| Title | Hilbert spaces of analytic functions between the Hardy and the Dirichlet space |
| Authors | Alexandru Aleman |
| Alternative Location | http://www.jstor.org/stable..., Restricted Access |
| Publication | Proceedings of the American Mathematical Society |
| Year | 1992 |
| Volume | 115 |
| Issue | 1 |
| Pages | 97 - 104 |
| Document type | Article |
| Status | Published |
| Quality controlled | Yes |
| Language | eng |
| Publisher | American Mathematical Society |
| Abstract 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 $w(r)=1-r$). <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, 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 $P_zg$ 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 $E(\max\{1,f\})\leq E(f)$. |
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