convolution:Unterstanding beginning of the proof

Question

I got $f \in L^1(\mathbb{R^n}), g \in L^p(\mathbb{R^n}), p \in [1, \infty]$

Then $$ \int_{R^n} |f(x-y)|| g(y)| d^n y < \infty$$

Proof: Let $ \frac{1}{q} +\frac{1}{p} =1$

Then: $ $$ \int_{R^n} |f(x-y)|| g(y)| d^n y = $$ \int_{R^n} |f(x-y)|^{\frac{1}{q}} \underbrace {| g(y)| |f(x-y)|^{\frac{1}{p}}}_{\in L^p (1)} d^n y$

Why does (1) hold?

Answer 1

Since that $f \in L^1(\mathbb{R}^n)$, $|f|=(|f|^{1/p})^p$ it's integrable, then $f^{1/p} \in L^p(\mathbb{R}^n)$ and therefore $g(y)f(x-y)^{1/p} \in L^p(\mathbb{R}^n)$,