Weighted norm inequalities in Lebesgue spaces with Muckenhoupt weights and some applications to operators

TL;DR

This paper introduces a new method for establishing weighted norm inequalities in Lebesgue spaces with Muckenhoupt weights, avoiding traditional extrapolation or interpolation techniques, and applies it to analyze difference operators and their properties.

Contribution

The paper presents a simple, alternative approach to weighted norm inequalities in Lebesgue spaces with Muckenhoupt weights, and demonstrates its application to difference operators and smoothness analysis.

Findings

Established weighted norm inequalities using the new method.

Analyzed properties of difference operators in weighted Lebesgue spaces.

Showed equivalence of difference norms to Peetre's K-functional.

Abstract

In the present work we give a simple method to obtain weighted norm inequalities in Lebesgue spaces $L_{p,\gamma }$ with Muckenhoupt weights $\gamma $. This method is different from celebrated Extrapolation or Interpolation Theory. In this method starting point is uniform norm estimates of special form. Then a procedure give desired weighted norm inequalities in $L_{p,\gamma }.$ We apply this method to obtain several convolution type inequalities. As an application we consider a difference operator of type $\Delta _{v}^{r}:=\left( \mathbb{I}-\mathfrak{T}_{v}\right) ^{r}$ where $\mathbb{I}$ is the identity operator, $r\in \mathbb{N}$ and \begin{equation*} \mathfrak{T}_{v}f\left( x\right) :=\frac{1}{v}\int\nolimits_{x}^{x+v}f\left(t\right) dt,\quad x\in \left[ -\pi ,\pi \right] ,\quad v>0,\quad \mathfrak{T}_{0}:=\mathbb{I}. \end{equation*} We obtain main properties of $\Delta _{v}^{r}f$ for functions $f$ given in $L_{p,\gamma }$, $1\leq p<\infty $, with weights $\gamma $ satisfying the Muckenhoupt's $A_{p}$ condition. Also we consider some applications of difference operator $\Delta _{v}^{r}$ in these spaces. In particular, we obtain that difference $\left\Vert \Delta _{v}^{r}f\right\Vert _{p,\gamma }$ is a useful tool for computing the smoothness properties of functions these spaces. It is obtained that $\left\Vert \Delta _{v}^{r}f\right\Vert_{p,\gamma }$ is equivalent to Peetre's K-functional.