The distribution function. Lorentz spaces. Elementary interpolation. Hardy's inequality. Coverings. The Vitali covering lemma(simplified version). Whitney coverings. The maximal function. Maximal inequalities. Lebesgue's differentiation theorem. Pointwise inequalities for convolutions. The Calderón-Zygmund decomposition. Hardy and bounded mean oscillations spaces. Substitutes of L1. The space LlogL. The Hardy space H1. More maximal functions. Transition from one admissible function to another. Proof of Theorem 12. Atomic decomposition. Atoms. Atomic decomposition. The substitute of L∞: BMO. Definition of BMO. H1 and BMO. BMO functions are almost bounded. Lp regularity for the Laplace operator. Preliminaries. Proof of Theorem17. An equation involving the Jacobian. Functions in Sobolev spaces. Improved Sobolev embeddings. An equivalent norm in Lorentz spaces. Properties of rearrangements. Rearrangement and convolutions. Improved Sobolev embeddings. The limiting case p=1. The limiting case p=N. Traces. Definition of the trace. Trace of W1,p,1
