Some Fundamental Tools for PDE, Harmonic Analysis and Function Space Theory
You can find an overview of the topics that will be covered in the course on the course catalogue.
You may find a first draft of the script here.
The proof of each of the statements below is part of the program for the examination.
| Week | Summary |
|---|---|
| Week 1 (17.02 and 20.02) |
Introduction to the course: The main goals from FA2. 1.1. The Fourier transforms of \(L^1\) functions. Theorem 1.1 the Fourier transform is mapping \(L^1\) functions into continuous functions converging to zero at infinity. Theorem 1.4: identification of the inverse Fourier transform. Proposition 1.5: Plancherel theorem. The Fourier Transform defines a linear isometry from \(L^2({\R}^n)\) into itself. Definition 1.6 of the Schwartz Space. The Schwartz Space is stable under multiplication by polynomials and under the action of partial differentiations (Proposition 1.7). The Fourier Transform is an isomorphism from the Scwartz Space into itself (Proposition 1.9). Definition 1.12 of a pseudo norm. Definition 1.13 of a Fréchet Space. Examples of Fréchet Spaces (\(L^p_{loc}({\R}^n)\), \(C^\infty_0(K)\) where \(K\) is compact, \(C^\infty({\R}^n)\), \({\mathcal S}({\R}^n))\). \(C^\infty_0({\R}^n)\) doesn't define a Fréchet space. Characterization of the convergence in a Fréchet space by the mean of the pseudo norms. Each pseudonorm is continuous (proposition 1.15). Characterisation of the continuity for linear maps between Fréchet Spaces (proposition 1.16). Banach Steinhaus Theorem for Fréchet Spaces (theorem 1.17). Definition of a Tempered distribution (definition 1.18). Characterization of a tempered distribution (proposition 1.19). Examples and counterexamples of tempered distributions: \(L^p\) functions, \(t\mapsto e^t\) is not a tempered distribution, \(1/|t|\) doesn't define a tempered distribution, definition of the principal value of \(1/t\) denoted \(pv(1/t)\) and proof that it defines a tempered distribution. |
| Week 2 (24.02 and 27.02) |
The space of slowly growing functions \(G({\R}^n)\) (definition 1.23). The continuity of the multiplication by elements of \(G({\R}^n)\) from \({\mathcal S}({\R}^n)\) into itself (proposition 1.24). Definition of the tempered distribution obtained by multiplying an arbitrary element of \({\mathcal S}'({\R}^n)\) by an element of \(G({\R}^n)\) (proposition 1.25). The weak convergence of sequences of distributions (definition 1.26). The weak convergence in \(L^p\) is an example of weak convergence in \({\mathcal S}'({\R}^n)\). Example of approximations of the Dirac mass. The derivative of a tempered distribution (proposition 1.27). Examples of derivatives: \(H'=\delta_0\), \((\log|x|)'=pv(1/x)\). Definition of the support of a tempered distribution (proposition 1.28). Existence of a partition of unity for any compact set (Lemma 1.29 and Lemma 1.30). The space \({\mathcal E}'({\R}^n)\) of compactly supported distributions. Examples of compactly supported distributions. The duality extension from \({\mathcal E}'({\R}^n)-{\mathcal S}({\R}^n)\) to \({\mathcal E}'({\R}^n)-C^\infty({\R}^n)\) (Proposition 1.32). The action of a distribution \(T\) of order \(p\) on a Schwartz function whose derivatives up to order \(p\) on the support of \(T\) vanish is zero (proposition 1.33). The Fourier transform of a tempered distribution (definition-proposition 1.34). Examples of explicit computations of Fourier transforms: \({\mathcal F}(\delta_a)\), \({\mathcal F}(\partial^\alpha \delta_a)\), \({\mathcal F}(x^\alpha)\), \({\mathcal F}(pv(1/t))\). The explicit form of the Fourier transform of a compactly supported distribution using the duality extension \({\mathcal E}'({\R}^n)-C^\infty({\R}^n)\) (theorem 1.37). Derivation inside the duality \({\mathcal E}'({\R}^n)-{\mathcal S}({\R}^n)\) for families of Schwartz functions depending smoothly on a parameter in \({\R}^n\) (Lemma 1.38). |
| Week 3 (03.03 and 06.03) |
The characterization of tempered distributions supported at a point (Schwartz Lemma: Proposition 1.39). Harmonic tempered distributions are polynomials (Theorem 1.41). The continuity of the convolution with a Schwartz function on \({\mathcal S}({\mathbb R}^n)\) (proposition 1.43). Convolution of a tempered distribution with a Schwartz function. The convolution between a compactly supported distribution and a Schwartz function is again a Schwartz function and the operation is continuous between \({\mathcal S}({\mathbb R}^n)\) and itself. The convolution between a compactly supported tempered distribution and a compactly supported \(C^\infty\) function is again an element of \(C^\infty_0({\mathbb R}^n)\), moreover the support of this convolution is included in the sum of the supports of the element in \({\mathcal E}'({\mathbb R}^n)\) and the initial compactly supported \(C^\infty\) function (proposition 1.44). An expression of the convolution between a tempered distribution and a Schwartz function using convolution in \({\mathcal S}({\mathbb R}^n)\) (proposition 1.45). Every tempered distribution is the limit (in the sense of \({\mathcal S}'({\R}^n)\)) of tempered distributions which have \(C^\infty\) representatives (corollary 1.46). The convolution between a compactly supported tempered distribution and an arbitrary tempered distribution. The commutativity of the convolution operation in \({\mathcal E}'({\mathbb R}^n)\times {\mathcal S}'({\mathbb R}^n)\) . The support of the convolution between two elements in \({\mathcal E}'({\mathbb R}^n)\) is included in the sum of the supports of these two elements in \({\mathcal E}'({\mathbb R}^n)\) (Proposition 1.51). Associativity for convolutions does not hold in general: A counter-example (remark 1.52). Associativity holds between three distributions if two of them are compactly supported (theorem 1.53). Definition of a convolution equation (definition 1.55). The formulation of arbitrary linear PDE with constant coefficients as a convolution equation. The definition of a fundamental solution to a convolution equation (Definition 1.56). The use of a fundamental solution to express a general solution to a convolution equation (theorem 1.57). |
| Week 4 (10.03 and 13.03) |
Computation of the fundamental solution of the Laplace operator (Lemma 1.58). The resolution of \(\Delta u=f\) on \({\mathbb R}^n\) for \(f\in{\mathcal E}'({\mathbb R}^n)\) (Theorem 1.59). The construction of a fundamental solution in \({\mathbb R}\times{\mathbb R}^3\) for \(\Box:=\partial^2_t-\Delta\) (proposition 1.60). The resolution of \(\Box u=f\) for \(f\in{\mathcal E}'({\R}^4)\) and uniqueness of the solution among the ones null in the past (theorem 1.61). The Fourier Transform of a convolution of two tempered distributions (one having compact support) is proportional to the product of the Fourier transforms of the two distributions (Theorem 1.62). The uniqueness of the solution to a convolution equation for a tempered distribution operator supported at one point and with invertible Fourier transform (theorem 1.63). The example of the Bessel operator (Example 1.64). A degenerate case with the Poisson equation (Example 1.65). The resolution, using the Fourier transform, of the homogeneous Cauchy Problem for the Heat Equation \(\partial_tu-\Delta u=0\) in \({\mathbb R}\times{\mathbb R}^n\) with arbitrary initial data \(u(0,\cdot)=f\in{\mathcal S}'({\R}^n)\). Resolution of the wave equation (Lemma 1.69) and Cauchy problem for initial data \(u\) and \(\partial_t u\) in \({\mathcal E}'({\mathbb R}^{n-1})\) for the wave equation (theorem 1.73). |
| Week 5 (17.03 and 20.03) |
Definition of Hilbert Sobolev Spaces of integer order. Characterization of Hilbert Sobolev Spaces of integer order using the Fourier transform (proposition 2.3). The definition of Hilbert Sobolev Spaces of arbitrary order. The scalar product on \(H^s({\mathbb{R}}^n)\) and the completeness of \(H^s({\mathbb{R}}^n)\) for this scalar product: \(H^s({\mathbb{R}}^n)\) defines an Hilbert Space (Proposition 2.6). The density of smooth compactly supported functions in \(H^s({\mathbb{R}}^n)\) (proposition 2.7). The dual of \(H^s({\mathbb{R}}^n)\) identifies to \(H^{-s}({\mathbb{R}}^n)\) (theorem 2.8). The space \(H^{s}({\mathbb{R}}^n)\) for \(s \gt n/2+k\) where \(k\in{\N}\) embeds continuously in \(C^k({\mathbb{R}}^n)\). The space \(H^{s}({\mathbb{R}}^n)\) for \(s \gt n/2\) defines an Algebra (theorem 2.10). The defintion of H\"older Spaces. The H\"older Spaces are Banach spaces (Exercise 2.12). For a bounded open set \(\Omega\subset{\mathbb R}^n\) the embedding of the space \(C^{k,\alpha}(\overline{\Omega})\) into \(C^{k,\beta}(\overline{\Omega})\) for \(0 \lt \beta \lt \alpha\le 1\) is compact (exercise 2.13). If \(s=n/2+k+\alpha\) where \(k\in{\mathbb N}\) and \(\alpha\in(0,1)\), the space \(H^s({\mathbb{R}}^n)\) embeds continuously into \(C^{k,\alpha}({\mathbb{R}}^n)\) (Theorem 2.14). There exists an element in \(H^{n/2}({\mathbb{R}}^n)\) which is not in \(L^\infty({\mathbb{R}}^n)\) (theorem 2.15). |
| Week 6 (24.03 and 27.03) |
Cauchy problems in Hilbert Sobolev Spaces for elliptic PDE. The Symbol of an elliptic operator (Definition 2.18). Cauchy Problem in \(H^s({\mathbb R}^n)\) for the Bessel operator \(-\Delta+I\) (Theorem I.19). The definition of \(H^s({\mathbb R}_+^n)\). The continuity of the trace operator from \(H^s({\mathbb R}^n)\) into \(H^{s-1/2}(\partial{\mathbb R}_+^n)\) for \(s \gt 1/2\) (theorem 2.20). The continuity of the trace operator from \(H^s({\mathbb R}_+^n)\) into \(H^{s-1/2}(\partial{\mathbb R}_+^n)\) for \(s \gt 1/2\) (proposition 2.23). The map \(L\) from \(H^{s+2}({\mathbb R}_+^n)\) into \(H^{s+3/2}(\partial{\mathbb R}_+^n)\times H^s({\mathbb R}_+^n)\) which to \(u\) assigns \((\operatorname{Trace}(u), -\Delta u+ u)\) is a continuous isomorphism. |
| Week 7 (31.03 and 03.04) |
Hölder inequality (Theorem 3.1). Littlewood inequality (Corollary 3.2). Minkowski inequality and the completeness of \(L^p(\Omega)\) (Theorem 3.3). Young inequality for the convolution of a \(L^1\) function with an \(L^p\) function (Theorem 3.4). General Young inequality for the convolution of a \(L^p\) function with an \(L^q\) function for \(p^{-1}+q^{-1}-1\ge 0\) (Theorem 3.5). Definition of a uniformly convex space (Definition 3.6). The Hilbert spaces are uniformly convex. Definition of a reflexive space (definition 3.7). A Banach space is reflexive if and only if its dual is reflexive (Proposition 3.8). Every uniformly convex space is reflexive (theorem 3.10). The first Clarkson inequality for \(2 \le p \lt +\infty\) and the uniform convexity and reflexivity of \(L^p(\Omega)\) for \(2 \le p \lt +\infty\) (lemma 3.11). \(L^p(\Omega)\) is reflexive for \(1 \lt p \lt +\infty\) (theorem 3.12). For \(1 \lt p \lt +\infty\) the dual of \(L^p(\Omega)\) is \(L^{p'}(\Omega)\) (theorem 3.13). The dual of \(L^1(\Omega)\) is \(L^\infty(\Omega)\) (theorem 3.15). The space \(L^1(\Omega)\) is not reflexive. Definition of a separable metric space. The spaces \(L^p(\Omega)\) for \(1\le p \lt +\infty\) are separable (theorem 3.18). The space \(L^\infty(\Omega)\) is not separable. For \(1 \le p \lt +\infty\) the space of smooth compactly supported functions is dense in \(L^p({\mathbb R}^n)\). |
| Week 8 (07.04 and 10.04) |
The Riesz--Thorin interpolation theorem (Theorem 3.23). The continuous embedding of \(L^p(\Omega)\) into \(L^{p_0}(\Omega)+L^{p_1}(\Omega)\) for \(1\le p_0 \lt p \lt p_1\le+\infty\) (remark 3.24). The Hausdorff--Young inequality about the continuity of the Fourier transform from \(L^p({\mathbb R}^n)\) into \(L^{p'}({\mathbb R}^n)\) for \(1\le p\le 2\) (theorem 3.26). The proof of the generalized Young inequality (theorem 3.5) using Riesz Thorin interpolation theorem. The definition of Marcinkiewicz weak \(L^p\) spaces and the notion of quasi-norm (remark 4.1). The definition of sub additive weak and strong \((p,q)\) operators (definition 4.1). The Marcinkiewicz interpolation theorem (first version : interpolation between weak (1,1) and weak (r,r) for \(1 \lt r \lt +\infty\) ) Theorem 4.2. General Marcinkiewicz interpolation theorem (Theorem 4.3). The Hardy Littlewood Sobolev theorem for fractional integration (Theorem 4.4). The definition of the fractional Laplacian (Remark 4.5). |
The new exercise lists will be posted here on Fridays. We expect you to look at the problems over the weekend and to prepare questions for the exercise class on Monday.
Exercises marked with a ♣ sign are intended as reminders of things you should be familiar with, and you should be able to solve them.
Exercises marked with a ★ sign, as well as their solutions, are part of the program for the examination.
| Exercise sheet | Due by | Upload link | Solutions |
|---|---|---|---|
| Serie 1 | Fri 28.02. | Submit Serie 1 | Solution |
| Serie 2 | Fri 07.03. | Submit Serie 2 | Solution |
| Serie 3 | Fri 14.03. | Submit Serie 3 | Solution |
| Serie 4 | Fri 21.03. | Submit Serie 4 | Solution |
| Serie 5 | Fri 28.03. | Submit Serie 5 | Solution |
| Serie 6 | Fri 04.04. | Submit Serie 6 | Solution |
| Serie 7 | Fri 11.04. | Submit Serie 7 | Solution |
| Serie 8 | Fri 25.04. | Submit Serie 8 |
| time | room | assistant | language |
|---|---|---|---|
| Mo 09-10 | HG E 33.3 | Alberto Pacati | English |
| Mo 09-10 | HG F 26.5 | Anthony Salib | English |