NORM OF THE COMPOSITION OPERATOR MAPPING BLOCH SPACE INTO HARDY OR BERGMAN SPACE

Title & Authors
NORM OF THE COMPOSITION OPERATOR MAPPING BLOCH SPACE INTO HARDY OR BERGMAN SPACE
Kwon, Ern-Gun; Lee, Jin-Kee;

Abstract
Let $1{\;}\leq{\;}p{\;}\infty{\;}and{\;}{\alpha}{\;}>{\;}-1$. If f is a holomorphic self-map of the open unit disc U of C with f(0) = 0, then the quantity $\int_U\;\{\frac{$\small{\mid}$f'(z)$\small{\mid}$}{1\;-\;$\small{\mid}$f(z)$\small{\mid}$^2}\}^p\;(1\;-\;$\small{\mid}$z$\small{\mid}$)^{\alpha+p}dxdy$ is equivalent to the operator norm of the composition operator $\small{C_f{\;}:{\;}B{\;}\rightarrow{\;}A^{p,{\alpha}}$ defined by $\small{C_fh{\;}={\;}h{\;}\circ{\;}f{\;}-{\;}h(0)}$, where B and $\small{A^{p,{\alpha}}$ are the Bloch space and the weighted Bergman space on U respectively.
Keywords
Bloch space;composition operator;$\small{A^{p,{\alpha}}$;$\small{spaceH^{p}}$ space;
Language
English
Cited by
1.
NORM OF THE COMPOSITION OPERATOR FROM BLOCH SPACE TO BERGMAN SPACE,;

대한수학회논문집, 2014. vol.29. 3, pp.409-413
1.
Composition operators between Bergman spaces of logarithmic weights, International Journal of Mathematics, 2015, 26, 09, 1550068
2.
NORM OF THE COMPOSITION OPERATOR FROM BLOCH SPACE TO BERGMAN SPACE, Communications of the Korean Mathematical Society, 2014, 29, 3, 409
References
1.
Pacific J. Math., 1983. vol.107. pp.267-278

2.
Indiana Univ. Math. J., 1987. vol.36. pp.131-148

3.
The theory of $H ^p$ functions, 1970.

4.
Pacific J. Math., 1968. vol.25. pp.463-494

5.
Bounded analytic functions, 1981.

6.
Proc. Amer. Math. Soc., 1996. vol.124-125. pp.1473-1480

7.
Math. Ann., 1991. vol.291. pp.591-606

8.
Trigonometric series, 1959.