Problem 1
Suppose that \(f_{1}, f_{2}, \ldots\), and \(f\) are in \(L_{\text {loc }}^{1}(U) .\) The conditions in (a) and (b) below imply that \(f_{n} \rightarrow f\) in \(\mathcal{D}^{\prime}(U)\), but the condition in (c) does not. a. \(f_{n} \in L^{P}(U)(1 \leq p \leq \infty)\) and \(f_{n} \rightarrow f\) in the \(L^{P}\) norm or weakly in \(L^{P}\). b. For all \(n,\left|f_{n}\right| \leq g\) for some \(g \in L_{\text {loe }}^{1}(U)\), and \(f_{n} \rightarrow f\) a.e. c. \(f_{n} \rightarrow f\) pointwise.
Problem 4
Suppose that \(U\) and \(V\) are open in \(R^{n}\) and \(\Phi: V \rightarrow U\) is a \(C^{\infty}\) diffeomorphism. Explain how to define \(F \circ \Phi \in \mathcal{D}^{\prime}(U)\) for any \(F \in \mathcal{D}^{\prime}(V)\).
Problem 10
Let \(f\) be a continuous function on \(\mathbb{R}^{n} \backslash\\{0\\}\) that is homogeneous of degree \(-n\) (i.e., \(\left.f(r x)=r^{-n} f(x)\right)\) and has mean zero on the unit sphere (i.e., \(\int f d \sigma=0\) where \(\sigma\) is surface measure on the sphere). Then \(f\) is not locally integrable near the origin (unless \(f=0)\), but the formula $$ \langle P V(f), \phi\rangle=\lim _{\epsilon \rightarrow 0} \int_{|x|>\epsilon} f(x) \phi(x) d x \quad\left(\phi \in C_{c}^{\infty}\right) $$ defines a distribution \(P V(f)-" P V "\) stands for "principal value" - that agrees with \(f\) on \(\mathbb{R}^{n} \backslash\\{0\\}\) and is homogeneous of degree \(-n\) in the sense of Exercise 9 .
Problem 11
Let \(F\) be a distribution on \(\mathbb{R}^{n}\) such that \(\operatorname{supp}(F)=\\{0\\}\). a. There exist \(N \in \mathbb{N}, C>0\) such that for all \(\phi \in C_{e}^{\infty}\). $$ |\langle F, \phi\rangle| \leq C \sum_{|a| \leq N} \sup _{|x| \leq 1}\left|\partial^{\alpha} \phi(x)\right| $$ b. Fix \(\psi \in C_{e}^{\infty}\) with \(\psi(x)=1\) for \(|x| \leq 1\) and \(\psi(x)=0\) for \(|x| \geq 2\). If \(\phi \in C_{e}^{\infty}\), let \(\phi_{k}(x)=\phi(x)[1-\psi(k x)]\). If \(\partial^{\alpha} \phi(0)=0\) for \(|\alpha| \leq N\), then \(\partial^{\alpha} \phi_{k} \rightarrow \partial^{\alpha} \phi\) uniformly as \(k \rightarrow \infty\) for \(|\alpha| \leq N\). (Hint: By 'Taylor's theorem. \(\left|\partial^{\alpha} \phi(x)\right| \leq C|x|^{N+1-|\alpha|}\) for \(\left.|\alpha| \leq N_{.}\right)\) c. If \(\phi \in C_{c}^{\infty}\) and \(\partial^{\alpha} \phi(0)=0\) for \(|\alpha| \leq N\), then \(\langle F, \phi\rangle=0\). d. There exist constants \(c_{\alpha}(|\alpha| \leq N)\) such that \(F=\sum_{|a| \leq N} c_{\alpha} \partial^{\alpha} \delta\).
