Morrey spaces and Hardy-Littlewood maximal function. (English) Zbl 0717.42023

Chiarenza, Filippo; Frasca, Michele

Rend. Mat. Appl., VII. Ser. 7, No. 3-4, 273-279 (1987).

The authors show that the Hardy-Littlewood maximal operator is bounded on any Morrey spaces \(L^{p,\lambda},\quad 1<p<\infty,\quad 0<\lambda <n.\) A weak type result is shown in the case \(p=1.\) \(L^{p,\lambda}\) is equivalent to the Morrey-Campanato space \({\mathcal L}^{p,\lambda}\). And the special case \(\lambda =n\) (BMO) is already known by DeVore and Sharpley. The authors give another proof of this case. They apply their results to Riesz potentials. They also apply their technique to the boundedness of the singular integral operators (given by J. Peetre). In this case their proof however should be corrected a little. They state that \(M\chi \in A_ 1,\) but this is not true. One must use \(M\chi^{1- \epsilon}\in A_ 1\quad (0<\epsilon <1).\) Then their arguments work. Their results are generalized to more general Morrey-Campanato spaces by E. Nakai (preprint).

Reviewer: K.Yabuta

MSC:

42B25	Maximal functions, Littlewood-Paley theory
42B20	Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B15	Multipliers for harmonic analysis in several variables

Keywords:

maximal function; singular integrals; Hardy-Littlewood maximal operator; Morrey spaces; Morrey-Campanato space; BMO; Riesz potentials; singular integral operators