While Studying Sobolev spaces, I have come across following two inequalities. For an element $\xi \in \mathbb{R^n}$and $N$,smallest integer greater than $n/2$, we have
\begin{equation} \bigg( 1+\sum^{n}_{j=1}|\xi_j|^{N}\bigg)^{-2} \le c|\xi|^{-n-1} \end{equation} (though a terrible estimate for small values of $|\xi|$) and \begin{equation} \sum^{n}_{j=1}|\xi _j|^{2N} \ge c|\xi|^{2N} \end{equation} I have no clue how to go about proving these inequalities for myself. Any help will be deeply acknowledged.
Two main ingredients are compactness and homogeneity.
Compactness
Claim: If $f$ and $g$ are continuous real functions on a compact set $K$, and $f>0$ everywhere on $K$, then there exists a constant $c>0$ such that $f(x) \ge cg(x)$ everywhere on $K$.
Proof: the function $g/f$ is continuous on $K$, therefore is bounded above by some constant $C$. This yields $f\ge C^{-1}g$ as claimed. $\quad\Box$
Homogeneity
Definition: A function $f$ on $\mathbb{R}^n$ is homogeneous of degree $d$ if $f(t\xi) = t^d f(\xi)$ for all $\xi\in\mathbb{R}^n$ and all $t>0$.
Claim: Suppose that $f$ is nonnegative and homogeneous of degree $d$, and $g$ is homogeneous of degree $d'\le d$. If there is a constant $c$ such that $f\ge cg$ on the unit sphere $S^{n-1}$, then $f(\xi)\ge cg(\xi)$ for all $|\xi|\ge 1$.
Proof: Given $\xi$ with $|\xi| > 1$, write it as $\xi=t\eta$ where $t=|\xi|>1$ and $|\eta|=1$. Then $$ f(\xi) = t^d f(\eta) \ge t^d c g(\eta) = t^{d-d'} cg(\xi) \ge cg(\xi) $$ as claimed. $\quad\Box$
Additional remark: if $d=d'$, then having inequality $f\ge cg$ on the sphere implies having it on the entire space, since we have $$ f(\xi) = t^d f(\eta) \ge t^d c g(\eta) = cg(\xi) $$ without the need for $t>1$.
With the above, the inequalities become obvious. For example, the first one is equivalent to $$ 1 + \sum^{n}_{j=1}|\xi_j|^{N} \ge C|\xi|^{(n+1)/2} \tag1$$ Considering both sides on the unit ball $|\xi|\le 1$, we see that such $C$ exists by compactness and the left hand side being positive. By the same reasoning, there is another $C$ such that $$ \sum^{n}_{j=1}|\xi_j|^{N} \ge C|\xi|^{(n+1)/2} \tag2$$ holds on the sphere $|\xi|=1$. The left hand side of (2) is homogeneous of degree $N$, the right hand side is homogeneous of degree $(n+1)/2$. Since $N\ge (n+1)/2$, it follows that (2) holds for $|\xi|>1$ as well.
The second is simpler: $$ \sum^{n}_{j=1}|\xi _j|^{2N} \ge c|\xi|^{2N} $$ holds on the unit sphere by compactness, and extends to the entire space by homogeneity (both sides are homogeneous of degree $2N$).