Extensions of Meyers-Ziemer results
Résumé
Let p is an element of (1, +infinity) and s is an element of (0, +infinity) be two real numbers, and let H-p(s) (R-n) denote the Sobolev space defined with Bessel potentials. We give a class A of operators, such that B-s,B-p-almost all points of R-n are Lebesgue points of T(f), for all f is an element of H-p(s)(R-n) and all T is an element of A (B-s,B-p denotes the Bessel capacity); this extends the result of Bagby and Ziemer (cf. [2], [15]) and Bojarski-Hajlasz [4], valid whenever T is the identity operator. Furthermore, we describe an interesting special subclass C of A (C contains the Hardy-Littlewood maximal operator, Littlewood-Paley square functions and the absolute value operator T: f --> \f\) such that, for every f is an element of H-p(s)(R-n) and every T is an element of C, T(f) is quasiuniformly continuous in R-n; this yields an improvement of the Meyers result [10] which asserts that every f is an element of H-p(s)(R-n) is quasicontinuous. However, T(f) does not belong, in general, to H-p(s)(R-n) whenever T is an element of C and s greater than or equal to 1 + 1/p (cf. Bourdaud-Kateb [5] or Korry [7)).