I'm reading up on weak $L^p$ spaces (a.k.a. Marcinkiewicz spaces, or $L^{p,\infty}$ spaces), and I have a little trouble seeing why the function $|x|^{-n/p}$ lies in $L^{p,\infty}(\mathbb{R}^n)$ but not in $L^p(\mathbb{R}^n)$. More specifically, the first part of that statement seems lost to me…
I assume it should be very easy to see it, but I cannot seem to grasp why. Could someone give me some guidance on why $|x|^{-n/p} \in L^{p,\infty}(\mathbb{R}^n)$?
For the record, I'm currently following Grafakos book Classical Fourier Analysis.
Denoting by $\lambda_n$ the $n$-dimensional Lebesgue measure, we have $$\lambda_n\{x\mid |f(x)|\geqslant t\}=\lambda_n\{x\mid |x|^{-n/p} \geqslant t\}=\lambda_n\{x\mid|x|^{n/p}\leqslant 1/t \} =\lambda_n\{ x\mid |x|\leqslant t^{-p/n} \},$$ and using the fact that $\lambda_n(cA)=c^n\lambda_n(A)$ for each $c\gt 0$ and each Borel subset $A$ of $\mathbb R^n$, we obtain $$\lambda_n\left\{x\mid |f(x)|\geqslant t\right\}=t^{n\cdot(-p/n)}\lambda_n\{ x\mid |x|\leqslant 1 \}=t^{-p}\lambda_n\{ x\mid |x|\leqslant 1 \}, $$ which show that $$\sup_{t\gt 0}t^p\lambda_n\{x\mid |f(x)|\geqslant t\}\leqslant \lambda_n\{ x\mid |x|\leqslant 1 \}, $$ hence $f$ belongs to $\mathbb L^{p,\infty}\left(\mathbb R^n\right)$.