Inequalities Involving Essential Norm Estimates of Product-Type Operators

Consider an open unit disk D � z ∈ C : | z | < 1 { } in the complex plane C , ξ a holomorphic function on D , and ψ a holomorphic self-map of D . For an analytic function f , the weighted composition operator is denoted and deﬁned as follows: ( W ξ , ψ f )( z ) � ξ ( z ) f ( ψ ( z )) . We estimate the essential norm of this operator from Dirichlet-type spaces to Bers-type spaces and


Introduction and Preliminaries
Consider an open unit disk D � z ∈ C: |z| < 1 { } in the complex plane C. Let H(D) denote the class of all analytic functions on D, S(D) be the class of all holomorphic selfmaps of D, and H ∞ be the space of all bounded holomorphic functions on D. Let ξ ∈ H(D) and ψ be a holomorphic selfmap of D. For z ∈ D, the composition operator and multiplication operator are, respectively, defined by (1) e weighted composition operator is denoted and defined as where W ξ,ψ is a product-type operator as W ξ,ψ � M ξ C ψ . Clearly, this operator can be seen as a generalization of the composition operator and multiplication operator. It can be easily seen that, for ξ ≡ 1, the operator reduced to C ψ . If ψ(z) � z, the operator gets reduced to M ξ . is operator is basically a linear transformation of H(D) defined by (W ξ,ψ f)(z) � ξ(z)f(ψ(z)) � (M ξ C ψ f)(z), for f in H (D) and z in D. e basic aim is to give the operator-theoretic characterization of these operators in terms of functiontheoretic characterization of their including functions. Various holomorphic function spaces on various domains have been studied for the boundedness and compactness of weighted composition operators acting on them. Moreover, a number of papers have been studied on these operators acting on different spaces of holomorphic functions on various domains. For more details, see [1][2][3][4][5][6][7][8][9][10][11][12][13][14] and the references therein. We say that a linear operator is bounded if the image of a bounded set is a bounded set. Moreover, a linear operator is said to be compact if it maps the bounded sets to those sets whose closure is compact. For each α > 0, the weighted Bloch space B is defined as follows: In this expression, seminormed is defined. is space forms a Banach space with the natural norm defined by For α � 1, this space gets reduced to classical Bloch space. A function ω: D ⟶ (0, ∞) is said to be a weight if it is continuous. For z ∈ D, the weight ω is said to be radial if ω(z) � ω(|z|). A weight ω is said to be a standard weight if lim |z|⟶1 − 1 ω(z) � 0. For a weight function ω, the Bloch-type space B ω is defined by e little Bloch-type space B ω,0 is the closure of the set of polynomials in B ω and is defined as follows: Both B ω and B ω,0 form a Banach space with the following norm: For more information about these spaces, one may refer [1-3, 5, 6, 15, 16] and the references therein. Likewise, for weight ω, the Bers-type space A ω is defined as follows: It is a nonseparable Banach space with the norm ‖ · ‖ A ω . e closure of the set of polynomials in A ω forms a separable Banach space. is set is denoted by A ω,0 and is defined as ese spaces and their properties are discussed in many papers; some of these are [3,15,16] and the references therein. e Dirichlet space is defined as follows: where dA(z) denotes the normalized Lebesgue area measure on D. With the following norm, it is a Hilbert space: Consider a function K: [0, ∞) ⟶ [0, ∞) which is right continuous and increasing. In this paper, we consider function K as a weight function. With a weight function K, the Dirichlet type space D K is given as follows: Clearly, space D K forms a Hilbert space with the norm ‖ · ‖ D K defined by Here, we have K(t) � t p , 0 ≤ p < ∞, and D K gives D p , that is, the usual Dirichlet-type space. is gives a classical Dirichlet space D for a case when p � 0 and for p � 1, and we gain the Hardy space H 2 . ese spaces have been studied widely in various papers. For example, Aleman, in [17], obtained that each element of D K can be written as a quotient of two bounded functions in D K . Kerman and Sawyer [18], by taking some conditions on weight function K, characterized Carleson measures and multipliers of D K in terms of maximal operator.
e Möbius invariant space generated by D K is denoted by Q K . e space Q K contains those functions f ∈ H(D) which satisfy the following: where σ a (z) � ((a − z)/(1 − az)) is the Möbius transformation of D. Wulan and Zhu, in [19], characterized Lacunary series in the Q K space under some conditions on weight function K. Furthermore, Wulan and Zhou [20] characterized space Q K in terms of fractional-order derivatives of function. ey also established a relationship between Morrey type spaces and Q K space in terms of fractional order derivatives. In the study of Q K spaces, the following two conditions play a very important role: where Let M(D K ) be the class of multipliers of D K , that is, Bao et al., in [21], characterized the interpolating sequences for M(D K ) of space D K , under certain conditions of weight function K. ey also obtained corona theorem, z -equation, and corona-type decomposition theorem on M(D K ). For more details, see [9,15,[21][22][23][24][25] and the references therein.
From [26], one can see that if K satisfies (1), then If K satisfies (16), then From condition (16), we get that K(2t) ≈ K(t) for 0 < t < 1. Also, there exist C > 0 sufficiently small for which t − C K 1 (t) is increasing and K 2 (t)t C− 1 is decreasing. For more information about weight function K, one can refer [19][20][21]. e criterion of boundedness as well as compactness has been discussed in many papers. Recently, Gürbüz, in [27], studied the boundedness of generalized commutators of rough fractional maximal and integral operators on generalized weighted Morrey spaces, respectively, and, in [28], he investigated the generalized weighted Morrey estimates for the boundedness of Marcinkiewicz integrals with rough kernel associated with Schrödinger operator and their commutators. Furthermore, in [29], Gürbuüz studied the behavior of multi-sublinear fractional maximal operators and rough multilinear fractional integral both on product L p and weighted L p spaces and, in [30], he obtained the boundedness of the variation and oscillation operators for the family of multilinear integrals with Lipschitz functions on weighted Morrey spaces. Among others in [3], we obtained the following results about boundedness and compactness of W ξ,ψ given as follows.

Theorem 1.
Let ω and K be two weight functions, ξ ∈ H(D), and ψ be a self-holomorphic map on D. en, the operator W ξ,ψ : D K ⟶ B ω is bounded if and only if the following conditions hold:

Theorem 2. Let ω be a standard weight, ξ ∈ H(D), and ψ be a self-holomorphic map on D. Let K be a weight function.
Assume that W ξ,ψ : D K ⟶ B ω is bounded. en, the operator W ξ,ψ : D K ⟶ B ω is compact if and only if the following conditions hold: Let ω be a weight and K be a weight function, ξ ∈ H(D), and ψ be a self-holomorphic map on D. en, the operator W ξ,ψ : D K ⟶ A ω is bounded if and only if the following condition holds: Theorem 4. Let ω be a standard weight, ξ ∈ H(D), and ψ be a self-analytic map on D. Let K be a weight function. Assume that the operator W ξ,ψ : e aim of this paper is to provide some estimates of essential norm of the operator W ξ,ψ : Assume that T: X 1 ⟶ X 2 is a bounded linear operator for Banach spaces X 1 and X 2 . e essential norm of operator T is denoted and defined as follows: where ‖ · ‖ X 1 ⟶ X 2 is the operator norm. In other words, the essential norm is the distance from compact operators E mapping X 1 into X 2 to the bounded linear operator T: X 1 ⟶ X 2 . If X 1 � X 2 , that is, the two Banach spaces are same, then the norm is simply denoted by ‖ · ‖ e . For unbounded linear operator T: As the class of all compact operators is contained in the class of all bounded operators, in fact, this subset is closed, which implies that the operator T is compact if and only if ‖T‖ e,X 1 ⟶ X 2 � 0. us, the estimate of essential norm leads to the compactness of the operator. Various results on the essential norm of different operators such as multiplication, composition, differentiation, weighted composition, generalized weighted composition, and their different combinations are studied in numerous research papers, and some of the references are [31][32][33][34][35][36][37]. is study is formulated in a systematic way. Introduction and literature part is kept in Section 1. In Section 2, we estimated the essential norm of operator W ξ,ψ : D K ⟶ B ω . Finally, in Section 3, we estimated the essential norm of operator W ξ,ψ : D K ⟶ A ω . roughout the paper, the notation a≲b, for any two positive quantities a and b, which means that a ≤ Cb, where C is some positive constant.
e value of constant C may change from one place to the other. We write a ≈ b if a≲b and b≲a.

Essential Norm of Weighted Composition
Operator from Dirichlet-Type Space to Bloch-Type Space Theorem 5. Let ω be a standard weight, ξ ∈ H(D), and ψ be a self-analytic map on D. Let K be a weight function. Assume that W ξ,ψ : D K ⟶ B ω is bounded. en, where

Journal of Mathematics
Proof. At first, we show that For z ∈ D, define a function where a 0 � (2 + ε/2) and a 1 � − (1 + ε/2). It can be easily checked that h z ∈ D K and, for all z ∈ D, ‖h z ‖ D K ≲1. On calculation, we have h z ′ (z) � 0 and ). Furthermore, on compact subsets of D, h z converges to zero as |z| ⟶ 1. Hence, for any compact operator E: D K ⟶ B ω and any (ζ n ) n∈D such that |ψ(ζ n )| ⟶ 1, we obtain In the above inequality, take lim sup |ψ(ζ n )| ⟶ 1 on both sides, and we obtain Again, for z ∈ D, define another function: where b 0 � 1 and b 1 � − 1. In the similar manner, we can check that k z ∈ D K and, for all z ∈ D, ‖k z ‖ D K ≲1. On calculation, we have k z (z) � 0 and . Furthermore, on compact subsets of D, k z converges to zero as |z| ⟶ 1. us, for any compact operator E: D K ⟶ B ω and any (ζ n ) n∈D such that |ψ(ζ n )| ⟶ 1, we obtain By taking lim sup |ψ(ζ n )| ⟶ 1 on both sides of the above inequality, we obtain On applying the definition of essential norm, we find that Next, we prove that For δ ∈ [0, 1), consider E δ : H(D) ⟶ H(D), defined as follows: Clearly, E δ is compact on D K and ‖E δ ‖ D K ⟶ D K ≤ 1. Consider a sequence δ n ⊂ (0, 1) satisfying δ n ⟶ 1 as n ⟶ ∞. en, for all n ∈ N, operator W ξ,ψ E δ n : D K ⟶ B ω is compact. By using the definition of essential norm, we obtain erefore, we only have to prove that Let f be a function in D K satisfying ‖f‖ D K ≤ 1; then, we have 4 Journal of Mathematics (39) Clearly, lim n⟶∞ |ξ(0)f(ψ(0)) − ξ(0)f(δ n ψ(0))| � 0. Furthermore, consider a large enough N ∈ N such that, for all n ≥ N, we have δ n ≥ 1/2. us, we obtain lim sup where Taking the operator W ξ,ψ to 1 and z and using its boundedness, it easily follows that ξ ∈ B ω and Also, on compact subsets of D, δ n f δ n ′ uniformly converges to f ′ as n ⟶ ∞; thus, we have Similarly, for ξ ∈ B ω and the fact that f δ n converges uniformly to f on compact subsets of D as n ⟶ ∞, we obtain Now, consider S 2 . We have S 2 ≤ lim sup n⟶∞ (P 1 + P 2 ), where First, we consider P 1 . As ‖f‖ D K ≤ 1, we obtain On taking limit as N ⟶ ∞, we obtain lim sup In the similar manner, we obtain lim sup On combining the above two inequalities, we obtain Next, consider S 4 . We have S 4 ≤ lim sup n⟶∞ (P 3 + P 4 ), where By similar calculation, we obtain On taking limit N ⟶ ∞, we obtain Journal of Mathematics 5 lim sup In the similar manner, we obtain lim sup Combining the above two inequalities, we obtain On combining (40), (43), (44), (49), and (54), we obtain us, inequalities (37) and (55) imply that Hence, inequalities (34) and (56) complete the theorem. e following corollary can be easily obtained from eorem 5. □ Corollary 1. Let ω be a standard weight and ψ be a selfanalytic map on D. Let K be a weight function. Assume that C ψ : D K ⟶ B ω is bounded. en,

Essential Norm of Weighted Composition Operator from Dirichlet-Type Space to Bers-Type Space
In this section, we consider the Bers-type spaces and estimated the essential norm of weighted composition operator from D K to A ω .

Theorem 6.
Let ω be a standard weight, ξ ∈ H(D), and ψ be a self-analytic map on D. Let K be a weight function. Assume that the operator W ξ,ψ : D K ⟶ A ω is bounded. en, Proof. Firstly, we prove that Consider a function f z ∈ D K such that ‖f z ‖ D K ≲1, and on compact subsets of D, f z converges to zero as |z| ⟶ 1.
us, for any compact operator E: D K ⟶ A ω and any (ζ n ) n∈D such that |ψ(ζ n )| ⟶ 1 − , we obtain Taking lim sup |ψ(ζ n a)| ⟶ 1 − on both sides, we obtain On applying the definition of essential norm, we find that 6 Journal of Mathematics Finally, we prove that For this, consider E δ : H(D) ⟶ H(D) with δ ∈ [0, 1) and a sequence δ n ⊂ (0, 1) satisfying δ n ⟶ 1 as n ⟶ ∞ defined in eorem 5. en, for all n ∈ N, the operator W ξ,ψ E δ n : D K ⟶ A ω is compact. By using the definition of essential norm, we obtain So, we only have to prove that (65) Let f be a function in D K satisfying ‖f‖ D K ≤ 1; then, we have Furthermore, consider a large enough N ∈ N such that, for all n ≥ N, we have δ n ≥ 1/2. us, we obtain lim sup where Similar to eorem 5, for ξ ∈ A ω and the fact that f δ n converges uniformly to f on compact subsets of D as n ⟶ ∞, we obtain Next, we consider A 2 . We have A 2 ≤ lim sup n⟶∞ (R 1 + R 2 ), where R 1 � sup |ψ(z)|>δ N ω(z)|ξ(z)‖f(ψ(z))|, On calculation, we obtain (75) us, inequalities (64) and (75) imply that Hence, inequalities (62) and (76) complete the theorem. e following corollary can be easily obtained from eorem 6. □ Corollary 2. Let ω be a standard weight and ψ be a selfanalytic map on D. Let K be a weight function. Assume that the operator C ψ : D K ⟶ A ω is bounded. en, (77)

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Conflicts of Interest
e authors declare that they have no conflicts of interest.