Why if $p\not = q$ we have $L^p(R^n) \not \subseteq L^q(R^n)$? This is a result present in my books, and I can't figure out really a nice proof about this.
An example say that the function $u(x) = (1+|x|)^{-n/p}$ is in all $L^q(R^n)$ with $q>p$ but not in $L^p(R^n)$.
Another example say that the function $u(x) = 1_B|x|^{-n/p}$ is in all $L^q(R^n)$ ($B=B_1(0)$) with $q<p$ but not in $L^p(R^n)$.
I can't figure out why those are verified. Also, $1_B$ is the ball $B=B_1(0)$? I've never used this notation on a function. Can someone explain me?
$1_B$ is the function defined by $1_B(x)=1$ if $x \in B$ and $0$ otherwise. To prove these facts you can use polar coordinates in $\mathbb R^{n}$. See Rudin's RCA for polar coordinates.