In this paper we prove necessary and sufficient conditions for the validity of the classical chain rule in the Sobolev space W 1,1 loc ( R ;R ) and in the space of functions of bounded variation BVloc ( R ;R ) .

A class of structures is said to have the homomorphism-preservation property just in case every firstorder formula that is preserved by homomorphisms on this class is equivalent to an existential-positive formula. It is known by a result of Rossman that the class of finite structures has this property and by previous work of Atserias et al. that various of its subclasses do. We extend the latte...

Recall that a function u is harmonic (superharmonic, subharmonic) in an open set U ⊂ Rn (n ≥ 1) if u ∈ C2(U) and Δu = 0 (Δu ≤ 0,Δu ≥ 0) on U . Denote by H(U) the space of harmonic functions in U and SH(U) (sH(U)) the subset of C2(U) consisting of superharmonic (subharmonic) functions in U . If A ⊂ Rn is Lebesgue measurable, L1(A) denotes the space of Lebesgue integrable functions on A and |A| d...