PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 86(100) (2009), 5–20 DOI: 10.2298/PIM0900005D

BOUNDEDNESS OF THE BERGMAN PROJECTIONS ON L^{p} SPACES WITH RADIAL WEIGHTS

Milutin Dostanić

Communicated by Stevan Pilipović

Abstract. Necessary as well as sufficient conditions are given for the Bergman projections to be bounded operators on L^{p} spaces on the unit disc.

Let w be a continuous positive function on [0, 1) such that lim_{r→1−0} w(r) = 0. Denote by dμ(z) the measure dμ(z) = w(|z|) dA(z), where dA(z) is Lebesgue’s measure (dA(z) = dx dy, z = x + i y) on the unit disc D. Denote by L^{p}(D, dμ) (or L^{p}(D) for short), 1 ⩽ p < ∞ the set of all complex measurable functions f for which ∥f∥_{p} = ( ∫_{D} |f(z)|^{p}dμ(z))^{1/p} < ∞ and by L_{a}^{p}(D, dμ) (or L_{a}^{p}(D) for short) the subspace of the space L^{p}(D) comprising the functions that are analytic on D.

If p = 2, L_{a}^{2}(D) is a Hilbert subspace of L^{2}(D) and it is called Bergman space. Let P denote the orthogonal projector of L^{2}(D) on L_{a}^{2}(D) (Bergman projection). Let {δ_{n}}_{n=0}^{∞} be defined by δ_{n} = (2π ∫_{0}^{1} r^{2n+1} w(r) dr)^{1/2} . Then, the sequence of functions {z^{n}/δ_{n}}_{n=0}^{∞} is an orthonormal basis of L_{a}^{2}(D) and so the corresponding Bergman reproducing kernel is given by K(z, ξ) = ∑ _{n=0}^{∞} z^{n} \overline{ξ}^{n}/δ_{n}^{2} (z, ξ ∈ D). Let I_{k}^{n} = [ \frac{k−1}{n} , \frac{k}{n} ] (k = 1, 2, …, n), Φ(λ) = ∫_{0}^{1} t^{2λ+1} w(t) dt (λ ∈ (0, +∞)), G(λ) = \frac{Φ(λ+1)}{Φ(λ)} and

H_{kν}^{n} (λ) = ∫_{I_{k}^{n} ×I_{ν}^{n}} (xy)^{λ}(xy − G(λ))(x w(x))^{1/p}(y w(y))^{1/q}dx dy ( p > 1, \frac{1}{p} + \frac{1}{q} = 1) .

Operator P : L^{2}(D) −→ L_{a}^{2}(D) acts in the following way:

P f(z) = ∫_{D} K(z, ξ) f(ξ) dμ(ξ), z ∈ D.

We will use the same notation: f ∈ L^{p}(D), P f = ∫_{D} K(·, ξ)f(ξ) dμ(ξ), 1 < p < ∞. The same notation, P, will be used for this mapping.

2000 Mathematics Subject Classification: 47B38.

Partially supported by MNZZS, grant ON144010.

In this paper we consider conditions for w so that P is a bounded operator on L^{p}(D) (1 < p < ∞).

The boundedness of the Bergman projection is a fact of fundamental importance. In the case of the unit disc, boundedness of the Beergman projection was studied in [3], [8] and it immediately gives the duality between the Bergman spaces. Also, the boundedness of the Bergman projection is used to establish corresponding theorems that concern duality and interpolation of analytic Besov spaces [8, Th. 5.3.6, Th. 5.3.8, pp. 94–97]

1. The Main Result

Let w ∈ C[0, 1), w > 0 in [0, 1), lim_{r→1−0} w(r) = 0 and let the function T : [0, 1) → ℝ be defined by T (r) = sup_{m⩾0} r^{m} /δ_{m}^{2}, r ∈ [0, 1). Assume that for some c_{0} ∈ (0, 1) all the functions H_{kν}^{n} have uniformly bounded number of zeroes on the interval (1, +∞) for all I_{k}^{n}, I_{ν}^{n} ⊂ [c_{0}, 1]. The main result is given by the following Theorem:

Theorem 1.1. (a) The necessary condition for operator P to be bounded on L^{p}(D) (1 < p < ∞) is:

(1.1) sup_{m⩾0} \frac{1}{δ_{m}^{2}} ( ∫_{0}^{1} x^{mp+1} w(x) dx)^{1/p}( ∫_{0}^{1} x^{mq+1} w(x) dx)^{1/q}< ∞, \frac{1}{p} + \frac{1}{q} = 1.

(b) If condition (1.1) holds, then the condition

(1.2)  \{ \begin{matrix}sup_{0⩽x<1} w(x)^{1/p} ∫_{0}^{1} T (xy)w(y)^{1/q} dy < +∞ \\ sup_{0⩽x<1} w(x)^{1/q} ∫_{0}^{1} T (xy)w(y)^{1/p} dy < +∞ \end{matrix}

is sufficient for the Bergman projection P to be a bounded operator on L^{p}(D) (1 < p < ∞).

Remark 1.1. From condition (1.1) it follows that ∀r ∈ [0, 1) and ∀m ⩾ 0:

\frac{1}{δ_{m}^{2}} ( ∫_{\sqrt{r}}^{1} r^{(mp+1)/2} w(x) dx)^{1/p} · ( ∫_{\sqrt{r}}^{1} r^{(mq+1)/2} w(x) dx^{)1/q}

⩽ \frac{1}{δ_{m}^{2}} ( ∫_{\sqrt{r}}^{1} x^{mp+1} w(x) dx)^{1/p} · ( ∫_{\sqrt{r}}^{1} x^{mq+1} w(x) dx)^{1/q} ⩽ M_{0} (< +∞)

i.e., T (r) · \sqrt{r} ∫_{√ \overline{r}}^{1} w(x) dx ⩽ M_{0} and so

(1.3)  T (r) = O ( ( ∫_{\sqrt{r}}^{1} w(x) dx)^{−1} ) as r → 1 − .

Therefore if condition (1.1) holds, then the sufficient condition (1.2) can be replaced (keeping in mind (1.3)) by more operative condition:

(1.2′)  \{ \begin{matrix}sup_{0⩽x<1} w(x)^{1/p} ∫_{0}^{1} _{\frac{w(y)^{1/q}}{∫_{\sqrt{xy}}^{1} w(t) dt}} dy < +∞ \\ sup_{0⩽x<1} w(x)^{1/q} ∫_{0}^{1} _{\frac{w(y)^{1/p}}{∫_{\sqrt{xy}}^{1} w(t) dt}} dy < +∞ \end{matrix}

Example 1.1. Let w(r) = (1 − r^{2})^{α} L( \frac{1}{1−r^{2}} ), α > 0, where L is a slowly

varying nondecreasing C^{1} function. In a similar way as in [1] we can show that for the function K(λ) = ∫_{0}^{1} x^{λ}w(x) dx, λ > 0, there holds K(λ) ≍ \frac{L(λ)}{λ^{α+1}} , λ → +∞ i.e., there exists constants C_{1}, C_{2} > 0 independent of λ such that C_{1} ⩽ K(λ)\frac{λ^{α+1}}{L(λ)} ⩽ C_{2} for λ ⩾ λ_{0}. From that it follows directly that the weight w satisfies the necessary condition of Theorem 1. Let us prove that w satisfies the sufficient conditions of Theorem 1. It is enough to check the conditions in (1.2′). We want to prove that

(1.4)  sup_{0⩽x<1} w^{1/p}(x) ∫_{0}^{1} w^{1/q}(y)( ∫_{\sqrt{xy}}^{1} w(t) dt)^{−1} dy < ∞.

In a similar way we can prove that

sup_{0⩽x<1} w^{1/q}(x) ∫_{0}^{1} w^{1/p}(y)( ∫_{\sqrt{xy}}^{1} w(t) dt)^{−1} dy < ∞.

Since

∫_{r}^{1} w(t) dt∼ \frac{1}{2} ∫_{r^{2}}^{1}(1 − t)^{α} L( \frac{1}{1 − t} ) dt

= \frac{1}{2} ∫_{1/(1−r^{2} )}^{+∞} \frac{L(u)}{u^{α+2}} du ∼ \frac{1}{2(α + 1)} (1 − r^{2}) ^{α+1} L( \frac{1}{1 − r^{2}} ) , r → 1−

(here we used the asymptotic formula ∫_{x}^{∞} \frac{L(u)}{u^{α+2}} du ∼ \frac{L(x)}{(α+1)x^{α+1}} ), we conclude that (1.4) is true if

(1.5)  sup_{0⩽x<1} w(x)^{1/p} ∫_{0}^{1} \frac{w(y)^{1/q}dy}{(1 − xy) ^{α+1} L(1/(1 − xy))} < ∞

is true. After the change of variables x = e^{−u} , y = e^{−v} in (1.5) we obtain

(1.6)  sup_{u>0}w (e^{−u} )^{1/p} ∫_{0}^{+∞} \frac{w (e^{−v} )e^{−v}}{(1 − e^{−(u+v)} ) ^{α+1} L (1/(1 − e^{−(u+v)} ))} dv < ∞.

Since L is a slowly varying function, we have

lim_{u→0+} \frac{L (1/(1 − e^{−2u} )) · (1 − e^{−2u} ) ^{α}}{(2u)^{α} L(1/u)} = 1

lim_{u→0+} \frac{L (1/(1 − e^{−u} )) · (1 − e^{−u} ) ^{α+1}}{u^{α+1} L(1/u)} = 1

and so, inequality (1.6) is satisfied if

(1.7)  sup_{0<u⩽1} (u^{α} L(1/u))^{1/p} ∫_{0}^{1} \frac{(v^{α} L ( \frac{1}{v}))^{1/q}}{(u + v)^{α+1} L ( \frac{1}{u+v} )} dv < ∞

is true. Let

A(u) = (u^{α} L(1/u))^{1/p} ∫_{0}^{1} \frac{(v^{α} L ( \frac{1}{v}))^{1/q}}{(u + v)^{α+1} L ( \frac{1}{u+v} )} dv.

It is clear that inequality (1.7) is true if the function A is bounded in a neighborhood of the point u = 0. After changing variables v = u · t, t ∈ (0, \frac{1}{u}) we obtain

A(u) = ∫_{0}^{1/u} \frac{t^{α/q}}{(1 + t)^{α+1}} \frac{(L ( \frac{1}{u} ))^{1/p} · (L ( \frac{1}{u} · \frac{1}{t}))^{1/q}}{L( \frac{1}{u} · \frac{1}{t+1} )}dt.

Let \frac{1}{u} = λ. Then the boundedness of function A in a neighborhood of the point u = 0 is equivalent to the boundedness of function λ ↦ A ( \frac{1}{λ} ) in a neighborhood of λ = +∞.

If t = \frac{1}{s}, we get

A( \frac{1}{λ}) = ∫_{1/λ}^{+∞} \frac{s^{α/p −1}}{(1 + s)^{α+1}} \frac{(L(λ))^{1/p} · (L(λs))^{1/q}}{L(λ · \frac{s}{s+1} )} ds = ∫_{\frac{1}{λ}}^{1}(·) ds + ∫_{1}^{+∞} (·) ds

= H_{1}(λ) + H_{2}(λ).

Now, we prove that H_{1} and H_{2} are bounded functions in a neighborhood of λ = +∞. From s ⩽ 1 it follows sλ ⩽ λ and we have L(λs) ⩽ L(λ) (L non-decreases) and

H_{1}(λ) ⩽ L(λ) ∫_{1/λ}^{1} \frac{t^{α/p −1}}{(1 + t)^{α+1}} \frac{ds}{L(λ · \frac{s}{s+1} )}.

After changing variables t = \frac{s}{s+1} , we obtain from the previous inequality

H_{1}(λ) ⩽ L(λ) ∫_{λ/(λ+1)}^{1/2} t^{α/p −1} (1 − t)^{α/q} \frac{dt}{L(λt)}

⩽ L(λ) ∫_{λ/(λ+1)}^{1/2} t^{α/p −1} \frac{dt}{L(λt)} (change λt = ξ)

= \frac{L(λ)}{λ^{α/p}} ∫_{λ/(λ+1)}^{\frac{λ}{2}} ξ^{α/p −1} \frac{dξ}{L(ξ)}

= \frac{L(λ)}{λ^{α/p}} ∫_{λ/(λ+1)}^{1} \frac{ξ^{α/p −1}}{L(ξ)} dξ + \frac{L(λ)}{λ^{α/p}} ∫_{1}^{λ/2} \frac{ξ^{α/p −1}}{L(ξ)} dξ = R_{1}(λ) + R_{2}(λ).

It is clear that lim_{λ→+∞} R_{1}(λ) = 0 and it follows that R_{1} is a bounded function in a neighborhood of λ = +∞. Since L is a slowly varying function we have

lim_{λ→+∞} \frac{λL′(λ)}{L(λ)} = 0

and hence

lim_{x→+∞} \frac{L(x)}{x^{α/p}} ∫_{1}^{x} \frac{t^{α/p −1}}{L(t)} dt = \frac{p}{λ}

From the previous equality follows that there exists lim_{λ→+∞} R_{2}(λ) and is finite. So, H_{1} is a bounded function in a neighborhood of λ = +∞.

If 1 ⩽ s < ∞, then \frac{1}{2} ⩽ \frac{s}{s+1} < 1 and we have L( \frac{λ}{2} ) ⩽ L(λ\frac{s}{s+1} ) and

H_{2}(λ) ⩽ \frac{(L(λ))^{1/p}}{L(λ/2)} ∫_{1}^{+∞} \frac{s^{α/p −1}}{(1 + s)^{α+1}} (L(λs))^{1/q}ds.

Having in mind that lim_{λ→∞} \frac{L(λ)}{L(λ/2)} = 1 and

lim_{λ→∞} ∫_{1}^{+∞} \frac{s^{α/p −1}}{(1 + s)^{α+1}} ( \frac{L (λs)}{L(λ)} )^{1/q}ds = ∫_{1}^{+∞} \frac{s^{α/p −1}}{(1 + s)^{α+1}} ds

[5, Th. 2.6, pp. 63–64], we obtain

lim_{λ→∞} \frac{(L(λ))^{1/p}}{L(λ/2)} ∫_{1}^{+∞} \frac{s^{α/p −1}}{(1 + s)^{α+1}} (L(λs))^{1/q}ds = ∫_{1}^{+∞} \frac{s^{α/p −1}}{(1 + s)^{α+1}} ds < ∞.

So, H_{2} is also a bounded function in a neighborhood of λ = +∞. (In the case L(λ) ≡ 1 the corresponding result is derived in [3, p. 10].)

If, for instance L(x) = (lnx)^{β} (β ⩾ 0, x ⩾ 1), then the functions H_{kν}^{n} (which correspond to the weights w(x) = (1 − x^{2}) ^{α} (ln(1/(1−x^{2})))^{β}) satisfy the condition on uniform boundedness of the number of zeroes on (1, +∞). (This follows from [2, Theorem 1.7, pp. 76–78].)

Example 1.2. Let w(r) = (1−r^{2})^{A} exp(−B/(1 − r^{2})^{α} ) , A ∈ ℝ, B > 0, α > 0.

In [1] it was demonstrated that for the function Φ(λ) = ∫_{0}^{1} r^{λ}w(r) dr the following asymptotic formula holds:

(1.8)  Φ(λ) ∼ C · λ^{D} exp(−E · λ^{α/(α+1)} ), λ → ∞.

(C, D, E are constants that depend only on A, B and α; E > 0).

From (1.8) it follows that the necessary condition (1.1) is not satisfied (except p = q = 2). This means that the corresponding Bergman projection is not bounded on L^{p}(D) for any p ≠ 2.

Corollary 1.1. If function w satisfies conditions (1.1) and (1.2) (i.e., (1.1) and (1.2′)), then the dual of the space L_{a}^{p}(D), (1 < p < ∞) is the space L_{a}^{q}(D), under the integral pairing

⟨f, g⟩ = ∫_{D} f(z) \overline{g(z)}dμ(z), f ∈ L_{a}^{p}(D), g ∈ L_{a}^{q}(D), ( \frac{1}{p} + \frac{1}{q} = 1)

Note that the identification isomorphism (L_{a}^{p}(D))^{∗} = L_{a}^{q}(D) need not be isometric for p ≠ 2.

2. Proof

In the proof of the main result we use the following two Theorems:

Theorem 2.1 (Marcinkiewicz’s Theorem [7, pp. 346–348]). Let λ_{0}, λ_{1} … be a sequence such that for a constant M holds: ∑ _{j=2^{ν}}^{2^{ν+1} −1} |λ_{j} −λ_{j+1} | ⩽ M and |λ_{ν}| ⩽ M for all ν = 0, 1, 2, . . .. Then for any p > 1

∥∑_{ν⩾0} λ_{ν} c_{ν} e^{iνx} ∥_{p} ⩽ M · A(p)∥∑_{ν⩾0} c_{ν} e^{iνx} ∥_{p}.

Here A(p) is a constant that depends only on p and ∥·∥_{p} is a norm in the space L^{p}(0, 2π).

Theorem 2.2 (Schur’s test [3, p. 9]). Denote by ℝ_{p}^{n} (p > 1) the space ℝ^{n} with the norm ∥x∥_{p} = \sqrt[p]{∑ _{i=1}^{n} |x_{i}t|^{p}}, x = (x_{1}, …, x_{n}) and let

C_{n} = ( \begin{matrix}c_{11}^{(n)} & c_{12}^{(n)} & ⋯ & c_{1n}^{(n)} \\ ⋮ & ⋮ & ... & ⋮ \\ c_{n1}^{(n)} & c_{n2}^{(n)} & ⋯ & c_{nn}^{(n)} \end{matrix} ) : ℝ_{p}^{n} → ℝ_{p}^{n}.

If there exists a constant λ (λ < ∞) which does not depend on n such that

max_{1⩽k⩽n} ∑_{ν=1}^{n} |c_{kν}^{(n)} | ⩽ λ, max_{1⩽ν⩽n} ∑_{k=1}^{n} |c_{kν}^{(n)} | ⩽ λ

then for any n ∈ ℕ and any x ∈ ℝ_{p}^{n} holds ∥C_{n}x∥_{p} ⩽ λ ∥x∥_{p} .

Lemma 2.1. Let {j_{mn} }_{n=1}^{∞} be the positive zeroes of the Bessel function of the first kind J_{m} (m = 0, ±1, ±2, …). Then, for any m ∈ ℤ and α, β ⩾ 0, the system of functions {J_{m}(j_{mn} r) · r^{α} · w(r)^{β}}_{n=1}^{∞} is complete in L^{p}(0, 1), (1 ⩽ p < ∞).

Proof. It is sufficient to prove the lemma for m ⩾ 0 since (−1)^{m}J_{−m} = J_{m} for all m ∈ ℤ. Suppose that m ⩾ 0 and that the system {J_{m}(j_{mn} r) · r^{α} · w(r)^{β}}_{n=1}^{∞} is not complete in L^{p}(0, 1). This would mean that there exists function g ∈ L^{q}(0, 1) ( \frac{1}{p} + \frac{1}{q} = 1) such that

(2.1)  ∫_{0}^{1} g(r) r^{α} w(r)^{β}J_{m}(j_{mn} r) dr = 0

for all n = 1, 2, 3, … and g ≠ 0 in L^{q}(0, 1). Let us show that g = 0 almost everywhere on [0, 1] (i.e., g = 0 in L^{q}(0, 1)). Define the function

G_{m}(λ) = \frac{1}{J_{m}(λ)} ∫_{0}^{1} g(r) r^{α} w(r)^{β}J_{m}(λr) dr, λ ∈ ℂ.

As all the zeros of the function λ ↦ J_{m}(λ)/λ^{m} are real and simple, from (2.1) it follows that G_{m} is an entire function. Furthermore, its order of growth is not bigger than 1. From the asymptotic formula (see [6])

J_{m}(z) = \frac{1}{\sqrt{πz}} [cos( z − \frac{mπ}{2} − \frac{π}{4}) + O( \frac{1}{z}) ], |z| → ∞, | argz| ⩽ π − ε

it follows that

(2.2)  lim_{r→±∞} G_{m}(re^{± iπ/4} ) = 0.

From (2.2), by the Phragmen–Lindelöf and Liouville Theorem, it follows that G_{m}(λ) ≡ 0 i.e.,

∫_{0}^{1} g(r) r^{α} w(r)^{β}J_{m}(λr) dr ≡ 0.

Now by representing function J_{m} by the power series and applying the Müntz–Szasz theorem, we get g(r) = 0 almost everywhere on [0, 1]. □

Lemma 2.2. Let z = re^{iθ} and f_{mn} (z) = J_{m}(j_{mn} r)e^{imθ} (m ∈ ℤ, n ∈ ℕ). For

any 1 < p < ∞ the system of functions {f_{mn} }_{m∈ℤ, n∈ℕ} is complete in L^{p}(D).

Proof. It follows by the standard method, from Lemma 1 and the completeness of the system {e^{imx} }_{m∈ℤ} in L^{q}(0, 2π). □

Let A_{m} : L^{p}(0, 1) → L^{p}(0, 1), (m ⩾ 0, 1<p<∞) be linear operators defined by

A_{m}f(x) = \frac{2π}{δ_{m}^{2}} x^{m+1/p} w(x)^{1/p} ∫_{0}^{1} y^{m+1/q} w(y)^{1/q}f(y) dy ( \frac{1}{p} + \frac{1}{q} = 1) .

Lemma 2.3. The Bergman projection P is a bounded operator on L^{p}(D) (1 < p < ∞) if and only if there exists a constant c_{p} (depending only on p) such that

(2.3)  ∫_{0}^{1} ∫_{0}^{2π}|∑_{m⩾0} e^{imx} A_{m}Φ_{m}(y)|^{p}dx dy ⩽ c_{p}^{p} ∫_{0}^{1} ∫_{0}^{2π}|∑_{m⩾0} e^{imx} Φ_{m}(y)|^{p}dx dy

for any (finite) choice of functions Φ_{m} ∈ L^{p}(0, 1).

Proof. By the direct evaluation one gets

(2.4)  (P f_{mn} )(z) = \{ \begin{matrix}0, & m < 0 \\ \frac{2π}{δ_{m}^{2}} z^{m} ∫_{0}^{1} r^{m+1} w(r) J_{m}(j_{mn} ) dr, & m ⩾ 0. \end{matrix}

Let

(2.5)  f = ∑_{m,n} c_{mn} f_{mn}

be a finite sum. Since, according to Lemma 2, the system {f_{mn} }_{m∈ℤ, n∈ℕ} is complete in L^{p}(D), it follows that P is a bounded operator if and only if there exists a constant B(p) (which depends only on p) such that the inequality

(2.6)  ∥P f∥_{p} ⩽ B(p) ∥f∥_{p}

holds for any choice of function f of type (2.5). Expanding (2.6) (keeping in mind (2.4) and (2.5)) we get

(2.7) ∫_{0}^{1} ∫_{0}^{2π}|∑_{n} ∑_{m⩾0} 2πc_{mn} \frac{r^{m}e^{imθ}}{δ_{m}^{2}} w(r)^{1/p} r^{1/p} ∫_{0}^{1} y^{m+1} w(y) J_{m}(j_{mn} y) dy|^{p}dr dθ

⩽ B(p)^{p} ∫_{0}^{1} ∫_{0}^{2π}|∑_{m} e^{imθ} ∑_{n} c_{mn} J_{m}(j_{mn} r) r^{1/p}w(r)^{1/p} |^{p}dr dθ.

Letting

(2.8)  Φ_{m}(r) = ∑_{n} c_{mn} r^{1/p}w(r)^{1/p}J_{m}(j_{mn} r),

we get

∑_{n} 2πc_{mn} \frac{r^{m}}{δ_{m}^{2}} w(r)^{1/p} r^{1/p} ∫_{0}^{1} y^{m+1} w(y) J_{m}(j_{mn} y) dy = A_{m}Φ_{m}(r)

and hence, the inequality (2.7) becomes

(2.9)  ∫_{0}^{1} ∫_{0}^{2π}|∑_{m⩾0} e^{imθ} A_{m}Φ_{m}(r)|^{p}dr dθ ⩽ B(p)^{p} ∫_{0}^{1} ∫_{0}^{2π}|∑_{m} e^{imθ} Φ_{m}(r)|^{p}dr dθ.

Since the system of functions {r^{1/p}w(r)^{1/p}J_{m}(j_{mn} r)} _{n=1}^{∞} is complete in L^{p}(0, 1) for every m ∈ ℤ (Lemma 1), then (2.6) implies (2.9) not only for functions of type (2.8) but also for arbitrary Φ_{m} ∈ L^{p}(0, 1) (because all the operators A_{m} are bounded and the sums in (2.9) are finite).

Conversely, if (2.9) holds for arbitrary (finite) choice of Φ_{m} ∈ L^{p}(0, 1), then by choosing Φ_{m} as in (2.8) we get that (2.6) holds for functions f of type (2.5). Since the system {f_{mn} }_{m∈ℤ, n∈ℕ} is complete in L^{p}(D), (by Lemma 2) (2.6) holds for every f ∈ L^{p}(D) and, hence, P is a bounded operator. Therefore, P is bounded on L^{p}(D) if and only if (2.9) holds for any (finite) choice of Φ_{m} ∈ L^{p}(0, 1). Let us now show that (2.9) is equivalent with the following inequality:

(2.10)  ∫_{0}^{1} ∫_{0}^{2π}|∑_{m⩾0} e^{imθ} A_{m}Φ_{m}(r)|^{p}dr dθ ⩽ c_{p}^{p} ∫_{0}^{1} ∫_{0}^{2π}|∑_{m⩾0} e^{imθ} Φ_{m}(r)|^{p}dr dθ.

for any (finite) choice of Φ_{m} ∈ L^{p}(0, 1) (where c_{p} depends only on p). By letting Φ_{m} = 0 for m < 0 in (2.9), we obtain (2.10) with c_{p} = B(p).

Conversely, suppose that (2.10) holds. By the Riesz Theorem [7], the following inequality holds

∫_{0}^{2π}|∑_{m⩾0} e^{i mθ} Φ_{m}(r)|^{p}dθ ⩽ K_{1}(p)^{p} ∫_{0}^{2π}|∑_{m} e^{i mθ} Φ_{m}(r)|^{p}dθ

(one may put K_{1}(p) = 1/ sin(π/p) based on Hollenbeck–Verbitsky result [4]), then by integrating (over r) the previous inequality we obtain

(2.11)  ∫_{0}^{1} ∫_{0}^{2π}|∑_{m⩾0} e^{imθ} Φ_{m}(r)|^{p}dr dθ ⩽ K_{1}(p)^{p} ∫_{0}^{1} ∫_{0}^{2π}|∑_{m} e^{imθ} Φ_{m}(r)|^{p}dr dθ.

So, from (2.10) and (2.11) it follows that (2.9) holds with B(p) = c_{p} · K_{1}(p) Thus, Lemma 3 is proven. □

Lemma 2.4. The sequence (θ_{mk}^{nν})_{m=0}^{∞} (n ∈ ℕ, m ⩾ 0, ν, k ∈ {1, 2, …, n}) has a uniformly bounded number of the intervals of monotonicity.

Proof. Since

θ_{m+1,k}^{nν} − θ_{m,k}^{nν} = n( ∫_{0}^{1} x^{2m+3} w(x) dx ∫_{0}^{1} x^{2m+1} w(x) dx) ^{−1} · Φ(m) · H_{kν}^{n} (m)

we have, by the hypothesis on zeroes of the functions H_{kν}^{n}, that the sequence (θ_{mk}^{nν})_{m=0}^{∞} has uniformly bounded number of the intervals of monotonicity, for all n, k, ν such that I_{k}^{n}, I_{ν}^{n} ⊂ [c_{0}, 1]. If at least one of the intervals I_{k}^{n}, I_{ν}^{n} is contained in [0, c_{0}], then

H_{kν}^{n} (λ) ⩽ (c_{0} − G(λ)) ∫_{I_{k}^{n} ×I_{ν}^{n}} (xy)^{λ}(x w(x))^{1/p}(y w(y))^{1/q}dx dy.

Having in mind that the function λ ↦ G(λ) is increasing and that lim_{λ→+∞} G(λ) = 1, we have H_{kν}^{n} (λ) < 0 for all λ ⩾ λ_{0} = λ_{0}(c_{0}). Hence, the sequence (θ_{mk}^{nν})_{m=0}^{∞} decreases for m ⩾ [λ_{0}] + 1 and so it has uniformly bounded number of the intervals of monotonicity □

Lemma 2.5. Let

θ_{mk}^{nν} = \frac{2π}{δ_{m}^{2}} · n · ∫_{(k−1)/n}^{k/n} x^{m+1/p} w(x)^{1/p}dx · ∫_{(ν−1)/n}^{ν/n} y^{m+1/q} w(y)^{1/q}dy

(n ∈ ℕ, m ⩾ 0, ν, k ∈ {1, 2, …, n}). Then

∥∑_{m⩾0} θ_{mk}^{nν} a_{m} e^{imx} ∥_{p} ⩽ c_{kν}^{(n)} ∥∑_{m⩾0} a_{m} e^{imx} ∥_{p} (p > 1),

where

c_{kν}^{(n)} = 4N_{0}πn · A(p) · ∫_{(k−1)/n}^{k/n} dx ∫_{(ν−1)/n}^{ν/n} T t(xy) w(x)^{1/p}w(y)^{1/q} dy

and A(p) is the constant from Theorem 2 (N_{0} does not depend on n, k, ν).

Proof. It is sufficient to show that the sequence {θ_{mk}^{nν} }_{m=0}^{∞} satisfies the conditions of Theorem 2. By Lemma 4, there is a positive number N_{0} such that sequence {θ_{mk}^{nν} }_{m=0}^{∞} , has the number of the intervals of monotonicity not greater than N_{0} for n ∈ ℕ and every k, ν ∈ {1, 2, …, n}.

Let s ∈ ℕ. Then the following holds

∑_{j=2^{s}}^{2^{s+1} −1} |θ_{jk}^{nν} − θ_{j+1,k}^{nν} | ⩽ 2N_{0} sup_{m⩾0} |θ_{mk}^{nν} |

= sup_{m⩾0} 4N_{0}πn ∫_{(k−1)/n}^{k/n} dx ∫_{(ν−1)/n}^{ν/n} \frac{(xy)^{m}}{δ_{m}^{2}} w(x)^{1/p}w(y)^{1/q}x^{1/p}y^{1/q}dy

⩽ 4N_{0}πn ∫_{(k−1)/n}^{k/n} dx ∫_{(ν−1)/n}^{ν/n} T (xy) w(x)^{1/p}w(y)^{1/q}dy.

BOUNDEDNESS OF THE BERGMAN PROJECTIONS 15

The proof of the Lemma is completed by applying Theorem 2 with

M = 4N_{0}πn ∫_{(k−1)/n}^{k/n} dx ∫_{(ν−1)/n}^{ν/n} T (xy) w(x)^{1/p}w(y)^{1/q}dy.  □

Lemma 2.6. Let {a_{mk}^{(n)} } be complex numbers (0 ⩽ m ⩽ N, 1 ⩽ k ⩽ n). If the condition (1.2) is satisfied, then there exists a constant B_{p} (which depends only on p) such that

(2.12)  ∑_{k=1}^{n} ∫_{0}^{2π}|∑_{ν=1}^{n} ( ∑_{m=0}^{N} θ_{mk}^{nν} a_{mν}^{(n)} e^{imx} ) |^{p}dx ⩽ B_{p}^{p} ∑_{ν=1}^{n} ∫_{0}^{2π}|∑_{m=0}^{N} a_{mν}^{(n)} e^{imx} |^{p}dx.

Proof. According to Lemma 5, the following holds

(2.13) ∥∑_{m=0}^{N} θ_{mk}^{nν} a_{mν}^{(n)} e^{imx} ∥_{p} ⩽ c_{kν}^{(n)} ∥∑_{m=0}^{N} a_{mν}^{(n)} e^{imx} ∥_{p},

(n ∈ ℕ, ν, k ∈ {1, 2, …, n}) and so we obtain

(2.14)

∑_{k=1}^{n} ∫_{0}^{2π}|∑_{ν=1}^{n} ( ∑_{m=0}^{N} θ_{mk}^{nν} a_{mν}^{(n)} e^{imx} ) |^{p}dx = ∑_{k=1}^{n} ∥∑_{ν=1}^{n} ( ∑_{m=0}^{N} θ_{mk}^{nν} a_{mν}^{(n)} e^{imx} ) ∥_{p}^{p}

according to Minkowski inequality

⩽ ∑_{k=1}^{n} ( ∑_{ν=1}^{n} ∥∑_{m=0}^{N} θ_{mk}^{nν} a_{mν}^{(n)} e^{imx} ∥_{p}) ^{p} ⩽ ∑_{k=1}^{n} ( ∑_{ν=1}^{n} c_{kν}^{(n)} ∥∑_{m=0}^{N} a_{mν}^{(n)} e^{imx} ∥_{p}) ^{p}

according to (2.13).

According to Theorem 3, putting x_{ν} = ∥∑ _{m=0}^{N} a_{mν}^{(n)} e^{imx} ∥_{p} (ν = 1, 2, …, n), one gets

(2.15) ∑_{k=1}^{n} ( ∑_{ν=1}^{n} c_{kν}^{(n)} ∥∑_{m=0}^{N} a_{mν}^{(n)} e^{imx} ∥_{p}) ^{p} ⩽ λ^{p} ∑_{ν=1}^{n} ∥∑_{m=0}^{N} a_{mν}^{(n)} e^{imx} ∥_{p}^{p},

where

λ = sup_{n⩾1}max{ max_{1⩽k⩽n} ∑_{ν=1}^{n} c_{kν}^{(n)} , max_{1⩽ν⩽n} ∑_{k=1}^{n} c_{kν}^{(n)} } .

Let us estimate λ. Since

∑_{k=1}^{n} c_{kν}^{(n)} = 4N_{0}πA(p) n ∫_{(ν−1)/n}^{ν/n} dy ∫_{0}^{1} T (xy) w(x)^{1/p}w(y)^{1/q} dx,

16 DOSTANIĆ

by the Mean Value Theorem (for integrals), we get

∫_{(ν−1)/n}^{ν/n} dy ∫_{0}^{1} T (xy) w(x)^{1/p}w(y)^{1/q}dx = \frac{1}{n} ∫_{0}^{1} T (xθ_{n}) w(x)^{1/p}w(θ_{n})^{1/q}dx,

where \frac{ν−1}{n} ⩽ θ_{n} ⩽ \frac{ν}{n}, and hence, it follows

∑_{k=1}^{n} c_{kν}^{(n)} = 4N_{0}πA(p) w(θ_{n})^{1/q} ∫_{0}^{1} T (xθ_{n}) w(x)^{1/p}dx

⩽ 4N_{0}πA(p) sup_{0⩽y<1} w(y)^{1/q} ∫_{0}^{1} T (xy) w(x)^{1/p}dx.

Similarly, we obtain

∑_{ν=1}^{n} c_{kν}^{(n)} ⩽ 4N_{0}πA(p) sup_{0⩽x<1} w(x)^{1/p} ∫_{0}^{1} T (xy) w(y)^{1/q}dy.

Therefore,

λ ⩽ B_{p} = 4N_{0}πA(p) max \{ \begin{matrix}sup_{x<1}w(x)^{1/p} ∫_{0}^{1} T (xy) w(y)^{1/q}dy \\ sup_{y<1}w(y)^{1/q} ∫_{0}^{1} T (xy) w(x)^{1/p}dx \end{matrix}\}

From (2.14) and (2.15) it follows that (2.12) holds with the constant

B_{p} = 4N_{0}πA(p) max \{ \begin{matrix}sup_{x<1}w(x)^{1/p} ∫_{0}^{1} T (xy) w(y)^{1/q}dy \\ sup_{y<1}w(y)^{1/q} ∫_{0}^{1} T (xy) w(x)^{1/p}dx \end{matrix}\}  □

3. Proof of Theorem 1 Necessity. If P : L^{p}(D) → L^{p}(D) (1 < p < ∞) is a bounded operator,

according to Lemma 3 inequality (2.3) holds for any (finite) choice of Φ_{m} ∈ L^{p}(0, 1). Letting in (2.3) all the functions Φ_{m}, except one, to be zero, one gets

∫_{0}^{1} ∫_{0}^{2π} |e^{ikx} A_{k}Φ_{k}(y)| ^{p} dx dy ⩽ c_{p}^{p} ∫_{0}^{1} ∫_{0}^{2π} |e^{ikx} Φ_{k}(y)| ^{p} dx dy

i.e., ∥A_{k}Φ_{k}∥_{p} ⩽ c_{p} ∥Φ_{k}∥_{p} for each Φ_{k} ∈ L^{p}(0, 1) (k = 0, 1, …). From this we obtain

(3.1)  ∥A_{k}∥_{p} ⩽ c_{p}

for each k = 0, 1, … (Here ∥S∥_{p} denotes the norm of operator S on space L^{p}(0, 1)). Since

∥A_{k}∥_{p} = \frac{2π}{δ_{k}^{2}} ( ∫_{0}^{1} x^{kp+1} w(x) dx) ^{1/p} · ( ∫_{0}^{1} x^{kq+1} w(x) dx) ^{1/q}

from (3.1), the necessary condition of Theorem 1 follows (i.e., inequality (1.1)).

Sufficiency. Let conditions (1.1) and (1.2) of Theorem 1 hold. To establish the boundedness of operator P it is sufficient to prove inequality (2.3) in the case when functions Φ_{m} are continuous on [0, 1]. (If (2.3) holds when Φ_{m} are continuous, then it will also hold for Φ_{m}, ∈ L^{p}(0, 1) because operators A_{m} are bounded and the space C[0, 1] is dense in L^{p}(0, 1).).

Consider functions G_{m} ∈ C[0, 1], m = 0, 1, …N and let

α = max_{0⩽m⩽N} max_{0⩽x⩽1} |G_{m}(x)|.

Let a_{mk}^{(n)} =^{def} G_{m}(\frac{k}{n}) and

Φ_{m}^{(n)} (x) = \{ a_{m1}^{(n)} , 0 ⩽ x < \frac{1}{n} ; a_{m2}^{(n)} , \frac{1}{n} ⩽ x < \frac{2}{n} ; ⋮ ; a_{mk}^{(n)} , \frac{k−1}{n} ⩽ x < \frac{k}{n} ; ⋮ ; a_{mn}^{(n)} , \frac{n−1}{n} ⩽ x ⩽ \frac{n}{n}

It is clear that the sequence of functions {Φ_{m}^{(n)} (x)}_{n=1}^{∞} converges uniformly on [0, 1] towards G_{m}(x). Note that max_{0⩽x⩽1} |Φ_{m}^{(n)} (x)| ⩽ α for n ⩾ 1 and m ∈ {0, 1, …, N}. Let

V_{mk}^{(n)} = \frac{2π}{δ_{m}^{2}} n ∫_{(k−1)/n}^{k/n} x^{m+1/p} w(x)^{1/p} dx ∫_{0}^{1} y^{m+1/q} w(y)^{1/q} Φ_{m}^{(n)} (y) dy.

Then the inequality (2.12) (Lemma 6) can be written as

(3.2)  ∑_{k=1}^{n} ∫_{0}^{2π}|∑_{m=0}^{N} e^{imx} V_{mk}^{(n)} |^{p}dx ⩽ B_{p}^{p} ∑_{k=1}^{n} ∫_{0}^{2π}|( ∑_{m=0}^{N} a_{mk}^{(n)} e^{imx} ) |^{p}dx.

Since

∫_{0}^{1} ∫_{0}^{2π}|∑_{m=0}^{N} e^{imx} A_{m}Φ_{m}^{(n)} (y)|^{p}dx dy = ∑_{k=1}^{n} ∫_{(k−1)/n}^{k/n} dy ∫_{0}^{2π}|∑_{m=0}^{N} e^{imx} A_{m}Φ_{m}^{(n)} (y)|^{p}dx

and since the continuity of function y ↦ ∫_{0}^{2π} |∑ _{m=0}^{N} e^{imx} A_{m}Φ_{m}^{(n)} (y)|^{p}dx (on [0, 1]) implies that there exist points ξ_{nk} such that \frac{k−1}{n} ⩽ ξ_{nk} ⩽ \frac{k}{n} (k = 1, 2, …, n) and

∫_{(k−1)/n}^{k/n} dy ∫_{0}^{2π}|∑_{m=0}^{N} e^{imx} A_{m}Φ_{m}^{(n)} (y)|^{p}dx = \frac{1}{n} ∫_{0}^{2π}|∑_{m=0}^{N} e^{imx} A_{m}Φ_{m}^{(n)} (ξ_{nk})|^{p}dx,

we obtain

(3.3) ∫_{0}^{1} ∫_{0}^{2π}|∑_{m=0}^{N} e^{imx} A_{m}Φ_{m}^{(n)} (y)|^{p}dx dy = ∑_{k=1}^{n} \frac{1}{n} ∫_{0}^{2π}|∑_{m=0}^{N} e^{imx} A_{m}Φ_{m}^{(n)} (ξ_{nk})|^{p}dx.

Functions x ↦ x^{m+1/p} w(x)^{1/p} /δ_{m}^{2} are continuous [0, 1] (and even uniformly continuous; we define w at x = 1 as w(1) = 0 for each m = 0, 1, …, N, and so, for a given ε > 0 there exists a natural number n_{0} such that for n ⩾ n_{0} (3.4)

| \frac{ξ_{nk}^{m+ \frac{1}{p}} w (ξ_{nk})^{1/p}}{δ_{m}^{2}} − \frac{n}{δ_{m}^{2}} ∫_{(k−1)/n}^{k/n} x^{m+1/p} w(x)^{1/p}dx| < ε( 2πα · ∫_{0}^{1} y^{1/q}w(y)^{1/q}dy)^{−1}

for each m = 0, 1, …, N and each k ∈ {1, 2, …, n}.

Since

A_{m}Φ_{m}^{(n)} (ξ_{nk}) = \frac{2π}{δ_{m}^{2}} · ξ_{nk}^{m+1/p} · w (ξ_{nk})^{1/p} ∫_{0}^{1} y^{m+1/q} w(y)^{1/q}Φ_{m}^{(n)} (y) dy

from (3.4) we get

(3.5)  |A_{m}Φ_{m}^{(n)} (ξ_{nk}) − V_{mk}^{(n)} | < ε

for n ⩾ n_{0}, m ∈ {0, 1, …, N} and k ∈ {1, 2, …, n}.

Let n ⩾ n_{0}. From

∑_{k=1}^{n} \frac{1}{n} ∫_{0}^{2π}|∑_{m=0}^{N} e^{imx} A_{m}Φ_{m}^{(n)} (ξ_{nk})|^{p}dx

= ∑_{k=1}^{n} \frac{1}{n} ∫_{0}^{2π}|∑_{m=0}^{N} e^{imx} V_{mk}^{(n)} + ∑_{m=0}^{N} e^{imx} ( A_{m}Φ_{m}^{(n)} (ξ_{nk}) − V_{mk}^{(n)} ) |^{p}dx,

BOUNDEDNESS OF THE BERGMAN PROJECTIONS 19

using the inequality (|a| + |b|)^{p} ⩽ 2^{p−1} (|a|^{p} + |b|^{p}), p ⩾ 1, we get

∑_{k=1}^{n} \frac{1}{n} ∫_{0}^{2π}|∑_{m=0}^{N} e^{imx} A_{m}Φ_{m}^{(n)} (ξ_{nk})|^{p}dx

⩽ 2^{p−1} [ ∑_{k=1}^{n} \frac{1}{n}( ∫_{0}^{2π}|∑_{m=0}^{N} e^{imx} V_{mk}^{(n)} |^{p}dx + ∫_{0}^{2π}|∑_{m=0}^{N} e^{imx} ( A_{m}Φ_{m}^{(n)} (ξ_{nk}) − V_{mk}^{(n)} ) |^{p}dx) ]

⩽ 2^{p−1} ∑_{k=1}^{n} \frac{1}{n} ∫_{0}^{2π}|∑_{m=0}^{N} e^{imx} V_{mk}^{(n)} |^{p}dx + (N + 1)^{p}ε^{p} · 2π · 2^{p−1}

(using the inequality (3.5) in the last line).

Therefore, for n ⩾ n_{0} we have:

(3.6) ∑_{k=1}^{n} \frac{1}{n} ∫_{0}^{2π}|∑_{m=0}^{N} e^{imx} A_{m}Φ_{m}^{(n)} (ξ_{nk})|^{p}dx

⩽ 2^{p}π ε^{p} (N + 1) ^{p} + 2^{p−1} ∑_{k=1}^{n} \frac{1}{n} ∫_{0}^{2π}|∑_{m=0}^{N} e^{imx} V_{mk}^{(n)} |^{p}dx

From (3.2) and (3.6) we obtain (for n ⩾ n_{0})

∑_{k=1}^{n} \frac{1}{n} ∫_{0}^{2π}|∑_{m=0}^{N} e^{imx} A_{m}Φ_{m}^{(n)} (ξ_{nk})|^{p}dx

⩽ 2^{p}π ε^{p} (N + 1) ^{p} + 2^{p−1} B_{p}^{p} ∑_{k=1}^{n} \frac{1}{n} ∫_{0}^{2π}|∑_{m=0}^{N} e^{imx} a_{mk}^{(n)} |^{p}dx

= 2^{p}π ε^{p} (N + 1) ^{p} + 2^{p−1} B_{p}^{p} ∫_{0}^{1} ∫_{0}^{2π}|∑_{m=0}^{N} e^{imx} Φ_{m}^{(n)} (y)|^{p}dx dy.

If n ⩾ n_{0}, the previous inequality and (3.3) give

(3.7) ∫_{0}^{1} ∫_{0}^{2π}|∑_{m=0}^{N} e^{imx} A_{m}Φ_{m}^{(n)} (y)|^{p}dx dy

⩽ 2^{p}π ε^{p} (N + 1) ^{p} + 2^{p−1} B_{p}^{p} ∫_{0}^{1} ∫_{0}^{2π}|∑_{m=0}^{N} e^{imx} Φ_{m}^{(n)} (y)|^{p}dx dy.

Since the sequence {Φ_{m}^{(n)} }_{n=1}^{∞} converges uniformly towards G_{m} on [0, 1], then, due to the boundedness of the operators A_{m}, we get A_{m}Φ_{m}^{(n)} (y) → A_{m}G_{m}(y), n → ∞, and so, taking the limit as n → ∞ in (3.7) we obtain

∫_{0}^{1} ∫_{0}^{2π}|∑_{m=0}^{N} e^{imx} A_{m}G_{m}(y)|^{p}dx dy

⩽ 2^{p}πε^{p}(N + 1)^{p} + 2^{p−1} B_{p}^{p} ∫_{0}^{1} ∫_{0}^{2π}|∑_{m=0}^{N} , e^{imx} G_{m}(y)|^{p}dx dy.

Keeping in mind that ε > 0 is arbitrary, then, as ε → 0^{+} we obtain

∫_{0}^{1} ∫_{0}^{2π}|∑_{m=0}^{N} e^{imx} A_{m}G_{m}(y)|^{p}dx dy ⩽ 2^{p−1} B_{p}^{p} ∫_{0}^{1} ∫_{0}^{2π}|∑_{m=0}^{N} e^{imx} G_{m}(y)|^{p}dx dy.

Consequently inequality (2.3) holds (with c_{p} = 2^{1−1/p} · B_{p}) if the functions Φ_{m} are continuous which proves Theorem 1. □

Remark 3.1. The necessary condition for the boundedness of the Bergman projection P on L^{p}(D) can be expressed as sup_{m⩾0} ∥Φ_{m}∥_{p} · ∥Φ_{m}∥_{q} < ∞ where Φ_{m}(z) = z^{m}/δ_{m}, and can be obtained if we apply operator P to the functions f_{m}(z) = |z|^{m/(p−1)} e^{imθ} , (z = re^{iθ}), m = 0, 1, 2, . . ..

Remark 3.2. It would be interesting to find a weight w that satisfies the necessary but does not satisfy the sufficient condition of Theorem 1. Such a weight should tend to 0 (when r → 1−) faster then r ↦ (1 − r^{2}) ^{A} but slower then (1 − r^{2}) ^{A} exp(−B (1 − r^{2}) ^{−α} ), A, B, α > 0. That is connected with nontrivial analysis of the asymptotic behavior (when λ → +∞) of the function λ ↦ ∫_{0}^{1} r^{λ}w(r) dr which will be the subject of further investigations.

References

[1] M. R. Dostanić, Unboundedeness of the Bergman projections on L^{p} spaces with exponential weights, Proc. Edinbourgh Math. Soc. 47 (2004), 111–117.

[2] M. V. Fedoryuk, Asymptotics Integrals and Series, Nauka, Moscow, 1987.

[3] H. Hadenmalm, B. Korenblum, K. Zhu, Theory of Bergman spaces, Springer-Verlag, Berlin, 2000.

[4] B. Hollenbeck, I. E. Verbitsky, Best constants for the Riesz projection, J. Func. Anal. 175 (2000), 370–392.

[5] E. Seneta, Regularly Varying Functions, Nauka, Moscow 1985.

[6] G. N. Watson A treatise of the theory of Bessel functions, 2-nd edition, Cambridge University Press, Cambridge, 1966.

[7] A. Zygmund, Trigonometric Series, Mir, Moscow, 1965.

[8] K. Zhy, Operator Theory in Function Spaces, Marcel Dekker, New York and Basel, 1990.

Matematički fakultet (Received 08 08 2008) Studentski trg 16 (Revised 16 04 2009) Beograd Serbia

    domi@matf.bg.ac.rs