I. INTRODUCTION
Let be the open unit disk in the complex plane. Let be the unit ball of , and its boundary. We will denote by the normalized Lebesgue measure on .
Recall that for the weighted Lebesgue measure is defined by
\[d v _ {\alpha} (z) = c _ {\alpha} (1 - | z | ^ {2}) ^ {\alpha} d v (z),\]
where
\[c _ {\alpha} = \frac {\Gamma (n + 1 + \alpha)}{n ! \Gamma (1 + \alpha)}\]
is a normalizing constant so that is a probability measure on B.
Let denotes the space of holomorphic functions on . Take .
Then is said to be in the weighted Bergman space if
\[\|f\|_{\mathbf{A}_{\alpha}^{p}}^p = \int_{\mathbb{B}} |f(z)|^p d v_{\alpha}(z) < \infty.\]
Let be an analytic self-mapping of , then the composition operator on is given by
\[C_{\varphi} f = f \circ \varphi.\]
Recently, there have been an increasing interest in studying composition operators acting on different spaces of analytic functions, for example, see for details about composition operators on classical spaces of analytic functions.
Let D be the differentiation operator defined by
\[D f = f ^ {\prime}, \quad f \in \mathbf {H} (\mathbb {D}).\]
Hibschweiler and Portnoy [3] defined the linear operators and and investigated the boundedness and compactness of these operators between Bergman spaces using Carleson-type measure. S. Ohno [4] discussed boundedness and compactness of between Hardy spaces. Recall the multiplication operator defined by
\[M_{\psi} f = \psi f,\quad f\in\mathbf{H}(\mathbb{D}).\]
A. K. Sharma defined [5] products of these operators in the following six ways:
\[(M_{\psi} C_{\varphi} D f)(z) = \psi(z) f^\prime(\varphi(z)),\]
\[(M_{\psi} D C_{\varphi} f)(z) = \psi(z) (\varphi^\prime(z)) f^\prime(\varphi(z)),\]
\[(C_{\varphi} M_{\psi} D f)(z) = \psi(\varphi(z)) f^\prime(\varphi(z)),\]
\[{(D M _ {\psi} C _ {\varphi} f) (z)} {= \psi^ {\prime} (z) f (\varphi (z)) + \psi (z) (\varphi^ {\prime} (z)) f ^ {\prime} (\varphi (z)),}\]
\[(C_{\varphi} D M_{\psi} f)(z) = \psi^{\prime}(\varphi(z)) f(\varphi(z)) + \psi(\varphi(z)) f^{\prime}(\varphi(z)),\]
\[(DC_{\varphi} M_{\psi} f)(z) = \psi^\prime(\varphi(z)) f(\varphi(z)) \varphi^\prime(z) + \psi(\varphi(z)) f^\prime(\varphi(z)) \varphi^\prime(z).\]
for and
There are a lot of papers researching these products, see . Since those results focus on D, naturally, we consider similar questions on B. Of course, the method we used is different from the case on D.
For , we define the differentiation operator on by radial derivative. Recall that for and ,
\[R f = \sum_ {j = 1} ^ {n} z _ {j} \frac {\partial f}{\partial z _ {j}} (z) = \lim _ {r \rightarrow 0} \frac {f (z + r z) - f (z)}{r}, r \in \mathbb {R}.\]
One can see that for ,
\[| R (f \circ \varphi) (z) | = \frac{| (R f) (\varphi (z)) \cdot R \varphi (z) |}{| \varphi (z) |}.\]
Then we also have six ways of products of these operators on the unit ball:
\[(M_{\psi} C_{\varphi} R)f(z)=\psi(z)\cdot(Rf)(\varphi(z)),\]
\[(C_{\varphi} M_{\psi} R f)(z)=\psi(\varphi(z))\cdot(Rf)(\varphi(z)),\]
\[\left| \left(M ^ {\psi} R C _ {\varphi} f\right) (z) \right| = \frac{\left| \psi (z) \cdot R \varphi (z) \cdot (R f) (\varphi (z)) \right|}{\left| \varphi (z) \right|}\]
\[(C_{\varphi} R M_{\psi} f)(z) = (R\psi)(\varphi(z)) \cdot f(\varphi(z)) + \psi(\varphi(z)) \cdot (Rf)(\varphi(z)),\]
\[(R M^{\psi} C_{\varphi} f)(z) = f(\varphi(z)) \cdot R\psi(z) + R(f(\varphi(z))),\]
\[(R C_{\varphi} M_{\psi} f)(z) = R(\varphi(z)) \cdot f(\varphi(z)) + R(f(\varphi(z))) \cdot \psi(\varphi(z))\]
for .
In this paper, we characterize the boundedness and compactness of and on the weighted Bergman spaces on the unit ball.
\[M_{\psi} R C_{\varphi}\]
For , we will denote the distance with the Bergman metric on . For r > 0, let the Bergman metric ball
\[D(a,r)=\{z\in\mathbb{B}:\beta(a,z)<r\}.\]
For a point and , the non-isotropic metric ball with center and radius is
\[Q_{t}(\zeta) = \{ z \in \mathbb{B} : |1 - \langle z, \zeta \rangle| < t \}.\]
The following Lemma is Theorm 50 of [9].
Lemma 2.1 Suppose , is real, and is a positive Borel maesure on B. Then for any nonnegative integer m with the following conditions are equivalent.
(a) There is a constant such that
\[\int_ {\mathbb {B}} | R ^ {m} f (w) | ^ {q} d \lambda (w) \leq C \| f \| _ {\mathbf {A} _ {\alpha} ^ {p}} ^ {q}\]
for all
(b) For each (or some) there is a constant such that
\[\int_ {\mathbb{B}} \frac{(1 - | z | ^ {2}) ^ {s}}{| 1 - \langle z , w \rangle | ^ {s + (n + 1 + \alpha + m p) q / p}} d \lambda (w) \leq C\]
for all
(c) There is a constant such that
\[\lambda(Q_{t}(\zeta)) \leq C t^{(n+1+\alpha+mp)q/p}\]
for all and .
(d) For each (or some) there is a constant such that
\[\lambda(D(a,r)) \leq C(1 - |a|^{2})^{(n+1+\alpha+mp)q/p}\]
for all
Theorem 2.2. Let and . Let be a holomorphic maps on and . Define a finite positive Borel measure on by
\[\mu(E) = \int_{\varphi^{-1}(E)} \left(\frac{|\psi(z) \cdot R\varphi(z)|}{|\varphi(z)|}\right)^q dv_\beta(z)\]
for all Borel sets of . Then the following are equivalent:
(2)
\[\mu\left(D\left(a,r\right)\right)=O\left(\left(1-|a|^{2}\right)^{\frac{q\left(n+1+\alpha+p\right)}{p}}\right)\text{as}|a|\to 1.\]
Proof. Suppose (1) holds. Since , by the definition of , we get (see [10, p.163])
\[\begin{array}{r c l} \| M _ {\psi} R C _ {\varphi} (f) \| _ {\mathbf {A} _ {\beta} ^ {q}} ^ {q} & = & \int_ {\mathbb {B}} \Big (\frac {| \psi (z) \cdot (R f) (\varphi (z)) \cdot R \varphi (z) |}{| \varphi (z) |} \Big) ^ {q} d v _ {\beta} (z) \\& = & \int_ {\mathbb {B}} | R f (w) | ^ {q} d \mu_ {(} w) \\& = & \| R f \| _ {\mathbf {L} ^ {q} (\mu)} ^ {q}. \end{array}\]
Since maps boundedly into ,
\[\| R f \|_{\mathbf{L}^q(\mu)}^q = \| M_\psi R C_\varphi (f) \|_{\mathbf{A}_\beta^q}^q \leq C \| f \|_{\mathbf{A}_\alpha^p}^q\]
holds for all . From Lemma 2.1, one can see that
\[\mu (D (a, r)) = O ((1 - | a | ^ {2}) ^ {\frac {q (n + 1 + \alpha + p)}{p}} a s | a | \rightarrow 1.\]
Conversely, if (2) holds, also by Lemma 2.1, we have
\[\| M _ {\psi} R C _ {\varphi} (f) \| _ {\mathbf {A} _ {\beta} ^ {q}} ^ {q} = \| R f \| _ {\mathbf {L} ^ {q} (\mu)} ^ {q} \leq C \| f \| _ {\mathbf {A} _ {\alpha} ^ {p}} ^ {q}.\]
Then, maps boundedly into .
The following lemmas were obtained in [11] and [9] respectively.
Lemma 2.3. let , , , then there is a constant such that
\[| f (z) | ^ {p} \leq \frac{C}{(1 - | z | ^ {2}) ^ {n + 1 + \alpha}} \int_ {D (z, r)} | f (w) | ^ {p} d v _ {\alpha} (w)\]
for all and all
Lemma 2.4. Suppose p > 0, , then there exists a constant C > 0 (depending on p and ) such that
\[| f (z) | \leq \frac{C \| f \| _ {\mathbf{A} _ {\alpha} ^ {p}}}{(1 - | z | ^ {2}) ^ {\frac{n + 1 + \alpha}{p}}}\]
for all in and .
Theorem 2.5. Let and . Let be a holomorphic maps on and . Define a finite positive Borel measure on by
\[\mu(E) = \int_{\varphi^{-1}(E)} \left(\frac{|\psi(z) \cdot R\varphi(z)|}{|\varphi(z)|}\right)^q dv_\beta(z)\]
for all Borel sets of . Then the following are equivalent:
\[\mu(D(a,r)) = o((1 - |a|^{2})^\frac{q(n+1+\alpha+p)}{p}) \text{as} |a| \to 1.\]
Proof. First suppose that maps compactly into . Let and consider function
\[f _ {a} (z) = \frac{(1 - | a | ^ {2}) ^ {\frac{n ^ {+ 1 +} \alpha}{p}}}{(1 - \langle z , a \rangle) ^ {\frac{2 (n + 1 + \alpha)}{p}}}.\]
Clearly and converges to zero uniformly on compact subsets of as . Since is compact, so for gives , we can find such that for . Thus
\[\varepsilon > \int_ {\mathbb {B}} | R f _ {a} (z) | ^ {q} d \mu (z) \geq \int_ {D (a, r)} | R f _ {a} (z) | ^ {q} d \mu (z)\]
for . Since when , so
\[| R f _ {a} (z) | = \frac{2 (n + 1 + \alpha) (1 - | a | ^ {2}) ^ {\frac{n + 1 + \alpha}{p}} \langle z , a \rangle}{p (1 - \overline{{a}} z) ^ {\frac{2 (n + 1 + \alpha) + p}{p}}} \cong \frac{2 (n + 1 + \alpha) | a | ^ {2}}{p (1 - | a | ^ {2}) ^ {\frac{n + 1 + \alpha + p}{p}}}.\]
Then
\[\mu\left(D\left(a,r\right)\right)=o\left(\left(1-|a|^{2}\right)^{\frac{q\left(n+1+\alpha+p\right)}{p}}\right)\]
as.
Conversely, assume that (2) holds. Let be a sequence in such that and uniformly on compact subsets of . To show that maps compactly into , it is sufficient to prove that
\[\|M\psi RC_{\varphi}(f_k)\|_{\mathbf{A}_{\beta}^q}^q = \|R f_k\|_{L^q(\mu)}^q \to 0 \text{as} k \to \infty\]
From Lemma 2.3,
\[\int_ {\mathbb{B}} | R f _ {k} | ^ {q} d \mu \leq C \int_ {\mathbb{B}} \frac{1}{(1 - | a | ^ {2}) ^ {n + 1 + \alpha}} \int_ {D (a, r)} | R f _ {k} (z) | ^ {q} d v _ {\alpha} (z) d \mu (a).\]
Note that and when . At the same time, if and only if , then by lemma 2.4,
\[| R f _ {k} (z) | \leq \frac{\| R f _ {k} \| _ {\mathbf{A} _ {\alpha + p} ^ {p}}}{(1 - | z | ^ {2}) ^ {\frac{n + 1 + \alpha + p}{p}}} \leq \frac{C \| f _ {k} \| _ {\mathbf{A} _ {\alpha} ^ {p}}}{(1 - | z | ^ {2}) ^ {\frac{n + 1 + \alpha + p}{p}}}.\]
Then, by an application of Fubini's theorem, we have
\[\begin{array}{r c l} \| M ^ {\psi} R C _ {\varphi} (f) \| _ {\mathbf{A} _ {\beta} ^ {q}} ^ {q} & \leq & C ^ {\prime} \int_ {\mathbb{B}} | R f _ {k} (z) | ^ {q} \frac{\mu (D (z , r))}{(1 - | z | ^ {2}) ^ {n + 1 + \alpha}} d v _ {\alpha} (z) \\& \leq & C ^ {\prime} \| f _ {k} \| _ {\mathbf{A} _ {\alpha} ^ {p}} ^ {q - p} \int_ {\mathbb{B}} | R f _ {k} (z) | ^ {p} \frac{\mu (D (z , r))}{(1 - | z | ^ {2}) ^ {\frac{q (n + 1 + \alpha + p) - p ^ {2}}{p}}} d v _ {\alpha} (z) \\& \leq & C ^ {\prime} M ^ {q - p} \Big (\int_ {| z | \leq r _ {0}} | R f _ {k} (z) | ^ {p} \frac{\mu (D (z , r))}{(1 - | z | ^ {2}) ^ {\frac{q (n + 1 + \alpha + p) - p ^ {2}}{p}}} d v _ {\alpha} (z) \\& + & \int_ {| z | > r _ {0}} | R f _ {k} (z) | ^ {p} \frac{\mu (D (z , r))}{(1 - | z | ^ {2}) ^ {\frac{q (n + 1 + \alpha + p) - p ^ {2}}{p}}} d v _ {\alpha} (z) \Big) \\& = & I + I I. \end{array}\]
Now (2) implies that for a give , there is such that
\[\begin{array}{r c l} I I & = & C ^ {\prime} M ^ {q - p} \int_ {| z | > r _ {0}} | R f _ {k} (z) | ^ {p} \frac{\mu (D (z , r))}{(1 - | z | ^ {2}) ^ {\frac{q (n + 1 + \alpha + p) - p ^ {2}}{p}}} d v _ {\alpha} (z) \\& \leq & \varepsilon C ^ {\prime} M ^ {q - p} \int_ {| z | > r _ {0}} | R f _ {k} (z) | ^ {p} (1 - | z | ^ {2}) ^ {p} d v _ {\alpha} (z) \\& \leq & \varepsilon C ^ {\prime} M ^ {q - p} \| f _ {k} \| _ {\mathbf{A} _ {\alpha} ^ {p}} ^ {p} \\& \leq & \varepsilon C ^ {\prime} M ^ {q}. \end{array}\]
Since uniformly on compact subsets of ,
\[\begin{array}{r c l} I & = & C ^ {\prime} M ^ {q - p} \int_ {| z | \leq r _ {0}} | R f _ {k} (z) | ^ {p} \frac{\mu (D (z , r))}{(1 - | z | ^ {2}) ^ {\frac{q (n + 1 + \alpha + p) - p ^ {2}}{p}}} d v _ {\alpha} (z) \\& \leq & \varepsilon C _ {1} C ^ {\prime} M ^ {q - p} \int_ {\mathbb{B}} \mu (D (z, r)) d v _ {\alpha} (z) \\& \leq & \varepsilon C _ {1} C _ {2} C ^ {\prime} M ^ {q - p} \int_ {\mathbb{B}} \mu (\mathbb{B}) d v _ {\alpha} (z) \\& = & \varepsilon C _ {1} C _ {2} C _ {3} C ^ {\prime} M ^ {q - p}. \end{array}\]
for large enough. Thus
\[\lim_{n\to\infty}\|M^\psi RC_\varphi f_k\|_{\mathbf{A}_\beta^q}^q=0,\]
and hece maps compactly into .
Lemma 2.6. [9, Theorem 54] Let and be any real number, and let be a positive Borel measure on . Then for any nonnegative integer with the following conditions are equivalent.
(a) There is a constant such that
\[\int_ {\mathbb {B}} | R ^ {m} f (w) | ^ {q} d \mu (w) \leq C \| f \| _ {\mathbf {A} _ {\alpha} ^ {p}} ^ {q}\]
for all .
(b) For any bounded sequence in with for every ,
\[\lim _ {j \to \infty} \int_ {\mathbb {B}} | R ^ {m} f _ {j} (z) | ^ {q} d \lambda (z) = 0.\]
(c) For any fixed , define the function
\[\widehat{\lambda} (z) = \frac{\mu (D (z , r))}{(1 - | z | ^ {2}) ^ {n + 1 + \alpha + m p}}, z \in \mathbb{B},\]
then .
(d) For any fixed , define the function
\[B (\lambda) (z) = \int_ {\mathbb{B}} \frac{(1 - | z | ^ {2}) ^ {s} d \lambda (w)}{| 1 - \langle z , w \rangle | ^ {n + 1 + s + m p}}, z \in \mathbb{B},\]
then
(d) For any fixed , define the function
\[B (\lambda) (z) = \int_ {\mathbb{B}} \frac{(1 - | z | ^ {2}) ^ {s} d \lambda (w)}{| 1 - \langle z , w \rangle | ^ {n + 1 + s + m p}}, z \in \mathbb{B},\]
then
Theorem 2.7. Let and . Let be a holomorphic maps on and . Define a finite positive Borel measure on by
\[\mu(E) = \int_{\varphi^{-1}(E)} \left( \frac{| (z) \cdot R\varphi(z) |}{| \varphi(z) |} \right)^q dv_\beta(z)\]
for all Borel sets of . Let . Then the following are equivalent:
(1) maps boundedly into .
(2) maps compactly into .
(3)
Proof. (1) (3). Suppose (1) holds. By the computation before,
\[\| M_{\psi} R C_{\varphi} f \|_{\mathbf{A}_{\beta}^{q}}^{q} = \| R f \|_{\mathbf{L}^q(\mu)}^q.\]
Since maps boundedly into , we can find a positive constant such that
\[\| R f \|_{\mathbf{L}^q(\mu)}^q \leq C \| f \|_{\mathbf{A}_{\alpha}^p}^q.\]
Then by Lemma 2.1 and Lemma 2.6, maps boundedly into if and only if .
It is clear that (2) implies (1).
It remains to verify that (3) implies (2). Assume that
\[\| f _ { k } \| _ { \mathbf{ A } _ { \alpha } ^ { p } } \leq C\]
and uniformly on compact subsets of . It is sufficient to show that
\[\lim_{n\to\infty} \|M^{\psi} R C_{\varphi} f_{k}\|_{\mathbf{A}_{\beta}^{q}}^{q} = 0.\]
By the computation in the Theorem 2.5, we have
\[\begin{array}{r c l} \| M _ {\psi} R C _ {\varphi} f _ {k} \| _ {\mathbf{A} _ {\beta} ^ {q}} ^ {q} & \leq & C \int_ {\mathbb{B}} | R f _ {k} (z) | ^ {q} \frac{\mu (D (z , r))}{(1 - | z | ^ {2}) ^ {n + 1 + \alpha}} d v _ {\alpha} (z) \\& = & C \int_ {\mathbb{B}} | R f _ {k} (z) | ^ {q} G (z) d v _ {\alpha + p} (z). \end{array}\]
Let . Then the hypothesis of (3) implies that there exists such that
\[\int_ {| z | > r _ {0}} (G (z)) ^ {\frac{p}{p - q}} d v _ {\alpha + p} (z) < \varepsilon^ {\frac{p}{p - q}}.\]
It follows by Holder's inequality that
\[\begin{array}{r l} & \int_ {| z | > r _ {0}} | R f _ {k} (z) | ^ {q} G (z) d v _ {\alpha + p} (z) \\\leq & \left(\int_ {\mathbb {B}} | R f _ {k} (z) | ^ {p} d v _ {\alpha + p} (z)\right) ^ {\frac {q}{p}} \left(\int_ {| z | > r _ {0}} \bigl (G (z) \bigr) ^ {\frac {p}{p - q}} d v _ {\alpha + p} (z)\right) ^ {\frac {p - q}{p}} \\\leq & \varepsilon \| R f _ {k} \| _ {\mathbf {A} _ {\alpha + p} ^ {p}} ^ {q} \\\leq & \varepsilon C \| f _ {k} \| _ {\mathbf {A} _ {\alpha} ^ {p}} ^ {q} \\\leq & C \varepsilon . \end{array}\]
Since uniformly on compact subsets of , by Cauchy's estimate, for all and for all . Thus
\[\int_ {| z | \leq r _ {0}} | R f _ {k} (z) | ^ {q} G (z) d v _ {\alpha + p} (z) \leq \varepsilon^ {q} \int_ {| z | \leq r _ {0}} G (z) d v _ {\alpha + p} (z).\]
for all . Since and thus
\[G (z) \leq C \mu (D (z, r)) \leq C \mu (\mathbb {B}) < \infty\]
thus
\[\int_ {| z | \leq r _ {0}} G (z) d v _ {\alpha + p} (z) \leq C \int_ {\mathbb {B}} \mu (D (z, r)) d v _ {\alpha + p} (z) \leq C.\]
Then
\[\int_ {| z | \leq r _ {0}} | R f _ {k} (z) | ^ {q} G (z) d v _ {\alpha + p} (z) \leq C \varepsilon\]
for . Hence, maps compactly into .
\[3. M _ {\psi} C _ {\varphi} R\]
Similar to the proof in section 2, we have the following results about , here we omit the details.
Theorem 3.1. Let and . Let be a holomorphic maps on and . Define a finite positive Borel measure on by
\[\mu (E) = \int_ {\varphi^ {- 1} (E)} | (z) | ^ {q} d v _ {\beta} (z)\]
for all Borel sets E of B. Then the following are equivalent:
\[\mu (D (a, r)) = O ((1 - | a | ^ {2}) ^ {\frac {q (n + 1 + \alpha + p)}{p}} a s | a | \to 1.\]
Theorem 3.2. Let and . Let be a holomorphic maps on and . Define a finite positive Borel measure on by
\[\mu (E) = \int_ {\varphi^ {- 1} (E)} | (z) | ^ {q} d v _ {\beta} (z)\]
for all Borel sets of . Then the following are equivalent:
(1) maps compactly into .
\[\mu (D (a, r)) = o ((1 - | a | ^ {2}) ^ {\frac {q (n + 1 + \alpha + p)}{p}} a s | a | \to 1.\]
Theorem 3.3. Let and . Let be a holomorphic maps on and . Define a finite positive Borel measure on by
\[\mu (E) = \int_ {\varphi^ {- 1} (E)} | (z) | ^ {q} d v _ {\beta} (z)\]
for all Borel sets of . Let . Then the following are equivalent:
(1) maps boundedly into .
(2) maps compactly into .
(3)
\[4. R C _ {\varphi} M _ {\psi}\]
In this section, we characterize the boundedness and compactness of by using Carleson measures.
Recall that a positive Borel measure on B is called Carleson measure for if there exists a constant C > 0 such that
\[\int_ {\mathbb {B}} | f | ^ {p} d \mu \leq C \int_ {\mathbb {B}} | f | ^ {p} d v _ {\alpha}\]
for all
Similarly, a positive Borel measure on is called a vanishing Carleson measure for if
\[\lim _ {k \to \infty} \int_ {\mathbb {B}} | f _ {k} | ^ {p} d \mu = 0\]
whenever is a bounded sequence in that converges to 0 uniformly on compact subsets of .
Theorem 4.1. Let , . Let be a holomorphic self-map of with and such that . Define a finite positive Borel measure on by
\[\mu_ {\varphi , \alpha} (E) = \int_ {\varphi^ {- 1} (E)} \left(\frac {| R \varphi (z) |}{| \varphi (z) |}\right) ^ {p} d v _ {\alpha} (z)\]
for all Borel sets of . Let . If for every (or some) , there is a constant such that
\[\mu (D (a, r)) \leq C (1 - | a | ^ {2}) ^ {n + 1 + \alpha + p}\tag{1}\]
holds for all , then is bounded on if and only if is a Carleson measure on .
Proof. First suppose that is a Carleson measure on . Then for , by the definition of , we get (see [10, p.163])
\[\begin{array}{r c l} \| R C _ {\varphi} M _ {\psi} (f) \| _ {\mathbf {A} _ {\alpha} ^ {p}} ^ {p} & = & \int_ {\mathbb {B}} \Big (\frac {| (R \psi) (\varphi (z)) \cdot R \varphi (z) \cdot f (\varphi (z)) | + | \psi (\varphi (z)) \cdot (R f) (\varphi (z)) \cdot R \varphi (z) |}{| \varphi (z) |} \Big) ^ {p} d v _ {\alpha} (z) \\& = & \int_ {\mathbb {B}} (| \psi (w) R f (w) | + | f (w) R \psi (w) |) ^ {p} d \mu_ {\varphi , \alpha} (w) \\& \leq & \int_ {\mathbb {B}} | \psi (w) | ^ {p} | R f (w) | ^ {p} d \mu_ {\varphi , \alpha} (w) + \int_ {\mathbb {B}} | f (w) | ^ {p} | R \psi (w) | ^ {p} d \mu_ {\varphi , \alpha} (w). \end{array}\]
Since is Carleson measure on , then
\[\int_ {\mathbb {B}} | f (w) | ^ {p} | R \psi (w) | ^ {p} d \mu_ {\varphi , \alpha} (w) \leq C \| f \| _ {\mathbf {A} _ {\alpha} ^ {p}} ^ {p};\]
On the other hand, for , there exists a constant such that
\[\mu (D (a, r)) \leq C (1 - | a | ^ {2}) ^ {n + 1 + \alpha + p}\]
holds for , then by Lemma 2.1,
\[\int_ {\mathbb {B}} | \psi (w) | ^ {p} | R f (w) | ^ {p} d \mu_ {\varphi , \alpha} (w) = \int_ {\mathbb {B}} | R f (w) | ^ {p} d \mu (w) \leq C \| f \| _ {\mathbf {A} _ {\alpha} ^ {p}} ^ {p},\]
thus
\[\| R C _ {\varphi} M _ {\psi} (f) \| _ {\mathbf {A} _ {\alpha} ^ {p}} ^ {p} \leq C \| f \| _ {\mathbf {A} _ {\alpha} ^ {p}} ^ {p},\]
Therefore, is bounded on .
For the converse, assume is bounded. Then there exists a constant C > 0 such that
\[\| R C _ {\varphi} M \psi (f) \| _ {\mathbf {A} _ {\alpha} ^ {p}} ^ {p} \leq C \| f \| _ {\mathbf {A} _ {\alpha} ^ {p}} ^ {p}\]
for all . Also, there exists a constant such that ,
\[\begin{array}{r c l} \| R C _ {\varphi} M \psi (f) \| _ {\mathbf {A} _ {\alpha} ^ {p}} ^ {p} & \geq & M \int_ {\mathbb {B}} | R (\psi f) (w) | ^ {p} d \mu_ {\varphi , \alpha} (w) \\& \geq & M \int_ {\mathbb {B}} | f (w) | ^ {p} | R \psi (w) | ^ {p} d \mu_ {\varphi , \alpha} (w) - M \int_ {\mathbb {B}} | (w) | ^ {p} | R f (w) | ^ {p} d \mu_ {\varphi , \alpha} (w) \\& \geq & M \int_ {\mathbb {B}} | f (w) | ^ {p} d \nu (w) - M \int_ {\mathbb {B}} | R f (w) | ^ {p} | (w) | ^ {p} d \mu_ {\varphi , \alpha} (w), \end{array}\]
where . From (1) and lemma 2.1, there exists a constant such that
\[\int_ {\mathbb {B}} | R f (w) | ^ {p} | \psi (w) | ^ {p} d \mu_ {\varphi , \alpha} (w) \leq C \| f \| _ {\mathbf {A} _ {\alpha} ^ {p}} ^ {p}.\]
then exists a constant K > 0 such that
\[\int_ {\mathbb {B}} | f (w) | ^ {p} d \nu (w) \leq K \| f \| _ {\mathbf {A} _ {\alpha} ^ {p}} ^ {p}.\]
Thus, is a Carleson measure on .
The proof of the following lemma follows on similar lines as in [1, Proposition 3.11].
Lemma 4.2. Suppose . Let . Let be a holomorphic mapping defined on and be such that is bounded. Then is compact if and only if whenever is a bounded sequence in converging to zero uniformly on compact subsets of , then .
Theorem 4.3. Let , . Let be a holomorphic self-map of with and such that . Define a finite positive Borel measure on by
\[\mu_ {\varphi , \alpha} (E) = \int_ {\varphi^ {- 1} (E)} \left(\frac {| R \varphi (z) |}{| \varphi (z) |}\right) ^ {p} d v _ {\alpha} (z)\]
for all Borel sets of . Let . If for every (or some) , there is a constant such that
\[\lim _ {| a | \to 1 ^ {-}} \frac {\mu (D (a , r))}{(1 - | a | ^ {2}) ^ {n + 1 + \alpha + p}} = 0\]
holds for all then is compact on if and only if is a vanishing Carleson measure on .
Proof. First suppose that is compact on . Then by using the similar argument as in Theorem 4.1, there exist a constant such that for ,
\[\| R C _ {\varphi} M _ {\psi} (f) \| _ {\mathbf {A} _ {\alpha} ^ {p}} ^ {p} \geq C \int_ {\mathbb {B}} | R (\psi f) (w) | ^ {p} d \mu_ {\varphi , \alpha} (w).\]
then
\[\begin{array}{l l} & \int_ {\mathbb {B}} | f (w) | ^ {p} | R \psi (w) | ^ {p} d \mu_ {\varphi , \alpha} (w) \\\leq & C \| R C _ {\varphi} M ^ {\psi} (f) \| _ {\mathbf {A} _ {\alpha} ^ {p}} ^ {p} + C \int_ {\mathbb {B}} | ^ {\psi} (w) | ^ {p} | R f (w) | ^ {p} d \mu_ {\varphi , \alpha} (w). \end{array}\]
In the above inequality, take , where
\[k _ {z} (w) = \frac {(1 - | z | ^ {2}) ^ {\frac {n + 1 + \alpha}{p}}}{(1 - \langle w , z \rangle) ^ {\frac {2 (n + 1 + \alpha)}{p}}},\]
then
\[\begin{array}{r l} & {\int_ {\mathbb {B}} | k _ {z} (w) | ^ {p} | R \psi (w) | ^ {p} d \mu_ {\varphi , \alpha} (w)} \\{\leq} & {C \| R C _ {\varphi} M _ {\psi} (f) \| _ {\mathbf {A} _ {\alpha} ^ {p}} ^ {p} + C \int_ {\mathbb {B}} | \psi (w) | ^ {p} | R k _ {z} (w) | ^ {p} d \mu_ {\varphi , \alpha} (w)} \\{=} & {C \| R C _ {\varphi} M _ {\psi} (f) \| _ {\mathbf {A} _ {\alpha} ^ {p}} ^ {p} + C \int_ {\mathbb {B}} | R k _ {z} (w) | ^ {p} d \mu .} \end{array}\]
Since is compact on and the unit vectors tends to 0 uniformly on compact subsets of B as , by lemma 4.2, as . On the other hand, since for every (or some) r > 0,
\[\lim _ {| a | \to 1 ^ {-}} \frac {\mu (D (a , r))}{(1 - | a | ^ {2}) ^ {n + 1 + \alpha + p}} = 0,\]
by lemma 2.1,
\[\int_ {\mathbb {B}} | R k _ {z} (w) | ^ {p} d \mu \leq \| k _ {z} \| _ {\mathbf {A} _ {\alpha} ^ {p}} ^ {p}.\]
Then, we have
\[\lim _ {| z | \rightarrow 1 ^ {-}} \int_ {\mathbb {B}} | k _ {z} (w) | ^ {p} | R \psi (w) | ^ {p} d \mu_ {\varphi , \alpha} (w) = 0.\]
thus, is a vanishing Carleson measure on .
Conversely, suppose that is a vanishing Carleson measure on . Let be a norm bounded sequence in such that and uniformly on compact subsets of . Now we prove that is compact on . By Lemma 4.2, it is enough to show that as . Using the similar argument as before, we have
\[\| R C _ {\varphi} M _ {\psi} (f _ {k}) \| _ {\mathbf {A} _ {\alpha} ^ {p}} ^ {p} \leq C \int_ {\mathbb {B}} | \psi (w) | ^ {p} | R f _ {k} (w) | ^ {p} d \mu_ {\varphi , \alpha} (w) + C \int_ {\mathbb {B}} | f _ {k} (w) | ^ {p} | R \psi (w) | ^ {p} d \mu_ {\varphi , \alpha} (w).\]
Since is a vanishing Carleson measure on , then
\[\lim _ {n \to \infty} \int_ {\mathbb {B}} | f _ {k} (w) | ^ {p} | R \psi (w) | ^ {p} d \mu_ {\varphi , \alpha} (w) = 0.\]
Using the similar argument as before, we have
\[\lim _ {n \to \infty} \int_ {\mathbb {B}} | (w) | ^ {p} | R f _ {k} (w) | ^ {p} d \mu_ {\varphi , \alpha} (w) = 0.\]
The proof is finished.
