Open AccessArticle Essential Norm of Generalized Integral-Type Operator from the Fractional Cauchy Transform Space into Weighted Bloch and Dirichlet Spaces by Mostafa Hassanlou Mostafa Hassanlou SciProfiles Scilit Preprints.org Google Scholar 1,*, Ebrahim Abbasi Ebrahim Abbasi SciProfiles Scilit Preprints.org Google Scholar 2,* and Maryam G. Alshehri Maryam G. Alshehri SciProfiles Scilit Preprints.org Google Scholar 3 1 Engineering Faculty of Khoy, Urmia University of Technology, Urmia 57166, Iran 2 Department of Mathematics, Mah. C., Islamic Azad University, Mahabad, Iran 3 Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi Arabia * Authors to whom correspondence should be addressed. Axioms 2026, 15(6), 418; https://doi.org/10.3390/axioms15060418 (registering DOI) Submission received: 7 April 2026 / Revised: 1 June 2026 / Accepted: 2 June 2026 / Published: 4 June 2026 (This article belongs to the Special Issue Recent Advances in Functional Analysis and Operator Theory, 2nd Edition) Download keyboard_arrow_down Download PDFDownload XML Versions Notes Abstract In this paper, we investigate the generalized integral-type operator acting between the fractional Cauchy transform space and two classical analytic function spaces: the weighted Bloch space and the weighted Dirichlet space. For the operator acting into the weighted Bloch space, we obtain two equivalent exact formulas for its operator norm. Furthermore, an estimate for its essential norm is provided, which leads to a necessary and sufficient condition for compactness. For the operator acting into the weighted Dirichlet space, we derive the exact operator norm and fully characterize its compactness. MSC: 47B38; 30H30 1. Introduction Let D be the open unit disk in C and H ( D ) be the set of analytic functions on D . Here, we use some subspaces of H ( D ) , which are known in the literature. If a function f is analytic in D , then the Cauchy formula gives f ( z ) = 1 2 π i ∫ T f ( ξ ) ξ − z d ξ , z ∈ D , (1) where T = { z ∈ C : | z | = 1 } . The above representation extends to more general functions belonging to the fractional Cauchy transform space. For α > 0 , a function f belongs to the space of fractional Cauchy transforms F α if it has a representation as f ( z ) = ∫ T 1 ( 1 − ξ ପ୍ତ z ) α d μ ( ξ ) , z ∈ D , (2) for some μ ∈ M , the space of all complex-valued Borel measures on T with the total variation norm. The representation ( 1) is a special case for α = 1 and d μ ( ξ ) = f ( ξ ) 2 π i d ξ . The family of fractional Cauchy transforms F α has been studied extensively; two standard references are [ 1, 2]. The space F α is a vector space with respect to the ordinary addition of functions and multiplication by complex numbers. The statement ∥ f ∥ F α = inf μ ∥ μ ∥ defines a norm on F α , which is a Banach space with this norm. Here, infimum is taken over measures in M for which ( 2) holds and ∥ μ ∥ is the total variation of μ . It can be shown that for every f ∈ F α , there is a μ ∈ M , such that ( 2) holds and ∥ f ∥ F α = ∥ μ ∥ . Now, we define weighted Bloch and Dirichlet spaces. Suppose that ω is a weight function on D , i.e., a positive continuous function on D . The weighted Bloch space B ω is the space of all analytic functions in D for which ∥ f ∥ B ω = | f ( 0 ) | + sup z ∈ D ω ( z ) | f ′ ( z ) | 0 , we obtain the Bloch-type space B α . See [ 3] for a more complete understanding of these spaces. For a radial weight ω , ω ( | z | ) = ω ( z ) , the weighted Dirichlet space D ω is defined as follows: D ω = { f ∈ H ( D ) : ∫ D | f ′ ( z ) | 2 ω ( z ) d A ( z ) 0 , ω be a weight, g ∈ H ( D ) , and φ ∈ S ( D ) . Then, ∥ C φ , g n ∥ F α → B ω = ∏ l = 0 n − 1 ( α + l ) sup ξ ∈ T sup z ∈ D ω ( z ) | g ( z ) | | 1 − ξ ପ୍ତ φ ( z ) | n + α . (4) Proof.For any ξ ∈ T , let f ξ ( z ) = 1 ( 1 − ξ ପ୍ତ z ) α . Then, f ξ ∈ F α , ∥ f ξ ∥ F α = 1 , and f ξ ( k ) ( z ) = ξ ପ୍ତ k ∏ l = 0 k − 1 ( α + l ) ( 1 − ξ ପ୍ତ z ) α + k , where k ∈ N [ 12]. The definition of the norm of the operator implies that ∥ C φ , g n ∥ F α → B ω ≥ ∥ C φ , g n ( f ξ ) ∥ B ω = | C φ , g n ( f ξ ) ( 0 ) | + sup z ∈ D ω ( z ) | ( C φ , g n f ξ ) ′ ( z ) | (5) Since C φ , g n ( f ξ ) ( 0 ) = 0 and ( C φ , g n f ξ ) ′ ( z ) = g ( z ) f ξ ( n ) ( φ ( z ) ) , we obtain ∥ C φ , g n ( f ξ ) ∥ B ω = ∏ l = 0 n − 1 ( α + l ) sup z ∈ D · ω ( z ) | g ( z ) | | 1 − ξ ପ୍ତ φ ( z ) | n + α . Taking the supremum over ξ ∈ T gives ∥ C φ , g n ∥ F α → B ω ≥ ∏ l = 0 n − 1 ( α + l ) sup ξ ∈ T sup z ∈ D ω ( z ) | g ( z ) | | 1 − ξ ପ୍ତ φ ( z ) | n + α . Conversely, for any f ∈ F α , there exists ν ∈ M such that ∥ ν ∥ = ∥ f ∥ F α and f ( z ) = ∫ T d ν ( ξ ) ( 1 − ξ ପ୍ତ z ) α ; see [ 12]. Moreover, f ( n ) ( z ) = ∏ l = 0 n − 1 ( α + l ) ∫ T ξ ପ୍ତ n ( 1 − ξ ପ୍ତ z ) n + α d ν ( ξ ) . Hence, for any f ∈ F α , we have ∥ C φ , g n f ∥ B ω = | ( C φ , g n f ) ( 0 ) | + sup z ∈ D ω ( z ) | ( C φ , g n f ) ′ ( z ) | = ∏ l = 0 n − 1 ( α + l ) sup z ∈ D ω ( z ) | g ( z ) | ∫ T ξ ପ୍ତ n ( 1 − ξ ପ୍ତ φ ( z ) ) n + α d ν ( ξ ) ≤ ∏ l = 0 n − 1 ( α + l ) sup z ∈ D ω ( z ) | g ( z ) | ∫ T 1 | 1 − ξ ପ୍ତ φ ( z ) ) | n + α d | ν | ( ξ ) ≤ ∏ l = 0 n − 1 ( α + l ) sup ξ ∈ T sup z ∈ D ω ( z ) | g ( z ) | | 1 − ξ ପ୍ତ φ ( z ) | n + α ∫ T d | ν | ( ξ ) ≤ ∏ l = 0 n − 1 ( α + l ) sup ξ ∈ T sup z ∈ D ω ( z ) | g ( z ) | | 1 − ξ ପ୍ତ φ ( z ) | n + α ∥ ν ∥ = ∏ l = 0 n − 1 ( α + l ) sup ξ ∈ T sup z ∈ D ω ( z ) | g ( z ) | | 1 − ξ ପ୍ତ φ ( z ) | n + α ∥ f ∥ F α = sup ξ ∈ T sup z ∈ D ∏ l = 0 n − 1 ( α + l ) ω ( z ) | g ( z ) | | 1 − ξ ପ୍ତ φ ( z ) | n + α ∥ f ∥ F α . Considering the above discussion with regard to inequality ( 5) yields ( 4). □ In the next theorem, we find another formula for the norm of the operator C φ , g n : F α → B ω . Theorem 2.Let n ∈ N 0 , α > 0 , ω be a weight, g ∈ H ( D ) , and φ ∈ S ( D ) . Then, ∥ C φ , g n ∥ F α → B ω = ∏ l = 0 n − 1 ( α + l ) sup z ∈ D ω ( z ) | g ( z ) | ( 1 − | φ ( z ) | ) n + α . Proof.For any w ∈ D , there exists 0 ≤ θ w 0 , ω be a weight, g ∈ H ( D ) , and φ ∈ S ( D ) . Then, the following statements are equivalent: 1. The operator C φ , g n : F α → B ω is bounded. 2. sup z ∈ D ω ( z ) | g ( z ) | ( 1 − | φ ( z ) | ) n + α 0 , ω be a weight, g ∈ H ( D ) and φ ∈ S ( D ) . If C φ , g n : F α → B ω is bounded, then∥ C φ , g n ∥ ϵ ≈ lim sup | φ ( z ) | → 1 ω ( z ) | g ( z ) | ( 1 − | φ ( z ) | 2 ) α + n . Proof.It is clear that when ∥ φ ∥ ∞ = sup z ∈ D | φ ( z ) | 0 , ω be a weight, g ∈ H ( D ) , and φ ∈ S ( D ) such that the operator C φ , g n : F α → B ω is bounded. Then, the operator C φ , g n : F α → B ω is compact if and only if lim sup | φ ( z ) | → 1 ω ( z ) | g ( z ) | ( 1 − | φ ( z ) | 2 ) α + n = 0 . 3. Boundedness and Compactness of C φ , g n : F α → D ω In this section, we investigate the norm, boundedness, and compactness of the operator C φ , g n : F α → D ω . First, we obtain the exact norm of the operator C φ , g n : F α → D ω . Theorem 4.Let n ∈ N , α > 0 , ω be a radial weight, g ∈ H ( D ) , and φ ∈ S ( D ) . Then,∥ C φ , g n ∥ F α → D ω = ∏ l = 0 n − 1 ( α + l ) sup ξ ∈ T ∫ D | g ( z ) | 2 | 1 − ξ ପ୍ତ φ ( z ) | 2 ( n + α ) ω ( z ) d A ( z ) 1 2 . Proof.Consider the function f ξ ( z ) = 1 ( 1 − ξ ପ୍ତ z ) α , ξ ∈ T . It is clear that ∥ f ξ ∥ F α = 1 . Therefore, ∥ C φ , g n ∥ F α → D ω ≥ ∥ C φ , g n f ξ ∥ D ω = | ( C φ , g n f ξ ) ( 0 ) | ︸ 0 + ∫ D | f ( n ) ( φ ( z ) ) | 2 | g ( z ) | 2 ω ( z ) d A ( z ) 1 2 = ( ∏ l = 0 n − 1 ( α + l ) ) 2 ∫ D | ξ | 2 n | 1 − ξ ପ୍ତ φ ( z ) | 2 ( α + n ) | g ( z ) | 2 ω ( z ) d A ( z ) 1 2 = ∏ l = 0 n − 1 ( α + l ) ∫ D | g ( z ) | 2 | 1 − ξ ପ୍ତ φ ( z ) | 2 ( α + n ) ω ( z ) d A ( z ) 1 2 . The above holds for every ξ ∈ T . Thus, ∏ l = 0 n − 1 ( α + l ) sup ξ ∈ T ∫ D | g ( z ) | 2 | 1 − ξ ପ୍ତ φ ( z ) | 2 ( α + n ) ω ( z ) d A ( z ) 1 2 ≤ ∥ C φ , g n ∥ F α → D ω . Now, we prove the other part of the inequality. Let f ∈ F α . Then, there exists a μ ∈ M such that ∥ μ ∥ = ∥ f ∥ F α and f ( z ) = ∫ T d μ ( ξ ) ( 1 − ξ ପ୍ତ z ) α . Moreover, f ( n ) ( z ) = ∏ l = 0 n − 1 ( α + l ) ∫ T ξ ପ୍ତ n ( 1 − ξ ପ୍ତ z ) n + α d μ ( ξ ) . Thus, Jensen’s inequality implies that | f ( n ) ( φ ( z ) ) | 2 | g ( z ) | 2 ω ( z ) = ∏ l = 0 n − 1 ( α + l ) 2 ∫ T ξ ପ୍ତ n ( 1 − ξ ପ୍ତ φ ( z ) ) n + α d μ ( ξ ) 2 | g ( z ) | 2 ω ( z ) ≤ ∏ l = 0 n − 1 ( α + l ) 2 ∥ μ ∥ 2 ∫ T | g ( z ) | 2 | 1 − ξ ପ୍ତ φ ( z ) | 2 ( n + α ) ω ( z ) d | μ | ( ξ ) ∥ μ ∥ . Set β = ∏ l = 0 n − 1 ( α + l ) . Then, based on Fubini’s Theorem, ∫ D | f ( n ) ( φ ( z ) ) | 2 | g ( z ) | 2 ω ( z ) d A ( z ) ≤ β 2 ∥ μ ∥ ∫ D ∫ T | g ( z ) | 2 | 1 − ξ ପ୍ତ φ ( z ) | 2 ( n + α ) ω ( z ) d | μ | ( ξ ) d A ( z ) = β 2 ∥ μ ∥ ∫ T ∫ D | g ( z ) | 2 | 1 − ξ ପ୍ତ φ ( z ) | 2 ( n + α ) ω ( z ) d A ( z ) d | μ | ( ξ ) ≤ β 2 ∥ μ ∥ sup ξ ∈ T ∫ D | g ( z ) | 2 | 1 − ξ ପ୍ତ φ ( z ) | 2 ( n + α ) ω ( z ) d A ( z ) ∫ T d | μ | ( ξ ) ≤ β 2 ∥ μ ∥ 2 sup ξ ∈ T ∫ D | g ( z ) | 2 | 1 − ξ ପ୍ତ φ ( z ) | 2 ( n + α ) ω ( z ) d A ( z ) = β 2 ∥ f ∥ F α 2 sup ξ ∈ T ∫ D | g ( z ) | 2 | 1 − ξ ପ୍ତ φ ( z ) | 2 ( n + α ) ω ( z ) d A ( z ) . Thus, ∥ C φ , g n f ∥ D ω = ∫ D | f ( n ) ( φ ( z ) ) | 2 | g ( z ) | 2 ω ( z ) d A ( z ) 1 2 ≤ β ∥ f ∥ F α sup ξ ∈ T ∫ D | g ( z ) | 2 | 1 − ξ ପ୍ତ φ ( z ) | 2 ( n + α ) ω ( z ) d A ( z ) 1 2 . Therefore, ∥ C φ , g n ∥ F α → D ω ≤ ∏ l = 0 n − 1 ( α + l ) sup ξ ∈ T ∫ D | g ( z ) | 2 | 1 − ξ ପ୍ତ φ ( z ) | 2 ( n + α ) ω ( z ) d A ( z ) 1 2 . The proof is complete. □ Through the application of Theorem 4, we obtain the following corollary. Corollary 3.Let n ∈ N , α > 0 , ω be a radial weight, g ∈ H ( D ) , and φ ∈ S ( D ) . Then, the operator C φ , g n : F α → D ω is bounded if and only if sup ξ ∈ T ∫ D | g ( z ) | 2 | 1 − ξ ପ୍ତ φ ( z ) | 2 ( n + α ) ω ( z ) d A ( z ) 0 , ω be a radial weight, g ∈ H ( D ) , and φ ∈ S ( D ) . If∫ D | g ( z ) | 2 ( 1 − | φ ( z ) | ) 2 ( n + α ) ω ( z ) d A ( z ) 0 , ω be a radial weight, g ∈ H ( D ) , and φ ∈ S ( D ) such that the operator C φ , g n : F α → D ω is bounded. Then, the operator C φ , g n : F α → D ω is compact if and only if lim r → 1 sup ξ ∈ T ∫ | φ ( z ) | > r | g ( z ) | 2 | 1 − ξ ପ୍ତ φ ( z ) | 2 ( n + α ) ω ( z ) d A ( z ) = 0 . (9) Proof.First, assume that ( 9) holds. To prove compactness, we show that for every bounded sequence { f j } j ∈ N in F α , which converges to zero uniformly on compact subsets of D , lim j → ∞ ∥ C φ , g n f j ∥ D ω = 0 ; see Lemma 2.10 [ 13]. Since for each j ∈ N , f j ∈ F α , there exists μ j ∈ M such that ∥ μ j ∥ = ∥ f j ∥ F α and f j ( z ) = ∫ T d μ j ( ξ ) ( 1 − ξ ପ୍ତ z ) α . In this case, we have f j ( n ) ( z ) = ∏ l = 0 n − 1 ( α + l ) ∫ T ξ ପ୍ତ n ( 1 − ξ ପ୍ତ z ) n + α d μ j ( ξ ) . Set β = ∏ l = 0 n − 1 ( α + l ) . Applying Jensen’s inequality, we have | f j ( n ) ( φ ( z ) ) | 2 | g ( z ) | 2 ω ( z ) = β 2 ∫ T ξ ପ୍ତ n ( 1 − ξ ପ୍ତ φ ( z ) ) n + α d μ j ( ξ ) 2 | g ( z ) | 2 ω ( z ) ≤ β 2 ∥ μ j ∥ 2 ∫ T | g ( z ) | 2 | 1 − ξ ପ୍ତ φ ( z ) | 2 ( n + α ) ω ( z ) d | μ j | ( ξ ) ∥ μ j ∥ . (10) It follows from ( 9) that for every ϵ > 0 , we can find 0 r | g ( z ) | 2 | 1 − ξ ପ୍ତ φ ( z ) | 2 ( n + α ) ω ( z ) d A ( z ) r | f j ( n ) ( φ ( z ) ) | 2 | g ( z ) | 2 ω ( z ) d A ( z ) : = Q 1 + Q 2 . (12) We know that f j ( n ) → 0 uniformly on compact subsets of D . Therefore, sup | φ ( z ) | ≤ r | f j ( n ) ( φ ( z ) ) | 2 r | f j ( n ) ( φ ( z ) ) | 2 | g ( z ) | 2 ω ( z ) d A ( z ) ≤ β 2 ∥ μ j ∥ ∫ | φ ( z ) | > r ∫ T | g ( z ) | 2 | 1 − ξ ପ୍ତ φ ( z ) | 2 ( n + α ) ω ( z ) d | μ j | ( ξ ) d A ( z ) = β 2 ∥ μ j ∥ ∫ T ∫ | φ ( z ) | > r | g ( z ) | 2 | 1 − ξ ପ୍ତ φ ( z ) | 2 ( n + α ) ω ( z ) d A ( z ) d | μ j | ( ξ ) ≤ β 2 ∥ μ j ∥ sup ξ ∈ T ∫ | φ ( z ) | > r | g ( z ) | 2 | 1 − ξ ପ୍ତ φ ( z ) | 2 ( n + α ) ω ( z ) d A ( z ) ∫ T d | μ j | ( ξ ) ≤ β 2 ∥ μ j ∥ 2 sup ξ ∈ T ∫ | φ ( z ) | > r | g ( z ) | 2 | 1 − ξ ପ୍ତ φ ( z ) | 2 ( n + α ) ω ( z ) d A ( z ) ⪯ ϵ . (14) From Equations ( 12)–( 14), we get ∥ C φ , g n f j ∥ D ω ⪯ 2 ϵ . Hence, the operator C φ , g n : F α → D ω is compact. To prove the converse, note that ∫ D | g ( z ) | 2 ω ( z ) d A ( z ) 0 , there is an r ∈ ( 0 , 1 ) such that ∫ | φ ( z ) | > r | g ( z ) | 2 ω ( z ) d A ( z ) 0 , there is a t ∈ ( 0 , 1 ) such that ∥ C φ , g n f t − C φ , g n f ∥ D ω 2 = ∫ D | f t ( n ) ( φ ( z ) ) − f ( n ) ( φ ( z ) ) | 2 | g ( z ) | 2 ω ( z ) d A ( z ) r | f ( n ) ( φ ( z ) ) | 2 | g ( z ) | 2 ω ( z ) d A ( z ) ≤ ∫ | φ ( z ) | > r | f t ( n ) ( φ ( z ) ) − f ( n ) ( φ ( z ) ) | 2 | g ( z ) | 2 ω ( z ) d A ( z ) + ∫ | φ ( z ) | > r | f t ( n ) ( φ ( z ) ) | 2 | g ( z ) | 2 ω ( z ) d A ( z ) ≤ ϵ + ∥ f t ( n ) ∥ ∞ 2 ϵ . Hence, for each f ∈ F α with ∥ f ∥ F α ≤ 1 , there is a δ 0 ∈ ( 0 , 1 ) and δ 0 = δ 0 ( f , ϵ ) such that for r ∈ ( δ 0 , 1 ) , ∫ | φ ( z ) | > r | f ( n ) ( φ ( z ) ) | 2 | g ( z ) | 2 ω ( z ) d A ( z ) 0 , there is a finite collection of functions f 1 , ⋯ , f k with ∥ f j ∥ F α ≤ 1 , 1 ≤ j ≤ k , such that for each f ∈ F α with ∥ f ∥ F α ≤ 1 , there is a j ∈ { 1 , 2 , ⋯ , k } such that ∫ D | f ( n ) ( φ ( z ) ) − f j ( n ) ( φ ( z ) ) | 2 | g ( z ) | 2 ω ( z ) d A ( z ) r | f j ( n ) ( φ ( z ) ) | 2 | g ( z ) | 2 ω ( z ) d A ( z ) r | f ( n ) ( φ ( z ) ) | 2 | g ( z ) | 2 ω ( z ) d A ( z ) ⪯ ϵ . (20) Now, consider the function f ( z ) = f ξ ( z ) = 1 ( 1 − ξ ପ୍ତ z ) α , ξ ∈ T and apply ( 20) to these functions; we obtain sup ξ ∈ T ∫ | φ ( z ) | > r | g ( z ) | 2 | 1 − ξ ପ୍ତ φ ( z ) | 2 ( n + α ) ω ( z ) d A ( z ) ⪯ ϵ . The proof is complete. □ Potential Applications Compact operators are important not only for their well-developed theory but also because they arise naturally in many applications, such as integral equations in mathematical physics. The compactness criteria established in Corollary 2.6 and 3.4 can be used to determine when the integral equation ∫ 0 z f ( n ) ( φ ( ζ ) ) g ( ζ ) d ζ = h ( z ) has a solution in the weighted Bloch space B ω or the weighted Dirichlet space D ω . Moreover, in the compact case, the operator C φ , g n can be approximated by finite-rank operators, which allows the use of numerical methods. These results also apply to special cases of the operator, including composition operators, integral operators, and differentiation-composition operators, which have been widely studied in the literature. 4. Conclusions In this paper, we systematically investigated the generalized integral-type operator C φ , g n acting between the fractional Cauchy transform space F α and two important classes of analytic function spaces: the weighted Bloch space B ω and the weighted Dirichlet space D ω . Our main objectives were to compute or estimate the operator and essential norms and characterize the boundedness and compactness of these operators. For the operator C φ , g n : F α → B ω , we established two equivalent exact formulas for the operator norm (see Theorems 1 and 2). These formulas are expressed in terms of the weight ω , functions g and φ , and parameters α and n, and they directly characterize boundedness (Corollary 1). Moreover, we obtained an estimate for the essential norm, showing that it is equivalent to a limit involving the weight and the symbol φ as | φ ( z ) | → 1 (Theorem 3). Consequently, we could derive a necessary and sufficient condition for the compactness of the operator (Corollary 2). In the second part, we studied the operator C φ , g n : F α → D ω . We provided an exact formula for the operator norm (Theorem 4) and, as a corollary, gave a boundedness criterion (Corollary 3). Furthermore, we characterized the compactness of this operator as a limiting condition on the integral kernel over the region where | φ ( z ) | is close to one (Theorem 5). The proof techniques applied here involved the use of test functions, measure representations for functions in F α , and standard compactness criteria in Banach spaces of analytic functions. The results presented here extend and unify several previous studies on composition operators, integral-type operators, and their generalizations on spaces of analytic functions. Exact norm formulas are obtained for both target spaces, and an essential norm estimate (in the sense of an equivalence) is provided for the Bloch case; for the Dirichlet case, a compactness criterion is established. Not only are these results of theoretical interest, but they also provide practical tools for verifying boundedness and compactness in concrete applications. Possible directions for future research include the study of similar operators on other function spaces, such as weighted Bergman or Hardy spaces, and the investigation of the essential norm for other operator classes. Author Contributions Conceptualization, M.H.; Methodology, M.H.; Software, E.A.; Validation, E.A. and M.G.A.; Formal analysis, M.G.A.; Investigation, M.H.; Resources, M.H.; Data curation, E.A.; Writing—original draft, E.A. and M.G.A.; Writing—review & editing, M.G.A.; Visualization, M.H.; Supervision, M.H.; Project administration, E.A.; Funding acquisition, E.A. and M.G.A. All authors have read and agreed to the published version of the manuscript. Funding This research received no external funding. Data Availability Statement The data that support the findings of this study are available on request from the corresponding author. The data are not publicly available due to privacy or ethical restrictions. Conflicts of Interest The authors declare no conflicts of interest. References Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. © 2026 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license. Share and Cite MDPI and ACS Style Hassanlou, M.; Abbasi, E.; Alshehri, M.G. Essential Norm of Generalized Integral-Type Operator from the Fractional Cauchy Transform Space into Weighted Bloch and Dirichlet Spaces. Axioms 2026, 15, 418. https://doi.org/10.3390/axioms15060418 AMA Style Hassanlou M, Abbasi E, Alshehri MG. Essential Norm of Generalized Integral-Type Operator from the Fractional Cauchy Transform Space into Weighted Bloch and Dirichlet Spaces. Axioms. 2026; 15(6):418. https://doi.org/10.3390/axioms15060418 Chicago/Turabian Style Hassanlou, Mostafa, Ebrahim Abbasi, and Maryam G. Alshehri. 2026. "Essential Norm of Generalized Integral-Type Operator from the Fractional Cauchy Transform Space into Weighted Bloch and Dirichlet Spaces" Axioms 15, no. 6: 418. https://doi.org/10.3390/axioms15060418 APA Style Hassanlou, M., Abbasi, E., & Alshehri, M. G. (2026). Essential Norm of Generalized Integral-Type Operator from the Fractional Cauchy Transform Space into Weighted Bloch and Dirichlet Spaces. Axioms, 15(6), 418. https://doi.org/10.3390/axioms15060418 Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here. Article Metrics Article metric data becomes available approximately 24 hours after publication online. 2. Norm and Essential Norm of C φ , g n : F α → B ω In this section, we obtain two exact formulas for the norm of the generalized integral-type operator from the Cauchy transform space into a weighted Bloch space. We also find an estimate for the essential norm of the operator C φ , g n : F α → B ω . As a result, we present equivalence conditions for the compactness of such operators. First, we state the following estimate of | f ( n ) ( z ) | for functions in F α : Lemma 1.If f ∈ F α and n ∈ N 0 , then| f ( n ) ( z ) | ≤ C ∥ f ∥ F α ( 1 − | z | 2 ) α + n , z ∈ D (3) where C is a positive constant independent of f. Proof.Since f ∈ F α , there exists a μ ∈ M such that f ( z ) = ∫ T 1 ( 1 − ξ ପ୍ତ z ) α d μ ( ξ ) . Taking the derivative of the above equation, we have f ( n ) ( z ) = ∏ l = 0 n − 1 ( α + l ) ∫ T ξ ପ୍ତ n ( 1 − ξ ପ୍ତ z ) α + n d μ ( ξ ) . Thus, | f ( n ) ( z ) | ≤ C ∫ T d | μ | ( ξ ) | 1 − ξ ପ୍ତ z | α + n ≤ C ∫ T d | μ | ( ξ ) ( 1 − | z | 2 ) α + n = C ∥ μ ∥ ( 1 − | z | 2 ) α + n . Taking the infimum over all measures in M , inequality ( 3) is obtained. □ In the following theorem, we obtain the first formula for the norm of the operator C φ , g n : F α → B ω . Theorem 1.Let n ∈ N 0 , α > 0 , ω be a weight, g ∈ H ( D ) , and φ ∈ S ( D ) . Then, ∥ C φ , g n ∥ F α → B ω = ∏ l = 0 n − 1 ( α + l ) sup ξ ∈ T sup z ∈ D ω ( z ) | g ( z ) | | 1 − ξ ପ୍ତ φ ( z ) | n + α . (4) Proof.For any ξ ∈ T , let f ξ ( z ) = 1 ( 1 − ξ ପ୍ତ z ) α . Then, f ξ ∈ F α , ∥ f ξ ∥ F α = 1 , and f ξ ( k ) ( z ) = ξ ପ୍ତ k ∏ l = 0 k − 1 ( α + l ) ( 1 − ξ ପ୍ତ z ) α + k , where k ∈ N [ 12]. The definition of the norm of the operator implies that ∥ C φ , g n ∥ F α → B ω ≥ ∥ C φ , g n ( f ξ ) ∥ B ω = | C φ , g n ( f ξ ) ( 0 ) | + sup z ∈ D ω ( z ) | ( C φ , g n f ξ ) ′ ( z ) | (5) Since C φ , g n ( f ξ ) ( 0 ) = 0 and ( C φ , g n f ξ ) ′ ( z ) = g ( z ) f ξ ( n ) ( φ ( z ) ) , we obtain ∥ C φ , g n ( f ξ ) ∥ B ω = ∏ l = 0 n − 1 ( α + l ) sup z ∈ D · ω ( z ) | g ( z ) | | 1 − ξ ପ୍ତ φ ( z ) | n + α . Taking the supremum over ξ ∈ T gives ∥ C φ , g n ∥ F α → B ω ≥ ∏ l = 0 n − 1 ( α + l ) sup ξ ∈ T sup z ∈ D ω ( z ) | g ( z ) | | 1 − ξ ପ୍ତ φ ( z ) | n + α . Conversely, for any f ∈ F α , there exists ν ∈ M such that ∥ ν ∥ = ∥ f ∥ F α and f ( z ) = ∫ T d ν ( ξ ) ( 1 − ξ ପ୍ତ z ) α ; see [ 12]. Moreover, f ( n ) ( z ) = ∏ l = 0 n − 1 ( α + l ) ∫ T ξ ପ୍ତ n ( 1 − ξ ପ୍ତ z ) n + α d ν ( ξ ) . Hence, for any f ∈ F α , we have ∥ C φ , g n f ∥ B ω = | ( C φ , g n f ) ( 0 ) | + sup z ∈ D ω ( z ) | ( C φ , g n f ) ′ ( z ) | = ∏ l = 0 n − 1 ( α + l ) sup z ∈ D ω ( z ) | g ( z ) | ∫ T ξ ପ୍ତ n ( 1 − ξ ପ୍ତ φ ( z ) ) n + α d ν ( ξ ) ≤ ∏ l = 0 n − 1 ( α + l ) sup z ∈ D ω ( z ) | g ( z ) | ∫ T 1 | 1 − ξ ପ୍ତ φ ( z ) ) | n + α d | ν | ( ξ ) ≤ ∏ l = 0 n − 1 ( α + l ) sup ξ ∈ T sup z ∈ D ω ( z ) | g ( z ) | | 1 − ξ ପ୍ତ φ ( z ) | n + α ∫ T d | ν | ( ξ ) ≤ ∏ l = 0 n − 1 ( α + l ) sup ξ ∈ T sup z ∈ D ω ( z ) | g ( z ) | | 1 − ξ ପ୍ତ φ ( z ) | n + α ∥ ν ∥ = ∏ l = 0 n − 1 ( α + l ) sup ξ ∈ T sup z ∈ D ω ( z ) | g ( z ) | | 1 − ξ ପ୍ତ φ ( z ) | n + α ∥ f ∥ F α = sup ξ ∈ T sup z ∈ D ∏ l = 0 n − 1 ( α + l ) ω ( z ) | g ( z ) | | 1 − ξ ପ୍ତ φ ( z ) | n + α ∥ f ∥ F α . Considering the above discussion with regard to inequality ( 5) yields ( 4). □ In the next theorem, we find another formula for the norm of the operator C φ , g n : F α → B ω . Theorem 2.Let n ∈ N 0 , α > 0 , ω be a weight, g ∈ H ( D ) , and φ ∈ S ( D ) . Then, ∥ C φ , g n ∥ F α → B ω = ∏ l = 0 n − 1 ( α + l ) sup z ∈ D ω ( z ) | g ( z ) | ( 1 − | φ ( z ) | ) n + α . Proof.For any w ∈ D , there exists 0 ≤ θ w 0 , ω be a weight, g ∈ H ( D ) , and φ ∈ S ( D ) . Then, the following statements are equivalent: 1. The operator C φ , g n : F α → B ω is bounded. 2. sup z ∈ D ω ( z ) | g ( z ) | ( 1 − | φ ( z ) | ) n + α 0 , ω be a weight, g ∈ H ( D ) and φ ∈ S ( D ) . If C φ , g n : F α → B ω is bounded, then∥ C φ , g n ∥ ϵ ≈ lim sup | φ ( z ) | → 1 ω ( z ) | g ( z ) | ( 1 − | φ ( z ) | 2 ) α + n . Proof.It is clear that when ∥ φ ∥ ∞ = sup z ∈ D | φ ( z ) | 0 , ω be a weight, g ∈ H ( D ) , and φ ∈ S ( D ) such that the operator C φ , g n : F α → B ω is bounded. Then, the operator C φ , g n : F α → B ω is compact if and only if lim sup | φ ( z ) | → 1 ω ( z ) | g ( z ) | ( 1 − | φ ( z ) | 2 ) α + n = 0 . 3. Boundedness and Compactness of C φ , g n : F α → D ω In this section, we investigate the norm, boundedness, and compactness of the operator C φ , g n : F α → D ω . First, we obtain the exact norm of the operator C φ , g n : F α → D ω . Theorem 4.Let n ∈ N , α > 0 , ω be a radial weight, g ∈ H ( D ) , and φ ∈ S ( D ) . Then,∥ C φ , g n ∥ F α → D ω = ∏ l = 0 n − 1 ( α + l ) sup ξ ∈ T ∫ D | g ( z ) | 2 | 1 − ξ ପ୍ତ φ ( z ) | 2 ( n + α ) ω ( z ) d A ( z ) 1 2 . Proof.Consider the function f ξ ( z ) = 1 ( 1 − ξ ପ୍ତ z ) α , ξ ∈ T . It is clear that ∥ f ξ ∥ F α = 1 . Therefore, ∥ C φ , g n ∥ F α → D ω ≥ ∥ C φ , g n f ξ ∥ D ω = | ( C φ , g n f ξ ) ( 0 ) | ︸ 0 + ∫ D | f ( n ) ( φ ( z ) ) | 2 | g ( z ) | 2 ω ( z ) d A ( z ) 1 2 = ( ∏ l = 0 n − 1 ( α + l ) ) 2 ∫ D | ξ | 2 n | 1 − ξ ପ୍ତ φ ( z ) | 2 ( α + n ) | g ( z ) | 2 ω ( z ) d A ( z ) 1 2 = ∏ l = 0 n − 1 ( α + l ) ∫ D | g ( z ) | 2 | 1 − ξ ପ୍ତ φ ( z ) | 2 ( α + n ) ω ( z ) d A ( z ) 1 2 . The above holds for every ξ ∈ T . Thus, ∏ l = 0 n − 1 ( α + l ) sup ξ ∈ T ∫ D | g ( z ) | 2 | 1 − ξ ପ୍ତ φ ( z ) | 2 ( α + n ) ω ( z ) d A ( z ) 1 2 ≤ ∥ C φ , g n ∥ F α → D ω . Now, we prove the other part of the inequality. Let f ∈ F α . Then, there exists a μ ∈ M such that ∥ μ ∥ = ∥ f ∥ F α and f ( z ) = ∫ T d μ ( ξ ) ( 1 − ξ ପ୍ତ z ) α . Moreover, f ( n ) ( z ) = ∏ l = 0 n − 1 ( α + l ) ∫ T ξ ପ୍ତ n ( 1 − ξ ପ୍ତ z ) n + α d μ ( ξ ) . Thus, Jensen’s inequality implies that | f ( n ) ( φ ( z ) ) | 2 | g ( z ) | 2 ω ( z ) = ∏ l = 0 n − 1 ( α + l ) 2 ∫ T ξ ପ୍ତ n ( 1 − ξ ପ୍ତ φ ( z ) ) n + α d μ ( ξ ) 2 | g ( z ) | 2 ω ( z ) ≤ ∏ l = 0 n − 1 ( α + l ) 2 ∥ μ ∥ 2 ∫ T | g ( z ) | 2 | 1 − ξ ପ୍ତ φ ( z ) | 2 ( n + α ) ω ( z ) d | μ | ( ξ ) ∥ μ ∥ . Set β = ∏ l = 0 n − 1 ( α + l ) . Then, based on Fubini’s Theorem, ∫ D | f ( n ) ( φ ( z ) ) | 2 | g ( z ) | 2 ω ( z ) d A ( z ) ≤ β 2 ∥ μ ∥ ∫ D ∫ T | g ( z ) | 2 | 1 − ξ ପ୍ତ φ ( z ) | 2 ( n + α ) ω ( z ) d | μ | ( ξ ) d A ( z ) = β 2 ∥ μ ∥ ∫ T ∫ D | g ( z ) | 2 | 1 − ξ ପ୍ତ φ ( z ) | 2 ( n + α ) ω ( z ) d A ( z ) d | μ | ( ξ ) ≤ β 2 ∥ μ ∥ sup ξ ∈ T ∫ D | g ( z ) | 2 | 1 − ξ ପ୍ତ φ ( z ) | 2 ( n + α ) ω ( z ) d A ( z ) ∫ T d | μ | ( ξ ) ≤ β 2 ∥ μ ∥ 2 sup ξ ∈ T ∫ D | g ( z ) | 2 | 1 − ξ ପ୍ତ φ ( z ) | 2 ( n + α ) ω ( z ) d A ( z ) = β 2 ∥ f ∥ F α 2 sup ξ ∈ T ∫ D | g ( z ) | 2 | 1 − ξ ପ୍ତ φ ( z ) | 2 ( n + α ) ω ( z ) d A ( z ) . Thus, ∥ C φ , g n f ∥ D ω = ∫ D | f ( n ) ( φ ( z ) ) | 2 | g ( z ) | 2 ω ( z ) d A ( z ) 1 2 ≤ β ∥ f ∥ F α sup ξ ∈ T ∫ D | g ( z ) | 2 | 1 − ξ ପ୍ତ φ ( z ) | 2 ( n + α ) ω ( z ) d A ( z ) 1 2 . Therefore, ∥ C φ , g n ∥ F α → D ω ≤ ∏ l = 0 n − 1 ( α + l ) sup ξ ∈ T ∫ D | g ( z ) | 2 | 1 − ξ ପ୍ତ φ ( z ) | 2 ( n + α ) ω ( z ) d A ( z ) 1 2 . The proof is complete. □ Through the application of Theorem 4, we obtain the following corollary. Corollary 3.Let n ∈ N , α > 0 , ω be a radial weight, g ∈ H ( D ) , and φ ∈ S ( D ) . Then, the operator C φ , g n : F α → D ω is bounded if and only if sup ξ ∈ T ∫ D | g ( z ) | 2 | 1 − ξ ପ୍ତ φ ( z ) | 2 ( n + α ) ω ( z ) d A ( z ) 0 , ω be a radial weight, g ∈ H ( D )
