On continuity of generalized maximal functions

Abstract We generalize the centered and uncentered Hardy-Littlewood maximal function in terms of shape of integration sets. Continuity of two global and two local generalized maximal functions in $$\textbf{R}^{n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>R</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:math> is studied. The global ones are continuous. One of the local ones is continuous, provided the integration set is star-shaped with respect to its center. For the second local one, we construct a counterexample.