Properties of Extremal Elements in the Duality Relation for Hardy Spaces

Burchaev, Kh. Kh.  , Ryabykh, G. Yu.
Vladikavkaz Mathematical Journal 2018. Vol. 20. Issue 4.

Abstract:
 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 \( 0<p<1 \). An article by L. Carleson and S. Jacobs (1972), investigated
 the problem of the properties of elements on which the infimum
 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 \(\nu>0\).

Keywords: linear functional, extremal element, approximation method, derivative.

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