Hilbert spaces of analytic functions between the Hardy and the Dirichlet space
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Abstract
Let $w$ be a positive on $[0,1)$ which is concave, decreasing and tends to $0$ at $1$. The space $H_w$ of analytic functions $f$ satisfying $\f\^2_w\coloneq f(0)^2+\int_{z<1}f'(z)^2w(z)dm(z)<\infty$ (where $dm(z)=dxdy$) is a Hilbert space lying between the usual Dirichlet space (where $w\equiv 1$) and the Hardy space (where $w(r)=1r$).
It is shown in this paper that every element of $H_w$ is a quotient of two bounded functions in $H_w$, proving a conjecture of S. Richter and A. Shields [\cita MR0939532 (89c:46039) \endcit Math. Z. 198 (1988), no. 2, 151159; MR0939532 (89c:46039)]. The proof involves first showing that $\f\^2_w=f(0)^2\frac 14\int_{z<1}\Delta(w(z))(P_z[f^2]f(z)^2)\,dm(z)$, where $P_z[g]$ denotes the Poisson integral of the boundary value of $g$. This is then used to show that the outer factor $F$ of $f$ belongs to $H_w$ when $f$ does. Finally, $F$ is truncated below and above in the usual way (take $\log^+f$ and $\log^f$ and use them to define outer functions on $z<1$). This last step requires two clever inequalities to prove that the resulting functions belong to $H_w$: Define $E(f)=\int_Xfd\mu\exp\int_X\log fd\mu$ for positive functions $f$ on a probability space $(X,\mu)$. Then $E(\min\{1,f\})\leq E(f)$ and $E(\max\{1,f\})\leq E(f)$.
For a large class of Hilbert spaces of analytic functions in the unit disc lying between the Hardy and the Dirichlet space we prove that each element of the space is the quotient of two bounded functions in the same space. It follows that the multiplication operator on these spaces is cellular indecomposable and that each invariant subspace contains nontrivial bounded functions.
It is shown in this paper that every element of $H_w$ is a quotient of two bounded functions in $H_w$, proving a conjecture of S. Richter and A. Shields [\cita MR0939532 (89c:46039) \endcit Math. Z. 198 (1988), no. 2, 151159; MR0939532 (89c:46039)]. The proof involves first showing that $\f\^2_w=f(0)^2\frac 14\int_{z<1}\Delta(w(z))(P_z[f^2]f(z)^2)\,dm(z)$, where $P_z[g]$ denotes the Poisson integral of the boundary value of $g$. This is then used to show that the outer factor $F$ of $f$ belongs to $H_w$ when $f$ does. Finally, $F$ is truncated below and above in the usual way (take $\log^+f$ and $\log^f$ and use them to define outer functions on $z<1$). This last step requires two clever inequalities to prove that the resulting functions belong to $H_w$: Define $E(f)=\int_Xfd\mu\exp\int_X\log fd\mu$ for positive functions $f$ on a probability space $(X,\mu)$. Then $E(\min\{1,f\})\leq E(f)$ and $E(\max\{1,f\})\leq E(f)$.
For a large class of Hilbert spaces of analytic functions in the unit disc lying between the Hardy and the Dirichlet space we prove that each element of the space is the quotient of two bounded functions in the same space. It follows that the multiplication operator on these spaces is cellular indecomposable and that each invariant subspace contains nontrivial bounded functions.
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Research areas and keywords  Subject classification (UKÄ) – MANDATORY

Original language  English 

Pages (fromto)  97104 
Journal  Proceedings of the American Mathematical Society 
Volume  115 
Issue number  1 
Publication status  Published  1992 
Publication category  Research 
Peerreviewed  Yes 
Externally published  Yes 