Section : Course record | FUNCTIONS THEORY 2026-2027 - SCQ0094119

Sélectionner l’activité L1 (28 sept.) Course presentation. Spaces of continuous functions and Ascoli Arzel\`a compactness thm

L1 (28 sept.) Course presentation. Spaces of continuous functions and Ascoli Arzel\`a compactness thm Fichier
Sélectionner l’activité L2 (30 sept): continuous and compact embeddings. The space of test functions. Test functions are dense in L^p, p

L2 (30 sept): continuous and compact embeddings. The space of test functions. Test functions are dense in L^p, p Fichier
Sélectionner l’activité L3 (5 oct): DuBois Reymond lemma. Geodesic distance.

L3 (5 oct): DuBois Reymond lemma. Geodesic distance. Fichier
Sélectionner l’activité L4 (7 oct): sets of class C^k. The outer normal. Partition of unit. Divergence thm

L4 (7 oct): sets of class C^k. The outer normal. Partition of unit. Divergence thm Fichier
Sélectionner l’activité L5 (8 oct): integration on spheres formula. Computation of the volume of the ball and area of the sphere. Introduction to measure theory.

L5 (8 oct): integration on spheres formula. Computation of the volume of the ball and area of the sphere. Introduction to measure theory. Fichier
Sélectionner l’activité L6 (12 oct): Absolutely continuous measures and singular measures. Lebesgue decomposition of measures. Measure theoretic boundary. Recap Riesz thm and weak compactness thms (Kakutani, Banach Alaoglu)

L6 (12 oct): Absolutely continuous measures and singular measures. Lebesgue decomposition of measures. Measure theoretic boundary. Recap Riesz thm and weak compactness thms (Kakutani, Banach Alaoglu) Fichier
Sélectionner l’activité L7 (14 oct): Riesz theorems for Radon measure and weak star compactness. Application to L^1. (no proofs)

L7 (14 oct): Riesz theorems for Radon measure and weak star compactness. Application to L^1. (no proofs) Fichier
Sélectionner l’activité L8 (15 oct): Hausdorff measures basic definition and properties. H^1=L^1 in R. Hausdorff dimension of a set. Example of the Cantor (ternary) set in R.

L8 (15 oct): Hausdorff measures basic definition and properties. H^1=L^1 in R. Hausdorff dimension of a set. Example of the Cantor (ternary) set in R. Fichier
Sélectionner l’activité L9 (19 oct): Steiner symmetrization of a set. Isodiametric inequality in R^n. Proof that H^n=H^n_d=Lebesgue in R^n.

L9 (19 oct): Steiner symmetrization of a set. Isodiametric inequality in R^n. Proof that H^n=H^n_d=Lebesgue in R^n. Fichier
Sélectionner l’activité L10 (21 oct): area formula and rectifiable sets. Definition of distributions and order of distributions.

L10 (21 oct): area formula and rectifiable sets. Definition of distributions and order of distributions. Fichier
Sélectionner l’activité L11 (22 oct): support of a distribution. Principal value distribution. Convergence in the sense of distributions. Convolution between distributions and test functions. Density of C^\infty_c(U) in the space of distributions (no proof).

L11 (22 oct): support of a distribution. Principal value distribution. Convergence in the sense of distributions. Convolution between distributions and test functions. Density of C^\infty_c(U) in the space of distributions (no proof). Fichier
Sélectionner l’activité L12 (2nov): derivative in the sense of distributions. Case of piecewise constant functions. Cantor Vitali function.

L12 (2nov): derivative in the sense of distributions. Case of piecewise constant functions. Cantor Vitali function. Fichier
Sélectionner l’activité Cantor Vitali function (Devil's staircase)

Cantor Vitali function (Devil's staircase) Fichier
Sélectionner l’activité L13 (4 nov): cantorian measures. Definition of weak derivatives. Fundamental solutions. Fundamental solution of the Laplacian.

L13 (4 nov): cantorian measures. Definition of weak derivatives. Fundamental solutions. Fundamental solution of the Laplacian. Fichier
Sélectionner l’activité L14 (5 nov): fundamental solutions. Hypoelliptic differential operators. Weyl lemma: the harmonic distributions are smooth.

L14 (5 nov): fundamental solutions. Hypoelliptic differential operators. Weyl lemma: the harmonic distributions are smooth. Fichier

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