A Generalization of Special Atom Spaces with Arbitrary Measure

Abstract

A brief historical account of the development of special atom spaces is presented followed by the introduction of two new function spaces, A ( ; ) and B ( ; ), which are generalizations of previous special atom spaces utilizing arbitrary measures rather than Lebesgue measure of intervals. Known definitions relating to normed vector spaces are extended to apply to the new function spaces of arbitrary measure. The properties of the new function spaces are discussed including the relationship between the spaces as well as the relationship of the spaces with well known function spaces such as Lebesgue spaces, Lp, Lip( ; ) and ( ; ). Major results include H older-type inequalities for both A ( ; ) and B ( ; ). In the case of B ( ; ), the dual of B ( ; ) is determined and a Representation Theorem for the weighted bounded linear functionals of B ( ; ) is presented in detail. However, for A ( ; ) we mention that the dual follows the same idea of the theorem for B ( ; ). That is, that we only need to estimate k AkA( ; ) for a -measurable set A. Indeed we show there is a positive constant M such that k AkA( ; ) M (A). The duality and representation theorems for A ( ; ) follow easily. Interpolation of Operators Theorems are presented on sublinear operators which map B( ; 1 p ) into weak Lp and A( ; 1 p ) into weak Lp spaces. Finally, we present the multiplication operator on A ( ; ) and B ( ; ) for (t) = t, and show under some conditions this operator is bounded on those spaces.