Gradient inequality in Morrey’s and Gagliardo-Nirenberg inequality proofs

Question

I'm approaching for the first time functional analysis and in general "advanced math". In the book "Lecture Notes on Functional Analysis" by A. Bressan, in the Sobolev space chapter, I get stuck in two inequalities involving $\nabla u$. $$$$ 1) In the proof of Morrey's inequality: be $u \in C^{1}(\mathbb{R}^{n})\cap W^{1,p}(\mathbb{R}^{n})$; in the proof, the following set of coordinates is defined: $(r,\xi)=(r,\xi_{2}…\xi_{n}) \in \left[0,\rho \right] \times B_{1}$, with $B_{\rho}:=\left( x=(x_{1},x_{2},…,x_{n});x_{1}=\rho, \ \sum_{i=2}^{n} \ x^2_{i}\leq \rho^{2}\right) $; to this system of coordinates, the point $x(r, \xi)=(r,r \xi)$ is associated (x belongs to a cone), moreover define $U(r,\xi)=u(r,r\xi)$. Here I got stuck (citing the text): since $|\xi|\leq 1$ the directional derivative of $u$ in the direction of the vector $(1,\xi_{2},\xi_{3},…,\xi_{n})$ is estimated by $$\left| \frac{\partial }{\partial r} U(r,\xi)\right|=\left| u_{x_{1}}+\sum_{i=2}^{n}\xi_{i}u_{x_{i}} \right| \leq 2|\nabla u(r,\xi)|.$$
$$ $$ 2) In the proof of Gagliardo-Nirenberg inequality: be $f \in C^{1}_{c}\left( \mathbb{R}^{n} \right)$ , I got stuck in the following inequality: $$ \prod _{i=1}^{n} \left( \int _{-\infty}^{\infty}…\int _{-\infty}^{\infty} \left | D _{ x _{i}} dx _{1} dx _{2}… dx _{n} \right | \right) ^{1/(n-1)} \leq \left ( \int _{\mathbb{R}^{n}}^{} |\nabla f|dx \right )^{n/(n-1)} $$ $$$$ It seems to me that I miss something becouse in the book it appears that such inequalities involving $\nabla u$ and $ \nabla f $ are so straightforward; I probalby lack the knowledge of some identity or inequality, perhaps from calculus. Could anyone give me some hints? Thank you for your kindness, I apologize if it might be trivial, but I’m a rookie!

Answer 1

The first one is Cauchy-Schwarz: $$ \big|\sum_{i=2}^n\xi_iu_{x_i}\big| \leq \big(\sum_{i=2}^n\xi_i^2\big)^{\frac12} \big(\sum_{i=2}^n|u_{x_i}|^2\big)^{\frac12} \leq|\xi||\nabla u| \leq|\nabla u|. $$