composition of Lebesgue integrable function with smooth function

Question

If I have a function $f\in L^{1}_{loc}(\mathbb{R})$ and I compose it with $g:\mathbb{R^2}\to\mathbb{R}$ which is smooth. Then is it always true that $f\circ g\in L^{1}_{loc}(\mathbb{R^2})$?

thank you

Answer 1

In general, no. Try $f (x) = |x|^{-\delta} $ for a suitable $\delta >0$ and $g (x_1, x_2) = x_1^{2n} $ for a large $n \in \Bbb {N} $ (depending on $\delta $).