As an exercice, I'm looking to find an inclusion or equality relationship between $L^2(\mathbb{R}_+)$ and $L^1_w(\mathbb{R}_+)$ when $w: x \to x^{-1/2}$.
Actually, I think that we have the inclusion $L^2(\mathbb{R}_+) \subset L^1_w(\mathbb{R}_+)$ but I am not able to prove it.
Could someone of you got a hint?
The inclusion $L^2\subset L^1_w$ does not hold: take $f(x):=\frac 1{\sqrt x\log x}\chi_{(1,\infty)}(x)$.
The converse reduces to ask whether $g\in L^1$ implies $x\mapsto xg(x)^2\in L^1$. Define $$g(x):=\sum_{j=1}^\infty c_j\cdot\chi(j^2-a_j,j^2+a_j)(x),$$ with the $a_j$ small enough. Then $g$ is integrable if and only if $\sum_{j=1}^\infty c_ja_j$ is convergent and $x\mapsto xg(x)^2\in L^1$ if and only if $\sum_{l=1}^\infty a_j j^2c_j^2$ is convergent.