Realization and characterization of modulus of smoothness in weighted Lebesgue spaces

A characterization is obtained for the modulus of smoothness of fractional order in the Lebesgue spaces $L_\omega^p$, $1<p<\infty$, with weights $\omega$ satisfying the Muckenhoupt $A_p$ condition. Also, a realization result and the equivalence between the modulus of smoothness and the Peetre $K$-functional are proved in $L_\omega^p$ for $1<p<\infty$ and $\omega\in A_p$.