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Mapping properties of the discrete fractional maximal operator in metric measure spaces. (English) Zbl 1280.42012

The authors study the boundedness of the fractional maximal operator in the context of doubling measure spaces. The Euclidean results find their metric counterparts but since the standard (fractional) Hardy-Littlewood maximal function might decrease the regularity of a function in the metric situation, the maximal operator is replaced by a discrete maximal operator in some of the results. The main results show that the fractional maximal operator is bounded on Sobolev, Hölder, Morrey, and Campanato spaces. In particular, the fractional maximal operator is shown to increase the smoothness of a function.

MSC:

42B25 Maximal functions, Littlewood-Paley theory
35J60 Nonlinear elliptic equations
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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