An integral operator on $H\sp p$
Författare
Summary, in 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.
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.
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.
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.
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.
Publiceringsår
1995
Språk
Engelska
Sidor
149-158
Publikation/Tidskrift/Serie
Complex Variables, Theory & Application
Volym
28
Issue
2
Dokumenttyp
Artikel i tidskrift
Förlag
New York ; Gordon and Breach, 1982-
Ämne
- Mathematics
Status
Published
ISBN/ISSN/Övrigt
- ISSN: 1563-5066