## Integral type operators from mixed-norm spaces to $$\alpha$$-Bloch spaces.(English)Zbl 1131.47031

Let $$B=\{z\in\mathbb C^n:| z| <1\}$$ and $$H(B)$$ be the class of all holomorphic functions on the unit ball. For $$f\in H(B)$$ with the Taylor expansion $$f(z)=\sum_{| \beta| \geq0}a_\beta z^\beta$$, let $$\operatorname{Re} f(z)=\sum_{|\beta|\geq0}|\beta| a_\beta z^\beta$$ be the radial derivative of $$f$$, where $$\beta=(\beta_1, \beta_2,\dots,\beta_n)$$ is a multi-index and $$z^\beta=z_1^{\beta_1}\ldots z_n^{\beta_n}$$. Let $$\alpha>0$$. The $$\alpha$$-Bloch space $$\mathcal{B}^\alpha=\mathcal{B}^\alpha(B)$$ is the space of all holomorphic functions $$f$$ on $$B$$ such that $$b_\alpha(f)=\sup_{z\in B}(1-| z| ^2)^\alpha|\operatorname{Re} f(z)|<\infty$$. Let $$\mathcal{B}_0^\alpha$$ be the subspace of $$\mathcal B$$ consisting of those $$f\in{\mathcal B}^\alpha$$ for which $$(1-| z|^2)^\alpha| \operatorname{Re} f(z)| \to0$$ as $$| z|\to1$$. This space is called the little $$\alpha$$-Bloch space. Let $$p,q\in[0,\infty)$$ and $$\gamma>-1$$. The mixed norm space $$H_{p,q,\gamma}(B)$$ consists of all holomorphic functions $$f$$ on the unit ball such that $\| f\| ^q_{H_{p,q,\gamma}}=\displaystyle\int_0^1 M_p^q(f, r)(1-r)^\gamma\,dr<\infty,$ where $$M_p(f, r)=(\int_{\partial B}| f(r\xi)| ^p\,d\sigma(\xi))^{1/p}$$ and $$\partial B=\{z\in\mathbb C^n:| z| =1\}$$. Let $$g: B\mapsto\mathbb C^1$$ be a holomorphic map of $$B$$. For any holomorphic function $$f: B\mapsto \mathbb C^1$$ and $$z\in B$$, define $T_gf(z)=\displaystyle\int_0^1 f(tz)\operatorname{Re} g(tz)\,\frac{dt}{t} \quad\text{and}\quad L_gf(z)=\displaystyle\int_0^1 \operatorname{Re} f(tz)g(tz)\frac{dt}{t}.$ In this paper, the authors obtain the boundedness and compactness of the operators $$T_g$$ and $$L_g$$ from $$H_{p,\,q,\,\gamma}$$ to the $$\alpha$$-Bloch space $$\mathcal B^\alpha(B)$$ and little $$\alpha$$-Bloch space $$\mathcal B_0^\alpha(B)$$.

### MSC:

 47B38 Linear operators on function spaces (general) 30H05 Spaces of bounded analytic functions of one complex variable 47G10 Integral operators

### Keywords:

integral operator; mixed-norm space; $$\alpha$$-Bloch space
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### References:

 [1] Zhu K., Graduate Texts in Mathematics 226 (2005) [2] Hu Z. J., Acta Mathematica Scientia. Series B. English Edition 23 pp 561– (2003) [3] DOI: 10.1016/j.jmaa.2004.01.045 · Zbl 1072.47029 [4] DOI: 10.1155/JIA.2005.81 · Zbl 1074.47013 [5] DOI: 10.1007/BF02567392 · Zbl 0369.30012 [6] Chang D. C., Taiwanese Journal of Mathematics 7 pp 423– (2003) [7] DOI: 10.1002/mana.200310013 · Zbl 1024.47014 [8] Stević S., Bulletin of the Institute of Mathematics Academia Sinica 31 pp 135– (2003) [9] DOI: 10.4171/ZAA/1138 · Zbl 1046.47026 [10] Rudin W., Function Theory in the Unit Ball of $$\mathbb{C}$$ n (1980) · Zbl 0495.32001 [11] Shields A. L., Transactions of the American Mathematical Society 162 pp 287– (1971) [12] Cowen C. C., Studies in Advanced Mathematics (1995) [13] DOI: 10.1216/rmjm/1181069993 · Zbl 1042.47018
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