Sobolev Embedding Theorems for Spaces W k p x 1

This paper gives a Sobolev-type embedding theorem for the generalized Lebesgue–Sobolev space W k p x , where is an open domain in RN N ≥ 2 with cone property, and p x is a Lipschitz continuous function defined on satisfying 1 < p − ≤ p + < N k .The main result can be stated as follows: for any measurable function q x x ∈ with p x ≤ q x ≤ p∗ x = Np x N − kp x there exists a continuous embedding from W k p x to Lq x .