Program
Mod. B
1. AC and BV functions. Nowhere differentiable functions. Lebesgue
theorem on differentiability a.e. of monotone functions. Functions with
bounded variation (BV). Absolutely continuous functions (AC),
fundamental theorem of calculus in Lebesgue framework.
2. Differentiation of measures. Hanh decomposition theorem,
Radon-Nikodim Theorem. Hardy-Littlewood maximal function. Symmetric
derivative of a measure, theorem on Lebesgue points for L^1 functions.
3. Distributions. Test functions. Distributions of finer and
infinite order. Derivatives in the sense of distributions. The "Théoreme
de structure". Distributon with compact support. Convolution of
distributions. Fourier transform of L^1 functions. Schwartz space,
Temperate distributions. Fourier transform of temperate distributions.
Plancherel theorem. Fourier-Laplace transform of a distribution with
compact support. Paley-Wiener theorem.
4. Sobolev spaces in 1 D. Definition and characterization of Sobolev
Spaces in 1 D. Results of extension and density. Sobolev embeddings,
Rellich theorem. Poincaré inequality. Boundary value problems in 1D.
Maximum principle in 1D.
5. Sobolev spaces in N D. Friedrichs lemma. BV functions in N D.
Extension of Sobolev functions defined in open sets.
Sobolev-Gagliardo-Nirenberg theorem, Morrey theorem. Rellich theorem.
Poincaré inequality in N D. Boundary value problems in N D.