On the variation of maximal operators of convolution type

In this paper we study the regularity properties of two maximal operators of convolution type: the heat flow maximal operator (associated to the Gauss kernel) and the Poisson maximal operator (associated to the Poisson kernel). In dimension d=1d=1 we prove that these maximal operators do not increase the LpLp-variation of a function for any p⩾1p⩾1, while in dimensions d>1d>1 we obtain the corresponding results for the L2L2-variation. Similar results are proved for the discrete versions of these operators.