Properties of extremal elements in the duality relation for Hardy spaces

Kh. Kh. Burchaev ; G. Yu. Ryabykh

Vladikavkazskij matematičeskij žurnal, Tome 20 (2018) no. 4, pp. 5-19

Résumé

Consider a Hardy space $H_p$ in the unit disk $D$, $p\geq1$. Let $l_\omega$ be a linear functional on $H_p$ determined by $\omega\in L_q$ $(T=\partial D,\ 1/p + 1/q=1)$ and let $F$ be an extremal function for $l_\omega$. Let $X\in H_q$ implements the best approximation of $\bar\omega$ in $L_q (T)$ by functions from $H_q^0 =\{y\in H_q: y(0)=0\}$. The functions $F$ and $X$ are called extremal elements (e. e.) for $l_\omega $. E. e. are related by the corresponding duality relation. We consider the problem of how certain properties of $ \omega $ will affect e. e. A similar problem is investigated in the case of $ 0$. An article by L. Carleson and S. Jacobs (1972), investigated the problem of the properties of elements on which the infimum $\inf\{\|\bar\omega-x\|_{L_\infty (T)}:\ x \in H_\infty ^0\}$ for a given $\omega\in L_q (T)$ is attained. The hypothesis of the authors that the relationship between extremal elements is similar to that of the function $\omega$ and its projection onto $H_q$ is partially confirmed in a paper by V. G. Ryabykh (2006). Some properties of e. e. for $l_\omega $, when $\omega$ is a polynomial, were studied in a paper by Kh. Kh Burchaev, G. Yu. Ryabykh V. G. Ryabykh (2017). In this paper, relying on the main result of the last article and using the method of successive approximations, the following is proved: if $\omega \in L_ {q^*}(T)$ and $q \le q^*\infty$, then $F\in H_{(p-1) q^*}$ and $X\in H_{q^*}$; if the derivative $\omega^{(n-1)}\in{\rm Lip}(\alpha,T)$ with $0\alpha 1$, then $F = Bf$, where $B$ is the Blaschke product, $f$ is an external function, with $(|f(t)|^p)^{(n-1)} \in {\rm Lip}(\alpha, T)$. If the function $\omega$ is analytic outside the unit circle, then e. e is analytic in the same circle. The listed results clarify and complement similar results obtained in an above mentioned paper by V. G. Ryabykh. It is also proved that the extremal function for $l_\omega\in (H_q)^* $ exists and has the same smoothness as the generator function $\omega$, whenever $1/(n + 1)\delta 1/n$, $\omega\in H_\infty \bigcap {\rm Lip}(\beta, T) $, $\beta=1/\delta-n +\nu 1$, and $\nu>0$.