*****Note: Parts A, C and D I managed. Only need help on part B now would really would appreciate the help on B
Hi, in my summer real analysis (or measures and real analysis as my instructor refers to it) I was presented with this question from Folland's real analysis second edition on distribution theory which has been slowly killing me. I really have no idea how to tackle it. It is question 9.11 which is:
By support of a distribution they obviously mean the complement of the maximal open set on which the distribution is identically zero. My problem is how to use the support being zero in order to proceed I cannot really see how to relate any parts of this question to something I already know Thank you all helpers
PROGRESS: After breaking my teeth I can figure all out except part b (really tough for me). Parts A, C and D I managed. Only need help on part B now would really appreciate answer on B
EDIT: Lastly I only require help on part B of this question as thanks to the helpful comments I can now solve everything but part B as I cannot incorporate the clue and express the indexed derivative
By the (multi-dimensional) Leibniz formula, we have
$$\begin{align*} \partial^{\alpha} \phi_k&= \sum_{\beta+\gamma = \alpha} \underbrace{c_{\beta,\gamma} (\partial^{\beta} \phi) \cdot (\partial^{\gamma} (1-\psi(k \bullet)))}_{=:S_{\beta,\gamma}} \end{align*}$$
for some constants $c_{\beta,\gamma}$ (which can be calculated expliticly). If $\gamma=0$, then $\alpha = \beta$ and
$$S_{\alpha,0}(x) = \underbrace{c_{\alpha,0}}_{1} (1-\psi(k x)) \partial^{\alpha} \phi(x).$$
If $\gamma \neq 0$, then
$$S_{\beta,\gamma}(x) = c_{\beta,\gamma} k^{|\gamma|} (\partial^{\beta} \phi(x)) (\partial^{\gamma} \psi(y) \big|_{y=kx}).$$
As the hint suggests, it follows directly from Taylor's formula that $|\partial^{\beta} \phi(x)| \leq C |x|^{N+1-|\beta|}$ for $|\beta| \leq N$. Recall that $\partial^{\gamma} \psi (x) = 0$ for all $|x| \geq 2$. Hence, $\partial^{\gamma} \psi(kx) = 0$ for all $|x| \geq 2/k$. This implies
$$\begin{align*} |S_{\beta,\gamma}(x)| &\leq c_{\beta,\gamma} k^{|\gamma|} C' \sup_{|y| \leq 2/k} |y|^{N+1-|\beta|} \\ &\leq c_{\beta,\gamma} C'' |k|^{|\gamma|+|\beta|-(N+1)}; \end{align*}$$
here, we have used that the derivatives of $\psi$ are bounded. Since $$|\gamma|+|\beta|-(N+1) = |\alpha| - (N+1) \leq -1,$$
we get
$$\sup_{x \in \mathbb{R}^d} |S_{\beta,\gamma}(x)| \to 0 \qquad \text{as $k \to \infty$}$$
for all $\gamma \neq 0$. Finally, we note that by the triangle inequality and the above observations
$$|\partial^{\alpha} \phi_k(x)-\partial^{\alpha}(x)| \leq |(1-\psi(kx)| |\partial^{\alpha} \phi(x)| + \bigg| \sum_{\substack{\beta+\gamma=\alpha \\ \gamma \neq 0}} S_{\beta,\gamma}(x) \bigg|.$$
We have already shown that the second terms converges uniformly to $0$. The uniform convergence of the first term follows directly from the properties of $\psi$ and the boundedness of $\partial^{\alpha} \phi$.