Realization and characterization of modulus of smoothness in weighted Lebesgue spaces

Abstract

We obtain a characterization of modulus of smoothnes of fractional order in the Lebesgue spaces \(L_{\omega}^{p}\), \(1 < p < \infty\), with weights \(\omega\) satisfying the Muckenhoupt’s \(A_{p}\) condition. Also, a realization result and equivalence between modulus of smoothness and the Peetre \(K\)-functional are proved in \(L_{\omega}^{p}\) for \(1 < p < \infty\) and \(\omega \in A_{p}\).