Sobolev extension by linear operators

Let L^m,pℝⁿ) be the Sobolev space of functions with m^th derivatives lying in L^p(ℝⁿ). Assume that n< p < ∞. For E ⊂ ℝⁿ, let L^m,p(E) denote the space of restrictions to E of functions in L^m,pℝⁿ). We show that there exists a bounded linear map T: L^m,p(E) → L^m,pℝⁿ) such that, for any f ∈ L^m,p(E), we have Tf = f on E. We also give a formula for the order of magnitude of {norm of matrix}f{norm of matrix}L^m,p(E) for a given f: E → ℝ when E is finite.