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

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[ -π,π\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 $L_{p,\gamma }$, $1\le p<\infty$, 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 ${\Vert {\Delta }_v^{r}f\Vert }_{p,\gamma }$ is a useful tool for computing the smoothness properties of functions these spaces. It is obtained that ${\Vert {\Delta }_v^{r}f\Vert }_{p,\gamma }$ is equivalent to Peetre's K-functional.

arXiv comment23 pages, Submitted