On the variation of maximal operators of convolution type

Abstract

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 $d=1$ we prove that these maximal operators do not increase the $L^p$-variation of a function for any $p\ge 1$, while in dimensions $d>1$ we obtain the corresponding results for the $L^2$-variation. Similar results are proved for the discrete versions of these operators.

References in corpus (4)

• On the regularity of maximal operators
• On the endpoint regularity of discrete maximal operators
• On a discrete version of Tanaka's theorem for maximal functions
• On regularity of the discrete Hardy-Littlewood maximal function

Cited by in corpus (14)

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• Endpoint Sobolev and BV continuity for maximal operators
• On the endpoint regularity of discrete maximal operators
• Regularity of fractional maximal functions through Fourier multipliers
• Regularity of maximal functions on Hardy-Sobolev spaces
• Gradient bounds for radial maximal functions
• Weak differentiability for fractional maximal functions of general $L^p$ functions on domains
• Regularity of maximal operators: recent progress and some open problems
• Sobolev contractivity of gradient flow maximal functions
• Regularity and continuity of local Multilinear Maximal type operator
• Lattice sums of $I$-Bessel functions, theta functions, linear codes and heat equations
• Regularity and continuity of the multilinear strong maximal operators
• The Variation of the Uncentered Maximal Operator with respect to Cubes
• Variation of the one-dimensional centered maximal operator on simple functions with gaps between pieces