On Hardy–Knopp Type Inequalities with Kernels via Time Scale Calculus

Abstract

In this paper, we study the inequalities of Hardy–Knopp type with kernel functions which have two nonnegative different weighted functions in two different spaces, in a general domain called a time scale calculus. A time scale calculus T is considered as a unification of the continuous calculus and the discrete calculus. We will prove these inequalities in a time scale calculus to avoid proving them twice once in the continuous case and the second in the discrete case. Also, as special cases of the time scale calculus, we can prove some new inequalities in new different domains. Our results will be proved by using the definition of a general Hardy operator on time scale. These inequalities (when T=ℕ) are essentially new.