Embedding of Besov Spaces and the Volterra Integral Operator

where σa(z) � ((a − z)/(1 − az)). /is space was first introduced by Zhao in [1]. F(2, 0, s) is the Qs space (see [2]). F(2, 0, 1) is the BMOA space. F(p, α, 0) is called the Dirichlet-type space, denoted byDpα . In particular, F(p, p − 2, 0) is the Besov space Bp. F(p, p, 0) is just the classical Bergman space A. When s> 1, from [1], we see that F(p, p − 2, s) is equivalent to the Bloch space, denoted byB, which consists of all f ∈ H(D) such that ‖f‖B � |f(0)| + sup z∈D 1 − |z| 􏼐 􏼑 f′(z) 􏼌􏼌􏼌 􏼌􏼌􏼌<∞. (3)


Introduction
Let D be the unit disk in the complex plane C and H(D) be the class of functions analytic in D. For 1 < p < ∞, the Besov space, denoted by B p , is the space of all functions f ∈ H(D) satisfying Let 0 < p < ∞, − 2 < q < ∞, and 0 ≤ s < ∞. e space F(p, q, s) is the space consisting of all f ∈ H(D) such that where σ a (z) � ((a − z)/(1 − az)). is space was first introduced by Zhao in [1]. F(2, 0, s) is the Q s space (see [2]). F(2, 0, 1) is the BMOA space. F(p, α, 0) is called the Dirichlet-type space, denoted by D p α . In particular, F(p, p − 2, 0) is the Besov space B p . F(p, p, 0) is just the classical Bergman space A p . When s > 1, from [1], we see that F(p, p − 2, s) is equivalent to the Bloch space, denoted by B, which consists of all f ∈ H(D) such that ‖f‖ B � |f(0)| + sup z∈D 1 − |z| 2 f ′ (z) < ∞. (3) For 1 < q < ∞ and 0 < s, t < ∞, let LF(q, q − 2, s, t) denote the space of all f ∈ H(D) such that e norm for f ∈ LF(q, q − 2, s, t) is given by e Volterra integral operator T g was introduced by Pommerenke in [3]. Here, In [3], Pommerenke showed that T g is bounded on H 2 if and only if g ∈ BMOA. Aleman and Siskakis showed that T g is bounded on H p (p ≥ 1) if and only if g ∈ BMOA in [4]. In [5], Aleman and Siskakis proved that T g : A p ⟶ A p is bounded (compact) if and only if g ∈ B(g ∈ B 0 ). Recently, the operator T g has been receiving much attention. See [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] and the references therein for more study of the operator T g .
For any arc I⊆zD, the boundary of D, let |I| � (1/2π) I |dζ| denote the normalized length of I and S(I) be the Carleson box defined by Let 0 < s < ∞ and μ be a positive Borel measure on D. We say that μ is a α-Carleson measure if When α � 1, it gives the classical Carleson measure. μ is said to be a vanishing α-Carleson measure if lim |I|⟶0 (μ(S (I))/|I| α ) � 0. e Carleson measure is very useful in the theory of function spaces and operator theory. e famous embedding theorem says that the inclusion mapping is bounded if and only if μ is a Carleson measure (see [19]). See [7,20] for the study of the inclusion mapping I d : B p ⟶ L p (dμ).
Let 0 < s, q < ∞, 0 ≤ t < ∞, and μ be a positive Borel measure on D. Let T q s,t (μ) denote the space of all μ-measurable functions f such that (see, e.g., [21]) e tent space T q s,t (μ) was introduced by Liu et al. in [21]. When t � 0, T q s,t (μ) will be denoted by T q s (μ) for the simplicity. In [21], Liu et al. studied the embedding of some Möbius invariant spaces, such as the Bloch space and the Q p space, into T 2 s,t . In [12], Pau and Zhao showed that the inclusion mapping is bounded if and only if μ is a p-logarithmic s-Carleson measure. In [9], Li et al. proved that the inclusion mapping I d : is bounded if and only if μ is a (s + 1)-Carleson measure. In [14], Qian and Li proved that the inclusion mapping I d : Motivated by [14,21], in this paper, we study the boundedness and compactness of the inclusion mapping and only if μ is an s-Carleson measure (resp. vanishing s-Carleson measure) under the assumption that 1 < p < q < ∞ and 0 < s < ∞. As an application, we study the boundedness of the operator T g : B p ⟶ LF(q, q − 2, s, q − (q/p)). Moreover, the compactness and essential norm of the operator T g : B p ⟶ LF(q, q − 2, s, q − (q/p)) are also investigated.
In this paper, the symbol f ≈ g means that f≲g≲f. We say that f≲g if there exists a constant C such that f ≤ Cg.

Embedding the Besov Space
We need the following equivalent description of α-Carleson measure (see Lemma 2.2 in [12]). Lemma 1. Let 0 < α, t < ∞ and μ be a positive Borel measure on D. en, μ is an α-Carleson measure if and only if Moreover, Using Lemma 3.10 in [22], we can easily obtain the following result. Proof. First, we assume that I d : B p ⟶ T q s,q(1− (1/p)) (μ) is bounded. For any given arc I⊆zD, set a � (1 − |I|)η, and η is the center point of I. It is easy to see that Let f a (z) � 1 log 2/1 − |a| 2 By Lemma 2, we see that f a ∈ B p . From the boundedness of I d : By the fact that |f a (z)| ≈ (log(2/|I|)) 1− (1/p) when z ∈ S(I), we get sup I⊂zD μ(S(I)) |I| s < ∞.

Journal of Mathematics
Conversely, assume that μ is an s-Carleson measure. Let f ∈ B p . For any given arc I⊆zD, set w � (1 − |I|)η, and η is the center point of I. en, where Since we get Now, we turn to estimate A. By eorem 1 in [7], we see that μ is an s-Carleson measure if and only if en, Since we deduce that A≲(W 1 + W 2 ) (q/p) , where Since we get that (25) Making the change of variable η � σ w (z) and combining with Proposition 4.2 in [22], we have which implies the desired result. e proof is completed. We say that the inclusion mapping I d : B p ⟶ T  Proof. First, we assume that I d : B p ⟶ T q s,q(1− (1/p)) (μ) is compact. Let I k be a sequence arc with lim k⟶∞ |I k | � 0. Set a k � (1 − |I k |)η k , where η k is the midpoint of arc I k . Take We see that f k ∈ B p , and f k converges to 0 uniformly on compact subsets of D when k ⟶ ∞. en, we get as k ⟶ ∞, which implies that μ is a vanishing s-Carleson measure.
Conversely, assume that μ is a vanishing s-Carleson measure. From [12], we see that Here, μ r (z) � μ(z) for |z| < r and μ r (z) � 0 for r ≤ |z| < 1. Let ‖f k ‖ B p ≲1 and f k converge to 0 uniformly on compact subsets of D. en, Letting k ⟶ ∞ and then r ⟶ 1, we have

Volterra Integral Operator
Proof. e proof is similar to that of Proposition 1 in [15]. us, we omit the details of the proof. □ Theorem 3. Let 1 < p < q < ∞ and 0 < s < ∞. en, T g : B p ⟶ LF(q, q − 2, s, q − (q/p)) is bounded if and only if g ∈ F(q, q − 2, s).
Next, we give an estimation for the essential norm of T g . First, we recall some definitions. e essential norm of T: X ⟶ Y, denoted by ‖T‖ e,X⟶Y , is defined by ‖T‖ e,X⟶Y � inf K ‖T − K‖ X⟶Y : K is compact from X to Y .
Here, X and Y are Banach spaces, and T: X ⟶ Y is a bounded linear operator. It is easy to see that T: X ⟶ Y is compact if and only if ‖T‖ e,X⟶Y � 0. Let A be a closed subspace of X. Given f ∈ X, the distance from f to A, denoted by dist X (f, A), is defined by □ Journal of Mathematics