A Note on the Regularity of the Two-Dimensional One-Sided Hardy-Littlewood Maximal Function

Abstract

We investigate the regularity properties of the two-dimensional one-sided Hardy-Littlewood maximal operator. We point out that the above operator is bounded and continuous on the Sobolev spaces , (R 2 ) for 0 ≤ ≤ 1 and 1 < < ∞. More importantly, we establish the sharp boundedness and continuity for the discrete two-dimensional one-sided Hardy-Littlewood maximal operator from ℓ 1 (Z 2 ) to BV(Z 2 ). Here BV(Z 2 ) denotes the set of all functions of bounded variation on Z 2 .