Sharp integral estimates for the fractional maximal function and interpolation

Abstract

We give sharp estimates for the fractional maximal function in terms of Hausdorff capacity. At the same time we identify the real interpolation spaces between L₁ and the Morrey space $\mathcal{L}^{1,\lambda}$. The result can be viewed as an analogue of the Hardy–Littlewood maximal theorem for the fractional maximal function.