Geodesic
Maximal functions measuring smooytness: counterexamples
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 42, Tome 397 (2011), pp. 53-72

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We consider two main maximal operators measuring smoothness. For all possible values of the parameters, we give simple examples of bounded functions with compact support that clearly show quite clearly the difference between these maximal operators. 
@article{ZNSL_2011_397_a2,
     author = {E. E. Lokharu},
     title = {Maximal functions measuring smooytness: counterexamples},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {53--72},
     publisher = {mathdoc},
     volume = {397},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2011_397_a2/}
}
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E. E. Lokharu. Maximal functions measuring smooytness: counterexamples. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 42, Tome 397 (2011), pp. 53-72. http://geodesic.mathdoc.fr/item/ZNSL_2011_397_a2/