A mean continuity type result for certain Sobolev spaces with variable exponent

For functions f in Sobolev spaces W-1,W-p(x) (Omega) with exponent lower semicontinuous, bounded away from 1 and oc, and with the property of the density of smooth functions, it is shown that for each open set omega subset ofsubset of Omega, for each h is an element of R-N such that omega + th subset of Omega For All t is an element of [0,1], the following inequality holds parallel tof(x + h) - f(x)parallel to(Lmin p(x, x + h) (omega)) less than or equal to (1 + \Omega\)\h\parallel to\delf\parallel to(Lp(x)(Omega)) where min p(x, x + h) denotes the minimum of p along the segment whose endpoints are x, x + h. As a consequence, if p(x) is also continuous, for mollifiers (rho(n))(nis an element ofN) the liminf and the limsup of [GRAPHICS] are respectively minorized and majorized by expressions equivalent to parallel to\delfparallel to\(Lp(x) (Omega)).