Boundedness of Operators on Certain Weighted Morrey Spaces Beyond the Muckenhoupt Range

We prove that for operators satistying weighted inequalities withA(p)weights the boundedness on a certain class of Morrey spaces holds with weights of the form |x|(alpha)w(x) forw is an element of A(p). In the case of power weights the shift with respect to the range of Muckenhoupt weights was observed by N. Samko for the Hilbert transform, by H. Tanaka for the Hardy-Littlewood maximal operator, and by S. Nakamura and Y. Sawano for Calderon-Zygmund operators and others. We extend the class of weights and establish the results in a very general setting, with applications to many operators. For weak type Morrey spaces, we obtain new estimates even for the Hardy-Littlewood maximal operator. Moreover, we prove the necessity of certainA(q)condition.