One of the aims of the present paper is to obtain some properties about logarithmic Bergman-type space on the unit ball. As some applications, the bounded and compact operators Sm→u,φ=∑mi=0MuiCφℜi from logarithmic Bergman-type space to weighted-type space on the unit ball are completely characterized.
Citation: Yan-fu Xue, Zhi-jie jiang, Hui-ling Jin, Xiao-feng Peng. Logarithmic Bergman-type space and a sum of product-type operators[J]. AIMS Mathematics, 2023, 8(11): 26682-26702. doi: 10.3934/math.20231365
[1] | Stevo Stević . Note on a new class of operators between some spaces of holomorphic functions. AIMS Mathematics, 2023, 8(2): 4153-4167. doi: 10.3934/math.2023207 |
[2] | Cheng-shi Huang, Zhi-jie Jiang, Yan-fu Xue . Sum of some product-type operators from mixed-norm spaces to weighted-type spaces on the unit ball. AIMS Mathematics, 2022, 7(10): 18194-18217. doi: 10.3934/math.20221001 |
[3] | Rabha W. Ibrahim, Dumitru Baleanu . Fractional operators on the bounded symmetric domains of the Bergman spaces. AIMS Mathematics, 2024, 9(2): 3810-3835. doi: 10.3934/math.2024188 |
[4] | Aydah Mohammed Ayed Al-Ahmadi . Differences weighted composition operators acting between kind of weighted Bergman-type spaces and the Bers-type space -I-. AIMS Mathematics, 2023, 8(7): 16240-16251. doi: 10.3934/math.2023831 |
[5] | Zhi-jie Jiang . Complex symmetric difference of the weighted composition operators on weighted Bergman space of the half-plane. AIMS Mathematics, 2024, 9(3): 7253-7272. doi: 10.3934/math.2024352 |
[6] | Xiaoman Liu, Yongmin Liu . Boundedness of the product of some operators from the natural Bloch space into weighted-type space. AIMS Mathematics, 2024, 9(7): 19626-19644. doi: 10.3934/math.2024957 |
[7] | Zhi-jie Jiang . A result about the atomic decomposition of Bloch-type space in the polydisk. AIMS Mathematics, 2023, 8(5): 10822-10834. doi: 10.3934/math.2023549 |
[8] | Houcine Sadraoui, Borhen Halouani . Commuting Toeplitz operators on weighted harmonic Bergman spaces and hyponormality on the Bergman space of the punctured unit disk. AIMS Mathematics, 2024, 9(8): 20043-20057. doi: 10.3934/math.2024977 |
[9] | Shuhui Yang, Yan Lin . Multilinear strongly singular integral operators with generalized kernels and applications. AIMS Mathematics, 2021, 6(12): 13533-13551. doi: 10.3934/math.2021786 |
[10] | Jinjin Liang, Liling Lai, Yile Zhao, Yong Chen . Commuting H-Toeplitz operators with quasihomogeneous symbols. AIMS Mathematics, 2022, 7(5): 7898-7908. doi: 10.3934/math.2022442 |
One of the aims of the present paper is to obtain some properties about logarithmic Bergman-type space on the unit ball. As some applications, the bounded and compact operators Sm→u,φ=∑mi=0MuiCφℜi from logarithmic Bergman-type space to weighted-type space on the unit ball are completely characterized.
Let C denote the complex plane and Cn the n-dimensional complex Euclidean space with an inner product defined as ⟨z,w⟩=∑nj=1zj¯wj. Let B(a,r)={z∈Cn:|z−a|<r} be the open ball of Cn. In particular, the open unit ball is defined as B=B(0,1).
Let H(B) denote the set of all holomorphic functions on B and S(B) the set of all holomorphic self-mappings of B. For given φ∈S(B) and u∈H(B), the weighted composition operator on or between some subspaces of H(B) is defined by
Wu,φf(z)=u(z)f(φ(z)). |
If u≡1, then Wu,φ is reduced to the composition operator usually denoted by Cφ. If φ(z)=z, then Wu,φ is reduced to the multiplication operator usually denoted by Mu. Since Wu,φ=Mu⋅Cφ, Wu,φ can be regarded as the product of Mu and Cφ.
If n=1, B becomes the open unit disk in C usually denoted by D. Let Dm be the mth differentiation operator on H(D), that is,
Dmf(z)=f(m)(z), |
where f(0)=f. D1 denotes the classical differentiation operator denoted by D. As expected, there has been some considerable interest in investigating products of differentiation and other related operators. For example, the most common products DCφ and CφD were extensively studied in [1,10,11,12,13,23,25,26], and the products
MuCφD,CφMuD,MuDCφ,CφDMu,DMuCφ,DCφMu | (1.1) |
were also extensively studied in [14,18,22,27]. Following the study of the operators in (1.1), people naturally extend to study the operators (see [5,6,30])
MuCφDm,CφMuDm,MuDmCφ,CφDmMu,DmMuCφ,DmCφMu. |
Other examples of products involving differentiation operators can be found in [7,8,19,32] and the related references.
As studying on the unit disk becomes more mature, people begin to become interested in exploring related properties on the unit ball. One method for extending the differentiation operator to Cn is the radial derivative operator
ℜf(z)=n∑j=1zj∂f∂zj(z). |
Naturally, replacing D by ℜ in (1.1), we obtain the following operators
MuCφℜ,CφMuℜ,MuℜCφ,CφℜMu,ℜMuCφ,ℜCφMu. | (1.2) |
Recently, these operators have been studied in [31]. Other operators involving radial derivative operators have been studied in [21,33,34].
Interestingly, the radial derivative operator can be defined iteratively, namely, ℜmf can be defined as ℜmf=ℜ(ℜm−1f). Similarly, using the radial derivative operator can yield the related operators
MuCφℜm,CφMuℜm,MuℜmCφ,CφℜmMu,ℜmMuCφ,ℜmCφMu. | (1.3) |
Clearly, the operators in (1.3) are more complex than those in (1.2). Since CφMuℜm=Mu∘φCφℜm, the operator MuCφℜm can be regarded as the simplest one in (1.3) which was first studied and denoted as ℜmu,φ in [24]. Recently, it has been studied again because people need to obtain more properties about spaces to characterize its properties (see [29]).
To reconsider the operator CφℜmMu, people find the fact
CφℜmMu=m∑i=0Cimℜi(ℜm−iu)∘φ,φ. | (1.4) |
Motivated by (1.4), people directly studied the sum operator (see [2,28])
Sm→u,φ=m∑i=0MuiCφℜi, |
where ui∈H(B), i=¯0,m, and φ∈S(B). Particularly, if we set u0≡⋯≡um−1≡0 and um=u, then Sm→u,φ=MuCφℜm; if we set u0≡⋯≡um−1≡0 and um=u∘φ, then Sm→u,φ=CφMuℜm. In [28], Stević et al. studied the operators Sm→u,φ from Hardy spaces to weighted-type spaces on the unit ball and obtained the following results.
Theorem A. Let m∈N, uj∈H(B), j=¯0,m, φ∈S(B), and μ a weight function on B. Then, the operator Sm→u,φ:Hp→H∞μ is bounded and
supz∈Bμ(z)|uj(φ(z))||φ(z)|<+∞,j=¯1,m, | (1.5) |
if and only if
I0=supz∈Bμ(z)|u0(z)|(1−|φ(z)|2)np<+∞ |
and
Ij=supz∈Bμ(z)|uj(z)||φ(z)|(1−|φ(z)|2)np+j<+∞,j=¯1,m. |
Theorem B. Let m∈N, uj∈H(B), j=¯0,m, φ∈S(B), and μ a weight function on B. Then, the operator Sm→u,φ:Hp→H∞μ is compact if and only if it is bounded,
lim|φ(z)|→1μ(z)|u0(z)|(1−|φ(z)|2)np=0 |
and
lim|φ(z)|→1μ(z)|uj(z)||φ(z)|(1−|φ(z)|2)np+j=0,j=¯1,m. |
It must be mentioned that we find that the necessity of Theorem A requires (1.5) to hold. Inspired by [2,28], here we use a new method and technique without (1.5) to study the sum operator Sm→u,φ from logarithmic Bergman-type space to weighted-type space on the unit ball. To this end, we need to introduce the well-known Bell polynomial (see [3])
Bm,k(x1,x2,…,xm−k+1)=∑m!∏m−k−1i=1ji!m−k−1∏i=1(xii!)ji, |
where all non-negative integer sequences j1, j2,…,jm−k+1 satisfy
m−k+1∑i=1ji=kandm−k+1∑i=1iji=m. |
In particular, when k=0, one can get B0,0=1 and Bm,0=0 for any m∈N. When k=1, one can get Bi,1=xi. When m=k=i, Bi,i=xi1 holds.
In this section, we need to introduce logarithmic Bergman-type space and weighted-type space. Here, a bounded positive continuous function on B is called a weight. For a weight μ, the weighted-type space H∞μ consists of all f∈H(B) such that
‖ |
With the norm \|\cdot\|_{H_{\mu}^{\infty}} , H_{\mu}^{\infty} becomes a Banach space. In particular, if \mu(z) = (1-|z|^{2})^{\sigma}\; (\sigma > 0) , the space H_{\mu}^{\infty} is called classical weighted-type space usually denoted by H_{\sigma}^{\infty} . If \mu\equiv 1 , then space H_{\mu}^{\infty} becomes the bounded holomorphic function space usually denoted by H^{\infty} .
Next, we need to present the logarithmic Bergman-type space on {\mathbb{B}} (see [4] for the unit disk case). Let dv be the standardized Lebesgue measure on \mathbb{B} . The logarithmic Bergman-type space A_{w_{\gamma, \delta}}^{p} consists of all f\in H({\mathbb{B}}) such that
\begin{align*} \|f\|_{A_{w_{\gamma,\delta}}^{p}}^{p} = \int_{\mathbb{B}}|f(z)|^{p}w_{\gamma,\delta}(z)dv(z) < +\infty, \end{align*} |
where -1 < \gamma < +\infty , \delta\leq 0 , 0 < p < +\infty and w_{\gamma, \delta}(z) is defined by
\begin{align*} w_{\gamma,\delta}(z) = \Big(\log\frac{1}{|z|}\Big)^{\gamma} \Big[\log\Big(1-\frac{1}{\log |z|}\Big)\Big]^{\delta}. \end{align*} |
When p\geq1 , A_{w_{\gamma, \delta}}^{p} is a Banach space. While 0 < p < 1 , it is a Fréchet space with the translation invariant metric \rho(f, g) = \|f-g\|_{A_{\omega_{\gamma, \delta}}^{p}}^{p} .
Let \varphi\in S({\mathbb{B}}) , 0\leq r < 1 , 0\leq\gamma < \infty , \delta\leq 0 , and a\in\mathbb{B}\backslash\{\varphi(0)\} . The generalized counting functions are defined as
N_{\varphi,\gamma,\delta}(r, a) = \sum\limits_{z_{j}(a)\in\varphi^{-1}(a)} w_{\gamma,\delta}\left(\frac{z_{j}(a)}{r}\right) |
where \left|z_{j}(a)\right| < r , counting multiplicities, and
N_{\varphi,\gamma,\delta}(a) = N_{\varphi,\gamma,\delta}(1,a) = \sum\limits_{z_{j}(a)\in\varphi^{-1}(a)}w_{\gamma,\delta}\left(z_{j}(a)\right). |
If {\varphi}\in S({\mathbb D}) , then the function N_{\varphi, \gamma, \delta} has the integral expression: For 1\leq\gamma < +\infty and \delta\leq 0 , there is a positive function F(t) satisfying
N_{\varphi,\gamma,\delta}(r,u) = \int_{0}^{r}F(t)N_{\varphi,1}(t,u)\mathrm{d}t,\quad r\in(0,1),\quad u\neq\varphi(0). |
When {\varphi}\in S({\mathbb D}) and \delta = 0 , the generalized counting functions become the common counting functions. Namely,
\begin{align*} N_{\varphi,\gamma}(r,a)& = \sum_{z\in\varphi^{-1}(a),|z| < r}\left(\log \frac{r}{|z|}\right)^{\gamma}, \end{align*} |
and
\begin{align*} N_{\varphi,\gamma}(a)& = N_{\varphi,\gamma}(1,a) = \sum_{z\in\varphi^{-1}(a)} \left(\log \frac{1}{|z|}\right)^{\gamma}. \end{align*} |
In [17], Shapiro used the function N_{\varphi, \gamma}(1, a) to characterize the compact composition operators on the weighted Bergman space.
Let X and Y be two topological spaces induced by the translation invariant metrics d_X and d_Y , respectively. A linear operator T: X\to Y is called bounded if there is a positive number K such that
\begin{align*} d_Y(Tf,0)\leq Kd_X(f,0) \end{align*} |
for all f\in X . The operator T: X\rightarrow Y is called compact if it maps bounded sets into relatively compact sets.
In this paper, j = \overline{k, l} is used to represent j = k, ..., l , where k, l\in\mathbb{N}_{0} and k\leq l . Positive numbers are denoted by C , and they may vary in different situations. The notation a\lesssim b (resp. a\gtrsim b ) means that there is a positive number C such that a\leq Cb (resp. a\geq Cb ). When a\lesssim b and b\gtrsim a , we write a\asymp b .
In this section, we obtain some properties on the logarithmic Bergman-type space. First, we have the following point-evaluation estimate for the functions in the space.
Theorem 3.1. Let -1 < \gamma < +\infty , \delta\leq0 , 0 < p < +\infty and 0 < r < 1 . Then, there exists a positive number C = C(\gamma, \delta, p, r) independent of z\in K = \{z\in\mathbb{B}:|z| > r\} and f\in A_{w_{\gamma, \delta}}^{p} such that
\begin{align} |f(z)|\leq\frac{C}{(1-|z|^{2})^{\frac{\gamma+n+1}{p}}} \left[\log\left(1-\frac{1}{\log|z|}\right)\right]^{-\frac{\delta}{p}} \|f\|_{A_{w_{\gamma,\delta}}^{p}}. \end{align} | (3.1) |
Proof. Let z\in\mathbb{B} . By applying the subharmonicity of the function |f|^{p} to Euclidean ball B(z, r) and using Lemma 1.23 in [35], we have
\begin{align} |f(z)|^{p}&\leq\frac{1}{v(B(z,r))}\int_{B(z,r)}|f(w)|^{p}dv(w) \leq\frac{C_{1,r}}{(1-|z|^{2})^{n+1}}\int_{B(z,r)}|f(w)|^{p}dv(w). \end{align} | (3.2) |
Since r < |z| < 1 and 1-|w|^{2}\asymp1-|z|^{2} , we have
\begin{align} \log\frac{1}{|w|}\asymp1-|w|\asymp1-|z|\asymp\log\frac{1}{|z|} \end{align} | (3.3) |
and
\begin{align} \log\left(1-\log\frac{1}{|w|}\right)\asymp\log\left(1-\log\frac{1}{|z|}\right). \end{align} | (3.4) |
From (3.3) and (3.4), it follows that there is a positive constant C_{2, r} such that w_{\gamma, \delta}(z)\leq C_{2, r}w_{\gamma, \delta}(w) for all w\in B(z, r) . From this and (3.2), we have
\begin{align} |f(z)|^{p} &\leq\frac{C_{1,r}C_{2,r}}{(1-|z|^{2})^{n+1}w_{\gamma,\delta}(z)} \int_{B(z,r)}|f(w)|^{p}w_{\gamma,\delta}(w)dv(w)\\ &\leq\frac{C_{1,r}C_{2,r}}{(1-|z|^{2})^{n+1}w_{\gamma,\delta}(z)} \|f\|_{A_{w_{\gamma,\delta}}^{p}}^{p}. \end{align} | (3.5) |
From (3.5) and the fact \log\frac{1}{|z|}\asymp1-|z|\asymp1-|z|^{2} , the following inequality is right with a fixed constant C_{3, r}
\begin{align*} |f(z)|^{p} \le \frac{C_{1,r} C_{2,r} C_{3,r} }{(1-|z|^2)^{n+1+\gamma } } \Big[\log\Big(1-\frac{1}{\log|z|}\Big)\Big]^{-\delta }\|f\|_{A_{w_{\gamma,\delta}}^{p}}^{p}. \end{align*} |
Let C = \frac{C_{1, r} C_{2, r} C_{3, r}}{p} . Then the proof is end.
Theorem 3.2. Let m\in\mathbb{N} , -1 < \gamma < +\infty , \delta\leq0 , 0 < p < +\infty and 0 < r < 1 . Then, there exists a positive constant C_m = C(\gamma, \delta, p, r, m) independent of z\in K and f\in A_{w_{\gamma, \delta}}^{p} such that
\begin{align} \Big|\frac{\partial^{m}f(z)}{\partial z_{i_{1}}\partial z_{i_{2} }\dots\partial z_{i_{m}}}\Big| \le\frac{C_m}{(1-|z|^{2})^{\frac{\gamma+n+1}{p}+m}}\Big[\log\Big(1-\frac{1} {\log|z|}\Big)\Big]^{-\frac{\delta}{p}}\|f\|_{A_{w_{\gamma,\delta}}^{p}}. \end{align} | (3.6) |
Proof. First, we prove the case of m = 1 . By the definition of the gradient and the Cauchy's inequality, we get
\begin{align} \Big|\frac{\partial f(z)}{\partial z_{i}}\Big|\le|\nabla f(z)|\le \tilde{C} _{1} \frac{\sup_{w\in B(z,q(1-|z|))}|f(w)|}{1-|z|}, \end{align} | (3.7) |
where i = \overline{1, n} . By using the relations
1-|z|\le 1-|z|^{2}\le2(1-|z|), |
(1-q)(1-|z|)\le 1-|w|\le(q+1)(1-|z|), |
and
\log\Big(1-\frac{1}{\log|z|}\Big)\asymp\log\Big(1-\frac{1}{\log|w|}\Big), |
we obtain the following formula
\begin{align*} |f(w)|\leq \frac{\breve{C}_{1}}{(1-|z|^{2})^{\frac{\gamma+n+1}{p}}} \left[\log\left(1-\frac{1}{\log|z|}\right)\right]^{-\frac{\delta}{p}} \|f\|_{A_{w_{\gamma,\delta}}^{p}} \end{align*} |
for any w\in B(z, q(1-|z|)) . Then,
\begin{align*} \sup_{w\in B(z,q(1-|z|))}|f(w)|\leq \frac{\breve{C}_{1}}{(1-|z|^{2})^{\frac{\gamma+n+1}{p}}} \left[\log\left(1-\frac{1}{\log|z|}\right)\right]^{-\frac{\delta}{p}} \|f\|_{A_{w_{\gamma,\delta}}^{p}}. \end{align*} |
From (3.1) and (3.2), it follows that
\begin{align} \Big|\frac{\partial f(z)}{\partial z_{i} } \Big| \leq \frac{\hat{C}_{1}}{(1-|z|^{2})^{\frac{\gamma+n+1}{p}+1}} \left[\log\left(1-\frac{1}{\log|z|}\right)\right]^{-\frac{\delta}{p}} \|f\|_{A_{w_{\gamma,\delta}}^{p}}. \end{align} | (3.8) |
Hence, the proof is completed for the case of m = 1 .
We will use the mathematical induction to complete the proof. Assume that (3.6) holds for m < a . For convenience, let g(z) = \frac{\partial^{a-1}f(z)}{\partial z_{i_{1}}\partial z_{i_{2} }\dots\partial z_{i_{a-1}}} . By applying (3.7) to the function g , we obtain
\begin{align} \Big|\frac{\partial g(z)}{\partial z_{i}}\Big|\le\tilde{C} _{1} \frac{\sup_{w\in B(z,q(1-|z|))}|g(w)|}{1-|z|}. \end{align} | (3.9) |
According to the assumption, the function g satisfies
\begin{align*} |g(z)|\le\frac{\hat{C}_{a-1}}{(1-|z|^{2})^{\frac{\gamma+n+1}{p}+a-1}} \Big[\log\Big(1-\frac{1}{\log|z|}\Big)\Big]^{-\frac{\delta}{p}} \|f\|_{A_{w_{\gamma,\delta}}^{p}}. \end{align*} |
By using (3.8), the following formula is also obtained
\begin{align*} \Big|\frac{\partial g(z)}{\partial z_{i}}\Big| \le\frac{\hat{C}_{a}}{(1-|z|^{2})^{\frac{\gamma+n+1}{p}+a}} \Big[\log\Big(1-\frac{1}{\log|z|}\Big)\Big]^{-\frac{\delta}{p}} \|f\|_{A_{w_{\gamma,\delta}}^{p}}. \end{align*} |
This shows that (3.6) holds for m = a . The proof is end.
As an application of Theorems 3.1 and 3.2, we give the estimate in z = 0 for the functions in A_{\omega_{\gamma, \delta}}^{p} .
Corollary 3.1. Let -1 < \gamma < +\infty , \delta\leq 0 , 0 < p < +\infty , and 0 < r < 2/3 . Then, for all f \in A_{w_{\gamma, \delta}}^{p} , it follows that
\begin{align} \left|f(0)\right| \leq \frac{C}{(1-r^{2})^{\frac{\gamma+n+1}{p}}} \left[\log \left(1-\frac{1}{\log r}\right)\right]^{-\frac{\delta}{p}}\|f\|_{A_{w_{\gamma, \delta}}^{p}}, \end{align} | (3.10) |
and
\begin{align} \Big|\frac{\partial ^{m}f(0)}{\partial z_{l_{1}}\dots\partial z_{l_{m}}}\Big| \leq C_m {(1-r^{2})^{\frac{\gamma+n+1}{p}+m}}\left[\log\left(1-\frac{1} {\log r}\right)\right]^{-\frac{\delta}{p}}\|f\|_{A_{w_{\gamma, \delta}}^{p}}, \end{align} | (3.11) |
where constants C and C_m are defined in Theorems 3.1 and 3.2, respectively.
Proof. For f \in A_{w_{\gamma, \delta}}^{p} , from Theorem 3.1 and the maximum module theorem, we have
\left|f(0)\right| \leq \max\limits_{|z| = r}\left|f(z)\right| \leq \frac{C}{(1-r^{2})^{\frac{\gamma+n+1}{p}}}\left[\log \left(1-\frac{1}{\log r}\right)\right]^{-\frac{\delta}{p}}\|f\|_{A_{w_{\gamma, \delta}}^{p}}, |
which implies that (3.10) holds. By using the similar method, we also have that (3.11) holds.
Next, we give an equivalent norm in A_{w_{\gamma, \delta}}^{p} , which extends Lemma 3.2 in [4] to \mathbb{B} .
Theorem 3.3. Let r_{0}\in[0, 1) . Then, for every f\in A_{w_{\gamma, \delta}}^{p} , it follows that
\begin{align} \|f\|_{A_{w_{\gamma,\delta}}^{p}}^{p}\asymp\int_{\mathbb{B}\setminus r_{0}\mathbb{B}} |f(z)|^{p}w_{\gamma,\delta}(z)dv(z). \end{align} | (3.12) |
Proof. If r_{0} = 0 , then it is obvious. So, we assume that r_{0}\in(0, 1) . Integration in polar coordinates, we have
\begin{align*} \|f\|_{A_{w_{\gamma,\delta}}^{p}}^{p} & = 2n\int_{0}^{1}w_{\gamma,\delta}(r)r^{2n-1}dr \int_{\mathbb{S}}|f(r\zeta)|^{p}d\sigma(\zeta). \end{align*} |
Put
\begin{align*} A(r) = w_{\gamma,\delta}(r)r^{2n-1} \quad \text{and} \quad M(r,f) = \int_{\mathbb{S}}|f(r\zeta)|^{p}d\sigma(\zeta). \end{align*} |
Then it is represented that
\begin{align} \|f\|_{A_{w_{\gamma,\delta}}^{p}}^{p}\asymp\int_{0}^{r_{0}} +\int_{r_{0}}^{1}M(r,f)A(r)dr. \end{align} | (3.13) |
Since M(r, f) is increasing, A(r) is positive and continuous in r on (0, 1) and
\begin{align*} \lim_{r\rightarrow0}A(r) = \lim_{x\rightarrow +\infty}x^{\gamma} \bigg[\log(1+\frac{1}{x})\bigg]^{\delta}e^{-(2n-1)x} = \lim_{x\rightarrow +\infty}\frac{x^{\gamma-\delta}}{e^{(2n-1)x}} = 0, \end{align*} |
that is, there is a constant \varepsilon > 0 \; (\varepsilon < r_{0}) such that A(r) < A(\varepsilon) for r\in(0, \varepsilon) . Then we have
\begin{align} \int_{0}^{r_{0}}M(r,f)A(r)dr&\leq\frac{2r_{0}}{1-r_{0}}\max_{\varepsilon\leq r\leq r_{0}}A(r) \int_{r_{0}}^{\frac{1+r_{0}}{2}}M(r,f)dr \\ &\leq\frac{2r_{0}}{1-r_{0}}\frac{\max_{\varepsilon\leq r\leq r_{0}}A(r)}{\min_{r_{0}\leq r\leq \frac{1+r_{0}}{2}}A(r)} \int_{r_{0}}^{\frac{1+r_{0}}{2}}M(r,f)A(r)dr \\ &\lesssim\int_{r_{0}}^{1}M(r,f)A(r)dr. \end{align} | (3.14) |
From (3.13) and (3.14), we obtain the inequality
\begin{align*} \|f\|_{A_{w_{\gamma,\delta}}^{p}}^{p}\lesssim\int_{r_{0}}^{1}M(r,f)A(r)dr. \end{align*} |
The inequality reverse to this is obvious. The asymptotic relationship (3.12) follows, as desired.
The following integral estimate is an extension of Lemma 3.4 in [4]. The proof is similar, but we still present it for completeness.
Lemma 3.1. Let -1 < \gamma < +\infty , \delta\leq0 , \beta > \gamma-\delta and 0 < r < 1 . Then, for each fixed w\in\mathbb{B} with |w| > r ,
\begin{align*} \int_{\mathbb{B}}\frac{\omega_{\gamma,\delta}(z)}{|1-\langle z,w\rangle|^{n+\beta+1}}dv(z) \lesssim\frac{1}{(1-|w|)^{\beta-\gamma}}\left[\log\left(1-\frac{1} {\log |w|}\right)\right]^{\delta}. \end{align*} |
Proof. Fix |w| with |w| > r_{0} ( 0 < r_{0} < 1 ). It is easy to see that
\begin{align} \log\frac{1}{r}\asymp1-r \quad\text{for}\; \; r_{0}\leq r < 1. \end{align} | (3.15) |
By applying Theorem 3.3 with
\begin{align*} f_{w}(z) = \frac{1}{(1-\langle z,w\rangle)^{n+\beta+1}} \end{align*} |
and using (3.15), the formula of integration in polar coordinates gives
\begin{align} &\int_{\mathbb{B}}\frac{1}{|1-\langle z,w\rangle|^{n+\beta+1}}\omega_{\gamma,\delta}(z)dv(z)\\ \lesssim&\int_{r_{0}}^{1}M(r,f_{w})(1-r)^{\gamma}\left[\log\left(1-\frac{1} {\log r}\right)\right]^{\delta}r^{2n-1}dr. \end{align} | (3.16) |
By Proposition 1.4.10 in [15], we have
\begin{align} M(r,f_{w})\asymp\frac{1}{(1-r^{2}|w|^{2})^{\beta+1}}. \end{align} | (3.17) |
From (3.16) and (3.17), we have
\begin{align*} &\int_{\mathbb{B}}\frac{1}{|1-\langle z,w\rangle|^{\beta+2n}}\omega_{\gamma,\delta}(z)dv(z)\nonumber\\ \lesssim&\int_{r_{0}}^{1}\frac{1}{(1-r^{2}|w|^{2})^{\beta+1}} (1-r)^{\gamma}\left[\log\left(1-\frac{1} {\log r}\right)\right]^{\delta}r^{2n-1}dr\nonumber\\ \lesssim&\int_{r_{0}}^{1}\frac{1}{(1-r|w|)^{\beta+1}} (1-r)^{\gamma}\left[\log\left(1-\frac{1} {\log r}\right)\right]^{\delta}r^{2n-1}dr\nonumber\\ \lesssim&\int_{r_{0}}^{|w|}\frac{1}{(1-r|w|)^{\beta+1}} (1-r)^{\gamma}\left[\log\left(1-\frac{1} {\log r}\right)\right]^{\delta}r^{2n-1}dr\nonumber\\ &+\int_{|w|}^{1}\frac{1}{(1-r|w|)^{\beta+1}} (1-r)^{\gamma}\left[\log\left(1-\frac{1} {\log r}\right)\right]^{\delta}r^{2n-1}dr \nonumber\\ = & I_{1}+I_{2}. \end{align*} |
Since [\log(1-\frac{1}{\log r})]^{\delta} is decreasing in r on [|w|, 1] , we have
\begin{align} I_{2} = &\int_{|w|}^{1}\frac{1}{(1-r|w|)^{\beta+1}} (1-r)^{\gamma}\left[\log\left(1-\frac{1} {\log r}\right)\right]^{\delta}r^{2n-1}dr \\ \lesssim&\frac{1}{(1-|w|)^{\beta+1}}\left[\log\left(1-\frac{1} {\log |w|}\right)\right]^{\delta}\int_{|w|}^{1}(1-r)^{\gamma}dr \\ \asymp&\frac{1}{(1-|w|)^{\beta-\gamma}}\left[\log\left(1-\frac{1} {\log |w|}\right)\right]^{\delta}. \end{align} | (3.18) |
On the other hand, we obtain
\begin{align*} I_{1} = &\int_{r_{0}}^{|w|}\frac{1}{(1-r|w|)^{\beta+1}} (1-r)^{\gamma}\left[\log\left(1-\frac{1} {\log r}\right)\right]^{\delta}r^{2n-1}dr \nonumber\\ \lesssim&\int_{r_{0}}^{|w|} (1-r)^{\gamma-\beta-1}\left(\log\frac{2}{1-r}\right)^{\delta}dr. \end{align*} |
If \delta = 0 and \beta > \gamma , then we have
\begin{align*} I_{1}(0)\lesssim(1-|w|)^{\gamma-\beta}. \end{align*} |
If \delta\neq0 , then integration by parts gives
\begin{align*} I_{1}(\delta) = &-\frac{1}{\gamma-\beta} (1-|w|)^{\gamma-\beta}\left(\log\frac{2}{1-|w|}\right)^{\delta}\\ &+\frac{1}{\gamma-\beta}(1-r_{0})^{\gamma-\beta} \left(\log\frac{2}{1-r_{0}}\right)^{\delta} +\frac{\delta}{\gamma-\beta}I_{1}(\delta-1). \end{align*} |
Since \delta < 0 , \gamma-\beta < 0 and
\begin{align*} \left(\log\frac{2}{1-r}\right)^{\delta-1}\leq\left(\log\frac{2} {1-r}\right)^{\delta} \quad \text{for}\; \; r_{0} < r < |w| < 1, \end{align*} |
we have
\begin{align*} I_{1}(\delta)\leq-\frac{1}{\gamma-\beta} (1-|w|)^{\gamma-\beta}\left(\log\frac{2}{1-|w|}\right)^{\delta} +\frac{\delta}{\gamma-\beta}I_{1}(\delta) \end{align*} |
and from this follows
\begin{align*} I_{1}(\delta)\lesssim(1-|w|)^{\gamma-\beta}\left(\log\frac{2} {1-|w|}\right)^{\delta}\asymp(1-|w|)^{\gamma-\beta}\left[\log\left(1-\frac{1} {\log |w|}\right)\right]^{\delta} \end{align*} |
provided \gamma-\beta-\delta < 0 . The proof is finished.
The following gives an important test function in A_{w_{\gamma, \delta}}^{p} .
Theorem 3.4. Let -1 < \gamma < +\infty , \delta\leq0 , 0 < p < +\infty and 0 < r < 1 . Then, for each t\geq0 and w\in\mathbb{B} with |w| > r , the following function is in A_{w_{\gamma, \delta}}^{p}
\begin{align*} f_{w,t}(z) = \left[\log\left(1-\frac{1}{\log|w|}\right)\right]^{-\frac{\delta}{p}} \frac{(1-|w|^{2})^{-\frac{\delta}{p}+t+1}}{(1-\langle z,w\rangle)^{\frac{\gamma-\delta+n+1}{p}+t+1}}. \end{align*} |
Moreover,
\begin{align*} \sup_{\{w\in\mathbb{B}:|w| > r\}}\|f_{w,t}\|_{A_{w_{\gamma,\delta}}^{p}}\lesssim 1. \end{align*} |
Proof. By Lemma 3.1 and a direct calculation, we have
\begin{align*} \|f_{w,t}\|_{A_{w_{\gamma,\delta}}^{p}}^{p} & = \int_{\mathbb{B}}\bigg|\left[\log\left(1-\frac{1} {\log|w|}\right)\right]^{-\frac{\delta}{p}} \frac{(1-|w|^{2})^{-\frac{\delta}{p}+t+1}} {(1-\langle z,w\rangle)^{\frac{\gamma-\delta+n+1}{p}+t+1}}\bigg|^{p} w_{\gamma,\delta}(z)dA(z)\\ & = (1-|w|^{2})^{p(t+1)-\delta}\left[\log\left(1-\frac{1} {\log|w|}\right)\right]^{-\delta} \\ &\quad\times\int_{\mathbb{B}}\frac{1}{|1-\langle z,w\rangle|^{\gamma-\delta+p(t+1)+n+1}}w_{\gamma,\delta}(z)dA(z) \\ &\lesssim1. \end{align*} |
The proof is finished.
In this section, for simplicity, we define
\begin{align*} B_{i,j}(\varphi(z)) = B_{i,j}(\varphi(z),\varphi(z),\ldots,\varphi(z)). \end{align*} |
In order to characterize the compactness of the operator \mathfrak{S}^m_{\vec{u}, {\varphi}}:A_{w_{\gamma, \delta}}^{p}\to H_{\mu}^{\infty} , we need the following lemma. It can be proved similar to that in [16], so we omit here.
Lemma 4.1. Let -1 < \gamma < +\infty , \delta\leq0 , 0 < p < +\infty , m\in\mathbb{N} , u_j\in H(\mathbb{B}) , j = \overline{0, m} , and \varphi\in S(\mathbb{B}) . Then, the bounded operator \mathfrak{S}^m_{\vec{u}, {\varphi}}:A_{w_{\gamma, \delta}}^{p}\to H_{\mu}^{\infty} is compact if and only if for every bounded sequence \{f_{k}\}_{k\in\mathbb{N}} in A_{w_{\gamma, \delta}}^{p} such that f_{k}\rightarrow 0 uniformly on any compact subset of \mathbb{B} as k\to\infty , it follows that
\begin{align*} \lim_{k\rightarrow \infty}\|\mathfrak{S}^m_{\vec{u},{\varphi}} f_{k}\|_{H_{\mu}^{\infty}} = 0. \end{align*} |
The following result was obtained in [24].
Lemma 4.2. Let s\geq 0 , w\in{\mathbb{B}} and
\begin{align*} g_{w,s}(z) = \frac{1}{(1-\langle z,w\rangle)^{s}}, \quad z\in\mathbb{B}. \end{align*} |
Then,
\begin{align*} \Re^{k}g_{w,s}(z) = s\frac{P_k(\langle z,w\rangle)}{(1-\langle z,w\rangle)^{s+k}}, \end{align*} |
where P_k(w) = s^{k-1}w^k+p_{k-1}^{(k)}(s)w^{k-1}+...+p_2^{(k)}(s)w^2+w , and p^{(k)}_j(s) , j = \overline{2, k-1} , are nonnegative polynomials for s .
We also need the following result obtained in [20].
Lemma 4.3. Let s > 0 , w\in{\mathbb{B}} and
\begin{align*} g_{w,s}(z) = \frac{1}{(1-\langle z,w\rangle)^{s}}, \quad z\in\mathbb{B}. \end{align*} |
Then,
\begin{align*} \Re^{k}g_{w,s}(z) = \sum_{t = 1}^{k}a_{t}^{(k)}\Big(\prod_{j = 0}^{t-1}(s+j)\Big) \frac{\langle z,w\rangle^{t}}{(1-\langle z,w\rangle)^{s+t}}, \end{align*} |
where the sequences (a_{t}^{(k)})_{t\in \overline{1, k}} , k\in\mathbb{N} , are defined by the relations
\begin{align*} a_{k}^{(k)} = a_{1}^{(k)} = 1 \end{align*} |
for k\in\mathbb{N} and
\begin{align*} a_{t}^{(k)} = ta_{t}^{(k-1)}+a_{t-1}^{(k-1)} \end{align*} |
for 2\leq t\leq k-1, k\geq3 .
The final lemma of this section was obtained in [24].
Lemma 4.4. If a > 0 , then
\begin{align*} D_{n}(a) = \left|\begin{array}{cccc} 1 & 1 & \cdots & 1 \\ a & a+1 & \cdots & a+n-1 \\ a(a+1) & (a+1)(a+2) & \cdots & (a+n-1)(a+n) \\ \vdots & \vdots & \cdots & \vdots \\ \prod\limits_{k = 0}^{n-2}(a+k) & \prod\limits_{k = 0}^{n-2}(a+k+1) & \cdots & \prod\limits_{k = 0}^{n-2}(a+k+n-1) \end{array}\right| = \prod_{k = 1}^{n-1} k !. \end{align*} |
Theorem 4.1. Let -1 < \gamma < +\infty , \delta\leq0 , 0 < p < +\infty , m\in\mathbb{N} , u_j\in H(\mathbb{B}) , j = \overline{0, m} , and \varphi\in S(\mathbb{B}) . Then, the operator \mathfrak{S}^m_{\vec{u}, {\varphi}}:A^p_{w_{\gamma, \delta}}\to H_\mu^\infty is bounded if and only if
\begin{align} M_{0}: = \sup_{z\in\mathbb{B}} \frac{\mu(z)|u_{0}(z)|}{(1-|\varphi(z)|^{2})^{\frac{\gamma +n+1}{p}}} \Big[\log\Big(1-\frac{1}{\log|\varphi(z)|}\Big)\Big]^{-\frac{\delta}{p}} < +\infty \end{align} | (4.1) |
and
\begin{align} M_{j}: = \sup_{z\in\mathbb{B}}\frac{\mu(z)|\sum_{i = j}^{m}u_{i}(z)B_{i,j} (\varphi(z))|}{(1-|\varphi(z)|^2)^{\frac{\gamma +n+1}{p}+j}}\Big[\log\Big(1-\frac{1}{\log|\varphi(z)|}\Big)\Big]^{-\frac{\delta}{p}} < +\infty \end{align} | (4.2) |
for j = \overline{1, m} .
Moreover, if the operator \mathfrak{S}^m_{\vec{u}, {\varphi}}:A^p_{w_{\gamma, \delta}}\rightarrow H_\mu^\infty is bounded, then
\begin{align} \|\mathfrak{S}^m_{\vec{u},{\varphi}}\|_{A^p_{w_{\gamma,\delta}}\rightarrow H_\mu^\infty} \asymp\sum_{j = 0}^{m}M_{j}. \end{align} | (4.3) |
Proof. Suppose that (4.1) and (4.2) hold. From Theorem 3.1, Theorem 3.2, and some easy calculations, it follows that
\begin{align} &\mu(z)\Big| \sum_{i = 0}^{m}u_{i}(z)\Re^{i}f(\varphi(z))\Big|\leq\mu(z)\sum_{i = 0}^{m}\big|u_{i}(z)\big|\big|\Re^{i} f(\varphi(z))\big|\\ & = \mu(z)|u_{0}(z)||f(\varphi(z))| \\ &\quad+\mu(z)\Big|\sum_{i = 1}^{m}\sum_{j = 1}^{i}\Big(u_{i}(z)\sum_{l_{1} = 1}^{n} \cdots\sum_{l_{j} = 1}^{n}\Big(\frac{\partial^{j} f}{\partial z_{l_{1}}\partial z_{l_{2}}\cdots\partial z_{l_{j}}}(\varphi(z)) \sum_{k_{1},\ldots,k_{j}}C_{k_{1},\ldots,k_{j}}^{(i)} \prod_{t = 1}^{j}\varphi_{l_{t}}(z)\Big) \Big)\Big|\\ & = \mu(z)|u_{0}(z)f(\varphi(z))| \\ &\quad+\mu(z)\Big|\sum_{j = 1}^{m}\sum_{i = j}^{m}\Big(u_{i}(z)\sum_{l_{1} = 1}^{n} \cdots\sum_{l_{j} = 1}^{n}\Big(\frac{\partial^{j} f}{\partial z_{l_{1}}\partial z_{l_{2}}\cdots\partial z_{l_{j}}}(\varphi(z)) \sum_{k_{1},\ldots,k_{j}}C_{k_{1},\ldots,k_{j}}^{(i)} \prod_{t = 1}^{j}\varphi_{l_{t}}(z)\Big) \Big)\Big|\\ &\lesssim\frac{\mu(z)|u_{0}(z)|}{(1-|\varphi(z)|^{2})^{\frac{\gamma +n+1}{p}}} \Big[\log\Big(1-\frac{1}{\log|\varphi(z)|}\Big)\Big]^{-\frac{\delta}{p}} \|f\|_{A^p_{w_{\gamma ,\delta}}}\\ &\quad+\sum_{j = 1}^{m}\frac{\mu(z)|\sum_{i = j}^{m}u_{i}(z)B_{i,j}(\varphi(z))|} {(1-|\varphi(z)|^2)^{\frac{\gamma+n+1}{p}+j}}\Big[\log\Big(1-\frac{1} {\log|\varphi(z)|}\Big)\Big]^{-\frac{\delta}{p}}\|f\|_{A^p_{w_{\gamma,\delta}}} \\ & = M_0\|f\|_{A^p_{w_{\gamma ,\delta}}}+\sum_{j = 1}^{m}{M_j\|f\|_{A^p_{w_{\gamma,\delta}}}}. \end{align} | (4.4) |
By taking the supremum in inequality (4.4) over the unit ball in the space A^p_{w_{\gamma, \delta}} , and using (4.1) and (4.2), we obtain that the operator \mathfrak{S}^m_{\vec{u}, {\varphi}}:A^p_{w_{\gamma, \delta}}\to H_\mu^\infty is bounded. Moreover, we have
\begin{align} \|\mathfrak{S}^m_{\vec{u},{\varphi}}\|_{A^p_{w_{\gamma ,\delta}}\rightarrow H_\mu^\infty} \leq C\sum_{j = 0}^{m}M_{j}, \end{align} | (4.5) |
where C is a positive constant.
Assume that the operator \mathfrak{S}^m_{\vec{u}, {\varphi}}:A^p_{w_{\gamma, \delta}}\rightarrow H_\mu^\infty is bounded. Then there exists a positive constant C such that
\begin{align} \|\mathfrak{S}^m_{\vec{u},{\varphi}}f\|_{H_{\mu}^{\infty}}\leq C\|f\|_{A^p_{w_{\gamma,\delta}}} \end{align} | (4.6) |
for any f\in{A^p_{w_{\gamma, \delta}}} . First, we can take f(z) = 1\in {A^p_{w_{\gamma, \delta}}} , then one has that
\begin{align} \sup_{z\in\mathbb{B}}\mu(z)|u_{0}(z)| < +\infty. \end{align} | (4.7) |
Similarly, take f_{k}(z) = z_{k}^{j}\in {A^p_{w_{\gamma, \delta}}} , k = \overline{1, n} and j = \overline{1, m} , by (4.7), then
\begin{align} \mu(z)\Big|u_{0}(z)\varphi_{k}(z)^{j}+\sum_{i = j}^{m}\Big(u_{i}(z) B_{i,j}(\varphi_{k}(z)))\Big)\Big| < +\infty \end{align} | (4.8) |
for any j\in\{1, 2, \ldots, m\} . Since \varphi(z)\in\mathbb{B} , we have |\varphi(z)|\leq1 . So, one can use the triangle inequality (4.7) and (4.8), the following inequality is true
\begin{align} \sup_{z\in\mathbb{B}}\mu(z)\Big|\sum_{i = j}^{m}u_{i}(z)B_{i,j}(\varphi(z))\Big| < +\infty. \end{align} | (4.9) |
Let w\in\mathbb{B} and d_{k} = \frac{\gamma +n+1}{p}+k . For any j\in\{1, 2, \ldots, m\} and constants c_{k} = c_{k}^{(j)} , k = \overline{0, m} , let
\begin{align} h_{w}^{(j)}(z) = \sum_{k = 0}^{m}c_{k}^{(j)}f_{w,k}(z), \end{align} | (4.10) |
where f_{w, k} is defined in Theorem 3.4. Then, by Theorem 3.4, we have
\begin{align} L_{j} = \sup_{w\in\mathbb{B}}\|h_{w}^{(j)}\|_{A^p_{w_{\gamma,\delta}}} < +\infty. \end{align} | (4.11) |
From (4.6), (4.11), and some easy calculations, it follows that
\begin{align} &L_{j}\|\mathfrak{S}^m_{\vec{u},{\varphi}}\|_{{A^p_{w_{\gamma ,\delta}}}\to H_{\mu}^{\infty}} \geq\|\mathfrak{S}^m_{\vec{u},{\varphi}} h_{\varphi(w)}^{(j)}\|_{H_{\mu}^{\infty}} \\ & = \sup_{z\in\mathbb{B}}\mu(z)\Big|\sum_{i = 0}^{m}u_{0}(z)h_{\varphi(w)}^{(j)} (\varphi(z))\Big|\\ &\geq\mu(w)\Big|u_{0}(w)h_{\varphi(w)}^{(j)}(\varphi(w))+ \sum_{i = 1}^{m}\Big(u_{i}(w)\Re^{i} h_{\varphi(w)}^{(j)}(\varphi(w))\Big)\Big|\\ & = \mu(w)\Big|u_{0}(w)h_{\varphi(w)}^{(j)}(\varphi(w))+\sum_{i = 1}^{m}u_{i}(w) \sum_{k = 0}^{m}{c_k^{(j)}f_{\varphi(w),k}(\varphi(w))}\Big|\\ & = \mu(w)\Big|u_{0}(w)\frac{c_{0}+c_{1}+\cdots+c_{m}} {(1-|\varphi(z)|^{2})^{\frac{\gamma +n+1}{p}}} +\big\langle\sum_{i = 1}^{m}u_{i}(w)B_{i,1}(\varphi(w)),\varphi(w)\big\rangle \frac{(d_{0}c_{0}+\cdots+d_{m}c_{m})}{(1-|\varphi(w)|^{2})^{\frac{\gamma +n+1}{p}+1}}+\cdots \\ &\quad+\big\langle\sum_{i = j}^{m}u_{i}(w)B_{i,j}(\varphi(w)), \varphi(w)^{j}\big\rangle\frac{(d_{0}\cdots d_{j-1}c_{0}+\cdots+d_{m} \cdots d_{m+j-1}c_{m})}{(1-|\varphi(w)|^{2})^{\frac{\gamma +n+1}{p}+j}} +\cdots\\ &\quad+\big\langle u_{m}(w)B_{m,m}(\varphi(w)),\varphi(w)^{m}\big\rangle\frac{(d_{0}\cdots d_{m-1}c_{0}+\cdots+d_{m} \cdots d_{2m-1}c_{m})}{(1-|\varphi(w)|^{2})^{\frac{\gamma +n+1} {p}+m}}\Big|\Big[\log\Big(1-\frac{1}{\log|\varphi(w)|}\Big)\Big]^{-\frac{\delta}{p}}. \end{align} | (4.12) |
Since d_{k} > 0 , k = \overline{0, m} , by Lemma 4.4, we have the following linear equations
\begin{equation} \left( \begin{array}{cccc} 1 & 1 &\cdots & 1 \\ d_{0} & d_{1} &\cdots & d_{m} \\ \vdots &\vdots &\ddots &\vdots \\ \prod\limits_{k = 0}^{j-1}d_{k}& \prod\limits_{k = 0}^{j-1} d_{k+m}&\cdots & \prod\limits_{k = 0}^{j-1}d_{k+m} \\ \vdots &\vdots &\ddots &\vdots \\ \prod\limits_{k = 0}^{m-1}d_{k}& \prod\limits_{k = 0}^{m-1} d_{k+m}&\cdots & \prod\limits_{k = 0}^{m-1}d_{k+m} \end{array} \right) \left( \begin{array}{cccc} c_{0}\\ c_{1}\\ \vdots\\ \quad\\ c_{j}\\ \quad\\ \vdots\\ \quad\\ c_{m} \end{array} \right) = \left( \begin{array}{cccc} 0\\ 0\\ \vdots\\ \quad\\ 1\\ \quad\\ \vdots\\ \quad\\ 0 \end{array} \right). \end{equation} | (4.13) |
From (4.12) and (4.13), we have
\begin{align} L_{j}\|\mathfrak{S}^l_{\vec{u},{\varphi}}\|_{{A^p_{w_\gamma,\delta}}\rightarrow H_{\mu}^{\infty}} &\geq\sup_{|\varphi(z)| > 1/2}\frac{\mu(z)|\sum _{i = j}^{m}u_{i}(z)B_{i,j} (\varphi(z))||\varphi(z)|^{j}}{(1-|\varphi(z)|^2)^{\frac{\gamma +n+1}{p}+j}} \Big[\log\Big(1-\frac{1}{\log|\varphi(z)|}\Big)\Big]^{-\frac{\delta}{p}}\\ &\gtrsim\sup_{|\varphi(z)| > 1/2}\frac{\mu(z)|\sum _{i = j}^{m}u_{i}(z)B_{i,j} (\varphi(z))|}{(1-|\varphi(z)|^2)^{\frac{\gamma +n+1}{p}+j}} \Big[\log\Big(1-\frac{1}{\log|\varphi(z)|}\Big)\Big]^{-\frac{\delta}{p}}. \end{align} | (4.14) |
On the other hand, from (4.9), we have
\begin{align} &\sup_{|\varphi(z)|\leq1/2}\frac{\mu(z)|\sum _{i = j}^{m}u_{i}(z)B_{i,j} (\varphi(z))|}{(1-|\varphi(z)|^2)^{\frac{\gamma +n+1}{p}+j}} \Big[\log\Big(1-\frac{1}{\log|\varphi(z)|}\Big)\Big]^{-\frac{\delta}{p}}\\ &\leq\sup_{z\in\mathbb{B}}\Big(\frac{4}{3}\Big)^{\frac{\gamma +n+1}{p}+j} \Big[\log\Big(1-\frac{1}{\log\frac{1}{2}}\Big)\Big]^{-\frac{\delta}{p}} \mu(z)\Big|\sum_{i = j}^{m}u_{i}(z)B_{i,j}(\varphi(z))\Big| < +\infty. \end{align} | (4.15) |
From (4.14) and (4.15), we get that (4.2) holds for j = \overline{1, m} .
For constants c_{k} = c_{k}^{(0)} , k = \overline{0, m} , let
\begin{align} h_{w}^{(0)}(z) = \sum_{k = 0}^{m}c_{k}^{(0)}f_{w,k}(z). \end{align} | (4.16) |
By Theorem 3.4, we know that L_{0} = \sup_{w\in\mathbb{B}}\|h_{w}^{(0)}\|_{A^p_{w_{\gamma, \delta}}} < +\infty . From this, (4.12), (4.13) and Lemma 4.4, we get
\begin{align*} L_{0}\|\mathfrak{S}^m_{\vec{u},{\varphi}}\|_{{A^p_{w_{\gamma,\delta}}}\rightarrow H_{\mu}^{\infty}} \geq \frac{\mu(z)|u_{0}(z)|}{(1-|\varphi(z)|^{2})^{\frac{\gamma +n+1}{p}}} \Big[\log\Big(1-\frac{1}{\log|\varphi(z)|}\Big)\Big]^{-\frac{\delta}{p}}. \end{align*} |
So, we have M_0 < +\infty . Moreover, we have
\begin{align} \|\mathfrak{S}^m_{\vec{u},{\varphi}}\|_{A^p_{w_{\gamma,\delta}}\rightarrow H_\mu^\infty} \geq\sum_{j = 0}^{m}M_{j}. \end{align} | (4.17) |
From (4.5) and (4.17), we obtain (4.3). The proof is completed.
From Theorem 4.1 and (1.4), we obtain the following result.
Corollary 4.1. Let m\in\mathbb{N} , u\in H(\mathbb{B}) , \varphi\in S(\mathbb{B}) and \mu is a weight function on \mathbb{B} . Then, the operator C_{{\varphi}}\Re^{m}M_{u}:A^p_{w_{\gamma, \delta}}\rightarrow H_\mu^\infty is bounded if and only if
\begin{align*} I_{0}: = \sup_{z\in\mathbb{B}} \frac{\mu(z)|\Re^mu \circ {\varphi}(z)|}{(1-|\varphi(z)|^{2})^{\frac{\gamma +n+1}{p}}} \Big[\log\Big(1-\frac{1}{\log|\varphi(z)|}\Big)\Big]^{-\frac{\delta}{p}} < +\infty \end{align*} |
and
\begin{align*} I_{j}: = \sup_{z\in\mathbb{B}}\frac{\mu(z)|\sum _{i = j}^{m}\Re^{m-i}u \circ {\varphi}(z)B_{i,j}(\varphi(z))|}{(1-|\varphi(z)|^2)^{\frac{\gamma +n+1}{p}+j}}\Big[\log\Big(1-\frac{1}{\log|\varphi(z)|}\Big)\Big]^{-\frac{\delta}{p}} < +\infty \end{align*} |
for j = \overline{1, m} .
Moreover, if the operator C_{{\varphi}}\Re^{m}M_{u}:A^p_{w_{\gamma, \delta}}\rightarrow H_\mu^\infty is bounded, then
\begin{align*} \|C_{{\varphi}}\Re^{m}M_{u}\|_{A^p_{w_{\gamma,\delta}}\rightarrow H_\mu^\infty} \asymp\sum_{j = 0}^{m}I_{j}. \end{align*} |
Theorem 4.2. Let -1 < \gamma < +\infty , \delta\leq0 , 0 < p < +\infty , m\in\mathbb{N} , u_j\in H(\mathbb{B}) , j = \overline{0, m} , and \varphi\in S(\mathbb{B}) . Then, the operator \mathfrak{S}^m_{\vec{u}, {\varphi}}:A^p_{w_{\gamma, \delta}}\to H_\mu^\infty is compact if and only if the operator \mathfrak{S}^m_{\vec{u}, {\varphi}}:A^p_{w_{\gamma, \delta}}\to H_\mu^\infty is bounded,
\begin{align} \lim_{|\varphi(z)|\rightarrow1}\frac{\mu(z)|\sum _{i = j}^{m}(u_{i}(z)B_{i,j} (\varphi(z))|}{(1-|\varphi(z)|^2)^{\frac{\gamma +n+1}{p}+j}}\Big[\log\Big(1-\frac{1}{\log|\varphi(z)|}\Big)\Big]^{-\frac{\delta}{p} } = 0 \end{align} | (4.18) |
for j = \overline{1, m} , and
\begin{align} \lim_{|\varphi(z)|\rightarrow1}\frac{\mu(z)|u_{0}(z)| }{(1-|\varphi(z)|^2)^{\frac{\gamma+n+1}{p}}}\Big[\log\Big(1-\frac{1}{\log|\varphi(z)|}\Big)\Big]^{-\frac{\delta}{p}} = 0. \end{align} | (4.19) |
Proof. Assume that the operator \mathfrak{S}^m_{\vec{u}, {\varphi}}:A^p_{w_{\gamma, \delta}}\rightarrow H_\mu^\infty is compact. It is obvious that the operator \mathfrak{S}^m_{\vec{u}, {\varphi}}:A^p_{w_{\gamma, \delta}}\rightarrow H_\mu^\infty is bounded.
If \|\varphi\|_{\infty} < 1 , then it is clear that (4.18) and (4.19) are true. So, we suppose that \|\varphi\|_{\infty} = 1 . Let \{z_{k}\} be a sequence in \mathbb{B} such that
\lim\limits_{k\rightarrow1}|\mu(z_k)|\to 1 \quad \mbox{and} \quad h_{k}^{(j)} = h_{\varphi(z_{k})}^{(j)}, |
where h_{w}^{(j)} are defined in (4.10) for a fixed j\in\{1, 2, \ldots, l\} . Then, it follows that h_{k}^{(j)}\rightarrow 0 uniformly on any compact subset of \mathbb{B} as k\rightarrow \infty . Hence, by Lemma 4.1, we have
\begin{align*} \lim_{k\to\infty}\|\mathfrak{S}^m_{\vec{u},{\varphi}} h_{k}\|_{H_{\mu}^{\infty}} = 0. \end{align*} |
Then, we can find sufficiently large k such that
\begin{align} &\frac{\mu(z_{k})|\sum_{i = j}^{m}(u_{i}(z_{k})B_{i,j}(\varphi(z_{k})) |}{{(1-|\varphi(z_k)|^2)^{\frac{\gamma+n+1}{p}+j}}}\Big[\log\Big(1-\frac{1} {\log|\varphi (z_k)|}\Big)\Big]^{-\frac{\delta}{p}} \leq L_k\|\mathfrak{S}^m_{\vec{u},{\varphi}} h_{k}^{(j)}\|_{H_{\mu}^{\infty}}. \end{align} | (4.20) |
If k\rightarrow \infty , then (4.20) is true.
Now, we discuss the case of j = 0 . Let h_{k}^{(0)} = h_{\varphi(z_{k})}^{(0)} , where h_{w}^{(0)} is defined in (4.16). Then, we also have that \|h_{k}^{(0)}\|_{A^p_{w_{\gamma, \delta}}} < +\infty and h_{k}^{(0)}\rightarrow 0 uniformly on any compact subset of \mathbb{B} as k\rightarrow \infty . Hence, by Lemma 4.1, one has that
\begin{align} \lim_{k\to\infty}\|\mathfrak{S}^m_{\vec{u},{\varphi}} h_{k}^{(0)}\|_{H_{\mu}^{\infty}(\mathbb{B})} = 0. \end{align} | (4.21) |
Then, by (4.21), we know that (4.18) is true.
Now, assume that \mathfrak{S}^m_{\vec{u}, {\varphi}}:A^p_{w_{\gamma, \delta}}\rightarrow H_\mu^\infty is bounded, (4.18) and (4.19) are true. One has that
\begin{align} \mu(z)|u_{0}(z)|\leq C < +\infty \end{align} | (4.22) |
and
\begin{align} \mu(z)\Big|\sum_{i = j}^{m}(u_{i}(z) B_{i,j}(\varphi(z)))\Big|\leq C < +\infty \end{align} | (4.23) |
for any z\in\mathbb{B} . By (4.18) and (4.19), for arbitrary \varepsilon > 0 , there is a r\in(0, 1) , for any z\in K such that
\begin{align} \frac{\mu(z)|u_{0}(z)| }{(1-|\varphi(z)|^2)^{\frac{\gamma+n+1}{p}}}\Big[\log\Big(1-\frac{1} {\log|\varphi(z)|}\Big)\Big]^{-\frac{\delta}{p}} < \varepsilon. \end{align} | (4.24) |
and
\begin{align} \frac{\mu(z)\Big|\sum_{i = j}^{m}(u_{i}(z)B_{i,j}(\varphi(z)))\Big| }{(1-|\varphi(z)|^2)^{\frac{\gamma+n+1}{p}+j}} \Big[\log\Big(1-\frac{1}{\log|\varphi(z)|}\Big)\Big]^{-\frac{\delta}{p}} < \varepsilon. \end{align} | (4.25) |
Assume that \{f_{s}\} is a sequence such that \sup_{s\in\mathbb{N}}\|f_{s}\|_{A^p_{w_{\gamma, \delta}}}\leq M < +\infty and f_{s}\rightarrow 0 uniformly on any compact subset of \mathbb{B} as s\rightarrow \infty . Then by Theorem 3.1, Theorem 3.2 and (4.22)–(4.25), one has that
\begin{align} \|\mathfrak{S}^m_{\vec{u},{\varphi}} f_{s}\|_{H_{\mu}^{\infty}(\mathbb{B})} & = \sup_{z\in\mathbb{B}}\mu(z)\Big|u_{0}(z)f(\varphi(z))+ \sum_{i = 1}^{m}u_{i}(z)\Re^{i} f(\varphi(z))\Big|\\ & = \sup_{z\in K}\mu(z)\Big|u_{0}(z)f(\varphi(z))+ \sum_{i = 1}^{m}u_{i}(z)\Re^{i} f(\varphi(z))\Big|\\ &\quad+\sup_{z\in\mathbb{B}\setminus K}\mu(z)\Big|u_{0}(z)f(\varphi(z))+ \sum_{i = 1}^{m}u_{i}(z)\Re^{i} f(\varphi(z))\Big|\\ &\lesssim \sup_{z\in K}\frac{\mu(z)|u_{0}(z)| }{(1-|\varphi(z)|^2)^{\frac{\gamma+n+1}{p}}}\Big[\log\Big(1-\frac{1} {\log|\varphi (z)|}\Big)\Big]^{-\frac{\delta}{p}}\|f_{s}\|_{A^p_{w_\gamma,\delta}} \\ &\quad+\sup_{z\in K}\frac{\mu(z)\Big|\sum_{i = j}^{m}(u_{i}(z) B_{i,j}(\varphi(z)))\Big| }{(1-|\varphi(z)|^2)^{\frac{\gamma+n+1}{p}+j}}\Big[\log\Big(1-\frac{1} {\log|\varphi (z)|}\Big)\Big]^{-\frac{\delta}{p}} \|f_{s}\|_{A^p_{w_\gamma,\delta}} \\ &\quad+\sup_{z\in\mathbb{B}\setminus K}\mu(z)|u_{0}(z)||f_{s}(\varphi(z))|\\ &\quad+\sup_{z\in\mathbb{B}\setminus K}\sum_{j = 1}^{m} \mu(z)\Big|\sum_{i = j}^{m}(u_{i}(z)B_{i,j}(\varphi(z)))\Big| \max_{\{l_{1},l_{2},\ldots,l_{j}\}}\Big|\frac{\partial^{j} f_{s}}{\partial z_{l_{1}} \partial z_{l_{2}}\cdots\partial z_{l_{j}}}(\varphi(z))\Big|\\ &\leq M\varepsilon+C\sup_{|w|\leq \delta}\sum_{j = 0}^{m} \max_{\{l_{1},l_{2},\ldots,l_{j}\}}\Big|\frac{\partial^{j} f_{s}}{\partial z_{l_{1}} \partial z_{l_{2}}\cdots\partial z_{l_{j}}}(w)\Big|. \end{align} | (4.26) |
Since f_{s}\rightarrow0 uniformly on any compact subset of \mathbb{B} as s\rightarrow \infty . By Cauchy's estimates, we also have that \frac{\partial^{j} f_{s}}{\partial z_{l_{1}}\partial z_{l_{2}}\cdots\partial z_{l_{j}}}\rightarrow 0 uniformly on any compact subset of \mathbb{B} as s\rightarrow \infty . From this and using the fact that \{w\in{\mathbb{B}}:|w|\leq\delta\} is a compact subset of \mathbb{B} , by letting s\rightarrow \infty in inequality (4.26), one get that
\begin{align*} \limsup_{s\rightarrow \infty}\|\mathfrak{S}^m_{\vec{u},{\varphi}} f_{s}\|_{H_{\mu}^{\infty}}\lesssim \varepsilon. \end{align*} |
Since \varepsilon is an arbitrary positive number, it follows that
\begin{align*} \lim_{s\rightarrow \infty}\|\mathfrak{S}^m_{\vec{u},{\varphi}} f_{s}\|_{H_{\mu}^{\infty}} = 0. \end{align*} |
By Lemma 4.1, the operator \mathfrak{S}^m_{\vec{u}, {\varphi}}:A^p_{w_{\gamma, \delta}}\rightarrow H_\mu^\infty is compact.
As before, we also have the following result.
Corollary 4.2. Let m\in\mathbb{N} , u\in H(\mathbb{B}) , \varphi\in S(\mathbb{B}) and \mu is a weight function on \mathbb{B} . Then, the operators C_{{\varphi}}\Re^{m}M_{u}:A^p_{w_{\gamma, \delta}}\rightarrow H_\mu^\infty is compact if and only if the operator C_{{\varphi}}\Re^{m}M_{u}:A^p_{w_{\gamma, \delta}}\rightarrow H_\mu^\infty is bounded,
\begin{align*} \lim_{|\varphi(z)|\rightarrow1}\frac{\mu(z)|\Re^mu \circ {\varphi}(z)| }{(1-|\varphi(z)|^2)^{\frac{\gamma+n+1}{p}}}\Big[\log\Big(1-\frac{1}{\log|\varphi (z)|}\Big)\Big]^{-\frac{\delta}{p}} = 0 \end{align*} |
and
\begin{align*} \lim_{|\varphi(z)|\rightarrow1}\frac{\mu(z)|\sum_{i = j}^{m}(\Re^{m-i}u \circ {\varphi}(z)B_{i,j}(\varphi(z))| }{\Big(1-|\varphi(z)|^2)^{\frac{\gamma +n+1}{p}+j}}\Big[\log(1-\frac{1}{\log|\varphi (z)|}\Big)\Big]^{-\frac{\delta}{p}} = 0 \end{align*} |
for j = \overline{1, m} .
In this paper, we study and obtain some properties about the logarithmic Bergman-type space on the unit ball. As some applications, we completely characterized the boundedness and compactness of the operator
\begin{align*} \mathfrak{S}^m_{\vec{u},{\varphi}} = \sum_{i = 0}^{m}M_{u_i}C_{\varphi}\Re^{i} \end{align*} |
from the logarithmic Bergman-type space to the weighted-type space on the unit ball. Here, one thing should be pointed out is that we use a new method and technique to characterize the boundedness of such operators without the condition (1.5), which perhaps is the special flavour in this paper.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
This work was supported by Sichuan Science and Technology Program (2022ZYD0010) and the Graduate Student Innovation Foundation (Y2022193).
The authors declare that they have no competing interests.
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