Self-Improving Properties of Continuous and Discrete Muckenhoupt Weights: A Unified Approach

Abstract

In this paper, we develop a new technique on a time scale T to prove that the self-improving properties of the Muckenhoupt weights hold. The results contain the properties of the weights when T=R and when T=N, and also can be extended to cover different spaces such as T=hN, T=qN, etc. The results will be proved by employing some new refinements of Hardy’s type dynamic inequalities with negative powers proven and designed for this purpose. The results give the exact value of the limit exponent as well as the new constants of the new classes.