# A second alternative approach for the study of the Muckenhoupt class $A_1(\mathbb{R})$

## TL;DR

This paper determines the precise range of p-values for which functions in the Muckenhoupt class A_1 on the real line are also in L^p, offering new proofs of existing results.

## Contribution

It provides an exact characterization of the p-range for A_1 functions to be in L^p, with alternative proofs of prior theorems.

## Key findings

- Identified the exact p-range for A_1 functions to belong to L^p.
- Provided alternative proofs for known results.
- Clarified the relationship between A_1 constants and L^p integrability.

## Abstract

We find the exact best possible range of those $p > 1$ for which any function which belongs to $A_1(\mathbb{R})$, with $A_1$-constant equal to $c$, must also belong to $L^p$. In this way we provide alternative proofs of the results in [2] and [10].