A class of integral operators from weighted integral transforms to Dirichlet spaces

dA(z) = 1 dxdy = 1 rdrd (z = x + iy = re ) the normalized area measure on , H the space of all bounded holomorphic functions on with the norm ‖f‖ ∞ = supz∈ �f (z)�, H( ) the class of all holomorphic functions on , and  the space of all complex Borel measures on . For α > 0, the family  of weighted integral transforms is the collection of functions f ∈ H( ) which admits a representation of the form


Introduction
Let be the open unit disk in the complex plane ℂ, (1) and varies over all measures in . The principal branch is used in the power function in Equation (2) and throughout the rest of the paper. The space  is a Banach space with respect to the norm where the infimum is taken over all complex Borel measures satisfying (1) and ‖ ‖ denotes the total variation of .
Let dA (z) = (1 + )(1 − |z| 2 ) dA(z) be the probability measure on , where ∈ (−1, ∞). For 0 < p < ∞ and ∈ (−1, ∞), the weighted Bergman space  p consists of functions f ∈ H( ) such that Also recall that the Dirichlet space  p is the collection of functions f ∈ H( ) for which For more about these type of spaces see [Girela and Peiaez (2006)] and references therein. If (X, ‖ ⋅ ‖ X ) and (Y, ‖ ⋅ ‖ Y ) are Banach spaces, and T is a linear operator from X to Y, then T is bounded if there exists a positive constant C such that ‖T(f )‖ Y ≤ C‖f ‖ X for all f ∈ X and the operator norm of T is defined as ‖T‖ X→Y = inf{C < 0 : ‖T(f )‖ Y ≤ C‖f ‖ X }. We denote by (X, Y) the set of all bounded linear operators from X to Y. If T ∈ (X, Y), then we say that T is a compact operator from X to Y if the image of every bounded set of X is relatively compact (that is, has compact closure) in Y. Equivalently, a linear operator T is a compact operator from X to Y if and only if for every bounded sequence {f m } in X, {T(f m )} has a convergent subsequence in Y. We will denote by (X, Y) the compact linear operators from X into Y.
In this paper, we consider a class of integral operators defined as follows where g ∈ H( ), a holomorphic self-map of and n ∈ ℕ ∪ {0} from  to  p . Operator (3) was first of all defined in Sharma and Sharma (2011) and is an extension of many operators appearing in the literature. If J (1) g, is so called generalized composition operator, which is a natural extension of the integral operator by Yoneda (2004) (see Cowen & MacCluer, 1995 for more about composition operators. In Hibschweiler and MacGregor (1989), Hibschweiler and MacGregor proved that if ≥ 1, then every holomorphic self-map of induces a bounded composition operator on  . In contrast with the situation when ≥ 1, a self-map of need not induce a bounded composition operator on  when 0 < < 1. In fact, the condition ∈  is necessary for C to be bounded on  . Hibschweiler and MacGregor (1989), constructed a self-map of with ∉  (0 < < 1). For some recent results in this area (see Choa & Kim, 2001, Hibschweiler, 1998, 2012Sharma & Sharma, 2014;Stevic & Sharma, 2011). Motivated by work in the above-cited articles, here we provide characterizations of . We also compute the norm of J (n) g, acting from  to  p . Throughout this paper constants are denoted by C, they are positive and not necessarily the same at each occurrence. (3)

Main results
Theorem 2.1 Let ≥ 0, > −1, n ∈ ℕ ∪ {0}, g ∈ H( ) and be a holomorphic self-map of . Then Proof First suppose that (4) holds. If f ∈  , then there is ∈ with ‖ ‖ = ‖f ‖  such that Thus, we have Replacing z in (6) by (z), using Jensen's inequality and multiplying such obtained inequality by |g(z)| p , we obtain Integrating (7) with respect to dA(z) and applying Fubini's theorem yield By (4), the inner integral in the second term of (8) is atmost M and so Thus, J (n) g, ∈ L( ,  p ) and Conversely, suppose that J (n) g, ∈ L( ,  p ). Then using the fact that ‖K x ‖  = 1 for each x ∈ T. Thus, by the boundedness of J (n) g, :  →  p , we have that Thus, (4) holds. Moreover, combining (9) and (10), we get (5), as desired. □ The next theorem is an easy consequence of the Theorem 2.1. We omit the details.
Theorem 2.2 Let n ∈ ℕ ∪ {0}, g ∈ H( ) and be a holomorphic self-map of . Then The next lemma can be found in Hibschweiler (2012).

Lemma 2.3
Let > 0 and f ∈ H( ). Then for f ∈  and z ∈ , we have that By Equation (11) of Lemma 2.3, we have that the unit ball of  is a normal family of holomorphic functions. A standard normal family argument then yields the proof of the the following lemma. See Proposition 3.11 of Cowen and MacCluer (1995) for details.
Lemma 2.4 Let n ∈ ℕ ∪ {0}, g ∈ H( ) and be a holomorphic self-map of . Then and only if for any sequence f j in  with ||f j ||  ≤ L and which converges to zero locally uniformly, we have lim j→∞ ||J (n) g, f j ||  p = 0.
(3) The family of measures { x : x ∈ } defined by is equi-absolutely continuous. That is, given > 0, there exists a > 0 such that x (E) < for all x ∈ whenever A(E) < .
(4) g ∈  p and Proof (1) ⇒ (2). Let x k ∈ with x k → x as k → ∞, and let K x k be defined as in Equation (2). Then ||K x k ||  = 1 and K x k → K x uniformly on compact subsets of . Since J (n) g, :  →  p is compact. By Lemma (2.4), we have (10) Using this along with Hölder's inequality, we have Thus, which shows the continuity of the integral in Equation (2).