| Title | An integral operator on $H\sp p$ |
| Authors | Alexandru Aleman, Aristomenis G Siskakis |
| Alternative Location | http://dx.doi.org/10.1080/1..., Restricted Access |
| Publication | Complex Variables, Theory and Application: An International Journal |
| Year | 1995 |
| Volume | 28 |
| Issue | 2 |
| Pages | 149 - 158 |
| Document type | Article |
| Status | Published |
| Quality controlled | Yes |
| Language | eng |
| Publisher | New York ; Gordon and Breach, 1982- |
| Abstract English | To every analytic function $g$ in the unit disk $\bold D$ one associates the integral operator $T_g(f)(z)\coloneq z^{-1}\int^z_0f(\xi)g'(\xi)d\xi$ on spaces of analytic functions in $\bold D$. Operators of this form appear naturally in complex analysis. For instance, the choice $g(z)\equiv z$ leads to the integration operator, and the choice $g(z)=\log(1/(1-z))$ leads to the Cesàro operator. <br> <br> The notation used in the paper is standard. For $0<p\leq\infty,\ H^p$ denotes the Hardy space, $B_p$ denotes the analytic Besov-$p$ space, and $S_p(H^2)$ denotes the Schatten-$p$ class of operators on $H^2$. BMOA and VMOA denote the spaces of analytic functions of bounded respectively, vanishing mean oscillation. <br> <br> The main results of the paper are as follows. Theorem 1: Let $g$ be an analytic function in $\bold D$ and let $1\leq p<\infty$. Then $T_g$ is bounded on $H^p$ if and only if $g\in{\rm BMOA}$. Theorem 2: Let $g$ be an analytic function in $\bold D$. (i) If $1<p<\infty$ then $T_g\in S_p(H^2)$ if and only if $g\in B_p$. (ii) If $0<p\leq 1$ then $T_g\in S_p(H^2)$ if and only if $g$ is constant. Theorem 1 generalizes an earlier result of Ch. Pommerenke Comment. Math. Helv. 52 (1977), no. 4, 591--602; MR0454017 (56 #12268) in the context of $H^2$. It also implies the following corollary: Let $g$ be an analytic function on $\bold D$ and let $1\leq p<\infty$. Then $T_g$ is compact on $H^p$ if and only if $g\in{\rm VMOA}$. The main tool in the proof of Theorem 2 is Luecking's results D. H. Luecking, J. Funct. Anal. 73 (1987), no. 2, 345--368; MR0899655 (88m:47046) on Cauchy transforms of Borel measures $\mu$ on $\bold D$. $$Q_\mu(f)(w)\coloneq\int_{\bf D}\frac{f(z)}{1-w\overline z}d\mu(z).$$ Let $\{R_j\}^\infty_{j=1}$ be disjoint ``Carleson cubes'' which cover $\bold D$, and let $l(R_j)$ be the distance from the center of $R_j$ to $\partial\bold D$. Then Luecking's theorem op. cit. says that $Q_µ\in S_p(H^2)$ if and only if $\sum^\infty_{j=1}(\mu(R_j)/l(R_j))^p<\infty$. The connection to Luecking's theorem is via the relation $T^\ast_gT_g=Q_\mu$ with $d\mu(z)=2|g'(z)|^2\log(1/|z|)dm(z)$, where $m$ is Lebesgue measure. |
| ISBN/ISSN/Other | ISSN: 0278-1077 |
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