Abstract
In this paper we investigate the regularity properties of one-sided fractional maximal functions, both in continuous case and in discrete case. We prove that the one-sided fractional maximal operators
$ \mathcal{M}_{\beta}^{+} $ and
$ \mathcal{M}_{\beta}^{-} $ map
$ W^{1,p}(\mathbb{R}) $ into
$ W^{1,q}(\mathbb{R}) $ with 1 <p <∞, 0≤β<1/p and q=p/(1-pβ), boundedly and continuously. In addition, we also obtain the sharp bounds and continuity for the discrete one-sided fractional maximal operators
$ M_{\beta}^{+} $ and
$ M_{\beta}^{-} $ from
$ \ell^{1}(\mathbb{Z}) $ to
$ {\rm BV}(\mathbb{Z}) $. Here
$ {\rm BV}(\mathbb{Z}) $ denotes the set of all functions of bounded variation defined on ℤ. The results we obtained represent significant and natural extensions of what was known previously.