Boundedness properties in a family of weighted Morrey spaces with emphasis on power weights

Abstract: We define a scale of weighted Morrey spaces which contains different weighted versions appearing in the literature. This allows us to obtain weighted estimates for operators in a unified way. In general, we obtain results for weights of the form $|x|^\alpha w(x)$ with $w\in A_p$ and nonnegative $\alpha$. We study particularly some properties of power-weighted spaces and in the case of the Hardy-Littlewood maximal operator our results for such spaces are sharp. By using extrapolation techniques the results are given in abstract form in such a way that they are automatically obtained for many operators.