Trace operators on bounded subanalytic manifolds

We prove that if M⊂ Rⁿ M ⊂ R n is a bounded subanalytic submanifold of Rⁿ R n such that B(x₀,ε )∩ M B ( x 0 , ε ) ∩ M is connected for every x₀∈ M̄ x 0 ∈ M ¯ and ε >0 ε > 0 small, then, for p∈ [1,∞ ) p ∈ [ 1 , ∞ ) sufficiently large, the space C^∞ ( M̄) C ∞ ( M ¯ ) is dense in the Sobolev space W1,p(M) W 1 , p ( M ) . We also show that for p large, if A⊂ M̄ M A ⊂ M ¯ \ M is subanalytic then the restriction mapping C^∞ ( M̄) u↦ u|A∈ Lᵖ(A) C ∞ ( M ¯ ) ∋ u ↦ u | A ∈ L p ( A ) is continuous (if A is endowed with the Hausdorff measure), which makes it possible to define a trace operator, and then prove that compactly supported functions are dense in the kernel of this operator. We finally generalize these results to the case where our assumption of connectedness at singular points of M̄ M ¯ is dropped.