This paper is a continuation of [4]; its aim is to extend the results of that paper to include abstract Sobolev spaces of higher order and even anisotropic spaces. Let Ω be a domain in ℝN, let X = X(Ω) and Y = Y(Ω) be Banach function spaces in the sense of Luxemburg (see [4] for details), and let W(X, Y) be the abstract Sobolev space consisting of all f ∈ X such that for each i ∈ {l, …, N} the distributional derivative
belongs to Y; equipped with the norm
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0305004100071991/resource/name/S0305004100071991_eqnU1.gif?pub-status=live)
W(X, Y) is a Banach space. Given any weight function w on Ω, the triple [w, X, Y] is said to support the Poincaré inequality on Ω. if there is a positive constant K such that for all u ∈ W(X, Y)
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0305004100071991/resource/name/S0305004100071991_eqn1.gif?pub-status=live)
the pair [X, Y] is said to support the Friedrichs inequality if there is a positive constant K such that for all u ∈ W0(X, Y) (the closure of
in W(X, Y))
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0305004100071991/resource/name/S0305004100071991_eqn2.gif?pub-status=live)