PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 98(112) (2015), 97–107 DOI: 10.2298/PIM150220023V

ON SOME CLASS OF INTEGRAL OPERATORS RELATED TO THE BERGMAN PROJECTION

Djordjije Vujadinović

Abstract. We consider the integral operator

C_{α} f(z) = ∫_{D} \frac{f(ξ)}{(1 − zξ̄)^{α}} dA(ξ), z ∈ D,

where 0 < α < 2 and D is the unit disc in the complex plane. and investigate boundedness of it on the space L^{p}(D, dλ), 1 < p < ∞, where dλ is the Möbius invariant measure in D. We also consider the spectral properties of C_{α} when it acts on the Hilbert space L^{2}(D, dλ), i.e., in the case p = 2, when C_{α} maps L^{2}(D, dλ) into the Dirichlet space.

1. Introduction and notation

Throughout the paper let D = {z : |z| < 1} be the open unit disc in complex plane ℂ and let dA(z) = \frac{1}{π} dx dy, z = x+iy stands for the normalized area measure in ℂ. For 1 < p < ∞ we consider the Besov space B_{p} of D, 1 < p < ∞, which is defined to be the space of all analytic functions f in D such that

∥f∥_{B_{p}} = ( ∫_{D} |f′(z)|^{p}(1 − |z|^{2})^{p}dλ(z))^{1/p} < ∞,

where dλ(z) = \frac{dA(z)}{(1−|z|^{2} )^{2}} is the Möbius invariant measure on D. It is known that ∥·∥_{B_{p}} is complete seminorm on B_{p}. It should be pointed that B_{p} is a Banach space with norm ∥f∥ = |f(0)| + ∥f∥_{B_{p}} . For p = 2 the space B_{2} is the classical Dirichlet space, and appropriate semi-inner product is given by the formula

(1.1)  ⟨f, g⟩ = ∫_{D} f′(z) \overline{g′(z)}dA(z), f, g ∈ B_{2}.

The weighted Bergman projection P_{s}, −1 < s < ∞ represents a central operator which appears in the research concerning the analytic function spaces. It is given by

P_{s}f(z) = (s + 1) ∫_{D} \frac{(1 − |ω|^{2})^{s}}{(1 − zω̄)^{2+s}} f(ω) dA(ω), z ∈ D.

2010 Mathematics Subject Classification: Primary 46E15,46E20.

Key words and phrases: Bergman projection, singular numbers of a compact operator.

Communicated by Stevan Pilipović.

Particulary, the ordinary Bergman projection P = P_{0} arises as the orthogonal projection from L^{2}(D; dA) onto an analytic function subspace. It connects B_{p} and L^{p}(D, dλ). This relation is expressed in the next theorem.

Theorem 1.1. Suppose f ∈ H(D) and 1 ⩽ p ⩽ ∞. Then

f ∈ B_{p} ⇔ f ∈ P (L^{p}(D, dλ)).

The inclusion operator V from B_{p} into L^{p}(D, dλ) is given by

V f(z) = 3(1 − |z|^{2})^{2} ∫_{D} \frac{f(ξ) dA(ξ)}{(1 − zξ̄)^{4}}, z ∈ D.

More precisely we have the following lemma (see [8]).

Lemma 1.1. The operator V is an embedding from B_{p} into L^{p}(D, dλ) for all 1 < p < ∞ if on B_{p} = P (L^{p}(D, dλ)) is given the quotient norm.

In this paper we consider the class of the operators

C_{α} f(z) = ∫_{D} \frac{f(ξ)}{(1 − zξ̄)^{α}} dξ, z ∈ D,

where 0 < α < 2. For α = 2 we have the Bergman projection. The norm of the Bergman projection from L^{p}(D, dλ) onto B_{p} was estimated in [7]. In Theorem 1.3 we prove that C_{α} is a bounded mapping from L^{p}(D, dλ) into B_{p} for all 0 < α < 2 and 1 < p < ∞. We investigate in the next section some of its spectral properties in the context of the Lebesgue space L^{2}(D, dλ) and the Besov space B_{2}.

By boundedness of an operator T : L^{p}(D, dλ) → B_{p} we mean that there exists a constant C > 0 such that ∥T f∥_{B_{p}} ⩽ C∥f∥_{L^{p} (D,dλ)} .

In this section we observe boundedness of C_{α} defined on L^{p}(D, dλ). We firstly state a technical lemma and a proposition (the Schur test).

Lemma 1.2. Suppose z ∈ D, c is real, t > −1, and

I_{c,t}(z) = ∫_{D} \frac{(1 − |ω|^{2})^{t}}{|1 − zω̄|^{2+t+c}} dA(ω).

Then we have (a) If c < 0, then I_{c,t}(z) is bounded in z.

(b) If c > 0, then I_{c,t}(z) ∼ \frac{1}{(1−|z|^{2} )^{c}} , |z| → 1^{−} .

(c) If c = 0, then I_{0,t}(z) ∼ log \frac{1}{1−|z|^{2}} , |z| → 1^{−} .

Proposition 1.1. Suppose K is a nonnegative measurable function on X ×X, where (X, μ) is a measure space. Let T be an integral operator induced by K, that is T f(x) = ∫_{X} K(x, y) f(y) dμ(y), where 1 < p < ∞ and \frac{1}{p} + \frac{1}{q} = 1. If there exist constants C_{1}, C_{2} > 0 and a positive measurable function h on X such that

∫_{X} K(x, y) h(y)^{q}dμ(y) ⩽ C_{1}h(x)^{q} for μ-almost every x ∈ X ,
∫_{X} K(x, y) h(x)^{p}dμ(x) ⩽ C_{2}h(y)^{p} for μ-almost every y ∈ X ,

then T is bounded on L^{p}(X, dμ) with the norm less than or equal to C_{1}^{1/q}C_{2}^{1/p}.

In the proof of Theorem 1.3 we will use the Gauss hypergeometric functions and its basic properties. Following [1] we recall some facts for the sake of easy reference.

The Gauss hypergeometric function _{2}F_{1}(a, b; c; z) is defined by

2F_{1}(a, b; c; z) = ∑_{n=0}^{∞} \frac{(a)_{n}(b)_{n}}{n!(c)_{n}}z^{n} for |z| < 1,

and by continuation elsewhere. Here (a)_{n} = a(a + 1) ⋯ (a + n − 1) denotes the shifted factorial, where a is any complex number.

The identity

(1.2)  _{2}F_{1}(a, b; c; z) = (1 − z^{2})^{c−a−b} _{2}F_{1}(c − a, c − b; c; z)

is known as Euler identity. The following properties of hypergeometric function are also going to be of interest

\frac{∂}{∂x}_{2}F_{1}(a, b; c; z) = \frac{ab}{c}_{2}F_{1}(a + 1, b + 1; c + 1; z), (1.3)

2F_{1}(a, b; c; 1) = \frac{Γ(c)Γ(c − a − b)}{Γ(c − a) Γ(c − b)} for Re(c − a − b) > 0.

It is known that _{2}F_{1}(a, b; c; z) diverges in general for z = 1 if Re(c − a − b) ⩽ 0. The next theorem, due to Gauss, describes the asymptotic behaviour of the hypergeometric functions as z → 1^{−} .

Theorem 1.2. If Re(c − a − b) < 0, then

lim_{x→1^{−}} \frac{_{2}F_{1}(a, b; c; z)}{(1 − z)^{c−a−b}} = \frac{Γ(c)Γ(a + b − c)}{Γ(a)Γ(b)} .

For c = a + b we have

lim_{x→1^{−}} \frac{_{2}F_{1}(a, b; c; z)}{log(\frac{1}{1−z} )} = \frac{Γ(a + b)}{Γ(a)Γ(b)} .

Theorem 1.3. For 0 < α < 2, C_{α} is a bounded mapping from L^{p}(D, dλ) into B_{p} (1 < p < ∞). The norm may be estimated by ∥C_{α} ∥_{L^{p} (D,dλ)→B_{p}} < αC_{1}^{1/q}C_{2}^{1/p}. Here

C_{1} = \{ \begin{matrix}\frac{pΓ(2+ \frac{1}{p} ) Γ(α−1− \frac{1}{p} )}{(p+1)Γ^{2} ( \frac{α+1}{2} )} , & α > 1 + \frac{1}{p}; \\ \frac{pqΓ(1+ \frac{1}{p} )}{eπ(p+1)} , & α = 1 + \frac{1}{p}; \\ \frac{pΓ(2+ \frac{1}{p} ) Γ(1+ \frac{1}{p} −α)}{(p+1)Γ^{2} ( \frac{3}{2} + \frac{1}{p} − \frac{α}{2} )} , & α < 1 + \frac{1}{p}, \end{matrix} and C_{2} = \{ \begin{matrix}\frac{qΓ(1+ \frac{1}{q} ) Γ(α− \frac{1}{q} )}{Γ^{2} ( \frac{α+1}{2} )} , & α > \frac{1}{q}; \\ \frac{pqΓ(1+ \frac{1}{q} )}{π(p+1)e} , & α = \frac{1}{q}; \\ \frac{qΓ(1+ \frac{1}{q} ) Γ( \frac{1}{q} −α)}{Γ^{2} ( \frac{1}{2} + \frac{1}{q} − \frac{α}{2} )} , & α < \frac{1}{q}, \end{matrix}

where \frac{1}{p} + \frac{1}{q} = 1.

Proof. First of all, it is easy to see that Cf is an analytic function for every f ∈ L^{p}(D, dλ). Further, for f ∈ L^{p}(D, dλ) we have

(Cf)′(z) = α ∫_{D} \frac{ξ̄}{(1 − zξ̄)^{α+1}} f(ξ) dA(ξ),

∥Cf∥_{B_{p}}^{p} = α^{p} ∫_{D} |(1 − |z|^{2})^{1−2/p} ∫_{D} \frac{ξ̄}{(1 − zξ̄)^{α+1}} f(ξ) dA(ξ)|^{p}dA(z).

Thus

∥Cf∥_{B_{p}}^{p} = α^{p} ∫_{D} |(1 − |z|^{2})^{1−2/p} ∫_{D} \frac{ξ̄}{(1 − zξ̄)^{α+1}} f(ξ) dA(ξ)|^{p}dA(z)

= α^{p} ∫_{D} |(1 − |z|^{2}) ∫_{D} \frac{ξ̄(1 − |ξ|^{2})^{2}}{(1 − zξ̄)^{α+1}} f(ξ) dλ(ξ)|^{p}dλ(z)

Therefore, we should consider the operator

T f(z) = (1 − |z|^{2}) ∫_{D} \frac{f(ξ)(1 − |ξ|^{2})^{2}}{|1 − zξ̄|^{α+1}} dλ(ξ)

on L^{p}(D, dλ). We will prove that it is bounded there and we will estimate its norm. Using the obvious relation ∥C_{α} ∥_{L^{p} (D,dλ)→B_{p}} ⩽ α∥T ∥_{L^{p} (D,dλ)→L^{p} (D,dλ)} we can estimate the norm of C_{α}.

We use Proposition 1.1 and test function h(z) = (1 − |z|^{2})^{1/pq} for the kernel K(z, ξ) = \frac{(1−|z|^{2} )(1−|ξ|^{2} )^{2}}{|1−zξ̄|^{α+1}} . We have to prove existence of the constants C_{1}, C_{2} such that

∫_{D} K(z, ξ) h^{q}(ξ) dλ(ξ) ⩽ C_{1}h^{q}(z), z ∈ D,
∫_{D} K(z, ξ) h^{p}(z) dλ(z) ⩽ C_{2}h^{p}(ξ), ξ ∈ D,

which is equivalent with

(1.4)  \begin{matrix}(1 − |z|^{2})^{1/q} ∫_{D} \frac{(1 − |ξ|^{2})^{1/p}}{|1 − zξ̄|^{α+1}} dA(ξ) & ⩽ C_{1}, z ∈ D, \\ (1 − |ξ|^{2})^{1+1/p} ∫_{D} \frac{(1 − |z|^{2})^{1/q−1}}{|1 − zξ̄|^{α+1}} dA(z) & ⩽ C_{2}, ξ ∈ D. \end{matrix}

From Lemma 1.2 we can easily check that both functions on the left-hand side in (1.4) are bounded and consequently relations (1.4) are true for some constants C_{1} and C_{2}. In the sequel we will determine the upper bounds for the constants C_{1} and C_{2}.

By using the uniform convergence and orthogonality we have

(1−|z|^{2})^{1/q} ∫_{D} \frac{(1 − |ξ|^{2})^{1/p}}{|1 − zξ̄|^{α+1}} dA(ξ) = (1−|z|^{2})^{1/q} ∑_{n=0}^{∞} \frac{Γ(\frac{1}{p} + 1)Γ^{2}(\frac{α+1}{2} + n)}{Γ(n + \frac{1}{p} + 2)Γ^{2}(\frac{α+1}{2} )n!} |z|^{2n}

= \frac{p}{1 + p} (1 − |z|^{2})^{1/q}_{2}F_{1}( \frac{α + 1}{2} , \frac{α + 1}{2} ; 2 + \frac{1}{p}, |z|^{2}) .

In a similar way we obtain that

(1 − |ξ|^{2})^{1+1/p} ∫_{D} \frac{(1 − |z|^{2})^{1/q−1}}{|1 − zξ̄|^{α+1}} dA(z)

= q(1 − |ξ|^{2})^{1+1/p} _{2}F_{1}( \frac{α + 1}{2} , \frac{α + 1}{2} ; 1 + \frac{1}{q}, |ξ|^{2}) .

Let us denote

C_{1} = \frac{p}{1 + p} sup_{|z|<1} (1 − |z|^{2})^{1/q}_{2}F_{1}( \frac{α + 1}{2} , \frac{α + 1}{2} ; 2 + \frac{1}{p}, |z|^{2}) ,

C_{2} = q sup_{|ξ|<1} (1 − |ξ|^{2})^{1+1/p} _{2}F_{1}( \frac{α + 1}{2} , \frac{α + 1}{2} ; 1 + \frac{1}{q}, |ξ|^{2}) .

Then the Schur test implies ∥C_{α} ∥_{L^{p} (dλ)→B_{p}} ⩽ αC_{1}^{1/q}C_{2}^{1/p}. By using the Euler transformation (1.2) for the hypergeometric functions, we obtain

C_{1} = \frac{p}{1 + p} sup_{|z|<1} (1 − |z|^{2})^{2−α} _{2}F_{1}( \frac{3}{2} + \frac{1}{p} − \frac{α}{2}, \frac{3}{2} + \frac{1}{p} − \frac{α}{2}; 2 + \frac{1}{p}, |z|^{2}) ,

C_{2} = q sup_{|ξ|<1} (1 − |ξ|^{2})^{2−α} _{2}F_{1}( \frac{1}{2} + \frac{1}{q} − \frac{α}{2}, \frac{1}{2} + \frac{1}{q} − \frac{α}{2}; 1 + \frac{1}{q}, |ξ|^{2}) .

Both functions

2F_{1}( \frac{3}{2} + \frac{1}{p} − \frac{α}{2}, \frac{3}{2} + \frac{1}{p} − \frac{α}{2}; 2 + \frac{1}{p}, |z|^{2}) ,

2F_{1}( \frac{1}{2} + \frac{1}{q} − \frac{α}{2}, \frac{1}{2} + \frac{1}{q} − \frac{α}{2}; 1 + \frac{1}{q}, |ξ|^{2})

are increasing in |z| and |ξ|, respectively (see (1.3)).

We distinguish the following five cases:

1) If α > 1 + \frac{1}{p}, then

C_{1} < \frac{p}{p + 1} _{2}F_{1}( \frac{3}{2} + \frac{1}{p} − \frac{α}{2}, \frac{3}{2} + \frac{1}{p} − \frac{α}{2}; 2 + \frac{1}{p}, 1)  = \frac{pΓ(2 + \frac{1}{p}) Γ(α − 1 − \frac{1}{p})}{(p + 1)Γ^{2}(\frac{α+1}{2} )} ,

C_{2} < q_{2}F_{1}( \frac{1}{2} + \frac{1}{q} − \frac{α}{2}, \frac{1}{2} + \frac{1}{q} − \frac{α}{2}; 1 + \frac{1}{q}, 1)  = \frac{qΓ(1 + \frac{1}{q}) Γ(α − \frac{1}{q})}{Γ^{2}(\frac{α+1}{2} )} .

2) If α < \frac{1}{q}, then according to Theorem 1.2 we have

C_{1} ⩽ \frac{p}{p + 1} lim sup_{|z|→1^{−}}(1 − |z|^{2})^{2−α} \frac{_{2}F_{1}( \frac{3}{2} + \frac{1}{p} − \frac{α}{2} , \frac{3}{2} + \frac{1}{p} − \frac{α}{2} ; 2 + \frac{1}{p}, 1)}{(1 − |z|^{2})^{α−1− \frac{1}{p}}}

< \frac{pΓ(2 + \frac{1}{p}) Γ(1 + \frac{1}{p} − α)}{(p + 1)Γ^{2}(\frac{3}{2} + \frac{1}{p} − \frac{α}{2} )} ,

C_{2} ⩽ q lim sup_{|ξ|→1^{−}}(1 − |ξ|^{2})^{2−α} \frac{_{2}F_{1}( \frac{1}{2} + \frac{1}{q} − \frac{α}{2} , \frac{1}{2} + \frac{1}{q} − \frac{α}{2} ; 1 + \frac{1}{q}, 1)}{(1 − |ξ|^{2})^{α−1/q}}

< \frac{qΓ(1 + \frac{1}{q}) Γ(\frac{1}{q} − α)}{Γ^{2}(\frac{1}{2} + \frac{1}{q} − \frac{α}{2} )} .

3) If α = 1 + \frac{1}{p}, then

C_{1} ⩽ \frac{p}{p + 1} lim sup_{|z|→1^{−}}(1 − |z|^{2})^{1/q} log \frac{1}{1 − |z|^{2}} \frac{_{2}F_{1}( \frac{3}{2} + \frac{1}{p} − \frac{α}{2} , \frac{3}{2} + \frac{1}{p} − \frac{α}{2} ; 2 + \frac{1}{p}, 1)}{log \frac{1}{1−|z|^{2}}}

< \frac{pq}{e(p + 1)} \frac{Γ(3 + \frac{2}{p} − α)}{Γ^{2}(\frac{3}{2} + \frac{1}{p} − \frac{α}{2} )} = \frac{pqΓ(1 + \frac{1}{p})}{eπ(p + 1)} , C_{2} < \frac{qΓ(1 + \frac{1}{q}) Γ(\frac{2}{p})}{Γ^{2}(\frac{α+1}{2} )} ,

since the maximal value of the function ϕ(x) = (1 − x)^{1/q} log \frac{1}{1−x} , x ∈ (0, 1) is \frac{q}{e}. 4) If α = \frac{1}{q}, then

C_{1} < \frac{pΓ(2 + \frac{1}{p}) Γ(\frac{2}{p})}{(p + 1)Γ^{2}(\frac{3}{2} + \frac{1}{p} − \frac{α}{2} )} , C_{2} < \frac{pqΓ(1 + \frac{1}{q})}{π(p + 1)e} .

5) If \frac{1}{q} < α < 1 + \frac{1}{p}, then
C_{1} < \frac{pΓ(2 + \frac{1}{p}) Γ(1 + \frac{1}{p} − α)}{(p + 1)Γ^{2}(\frac{3}{2} + \frac{1}{p} − \frac{α}{2} )} , C_{2} < \frac{qΓ(1 + \frac{1}{q}) Γ(α − \frac{1}{q})}{Γ^{2}(\frac{α+1}{2} )} .  □

2. Hilbert case and spectral properties

Folowing [6] let us recall some basic facts from spectral-operator theory. Let us firstly recall that for the bounded measurable function A(z, ξ) the operator

Af(z) = ∫_{D} \frac{A(x, ξ)f(ξ)}{|z − ξ|^{α}} dA(ξ), z ∈ D

is compact on L^{2}(D, dA), where 0 < α < 2.

For a compact operator T defined on a separable Hilbert space H, let s_{n}(T), n ⩾ 1 denote the eigenvalues of the operator (T^{∗}T)^{1/2} arranged in nondecreasing order [4]. In general, if T is a compact operator on a separable Hilbert space H, then there exist orthonormal sets {e_{n}} and {σ_{n}} in H such that

T x = ∑_{n} λ_{n}⟨x, e_{n}⟩σ_{n}, x ∈ H,

where λ_{n} is n-th singular value of T.

For 0 < p < ∞, we define the Schattene p-class of H denoted by S_{p}(H), or simply S_{p}, to be the space of all compact operators T on H with singular value sequence {λ_{n}} belonging to l^{p} (p-summable sequence space). The Schattene class S_{p} is a Banach space for the range 1 ⩽ p < ∞, and appropriate norm of the operator T ∈ S_{p} is given by ∥T ∥_{p} = ( ∑ _{n} |λ_{n}|^{p})^{1/p}.

Theorem 2.1. Let A be a compact operator. Then A has the norm convergent expansion

(2.1)  A = ∑_{n=1}^{N} μ_{n}(A)⟨ϕ_{n}, ·⟩ψ_{n}

(where N is a finite non-negative integer or infinity), each μ_{n}(A) > 0, μ_{1}(A) ⩾ μ_{2}(A) ⩾ ⋯, and (ϕ_{n}) and (ψ_{n}) are (not necessarily complete) orthonomal sets. Moreover, μ_{n}(A) are uniquely determined and ϕ’s and ψ’s are essentially uniquely determined.

Here μ_{n}(A) are singular values of A and formula (2.1) is called canonical expansion for A.

Now we state a known result related to minimax properties of eigenvalues for compact nonnegative operators [4].

Theorem 2.2. Let A (≠ 0) be a nonnegative compact operator and let φ_{j} (j = 1, 2, …) be an orthonomal system of its eigenvalues which is complete in the range of A, so that Aφ_{j} = λ_{j} (A)φ_{j}, j = 1, 2, … where λ_{1}(A) ⩾ λ_{2}(A) ⩾ · · · . Then its eigenvalues have the following minimax properties

(2.2)  λ_{1}(A) = max_{φ∈H} \frac{⟨Aφ, φ⟩}{⟨φ, φ⟩}

where the maximum in (2.2) is attained only for those eigenvalues of A that correspond to λ_{1}(A).

(2.3)  λ_{j+1} (A) = min_{L∈N_{j}} max_{φ∈L^{T}} \frac{⟨Aφ, φ⟩}{⟨φ, φ⟩} , j = 1, 2, …

where N_{j} is the set of all j-dimensional lineals of H, and the minimum in (2.3) is attained when L coincides with the linear hull L_{j} of the eigenvectors φ_{1}, φ_{2}, …, φ_{j} , … so that

λ_{j+1} (A) = max_{φ∈L_{j}^{T}} \frac{⟨Aφ, φ⟩}{⟨φ, φ⟩} .

We note that

(2.4)  s_{1}(A) = ∥A∥.

Dostanić [3] investigated the singular values of the operator S : L^{2}(D) → L^{2}(D) defined by

Sf(z) = \frac{1}{π} ∫_{D} \frac{ξ̄}{1 − zξ̄} m(ξ) f(ξ) dA(ξ),

where m ∈ C(D̄). He obtained that s_{n}(S) ∼ \frac{1}{2nπ} ∫_{0}^{2π} |m(e^{iθ})| dθ.

The next theorem is our second main result and is related to finding singular numbers of the operator V C_{α}.

Theorem 2.3. The operator V C_{α} : L^{2}(D, dλ) → L^{2}(D, dλ) is compact for 0 < α < 2. The following asymptotic formula holds

s_{n}(C_{α} |_{L^{2} (D,dλ)} ) = s_{n}(V C_{α}) = \frac{Γ(n + α)}{12Γ(α) Γ(n + 2)} ∼ \frac{1}{n^{2−α}} , n → ∞.

In the proof we will need the following inequalities for Γ function (see [2]).

Proposition 2.1. Let m, p and k be real numbers with m, p>0 and p>k >−m . If k(p − m − k) ⩾ 0 (⩽ 0), then Γ(p)Γ(m) ⩾ (⩽)Γ(p − k) Γ(m + k).

Proof. From Theorem 1.3 and properties of the operator V we have that V C_{α} maps L^{2}(D, dλ) into itself. Let H(·, ·) be appropriate kernel of V C_{α}, i.e.,

V C_{α} f(z) = ∫_{D} H(z, ξ) f(ξ) dA(ξ).

The the kernel H(·, ·) is given by

H(z, ξ) = (1 − |z|^{2})^{2} ∫_{D} \frac{dA(t)}{(1 − zt̄)^{4}(1 − ξ̄t)^{α}} .

On the other hand

H(z, ξ) = (1 − |z|^{2})^{2} ∫_{D} \frac{dA(t)}{(1 − zt̄)^{4}(1 − ξ̄t)^{α}}

= (1 − |z|^{2})^{2} ∫_{D} ∑_{n=0}^{∞} \frac{Γ(n + 4)}{Γ(4)n!} (zt̄)^{n} ∑_{k=0}^{∞} \frac{Γ(α + k)}{Γ(α)k!} (ξ̄t)^{k}dA(t)

= (1 − |z|^{2})^{2} ∑_{n=0}^{∞} \frac{Γ(n + 4)Γ(n + α)}{12Γ(α)n!(n + 1)!} z^{n} ξ̄^{n}.

So,

V C_{α} f(z) = ∑_{n=0}^{∞} \frac{Γ(n + 4)Γ(n + α)}{12Γ(α)n!(n + 1)!} (1 − |z|^{2})^{2}z^{n} ∫_{D} f(ξ)ξ̄^{n}dA(ξ)

= ∑_{n=0}^{∞} \frac{Γ(n + 4)Γ(n + α)}{12Γ(α)n!(n + 1)!} (1 − |z|^{2})^{2}z^{n} ∫_{D} f(ξ)(1 − |ξ|^{2})^{2} ξ̄^{n}dλ(ξ).

Let us note that e_{n}(z) = \sqrt{\frac{1}{2}(n + 3)(n + 2)(n + 1)} (1 − |z|^{2})^{2}z^{n}, n = 0, 1, 2, … represents orthonomal set in L^{2}(D, dλ), which implies that

V C_{α} f(z) = ∑_{n=0}^{∞} \frac{Γ(n + α)}{6Γ(α)Γ(n + 2)} e_{n}(z)⟨f, e_{n}⟩, z ∈ D.

Since, Stirling’s formula implies s_{n}(V C_{α}) ∼ \frac{1}{n^{2−α}} , as n → ∞, and by using the fact

∥V C_{α} f∥_{2}^{2} = ∑_{n=0}^{∞} ( \frac{Γ(n + α)}{6Γ(α)Γ(n + 2)} )^{2}|⟨f, e_{n}⟩|^{2}, f ∈ L^{2}(D, dλ),

we conclude that V C_{α} is compact for 0 < α < 2. The sequence ( \frac{Γ(α+n)}{Γ(α)(n+1)!} ) is decreasing in n, and this is a consequence of Proposition 2.1 with p = α + n + 1, m = n + 2, k = 1. Then by Theorem 2.1 we get s_{n}(V C_{α}) = \frac{Γ(n+α)}{6Γ(α)Γ(n+2)} . □

The next corollary is a direct consequence of the previous theorem.

Corollary 2.1. For 0 < α < 2, V C_{α} ∈ S_{p} holds, where p > \frac{1}{2−α} and

∥V C_{α} ∥_{p} = \frac{1}{12Γ(α)} ( ∑_{n=0}^{∞} ( \frac{Γ(n + α)}{Γ(n + 2)} )^{p}) ^{1/p}.

According to (2.4) and Theorem 2.3 we easily obtain the following result.

Theorem 2.4. If V C_{α} : L^{2}(D, dλ) → L^{2}(D, dλ), 0 < α < 2, then

∥V C_{α} ∥_{L^{2} (dλ)→L^{2} (dλ)} = \frac{Γ(1 + α)}{12Γ(α)} .

In the next theorem we will consider the operator C_{α} defined on the Dirichlet space B_{2}.

Theorem 2.5. The operator C_{α} : B_{2} → B_{2} is compact for 0 < α < 2 and

s_{n}(C|_{B_{2}} ) = \frac{Γ(α + n)}{Γ(α)(n + 1)!} ∼ \frac{1}{n^{2−α}} , n → ∞.

Proof. Let us note that the sequence e_{n}(z) = \frac{z^{n}}{\sqrt{n}}, n ⩾ 1 is orthonomal in B_{2} according to the invariant integral pairing defined in (1.1). Then, for the function f ∈ B_{2}, f(z) = ∑ _{n=0}^{∞} a_{n}z^{n}, where we can add the condition a_{0} = 0, we have

⟨f, e_{n}⟩ = \sqrt{n} ∫_{D} ∑_{k=1}^{∞} ka_{k}z^{k−1} z̄^{n−1} dA(z) = \sqrt{n}a_{n},

C_{α} f(z) = ∫_{D} \frac{f(ξ)}{(1 − zξ̄)^{α}} dA(ξ) = ∑_{n=1}^{∞} \frac{Γ(α + n)a_{n}}{Γ(α)(n + 1)!} z^{n} = ∑_{n=1}^{∞} \frac{Γ(α + n)e_{n}(z)}{Γ(α)(n + 1)!} ⟨f, e_{n}⟩.

By Stirling’s formula we obtain \frac{Γ(n+α)}{Γ(α)Γ(n+2)} ∼ \frac{1}{n^{2−α}} , n → ∞. On the other hand,

∥C_{α} f∥_{B_{2}}^{2} = ∑_{n=0}^{∞} ( \frac{Γ(n + α)}{Γ(α) Γ(n + 2)} ) ^{2}|⟨f, e_{n}⟩|^{2}

So, we conclude that C_{α} is a compact operator on B_{2} and Theorem 2.1 implies s_{n}(C_{α} |_{B_{2}} ) = \frac{Γ(α+n)}{Γ(α)(n+1)!} . □

Corollary 2.2. For the operator C_{α} : B_{2} → B_{2}, (0 < α < 2) holds C_{α} ∈ S_{p},

where p > \frac{1}{2−α} , and

∥C_{α} ∥_{p} = \frac{1}{Γ(α)} ( ∑_{n=0}^{∞} ( \frac{Γ(n + α)}{Γ(n + 2)} )^{p}) ^{1/p}.

A direct consequence of Theorem 2.5 is that ∥C_{α} ∥_{B_{2} →B_{2}} = s_{1}(C_{α} |_{B_{2}} ), i.e.,

∥C_{α} ∥_{B_{2} →B_{2}} = \frac{Γ(α + 1)}{2Γ(α)} .

However, we present here a direct way for finding the norm of C_{α} on B_{2} without using singular numbers.

Theorem 2.6. The operator C_{α} : B_{2} → B_{2}, 0 < α < 2 is bounded and

∥C_{α} ∥_{B_{2} →B_{2}} = \frac{Γ(α + 1)}{2Γ(α)} .

Proof. Every function in B_{2} can be approximated in norm by a sequence of polynomials. It is enough to find the norm of C_{α} on the set of polynomials p_{m}(z) = ∑ _{k=0}^{m} a_{k}z^{k}, where m is a nonnegative integer. From the proof of Theorem 2.5 we get

C_{α} f(z) = ∑_{n=1}^{∞} \frac{Γ(α + n)e_{n}(z)}{Γ(α)(n + 1)!} ⟨f, e_{n}⟩.

Thus,

C_{α} p_{m}(z) = ∑_{k=1}^{m} \frac{Γ(α + n)a_{n}z^{n}}{Γ(α)(n + 1)!} ,

∥C_{α} p_{m}∥_{B_{2}}^{2} = ∑_{n=1}^{m} \frac{n|Γ(α + n)|^{2}|a_{n}|^{2}}{|Γ(α)(n + 1)!|^{2}} , ∥p_{m}∥_{B_{2}}^{2} = ∑_{n=1}^{m} n|a_{n}|^{2}.

We want to find the minimal constant A such that

(2.5)  ∑_{n=1}^{m} \frac{n|Γ(α + n)|^{2}|a_{n}|^{2}}{|Γ(α)(n + 1)!|^{2}} ⩽ A^{2} ∑_{n=1}^{m} n|a_{n}|^{2},

for every polynomial p_{m}. In the above inequality we can treat the sequences (n|a_{n}|^{2}) and ( \frac{|Γ(α+n)|^{2}}{|Γ(α)(n+1)!|^{2}} ) as elements of l^{1} and l^{∞} , respectively, so (2.5) can be rewritten as

⟨ (n|a_{n}|^{2}), ( \frac{|Γ(α + n)|^{2}}{|Γ(α)(n + 1)!|^{2}} ) ⟩ ⩽ A^{2}∥(n|a_{n}|^{2})∥_{l^{1}} ,

where ⟨(ξ_{n}), (η_{n})⟩ = ∑ _{n=1}^{∞} ξ_{n}η_{n}. By using the duality argument (l^{1})^{∗} = l^{∞} we obtain

(2.6)  A^{2} = sup_{n⩾1} \frac{Γ^{2}(α + n)}{Γ^{2}(α)((n + 1)!)^{2}} = \frac{Γ^{2}(α + 1)}{4Γ(α)^{2}} ,

i.e.,

∥C_{α} ∥_{B_{2} →B_{2}} = \frac{Γ(α + 1)}{2Γ(α)} .

In (2.6) we used again the fact that the sequence ( \frac{Γ(α+n)}{Γ(α)(n+1)!} ) is decreasing in n. □

References

1. G. Andrews, R. Askey, R. Roy, Special Functions, Cambridge University Press, 2000.

2. S. S. Dragomir, R. P. Agarwal, N. S. Barnett, Inequalities for Beta and Gamma functions via some classical and new integral inequalities, J. Inequal. Appl. 5(2) (2000), 103–165.

3. M. Dostanić Spectral properties of the Cauchy operator and its product with Bergmans’s projection on a bounded domain, Proc. Lond. Math. Soc. 76 (1998), 667–684.

4. I. C. Gohberg, M. G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, Trans. Math. Monogr. 18, Am. Math. Soc., Providence, RI, 1969.

5. H. Hedenmalm, B. Korenblum, K. Zhu, Theory of the Bergman Spaces, Grad. Text Math., Springer, 2000.

6. B. Simon, Trace Ideals and Their Applications, 2nd ed., Math. Surv. Monogr. 120, Am. Math. Soc., Providence, RI, 2005.

7. Dj. Vujadinović, Some estimates for the norm of the Bergman projection on Besov Spaces, Integral Equations Oper. Theory 76 (2013), 213–224.

8. K. Zhu, Operator Theory in Function Spaces, Pure Appl. Math., Marcel Dekker 139, 1990.

Faculty of Natural Sciences and Mathematics (Received 19 05 2014) University of Montenegro (Revised 28 10 2014) Podgorica Montenegro

djordjijevuj@t-com.me