Hilbert spaces of analytic functions between the Hardy and the Dirichlet space
(1992) In Proceedings of the American Mathematical Society 115(1). p.10497 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... (More)  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. (Less)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/record/1467368
 author
 Aleman, Alexandru ^{LU}
 publishing date
 1992
 type
 Contribution to journal
 publication status
 published
 subject
 in
 Proceedings of the American Mathematical Society
 volume
 115
 issue
 1
 pages
 104  97
 publisher
 American Mathematical Society (AMS)
 external identifiers

 Scopus:0002628287
 ISSN
 10886826
 language
 English
 LU publication?
 no
 id
 20640b9156c6446e943f7961397ee549 (old id 1467368)
 alternative location
 http://www.jstor.org/stable/pdfplus/2159570.pdf
 date added to LUP
 20090916 13:55:47
 date last changed
 20161013 04:47:26
@misc{20640b9156c6446e943f7961397ee549, 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 $w(r)=1r$). <br/><br> <br/><br> 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 $E(\max\{1,f\})\leq E(f)$.<br/><br> <br/><br> 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.}, author = {Aleman, Alexandru}, issn = {10886826}, language = {eng}, number = {1}, pages = {10497}, publisher = {ARRAY(0x9dca5f0)}, series = {Proceedings of the American Mathematical Society}, title = {Hilbert spaces of analytic functions between the Hardy and the Dirichlet space}, volume = {115}, year = {1992}, }