Volterra type operators on Bergman spaces with exponential weights

In this paper we characterize the boundedness, compactness and membership in Schatten p-classes of Volterra type operators on Bergman spaces with exponential weights.


Introduction and main results
Let D be the unit disc in the complex plane, dm(z) = dx dy π be the normalized area measure on D, and denote by H(D) the space of all analytic functions in D. If g ∈ H(D), we consider the linear operator J g defined by This operator was introduced by C. Pommerenke in [9] as a tool in his study of BM OA functions.The operator J g has received many names in the literature: the Pommerenke operator, a Volterra type operator (since the choice g(z) = z gives the usual Volterra operator), the generalized Cesàro operator (since the usual Cesàro operator appears with the choice g(z) = − log(1 − z)), a Riemann-Stieltjes type operator, or simply called an integration operator.It not was until the works of Aleman and Siskakis in [2] and [3] that the operator J g began to be extensively studied.The operator J g is related to the multiplication operator M g (f ) = gf by the formula M g (f ) = f (0)g(0) + J g (f ) + I g (f ), where I g is another integration operator defined by We refer to [1] and [11] for surveys on the operator J g acting in several spaces of analytic functions.We are mainly interested on the operator J g acting on weighted Bergman spaces, so let's recall the definition.
The following result describes the boundedness and compactness of the operator J g on A p (w γ,α ) in terms of the growth of the maximum modulus of g , for the exponential type weights w γ,α .
We note that Theorem 1.1 answers the question which appears in [3, p. 353].The case p = 2, c > 0 and α ∈ (0, 1] was proved by Dostanić in [4], while the general case is proved by the authors in [7], where a characterization is also obtained for a general class of radial rapidly decreasing weights.It is our aim in the first part of this note to provide a different proof of Theorem 1.1 using the test functions considered by Dostanic when α ∈ (0, 1], and Oleinik's description [6] of the Carleson measures for A p (w α ) when α > 1, where w α are the exponential weights One of the main tools in order to prove Theorem 1.1 is a description of the weighted Bergman spaces in terms of derivatives obtained in [8].The version proved in [8] is much more general than the one we state next, and uses a suitable distorsion function.
Theorem A. Let 0 < p < ∞, and g ∈ H(D).Then Let H be a separable Hilbert space.Given 0 < p < ∞, let S p (H) denote the Schatten p-class of operators on H. S p (H) contains those compact operators T on H whose sequence of characteristic (or singular) numbers λ n belongs to p , the p-summable sequence space.The singular numbers of an operator T are defined by Thus finite rank operators belong to every S p (H), and the membership of an operator in S p (H) measures in some sense the size of the operator.If 1 ≤ p < ∞, S p (H) is a Banach space with the norm T p = {λ n } p .We refer to [12, Chapter 1] for more information about S p (H).
Our next result will be a characterization, in terms of the symbol g, of the membership of the operator J g in the Schatten p-classes of A 2 (w γ,α ).In order to state our result, we recall the definition of another class of analytic function spaces, the so called Besov type spaces B p σ .Let 0 < p < ∞, and σ ≥ 0. The space B p σ consists of those analytic functions on D with , and consider the weights w γ,α defined by ( 1.1).Then J g ∈ S p (A 2 (w γ,α )) if and only if g ∈ B p α(p−1) .
This result was also proved by the authors in [7] for more general weights.However, here we will present a different proof.
The paper is organized as follows: Section 2 is devoted to some preliminaries needed for the proofs of the main results.We prove Theorem 1.1 in Section 3 and Theorem 1.2 in Section 4.
Throughout the paper, the letter C will denote an absolute constant whose value may change at different occurrences.We also use the notation a b to indicate that there is a constant C > 0 with a ≤ Cb, and the notation a b means that a b and b a.

Preliminary results
In this section we shall prove a few preliminary results which are used for the proofs of the main results of the paper.
From now on, we will always use the following notations: D(z, r) is the Euclidean disc centered at z with radius r > 0; For fixed α > 0, the function τ α is defined by If there is no confusion and for easy of notation, we shall write τ α = τ and for any δ > 0, D(δτ (z)) for the disc D(z, δτ (z)).
Lemma 2.1.Let γ ≥ 0 and 0 < p, α < ∞.Then there exist constants An immediate consequence of Lemma 2.1 is that the point evaluations are bounded linear functionals on A p (w γ,α ).In particular, A 2 (w γ,α ) is a reproducing kernel Hilbert space: there are reproducing kernels Lemma 3.5] (see also [7,Corollary 1]) that this is the corresponding growth of the reproducing kernel, that is, Next, bearing in mind Lemma 4 of Dostanic's paper [4], the following "test functions" are constructed in order to prove Theorem 1.
Finally, we remind the reader a description of Carleson measures for A p (w α ) due to Oleinik (see [6,Theorem 3.3]), for α > 1.

Proof of Theorem 1.1.
Throughout this section, for each z ∈ D and g ∈ H(D), we will use the notation: Proof of (i).Suppose first that sup z∈D B g (z) < ∞, and let f ∈ A p (w γ,α ).Since (J g f ) (z) = f (z)g (z), from Theorem A we obtain and it follows that Suppose now that J g is bounded on A p (w γ,α ) and choose δ > 0 sufficiently small.We shall split the proof of this implication in two cases.
) and a is any point of D, then by Lemma 2.1 we have In the last inequality we have used the fact that (1−|a|) (1−|z|) for z ∈ D(δτ (a)).Since (J g f ) (a) = f (a)g (a), then it follows from Theorem A and the boundedness of J g that Now, consider the test function F a (z) defined in Lemma 2.2.Since F a (z) has no zeros on D, then the function H a (z) = (F a (z)) 2/p belongs to A p (w γ,α ) with and then, bearing in mind that a is arbitrary, we have This finishes the proof for the case 0 < α ≤ 1.
Case α > 1.It follows from Theorem A and the boundedness of whenever f belongs to A p (w γ,α ).Next, note that if h is a function in A p (w α ), then . Consequently, if we write bearing in mind (3.4), we deduce that where C is a constant independent of ζ.That is, the operators Finally, bearing in mind (3.5), this gives sup a∈D B g (a) J g .
Thus, the proof is complete.
Before going into the proof of the compactness part, some previous results will be needed.Using the fact that the point evaluation functionals are bounded on A p (w γ,α ), the proof of the following result is standard, and we omit it here.Lemma 3.1.Let 0 < p < ∞ and g ∈ H(D).Then J g is compact on A p (w γ,α ) if and only if whenever {f n } is a bounded sequence in A p (w γ,α ) such that f n → 0 uniformly on compact subsets of D, then J g f n → 0 in A p (w γ,α ).Now we choose the appropriate test functions to study the compactness.Lemma 3.2.Let 0 < α ≤ 1, γ ≥ 0, and let λ = 1 + α/2 + γ/4.For each a ∈ D, consider the functions where F a is the function defined in Lemma 2.2.Then f a A 2 (w γ,α ) ≤ C, where C > 0 does not depend on the point a, and Therefore the result follows from the fact that Proof of (ii).Suppose first that g ∈ H(D) satisfies (3.7) lim |z|→1 − B g (z) = 0, and let {f n } be a bounded sequence of functions in A p (w γ,α ) such that f n → 0 uniformly on compact subsets of D. Fixed ε > 0, by (3.7) there is r ∈ (0, 1) such that B g (z) p < ε, for all z ∈ {ξ ∈ D : r ≤ |ξ| < 1}.Moreover, since f n → 0 uniformly on compact subsets of D, there is n 0 ∈ N such that |f n (z)| p < ε, for all n ≥ n 0 and z ∈ {ξ : |ξ| < r}.Since (3.7) gives that sup z∈D (1−|z|) (1+α) |g (z)| < ∞, by Theorem A the function g belongs to A p (w γ,α ).Using again Theorem A, for n ≥ n 0 we obtain ≤ Cε, for all n ≥ n 0 , that is, lim n→∞ J g (f n ) p A p (w γ,α ) = 0.So by Lemma 3.1, J g is compact.Conversely, suppose that J g is compact.We shall split the proof of this implication in two cases.
Case 0 < α ≤ 1.Consider the function f a from Lemma 3.2.Since f a (z) never vanishes on D, then, by Lemma 3.2, the function h a (z) = (f a (z)) 2/p belongs to A p (w γ,α ) with h a p A p (w γ,α ) = f a 2 A 2 (w γ,α ) ≤ C, and h a → 0 as |a| → 1 uniformly on compact subsets of D. Since J g is compact, Lemma 3.1 implies that For f ∈ A p (w γ,α ), proceeding as in the proof of the boundedness part (see equation (3.1)), we obtain Recall that τ (a) 2 = (1 − |a| 2 ) 2+α .On the other hand, which together with (3.9) (with f = h a ) and (3.8) gives that This finishes the proof of this case.
Case α > 1.This can be proved with similar arguments as in the boundedness part using Theorem C. We left the details to the interested reader.

Schatten p-classes
In this section we will prove Theorem 1.2.For easy of notation, throughout this section we denote S p := S p (A 2 (w γ,α )), the norm • is the norm in A 2 (w γ,α ), and •, • is the inner product in A 2 (w γ,α ).First, we need several definitions and preparatory results that can be of independent interest.Let F = {f n } be a sequence of analytic functions on D. We denote , z ∈ D, and for 0 < p < ∞, consider the p-integral means If ω is a weight function, following Siskakis [10], we define the distorsion function of ω as Now, the proof of the following lemma is analogue to the case of one function (see [10,Lemma 2.1].We shall give an sketch of the proof for the sake of completeness.
and let F = {f n }.Then for any weight function ω one has where the constant C depends only on p and the weight ω.
Proof.First, we will show that (4.1) If F = 0, (4.1) is clear.If F = 0, at points z ∈ D where F is not zero, by Cauchy-Schwarz inequality and consequently Thus (4.1) holds for p = 1.If p > 1 apply Hölder's inequality to obtain (4.1).From now, the proof can be mimicked from that of [10, Lemma 2.1].
We also need the fact that for any orthonormal set {e n } of A 2 (w γ,α ), one has with equality if {e n } is also an orthonormal basis.
The following Proposition gives the sufficiency in Theorem 1.2.
This finishes the proof for p ≥ 2.
This completes the proof of the Proposition.
For the necessity we need first some lemmas.
Lemma A (Oleinik [6]).Let τ There is a number δ 0 and a sequence of points {z j } ⊂ D, such that for each δ ∈ (0, δ 0 ) one has: Let k z = K z / K z be the normalized reproducing kernels of A 2 (w γ,α ).Lemma 4.3.Let {z j } be the sequence given in Lemma A. Then for every ortonormal sequence {e j } in A 2 (w γ,α ), the operator B taking e j to k z j is bounded.
For any g ∈ A 2 (w γ,α ), we have . Now the result follows from the fact that, by (2.1), Lemma 2.1 and Lemma A The next result gives the necessity in Theorem 1.2 completing the proof of that Theorem.
We consider first the case p ≥ 2. Suppose that J g is in S p , and let {e k } be an orthonormal set in A 2 (w γ,α ).By Lemma 4.3, the operator B taking e j to the normalized reproducing kernels k z j is bounded on A 2 (w γ,α ), where {z j } is the sequence from Lemma A. Since S p is a two-sided ideal in the space of bounded linear operators on A 2 (w γ,α ), then J g B belongs to S p (see [12, p.27]).Thus, by [12,Theorem 1.33 Now, using the subharmonicity of |g | 2 and Lemma A we obtain This completes the proof for the case p ≥ 2.
If 0 < p < 2 we follow the argument in [12,Proposition 7.15].If J g ∈ S p then the positive operator J * g J g belongs to S p/2 .Without loss of generality we may assume that g = 0. Suppose J * g J g f = n λ n f, e n e n is the canonical decomposition of J * g J g .Then {e n } is also an orthonormal basis.Indeed, if there is an unit vector e ∈ A 2 (w γ,α ) such that e ⊥ e n for all n ≥ 1, then by Theorem A, J g e 2 = J * g J g e, e = 0 because J * g J g is a linear combination of the vectors e n .This would give g ≡ 0. Now (2.1), the fact that equality holds in (4.2) (since {e n } is an orthonormal basis), and Hölder's inequality yields The last inequality is due to Theorem A. This completes the proof.
Corollary 4.5.Let 0 < p ≤ 1.Then J g ∈ S p if and only if g is constant.
Proof.The sufficiency is obvious, and the necessity follows from Proposition 4.4, since B p α(p−1) contains only constant functions for 0 < p ≤ 1.