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

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 $L_{p,\gamma }\mathrm{.}$ We apply this method to obtain several convolution type inequalities. As an application we consider a difference operator of type ${\mathrm{\Delta }}_v^r:={(\mathbb{I}-{\mathfrak{T}}_v)}^r$ 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 ${\mathrm{\Delta }}_v^{r}f$ for functions $f$ given in $L_{p,\gamma }$, $1\le p<\mathrm{\infty }$, with weights $\gamma$ satisfying the Muckenhoupt’s $A_p$ condition. Also we consider some applications of difference operator ${\mathrm{\Delta }}_v^r$ in these spaces. In particular, we obtain that difference ${\parallel {\mathrm{\Delta }}_v^{r}f\parallel }_{p,\gamma }$ is a useful tool for computing the smoothness properties of functions these spaces. It is obtained that ${\parallel {\mathrm{\Delta }}_v^{r}f\parallel }_{p,\gamma }$ is equivalent to Peetre’s K-functional.