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

Résumé

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

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