Function spaces and composition properties of holomorphic functions on the unit ball of Cn

Abstract

The purpose of this thesis is to study the functional analytic and function theoretic properties of certain class of function spaces which consist of holomorphic functions on $B_n$ and to deal with related problems. In Chapter 2 we compute the dual spaces of a Hardy space $H^p(B_n)$ ($0 < p <1$) and a weighted Bergman space $A^p(B_n)$ ($0 < P <1$, $\alpha > -1$). The crucial points in our analysis of these spaces are explicit determinations of their Mackey topologies using an "atomic decomposition" of the same sort discussed by Shapiro in [18]. The corresponding problems for the case n = 1 are settled by Duren, Romberg and Shields[10] Shapiro[18] and Ahern and Jevticc[2]. In Chapter 3 we investigate the composition preoperty of a nonhomogeneous bounded holomorphic function. The homogeneous polynomials $n^{n/2}z_1z_2\cdots{z_n},z_1^2+z_2^2+\cdots+z_n^2$ and $b_\alpha{z_1^\alpha1}z_2^\alpha2\cdots{z_p^\alpha{p}}$ ($b_\alpha$ properly chosen) have been known to pull a Bloch function in the unit disc U back to a holomorphic function in the unit ball $B_n$ of bounded mean oscillation. We show that the nonhomogeneous holomorphic function $\pi$n,m(z) $\frac{z_m^2+1+\cdots+z_n^2}{1-(z_1^2+\cdots+z_n^2}$ pulls the Bloch space B(U) back to $\cap_{0<p<\infty}H^P(B_n)$. Chapter 4 is concerned with Cauchy integral equalities(CIE). We find a concrete characterization of $\pi\in{P_{2,k}}$(homogeneous polynomials in $C^n$ of degree k) satisfying CIE in some special cases. If $\pi\in{P_{2,k}}$ with k $\leq$ 3 or $\pi\in{P_{2,k}}$ with real coefficients satisfies CIE, then it is shown that $\pi$ can be transformed into a monomial by a unitary change of variables. When $\pi\in{P_{3,2}}$ a slight different characterization is obtained. In this case it is shown that $\pi$ can be transformed, by a unitary change of variables, into a monomial or a sum-of-squares, $az_1^2\,+\,bz_2^2\,+\,cz_3^2$, where $\mid{a}\mid$ = $\mid{b}\mid$ = $\mid{c}\mid$ = 1. In Chapter 5, we deal with a cert...