Generalization of one theorem of F. Riesz to some other spaces

Abstract

It is known from the analysis course that in order a function to serve as an undefined integral of a summable function, it is necessary and sufficient that it be absolutely continuous. Therefore, it is natural to raise the question of the characteristic of a function which is an undefined integral of the function included in Lp,p > 1 . The answer is well known theorem of F.Riesz concerning the conditions of representability of a given function in the form of an integral with variable upper limit on the functions of Lebesgue spaces. In the one - dimensional and multi - dimensional case, many mathematicians have generalized this theorem for Lebesgue and Orlicz spaces. In this work we will prove theorem of F.Riesz for other functional spaces. Generalization of the theorem of F.Riesz to the case when subintegral function from the weighted Lebesgue spaces is obtained. Also, we prove a necessary condition for the above representation of a function f ∈ Lpϕ (Lp).

Keywords

• function
• functional spaces
• integral
• theorem of F.Riesz
• weighted Lebesgue space