Assess graduate students' understanding of Sobolev spaces W^{k,p}(Ω): precise definitions (weak derivatives, norms/seminorms), basic properties (Banach/Hilbert structure, density of C_c^∞, trace and Poincaré inequalities), and Sobolev embedding theorems (continuous and compact embeddings, critical exponents, embeddings into L^q and Hölder spaces). Include short proofs, verification on examples (varying k, p, and space dimension), and statements of sharpness or counterexamples; assume standard domain regularity as specified in each question.