Characterisation of zero trace functions in higher-order spaces of Sobolev type

Let Ω be a bounded open subset of Rn with a mild regularity property, let m∈N and p∈(1,∞), and let Wm,p(Ω) be the usual Sobolev space of order m based on Lp(Ω); the closure in Wm,p(Ω) of the smooth functions with compact support is denoted by W0m,p(Ω). A special case of the results given below is that u∈W0m,p(Ω) if and only if all distributional derivatives of u of order m belong to Lp(Ω) and u/dm∈L1(Ω), where d(x)=dist(x,∂Ω). In fact what is proved is the analogous result when the Sobolev space is based on a member of a class of Banach function spaces that includes both Lp(Ω) and Lp(⋅)(Ω), the Lebesgue space with variable exponent p(⋅) satisfying natural conditions.