Structure of a generalized class of weights satisfy weighted reverse Hölder’s inequality

Abstract

Abstract In this paper, we will prove some fundamental properties of the power mean operator M p g ( t ) = ( 1 ϒ ( t ) ∫ 0 t λ ( s ) g p ( s ) d s ) 1 / p , for t ∈ I ⊆ R + , $$ \mathcal{M}_{p}g(t)= \biggl( \frac{1}{\Upsilon(t)} \int _{0}^{t} \lambda (s)g^{p} ( s ) \,ds \biggr) ^{1/p},\quad\text{for }t\in \mathbb{I}\subseteq \mathbb{R}_{+}, $$ of order p and establish some lower and upper bounds of the compositions of operators of different powers, where g, λ are a nonnegative real valued functions defined on I $\mathbb{I}$ and ϒ ( t ) = ∫ 0 t λ ( s ) d s $\Upsilon(t)=\int _{0}^{t}\lambda ( s ) \,ds$ . Next, we will study the structure of the generalized class U p q ( B ) $\mathcal{U}_{p}^{q}(B)$ of weights that satisfy the reverse Hölder inequality M q u ≤ B M p u , $$ \mathcal{M}_{q}u\leq B\mathcal{M}_{p}u, $$ for some p 1 $B>1$ is a constant. For applications, we will prove some self-improving properties of weights in the class U p q ( B ) $\mathcal{U}_{p}^{q}(B)$ and derive the self improving properties of the weighted Muckenhoupt and Gehring classes.

Keywords

• Power mean operators
• Discrete Muchenhoupt’s class
• Discrete Gehring’s class
• Reverse Hölder’s inequality
• Self-improving properties