Hilbert spaces of analytic functions between the Hardy and the Dirichlet space
Author
Summary, in English
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 $w(r)=1-r$).
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, 151--159; 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, 151--159; 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.
Publishing year
1992
Language
English
Pages
97-104
Publication/Series
Proceedings of the American Mathematical Society
Volume
115
Issue
1
Document type
Journal article
Publisher
American Mathematical Society (AMS)
Topic
- Mathematics
Status
Published
ISBN/ISSN/Other
- ISSN: 1088-6826