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DOI: 10.23671/VNC.2018.4.23383

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
\(\inf\{\|\bar\omega-x\|_{L_\infty (T)}:\ x \in H_\infty ^0\}\) 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 \(\nu>0\).
Keywords: linear functional, extremal element, approximation method, derivative.
Language: Russian Download the full text  
For citation: Burchaev, Kh. Kh. and Ryabykh, G. Yu. Properties of Extremal Elements in the Duality Relation for Hardy Spaces, Vladikavkaz Math. J., 2018, vol. 20, no. 3, pp. 5-19 (in Russian). DOI 10.23671/VNC.2018.4.23383
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