Israel Journal of Mathematics

, Volume 133, Issue 1, pp 357–367

Extensions of Meyers-Ziemer results

Article

DOI: 10.1007/BF02773074

Cite this article as:
Korry, S. Isr. J. Math. (2003) 133: 357. doi:10.1007/BF02773074
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Abstract

Letp∈(1, +∞) ands ∈ (0, +∞) be two real numbers, and letHps(ℝn) denote the Sobolev space defined with Bessel potentials. We give a classA of operators, such thatBs,p-almost all points ℝn are Lebesgue points ofT(f), for allfHps(ℝn) and allTA (Bs,p denotes the Bessel capacity); this extends the result of Bagby and Ziemer (cf. [2], [15]) and Bojarski-Hajlasz [4], valid wheneverT is the identity operator. Furthermore, we describe an interesting special subclassC ofA (C contains the Hardy-Littlewood maximal operator, Littlewood-Paley square functions and the absolute value operatorT: f→|f|) such that, for everyfHps(ℝn) and everyTC, T(f) is quasiuniformly continuous in ℝn; this yields an improvement of the Meyers result [10] which asserts that everyfHps(ℝn) is quasicontinuous. However,T (f) does not belong, in general, toHps(ℝn) wheneverTC ands≥1+1/p (cf. Bourdaud-Kateb [5] or Korry [7]).

Copyright information

© Hebrew University 2003

Authors and Affiliations

  1. 1.Equipe d’Analyse et de Mathématiques Appliquées, bâtiment CopernicUniversité de Marne-La-ValléeMarne-La-Vallée Cedex 2France

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