If $f\in L^1_{loc}(\mathbb{R}^n)$ and $g\in C^{\infty}_c(\mathbb{R}^n)$, it is true that $fg\in L^2(\mathbb{R}^n)?$

Question

Let $n\geq 2$. Consider $f\in L^1_{loc}(\mathbb{R}^n)$ and $g\in C^{\infty}_c(\mathbb{R}^n)$. It is true that $$fg\in L^2(\mathbb{R}^n)?$$ If it is true, could anyone explain me why? Or, if it false, could anyone give me a counterexample?

Thank you in advance!

Answer 1

Of course it’s false. Take Your favourite function that is $L^1\setminus L^2(B_1(0))$. Extend it by zero. Multiply by a test function $g$ that is identically one on $B_1(0)$. Then $fg=f$ is not $L^2$

PS the dimension played no role at all

Answer 2

This is false when $n=1$ and $g \in C_c^{\infty}(\mathbb R)$ with $g(x)=1$ for $|x| \leq 1$. Let $f(x)=\frac 1 {\sqrt {|x|}}$. Then $f \in L_{loc}^{1} $ and $\int |fg|^{2}=\infty$. Can you construct a similar example with $n=2$?