A MEAN CONTINUITY TYPE RESULT FOR CERTAIN SOBOLEV SPACES WITH VARIABLE EXPONENT

Alberto Fiorenza

Dipartimento di Costruzioni e Metodi Matematici in Architettura

Universitá di Napoli via Monteoliveto 3, 80134 Napoli, Italia

                       fiorenza@unina.it

July 16, 2001

Abstract

For functions f in Sobolev spaces W^{1,p(x)}(Ω) with exponent lower semicontinuous, bounded away from 1 and ∞ and with the property of the density of smooth functions, it is shown that for each open set ω ⊂⊂ Ω, for each h ∈ R^{N} such that ω + th ⊂ Ω ∀ t ∈ [0, 1], the following inequality holds

∥f(x + h) − f(x)∥_{L^{minp(x,x+h)}(ω)} ≤ (1 + |Ω|)|h|∥|∇f|∥_{L^{p(x)}(Ω)}

where minp(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 (ρ_{n})_{n∈N} the liminf and the limsup of

(∥\frac{|f(x) − f(y)|}{|x − y|}[ρ_{n}(x − y)]^{\frac{1}{minp(x,y)}} ∥_{L^{minp(x,y)}(Ω×Ω)})_{n∈N}

are respectively minorized and majorized by expressions equivalent to ∥|∇f|∥_{L^{p(x)}(Ω)}.

AMS Class: 46E35 Keywords: Generalized Sobolev spaces, variable exponents, p-mean continuity, den-

sity of smooth functions, equivalent norms.

1 Notation

Throughout the paper Ω will be a non-empty, open cube in R^{N}, N ∈ N and P(Ω) will be the class of all lower semicontinuous functions p = p(x) on Ω such that

p_{∗} = inf_{x∈Ω} p(x) > 1

p^{∗} = sup_{x∈Ω} p(x) < ∞

and satifying the following condition:

(C) For every x ∈ Ω there exist numbers 0 < r(x) ≤ 1, ζ(x) > 0 and a vector ξ(x) ∈ R^{N}\ {0} such that

ζ(x) < |ξ(x)| ≤ 1