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For $-2<\alpha<\infty$ and $0<p<\infty$, the $\mathcal{Q}_K$-type space is the space of all analytic functions on the open unit disk $\mathbb{D}$ satisfying $$\sup_{a\in\mathbb{D}}\int_\mathbb{D}|f'(z)|^p(1-|z|^2)^\alpha K(g(z,a))dA(z)<\infty,$$ where $g(z,a)=\log\frac{1}{|\sigma_a(z)|}$ is the Green's function on $\mathbb{D}$ and $K:[0,\infty)\rightarrow[0,\infty)$, is a right-continuous and non-decreasing function. For $0<s<\infty$, the space $\mathcal{Q}_s$ consists of all analytic functions on $\mathbb{D}$ for which $$\sup_{a\in\mathbb{D}}\int_\mathbb{D}|f'(z)|^2(g(z,a))^sdA(z)<\infty.$$ Boundedness and compactness of composition operators $C_\varphi$ acting on $\mathcal{Q}_K$-type spaces and $\mathcal{Q}_s$ spaces is characterized in terms of the norms of $\varphi^n$. Thus the author announces a solution to the problem raised by Wulan, Zheng and Zhou.
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참고문헌 (26)
참고문헌 신청
A. Baernstein II / 1980 / Aspects of contempo-rary complex analysis (Proc. NATO Adv. Study Inst., Univ. Durham, Durham, 1979) / Academic Press : 3 ~ 36
A. Montes-Rodriguez / 1999 / The essential norm of a composition operator on Bloch spaces / Pacific J. Math 188 (2) : 339 ~ 351
C. Fefferman / 1971 / Characterizations of bounded mean oscillation / Bull. Amer. Math. Soc 77 : 587 ~ 588
C. Ouyang / 1998 / Mobius invariant Qp spaces associated with the Green,s function on the unit ball of Cn / Pacific J. Math 182 (1) : 69 ~ 99
C. Pommerenke / 1992 / Boundary Behaviour of Conformal Maps / Speringer-Verlag
A. Montes-Rodriguez / 1999 / The essential norm of a composition operator on Bloch spaces / Pacific J. Math 188 (2) : 339 ~ 351
C. Fefferman / 1971 / Characterizations of bounded mean oscillation / Bull. Amer. Math. Soc 77 : 587 ~ 588
C. Ouyang / 1998 / Mobius invariant Qp spaces associated with the Green,s function on the unit ball of Cn / Pacific J. Math 182 (1) : 69 ~ 99
C. Pommerenke / 1992 / Boundary Behaviour of Conformal Maps / Speringer-Verlag
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