i),(ii) and (iii), for instance

p_{m}(z) = r_{m} exp(|x_{1}|^{γ_{1}} + |y_{1}|^{δ_{1}} + ⋯ + |x_{n}|^{γ_{n}} + |y_{n}|^{δ_{n}}),

where (r_{m})_{m=1}^{∞} is a decreasing sequence of positive real numbers and 0 < γ_{j}, δ_{j} < 1, 1 ≤ j ≤ n.

(c) Let P = (p_{m})_{m=1}^{∞} be a decreasing family of convex functions with the property that for each m ∈ ℕ there exists a number k ∈ ℕ such that

(iv)’ 2p_{k}(z) ≤ p_{m}(z) + Const.,

for each z ∈ ℂ^{n}.

It is easily seen that the corresponding Fréchet space F_{P} is an algebra. If , in addition,

p_{m}(z + w) ≤ C_{m}(p_{m}(z) + p_{m}(w)),

for each z, w ∈ ℂ^{n}, C_{m} > 0 being a constant only depending on m, then condition (iv) follows from (iv)’.

Examples of convolution operators. Let T ∈ F′_{P} be a continuous linear functional which is also continuous in the topology of uniform convergence on the compact subsets of ℂ^{n}, i.e.

|T (f)| ≤ Const. sup{|f(z)| : z ∈ K},

for each f ∈ F_{P}, where K is a compact subset of ℂ^{n}. Using the Taylor series expansion of an entire function it comes out that the corresponding convolution operator can be regarded as a linear differential operator of infinite order with constant coefficients (see [15] ).

In the last example it is essential to suppose that the functional T is also continuous in the weaker topology of uniform convergence on compact subsets of ℂ^{n}, because, in general, the Taylor series expansion does not converge in the stronger topology determined by the system of norms (∥.∥)_{m=1}^{∞} (see [6] ). Hence, on F_{P}, there exist convolution operators which cannot be regarded as differential operators of infinite order, especially when the functional T is not continuous in the weaker topology. When T̂ is a polynomial, the corresponding convolution operator T_{T} is a finite order differential operator with constant coefficients and T_{T} is always surjective (see [13] ,

[14] , [15]).

2. Convolution operators.

We want to find necessary conditons and sufficient conditions for a convolution operator T_{T} to be surjective. For this aim we first consider the adjoint operator of T_{T} and use several facts from functional analysis.

Let f ∈ F_{P} and S, T ∈ F′_{P}. We have

⟨T_{T} (f), S⟩ = ⟨f, T_{T}^{∗}(S)⟩,

we also write

T_{T}^{∗}(S) = T ∗ S.

Let ψ_{1}, ψ_{2} : ℂ^{n} ⟶ ℂ be continuous functions representing the functionals S and T in the form

S(f) = ∫_{ℂ^{n}} f(z)ψ_{1}(z) dλ(z) , T (f) = ∫_{ℂ^{n}} f(z)ψ_{2}(z) dλ(z)

(see [6] ). Applying the Fourier–Borel transform we obtain

{ \atop T_{T}^{∗}(S)}(w) = { \atop T ∗ S}(w) = T̂(w)Ŝ(w),

for w ∈ C^{n}.

2.1.Proposition. Let P = (p_{m})_{m=1}^{∞} satisfy the conditions (i)–(iv) , and let T ∈ F′_{P} . Then the convolution operator

T_{T} : F_{P} ⟶ F_{P}

is surjective if and only if for each m ∈ ℕ there exists a number l ∈ ℕ such that

∥Ŝ∥_{l}^{∗} ≤ Const.∥T̂Ŝ∥_{m}^{∗},

for each S ∈ F′_{P}.

Proof. We use several standard methods from functional analysis (see for instance H.H. Schaefer [21] ). So T_{T} is surjective if and only if T_{T}^{∗} is injective and has closed range. Since T_{T} is continuous (by Proposition 1.2.), T_{T} is surjective if and only if T_{T}^{∗} is injective and open. Now we use the fact that F′_{P} is topologically isomorphic to G_{P^{∗}} via Fourier–Borel transform. It is clear that T_{T}^{∗} is injective and it is easily seen that T_{T}^{∗} is open if and only if for each m ∈ ℕ there exists a number l ∈ ℕ such that

∥Ŝ∥_{l}^{∗} ≤ Const.∥T̂Ŝ∥_{m}^{∗},

for each S ∈ F′_{P} (see [21] ).

From the last result it is quite easy to derive a necessary condition for surjectivity.

2.2.Proposition. Suppose that the family P = (p_{m})_{m=1}^{∞} satisfies the conditions (i)–(iv) and let T ∈ F′_{P}. Now assume that the corresponding convolution operator is surjective. Then for each m ∈ ℕ there exists a number l ∈ ℕ such that

(∗)  sup{|T̂(w)| exp(ℜ(z.w) − p_{m}^{∗}(w)) : w ∈ ℂ^{n}} ≥ Const. exp(p_{l}(z))

for each z ∈ ℂ^{n}.

Proof. We apply Proposition 2.1. taking for the functionals S the point evaluations at z ∈ ℂ^{n}. The Fourier–Borel transforms of these point evaluations are just the exponential functions

E_{z} : w ↦ exp(z.w).

Hence we obtain

exp(p_{l}(z)) = sup{exp(ℜ(z.w) − p_{l}^{∗}(w)) : w ∈ ℂ^{n}}
= ∥E_{z}∥_{l}^{∗} ≤ Const.∥T̂E_{z}∥_{m}^{∗}
= Const. sup{|T (w)| exp(ℜ(z.w) − p_{m}^{∗}(w)) : w ∈ ℂ^{n}}

for each z ∈ ℂ^{n}, which proves the proposition.

Since p_{m}^{∗∗} = p_{m}, condition (∗) means that the procedure to form the convex conjugate function of p_{m}^{∗} must not be disturbed too much by the entire function T̂.

For radial weights it can be shown directly that condition (∗) is always satisfied for a nontrivial functional T ∈ F′_{P} , for more details see [7] .

3. Nonsurjective convolution operators and the division problem.

In this section we determine continuous linear functionals T with the property that the corresponding convolution operator T_{T} is not surjective. We use the duality and certain properties of the function z ↦ exp(z^{2}).

J.J. Metzger [17] found counterexamples to the division problem for the special case of weight functions of the form

p_{m}(z) = \frac{1}{m}(|x|^{α} + |y|^{β}),

where m ∈ ℕ, α, β > 1 , α ≠ β, z = x + iy. His method focusses on a skillful use of the existence of entire functions with a prescribed indicator.

In the sake of simplicity we consider entire functions of one complex variable, but the result can easily be carried over to entire functions of several variables.

3.1.Proposition. Suppose that the family P = (p_{m})_{m=1}^{∞} satisfies the conditions (i)–(iv). Furthermore assume that

p_{m}^{∗}(w) = ρ_{m}^{∗}(u) + σ_{m}^{∗}(v),

where w = u + iv and

ρ_{m}^{∗}(u) ≥ C_{m}|u|^{2}, σ_{m}^{∗}(v) ≤ C′_{m}|v|^{γ_{m}},

for each u, v ∈ ℝ, where C_{m}, C′_{m} > 1 are constants, and 1 < γ_{m} < 2 for each m ∈ ℕ. Then the convolution operator T_{T} : F_{P} ⟶ F_{P} , where

T̂(w) = exp(w^{2}),

is not surjective.

Proof. It suffices to show that the entire function

T̂(w) = exp(w^{2})

does not satisfy condition (*) of Proposition 2.2. First note that T̂ ∈ G_{P^{∗}}, because

∥T̂∥_{m}^{∗} = sup{exp(u^{2} − v^{2} − ρ_{m}^{∗}(u) − σ_{m}^{∗}(v)) : w ∈ ℂ} < ∞,

by the assumption on ρ_{m}^{∗}. Now we have

sup{exp(u^{2} − v^{2} + xu − yv − ρ_{m}^{∗}(u) − σ_{m}^{∗}(v)) : w ∈ ℂ}
= sup{|T̂(w)| exp(ℜ(zw) − p_{m}^{∗}(w)) : w ∈ ℂ}.

We consider the imaginary part only, and take −v instead of v, and obtain

yv − σ_{m}^{∗}(v) − v^{2} ≤ yv − v^{2},

for each v ∈ ℝ, hence

sup{yv − σ_{m}^{∗}(v) − v^{2} : v ∈ ℝ} ≤ sup{yv − v^{2} : v ∈ ℝ} = O(y^{2}),

for each v ∈ ℝ. But

p_{l}(z) = ρ_{l}(x) + σ_{l}(y)

and, by assumption, σ_{l} grows at least as fast as the function

y ↦ |y|^{γ′_{l}},

where

\frac{1}{γ_{l}} + \frac{1}{γ′_{l}} = 1,

and since γ_{l} < 2, it follows that γ′_{l} > 2, for each l ∈ ℕ. This shows that condition (*) is violated.

Next we investigate the relationship between convolution operators and the problem of division in spaces of entire functions.

3.2.Proposition. Let T̂ ∈ G_{P^{∗}} and suppose that for for each F ∈ G_{P^{∗}} such that F/T̂ is an entire function, the quotient F/T̂ lies in G_{P^{∗}}, which means that the division problem for T̂ in G_{P^{∗}} is solvable. Then the convolution operator T_{T} : F_{P} ⟶ F_{P} is surjective.

Proof. The assumption implies that the ideal I(T̂), which is generated by T̂ in G_{P^{∗}}, is closed in G_{P^{∗}}. Hence by a result of Dieudonné and Schwartz (see Horvath [9], pg.308) about duality of mappings, the convolution operator T_{T} is surjective.

3.3.Proposition. Suppose that the weights p_{m}^{∗} have the form

p_{m}^{∗}(w) = m (|u|^{α} + |v|^{β}) ,

where w = u + iv and α, β > 1. Suppose further that the convolution operator T_{T} is surjective. Then the corresponding division problem for T̂ is solvable in G_{P^{∗}}.

Proof. Again by the theorem of Dieudonné and Schwartz we obtain that the ideal (T̂) is closed in G_{P^{∗}}. Now by a result of Metzger [18] we have

I^{−}(T̂) = I_{loc}(T̂),

which means that the division problem for T̂ is solvable in G_{P^{∗}}.

Remark. The conclusion of the last Proposition essentially uses the assumption about the weights and does not hold for the general case.

3.4.Proposition. Suppose that the family P = (p_{m})_{m=1}^{∞} satisfies the conditions (i)–(iv). Furthermore assume that

p_{m}^{∗}(w) = ρ_{m}^{∗}(u) + σ_{m}^{∗}(v),

where w = u + iv and

ρ_{m}^{∗}(u) = C_{m}|u|^{2}, σ_{m}^{∗}(v) = C′_{m}|v|^{γ_{m}},

for each u, v ∈ ℝ, where C_{m}, C′_{m} > 1 are constants , and 1 < γ_{m} < 2 for each m ∈ ℕ. Let Φ(w) = exp(w^{2}) and F ∈ G_{P^{∗}} be such that ΦF ∈ G_{P^{∗}}.

Then the convolution operator T_{T} : F_{P} ⟶ F_{P} , where T̂ = ΦF, is not surjective.

Proof. By assumption we have T̂ = ΦF ∈ G_{P^{∗}}, and \frac{F}{ΦF} = \frac{1}{Φ} is an entire function , but

∥1/ P hi∥_{m}^{∗} = sup{exp(−u^{2} + v^{2} − ρ_{m}^{∗}(u) − σ_{m}^{∗}(v)) : w ∈ ℂ} = ∞,

for each m ∈ N. This means that the division problem for F and ΦF is not solvable in G_{P^{∗}} and hence the corresponding convolution operator T_{T} is not surjective by Proposition 3.3.

The main points of the conclusions in Propositions 3.1. and 3.4. lie in the outstanding properties of the harmonic, positively homogeneous function

(u, v) ↦ u^{2} − v^{2}.

Proposition 3.1. and 3.4. can easily be generalized to the case of entire functions of several variables taking the function

Φ(w_{1},…, w_{n}) = exp(w_{1}^{2} + ⋯ + w_{n}^{2}).

4. A sufficient condition related with conformal mappings.

Throughout this chapter we suppose that n = 1. We consider the case where the conjugated weight functions p_{m}^{∗} are of the form

p_{m}^{∗}(w) = mp^{∗}(w), w ∈ ℂ,

where

p^{∗}(w) = |u|^{α} + |v|^{β}, w = u + iv, α, β > 1.

Then the corresponding space G_{P^{∗}} is an algebra. We want to find a growth condition for an entire function T̂ ∈ G_{P^{∗}}, T̂ ≢ 0, which implies that the corresponding convolution operator

T_{T} : F_{P} ⟶ F_{P}

is surjective.

For this purpose we consider the conformal mappings

ϕ_{R} : D_{R} ⟶ E_{R},

where

D_{R} = {z : |z| < R} , E_{R} = {z : p^{∗}(z) < R};

ϕ_{R} is uniquely determined by the assertions

ϕ_{R}(0) = 0 and ϕ′_{R}(0) > 0.

We can now formulate the main result of this section.

4.1.Proposition. Assume that the family P^{∗} satisfies the above assumptions and let T ∈ F′_{P}, such that T̂ ≢ 0. Suppose that for each R > 0 and for each ζ ∈ E_{R} the inequality

(1)  ∫_{D_{R}} |log |T̂(ϕ_{R}(z) + ζ)|| dλ(z) ≤ A ∫_{D_{R}} |log |T̂(ϕ_{CR}(Cz))|| dλ(z)

holds, where C > 1 and A > 0 are constants independent on R and ζ.

Then the corresponding convolution operator

T_{T} : F_{P} ⟶ F_{P}

is surjective.

Proof. We consider an entire function F ∈ G_{P^{∗}} such that G = F/T̂ is entire. By Proposition 3.2. we have to show that G ∈ G_{P^{∗}}.

It suffices to prove that

sup{log |G(ζ)| : ζ ∈ E_{R}} ≤ Const.R + C_{1},

for each R > 0 , C_{1} > 0 being a constant independent of R.

Set ϕ_{R,ζ}(z) = ϕ_{R}(z − ζ) + ζ, then ϕ_{R,ζ}(ζ) = ζ and

ϕ_{R,ζ} : D_{R,ζ} ⟶ E_{R,ζ},

where D_{R,ζ} = {z : |z − ζ| < R} and E_{R,ζ} = {z : p^{∗}(z − ζ) < R}.

Then, by the mean value property of the subharmonic function log |G ∘ ϕ_{R,ζ}|, we get

log |G(ζ)| = log |G(ϕ_{R,ζ}(ζ))| ≤ \frac{1}{|D_{R}|} ∫_{D_{R,ζ}} log |G(ϕ_{R,ζ}(z))| dλ(z),

where |D_{R}| denotes the Lebesgue measure of D_{R}.

Next we apply the integral transformation induced by the conformal mapping ϕ_{R,ζ} and obtain

(2)  log |G(ζ)| ≤ \frac{1}{|D_{R}|} ∫_{E_{R,ζ}} |ϕ′_{R,ζ}(ϕ_{R,ζ}^{−1}(z))|^{−2} log |G(z)| dλ(z).

We remark that the Jacobian |ϕ′_{R,ζ}(z)|^{2} satisfies

(3)  0 < c_{1,R} ≤ |ϕ′_{R,ζ}(z)|^{2} ≤ c_{2,R},

for each z ∈ D_{R,ζ} , where c_{1,R} and c_{2,R} are constants, depending only on R. This follows from a result due to S.E. Warschawski (see [23] , [5] , [20] ).

Now we get

log |G(ζ)| ≤ \frac{1}{|D_{R}|} ∫_{E_{R,ζ}} |ϕ′_{R,ζ}(ϕ_{R,ζ}^{−1}(z))|^{−2} [log |F (z)| − log |T̂(z)|] dλ(z) ≤

\frac{1}{|D_{R}|} [ sup_{z∈E_{R,ζ}} log |F (z)|] ∫_{E_{R,ζ}} |ϕ′_{R,ζ}(ϕ_{R,ζ}^{−1}(z)|^{−2} dλ(z)+

\frac{1}{|D_{R}|} ∫_{E_{R,ζ}} |ϕ′_{R,ζ}(ϕ_{R,ζ}^{−1}(z))|^{−2} |log |T̂(z)|| dλ(z).

For ζ ∈ E_{R} and z ∈ E_{R,ζ} we get

p^{∗}(z) = p^{∗}(z − ζ + ζ) ≤ C̃(p^{∗}(z − ζ) + p^{∗}(ζ)) ≤ 2C̃R,

further, it is easily seen that

∫_{E_{R,ζ}} |ϕ′_{R,ζ}(ϕ_{R,ζ}^{−1}(z))|^{−2} dλ(z) = |D_{R}|,

and we transform the second integral of the above expression again back, so that we arrive at

sup{log |G(ζ)| : ζ ∈ E_{R}} ≤

sup_{z∈E_{2C̃R}} log |F (z)| + \frac{1}{|D_{R}|} ∫_{D_{R,ζ}} |log |T̂(ϕ_{R}(z − ζ) + ζ)|| dλ(z).

Since F ∈ G_{P^{∗}} we can estimate

sup_{z∈E_{2C̃R}} log |F (z)| ≤ Const.R,

for each R > 0. In the above integral we set z for z − ζ and use assumption (1) to obtain

sup{log |G(ζ)| : ζ ∈ E_{R}} ≤

Const.R + \frac{1}{|D_{R}|}Const. ∫_{D_{CR}} |log |T̂(ϕ_{CR}(z))|| dλ(z).

Assuming as we may that T̂(0) ≠ 0, it suffices to show that

\frac{1}{|D_{R}|} ∫_{D_{R}} |log |T̂(ϕ_{R}(z))|| dλ(z) ≤ Const.R − log |T̂(0)|,

then the desired estimate can easily be derived.

For this last step observe that

log |T̂(z)| ≤ Const.p^{∗}(z) + A,

for each z ∈ ℂ ,where A > 0 is a constant, and apply again the integral transformation induced by the conformal mapping ϕ_{R} to get

\frac{1}{|D_{R}|} ∫_{D_{R}} |log |T̂(ϕ_{R}(z))|| dλ(z) =

\frac{1}{|D_{R}|} ∫_{E_{R}} |ϕ′_{R}(ϕ_{R}^{−1}(z))|^{−2} |log |T̂(z)|| dλ(z) ≤

\frac{1}{|D_{R}|} ∫_{E_{R}} |ϕ′_{R}(ϕ_{R}^{−1}(z))|^{−2} (Const.p^{∗}(z) + A − log |T̂(z)|) dλ(z)+

\frac{1}{|D_{R}|} ∫_{E_{R}} |ϕ′_{R}(ϕ_{R}^{−1}(z))|^{−2} (Const.p^{∗}(z) + A) dλ(z) ≤

Const.R − \frac{1}{|D_{R}|} ∫_{E_{R}} |ϕ′_{R}(ϕ_{R}^{−1}(z))|^{−2} log |T̂(z)| dλ(z).

Finally apply formula (2) for the function T̂ and for ζ = 0 to obtain

\frac{1}{|D_{R}|} ∫_{D_{R}} |log |T̂(ϕ_{R}(z))|| dλ(z) ≤ Const.R − log |T̂(0)|,

which proves the proposition.

Remark. It is not known to the author whether condition (1) is also necessary for surjectivity of the corresponding convolution operator. S. Momm [19] recently obtained a complete characterization of surjective convolution operators by different methods.

In the following corollary we describe a situation in which each nontrivial convolution operator on F_{P} is surjective. The sufficient condition is formulated in terms of the derivative of the conformal mapping ϕ_{R} only ; the constants c_{1,R} and c_{2,R} appearing in formula (3) play an important role in this context.

4.2.Corollary. Suppose that for each R > 0 the inequality

(4)  max_{z∈D_{CR}} |ϕ′_{CR}(z)| ≤ A min_{z∈D_{R}} |ϕ′_{R}(z)|

holds, where A > 0 is a constant, independent of R, and C ≥ 1 satisfies

p^{∗}(z + w) ≤ C(p^{∗}(z) + p^{∗}(w)),

for each z, w ∈ ℂ. Then each nontrivial convolution operator T_{T} : F_{P} ⟶ F_{P} is surjective.

Proof. It is easily seen that condition (4) is equivalent to

(5)  max_{w∈E_{R}} |ϕ′_{R}(ϕ_{R}^{−1}(w))|^{−2} ≤ A min_{w∈E_{CR}} |ϕ′_{CR}(ϕ_{CR}^{−1}(w))|^{−2}

We now show that condition (1) of Proposition 4.1. is satisfied. First we apply the integral transformation corresponding to the conformal mapping ϕ_{R} :

∫_{D_{R}} |log |T̂(ϕ_{R}(z) + ζ)|| dλ(z) = ∫_{E_{R}} |log |T̂(z + ζ)|| |ϕ′_{R}(ϕ_{R}^{−1}(z))|^{−2} dλ(z) ≤

∫_{E_{R}} |log |T̂(z + ζ)|| dλ(z) max_{w∈E_{R}} |ϕ′_{R}(ϕ_{R}^{−1}(w))|^{−2}.

By (5) and by the fact that E_{R} + ζ ⊆ E_{CR}, for each ζ ∈ E_{R} and for each R > 0, we obtain

∫_{E_{R}} |log |T̂(z + ζ)|| dλ(z) max_{w∈E_{R}} |ϕ′_{R}(ϕ_{R}^{−1}(w))|^{−2} ≤

A ∫_{E_{CR}} |log |T̂(z)|| dλ(z) min_{w∈E_{CR}} |ϕ′_{CR}(ϕ_{CR}^{−1}(w))|^{−2} ≤

A ∫_{E_{CR}} |log |T̂(z)|| |ϕ′_{CR}(ϕ_{CR}^{−1}(z))|^{−2} dλ(z) = A ∫_{D_{CR}} |log |T̂(ϕ_{CR}(z))|| dλ(z),

where, in the last step, the inverse integral transformation corresponding to the conformal mapping ϕ_{CR} was carried out. Hence condition (1) of Proposition 4.1. is satisfied, and we are done.

Next suppose that the weight function p^{∗} is radial. We write

p^{∗}(z) = ρ(|z|),

and assume that the function

ρ : ℝ_{+} ⟶ ℝ_{+}

is strictly increasing with inverse function ρ^{−1}. Then we can write the sets E_{R} in the form

E_{R} = {z : |z| < ρ^{−1}(R)}.

The conformal mapping

ϕ_{R} : D_{R} ⟶ E_{R}

has then the form

ϕ_{R}(z) = \frac{ρ^{−1}(R)}{R}z

and

ϕ′_{R}(z) = \frac{ρ^{−1}(R)}{R},

which is independent of z.

In this case condition (4) can be expressed in the following way :

4.3.Corollary. Suppose that the weight function p^{∗} is a radial function ρ and suppose that there exist constants D_{1} > 1 and D_{2} > 1 such that

ρ(R) ≤ \frac{1}{D_{1}} ρ(D_{2}R),

for each R > 0. Then each convolution operator T_{T}, where T̂ ≢ 0, is surjective.

Compare also with [1] and [13] .

As there are examples of nonsurjective convolution operators, condition (4) cannot always be fulfilled. We consider nonradial weight functions, which appeared in connection with the construction of nonsurjective convolution operators of section 3. and get the following result on conformal mappings :

4.4.Corollary. Suppose the weight functions

p_{m}^{∗} = mp^{∗}, m ∈ N,

satisfies the assumptions of Proposition 4.1. Let C > 1 be such that

p^{∗}(z + w) ≤ C(p^{∗}(z) + p^{∗}(w)),

for each z, w ∈ ℂ. Then

inf_{R>0} \frac{min\{|ϕ′_{R}(z)| : z ∈ D_{R}\}}{max\{|ϕ′_{CR}(z)| : z ∈ D_{CR}\}} = 0.

Proof. If the above infimum is greater than zero, then condition (4) would be satisfied and hence each nontrivial convolution operator would be surjective, contradicting Proposition 3.1. or 3.4.

5. Some remarks on a variational principle for conformal mappings.

The above results show that a slight modification of the function p^{∗} in the sense of Propositon 3.1. causes a completely different behavior of the first derivatives of the conformal mapping ϕ_{R}. If one takes

p^{∗}(w) = u^{2} + v^{2} ,

then the first derivative of the corresponding conformal mapping ϕ_{R} is constant in z, in fact we have

ϕ′_{R}(z) = R^{− \frac{1}{2}}

and the infimum appearing in Corollary 4.4. is equal to C^{− \frac{1}{2}} .

If

p^{∗}(w) = u^{2} + |v|^{γ} , for 1 < γ < 2 ,

then Corollary 4.4. shows that the corresponding infimum must be zero.

This is also of some interest in the connection with a variational principle for conformal mappings (see [11] or [12] ).

As above let

p^{∗} : ℂ ⟶ ℝ_{+}

denote a convex function, and let E_{R} be the the level sets

E_{R} = {z : p^{∗}(z) < R} .

We suppose that 0 ∈ E_{R}. Let further

ψ_{R} : E_{R} ⟶ D

be the conformal mappings of E_{R} to the unit disk D, with the property

ψ_{R}(0) = 0 .

There are infinitely many functions having this property, but all of them differ only by a factor e^{iθ}, where θ ∈ ℝ is arbitrary.

We assume now that the boundary curves of the sets E_{R} have representations in polar coordinates around 0. Let r = r(φ) be a polar representation for the boundary curve of E_{R} and r̃ = r̃(φ) a polar representation for the boundary curve of ER̃ , 0 < R̃ < R. Each point z_{0} = r_{0}e^{iφ_{0}} of ∂E_{R} and the corresponding point z̃_{0} = r̃_{0}e^{iφ_{0}} of ∂ER̃, where the value of

|log \frac{r̃(φ)}{r(φ)}|

attains a maximum is called a point of largest deformation and the number

λ = \frac{r̃(φ_{0})}{r(φ_{0})}

is called the largest deformation of the boundary curve of E_{R} with respect to the boundary curve of ER̃.

Now it follows that

|ψ′ ; R̃(z̃_{0})| ≥ \frac{1}{λ}|ψ′_{R}(z_{0})|

(see [11] or [12] ).

If C > 1, we get

|\frac{ψ′_{CR}(z_{0})}{ψ′_{R}(z̃_{0})}| ≤ λ ,

where λ is the largest deformation. In the case of the functions

p^{∗}(z) = |x|^{α} + |y|^{β} ,

where α, β > 1 , α < β , we get for the largest deformation of the boundary curve of E_{CR} with respect to the boundary curve of E_{R}

λ = C^{\frac{1}{α}} ,

which is independent of R.

For the cases of functions p^{∗} appearing in Corollary 4.4. we can determine the behavior of the expression

\frac{min\{|ψ′_{CR}(z)| : z ∈ E_{CR}\}}{max\{|ψ′_{R}(z)| : z ∈ E_{R}\}}

as a function of R. It follows that this expression tends to zero when R tends to infinity.

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INSTITUT FÜR MATHEMATIK, UNIVERSITÄT WIEN, STRUDLHOFGASSE 4, A–1090 WIEN, AUSTRIA.