FUNCTIONS THEORY 2026-2027 - SCQ0094119 , Sezione: Course record

Seleziona attività 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 File
Seleziona attività 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 File
Seleziona attività L3 (5 oct): DuBois Reymond lemma. Geodesic distance.

L3 (5 oct): DuBois Reymond lemma. Geodesic distance. File
Seleziona attività 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 File
Seleziona attività 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. File
Seleziona attività 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) File
Seleziona attività 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) File
Seleziona attività 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. File
Seleziona attività 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. File
Seleziona attività 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. File
Seleziona attività 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). File
Seleziona attività 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. File
Seleziona attività Cantor Vitali function (Devil's staircase)

Cantor Vitali function (Devil's staircase) File
Seleziona attività 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. File
Seleziona attività 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. File

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