A necessary and sufficient condition for existence of extremal functions of a linear functional on $H_1$

V. G. Ryabykh

Rostov State University

Abstract: We consider the following well-known old problem: 1) find a necessary and sufficient condition for existence of the functions on the unit sphere of the Hardy space ($p=1$) at which the norm of a linear functional is attained; 2) obtain parametric description of the set of these functions; 3) find conditions for uniqueness of the extremal functions. We prove that the answers to these questions follow from existence and uniqueness of a solution to a homogeneous linear integral equation whose kernel is explicit in terms of the function determining the analytical representation of the indicated linear functional. Its extremal functions can be obtained from solutions to this integral equation.

Keywords: Hardy space, extremal problem, extremal function, exposed point.

UDC: 517.53/.57

Received: 13.03.2006
Revised: 20.02.2007

 English version: Siberian Mathematical Journal, 2007, 48:6, 1085–1092

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