Estimates of the singular numbers of the Carleson imbedding operator

O. G. Parfenov

Sbornik. Mathematics, Tome 59 (1988) no. 2, pp. 497-514

Résumé

Let $H^2$ be the Hardy class in the unit disc $D$ and $\mu$ a finite Borel measure in $D$. Carleson's theorem describes conditions on $\mu$ under which the corresponding imbedding operator $J\colon H^2\to L_2(\mu)$ (the Carleson operator) is bounded. From this theorem follows a criterion for compactness of $J$ in terms of $\mu$. This paper is devoted to further study of the Carleson operator. Almost sharp upper bounds on the singular numbers of $J$ are presented in terms of the intensity of $\mu$. For measures whose support is a set of nonzero linear measure adjacent to the unit circle (and when certain other conditions), an asymptotic formula is obtained. A study is begun of measures whose support has just one point on the unit circle. A solution of a problem from the theory of rational approximation, posed by A. A. Gonchar, is also presented. Bibliography: 17 titles.