A refinement of a Hardy type inequality for negative exponents, and sharp applications to Muckenhoupt weights on R

Abstract

We prove a sharp integral inequality that generalizes the well known Hardy type integral inequality for negative exponents. We also give sharp applications in two directions for Muckenhoupt weights on R. This work refines the results that appear in [9].

In this article we refine the Hardy inequality that appears in [9] by inserting the $L^1$ norm of the related function as a variable. By using this we refine the results for the Muckenhoupt weights on R even further