, Volume 133, Issue 1, pp 357-367

Extensions of Meyers-Ziemer results

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Abstract

Letp∈(1, +∞) ands ∈ (0, +∞) be two real numbers, and letH p s (ℝ n ) denote the Sobolev space defined with Bessel potentials. We give a classA of operators, such thatB s,p -almost all points ℝ n are Lebesgue points ofT(f), for allfH p s (ℝ n ) and allTA (B s,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 everyfH p s (ℝ n ) and everyTC, T(f) is quasiuniformly continuous in ℝ n ; this yields an improvement of the Meyers result [10] which asserts that everyfH p s (ℝ n ) is quasicontinuous. However,T (f) does not belong, in general, toH p s (ℝ n ) wheneverTC ands≥1+1/p (cf. Bourdaud-Kateb [5] or Korry [7]).