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1997 | 124 | 2 | 107-132
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Distribution and rearrangement estimates of the maximal function and interpolation

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There are given necessary and sufficient conditions on a measure dμ(x)=w(x)dx under which the key estimates for the distribution and rearrangement of the maximal function due to Riesz, Wiener, Herz and Stein are valid. As a consequence, we obtain the equivalence of the Riesz and Wiener inequalities which seems to be new even for the Lebesgue measure. Our main tools are estimates of the distribution of the averaging function f** and a modified version of the Calderón-Zygmund decomposition. Analogous methods allow us to obtain K-functional formulas in terms of the maximal function for couples of weighted $L_p$-spaces.
  • Department of Mathematics, Yaroslavl' Pedagogical University, Respublikanskaya 108, 150 000 Yaroslavl', Russia
  • Department of Mathematics, Yaroslavl' State University, Sovetskaya 14, 150 00 Yaroslavl', Russia
  • Department of Mathematics, Luleå University, S-971 87 Luleå, Sweden
  • Department of Mathematics, Luleå University, S-971 87 Luleå, Sweden
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