The concept of t-basis and vector-valued Hardy classes

Abstract

This paper introduces the concept of a t-basis generated by some bilinear mapping t (; ). It is considered the vector-valued class L-p (X) =: L-p (J; X), 1 <= p < +infinity, where J = [-pi, pi] and X is a Banach space with the UMD property, and it is proven that the classical system of exponents {(eint)}(n is an element of Z) forms a t-basis for L-p (X ), 1 < p < +infinity. Using this fact, the Hardy vector classes nH(p)(+/-) (X ), 1 < p < +infinity, different from the classical ones, are defined, and an equivalent definition of these classes is given and some of their properties are studied. In addition, the concept of t-Riesz property of a system of exponentials is introduced in L-p (X ), 1 < p < +infinity, and it is proved that this system has the t-Riesz property. A new method is given for establishing the Plemelj-Sokhotski formulas for X-valued Cauchy type integrals when X has the UMD property. An abstract analogue of the 1/4-Kadets theorem is obtained for L-2 (H), where H is a Hilbert space.