Extensions of Meyers-Ziemer results

Abstract

Letp∈(1, +∞) ands ∈ (0, +∞) be two real numbers, and letH ^s _p (ℝⁿ) denote the Sobolev space defined with Bessel potentials. We give a classA of operators, such thatB _s,p -almost all points ℝⁿ are Lebesgue points ofT(f), for allf ∈H ^s _p (ℝⁿ) and allT ∈A (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 everyf ∈H ^s _p (ℝⁿ) and everyT ∈C, T(f) is quasiuniformly continuous in ℝⁿ; this yields an improvement of the Meyers result [10] which asserts that everyf ∈H ^s _p (ℝⁿ) is quasicontinuous. However,T (f) does not belong, in general, toH ^s _p (ℝⁿ) wheneverT ∈C ands≥1+1/p (cf. Bourdaud-Kateb [5] or Korry [7]).