I am trying to understand weighted Lebesgue spaces. For $a\geq 0$, let consider the space $W_a$ of all measurable vector fields $f$ in $\mathbb{R}^3$, such that $$ \| f \|_{W_a} = \mathrm{ess \, sup}_{x\in \mathbb{R^3}} |x|^a f(x) < \infty. $$ Some questions that I am trying to solve are:
For which values of $a\geq 0$ the quantity $\| f \|_{W_a}$ is finite ?
If $f\in L^\infty(\mathbb{R}^3)$, for which values of $a\geq 0$ the quantity $\| f \|_{W_a}$ is finite ?
is it possible to prove that if $a>b$, then $W_a \subset W_b$ ?
Until now I tried to attack 3 considering that if $x\in \mathbb{R}^3$, then $$ |x|^a=|x|^b|x|^{r}, $$ where $r$ is such that $a=b+r$.
Do you have any reference where I can read more about this kind of spaces ?
Notice first that $W_a$ is just the space of functions $f$ such that $f(x) \leq C/|x|^a$ for some constant $C$.
This question is badly formulated, it will depend on $f$. For example if $f$ is bounded and compactly supported, then $\|f\|_{W^a}$ is always finite, but if $f=1$, then $f$ is infinite as soon as $a>0$.
Again, $\|f\|_{W^a}$ is finite for any $f\in L^\infty$ only if $a=0$, as you can notice by taking $f=1$.
No, in general, $f(x) \leq C/|x|^a$ does not imply $f(x) \leq C/|x|^b$ for another value $b\neq a$. Take for example $f(x) = 1/|x|^a$, then $f\in W_a$ but not in $W_b$ for any other $b$.