I am trying to follow a step in a proof which uses the Hölder Inequality.

The setting is: $f\in L^{p}(\mathbb{R}^{d})$, $g\in L^{1}(\mathbb{R}^{d})$, with $1\leq p\leq \infty$.

The claim is that: By the Hölder Inequality, for $1\leq p < \infty$,

$$\left|\int\_{\mathbb{R}^{d}}|f(x-y)g(y)|dy\right|^{p} \leq \left(\int\_{\mathbb{R}^{d}}|f(x-y)|^{p}|g(y)|dy\right)\left(\int\_{\mathbb{R}^{d}}|g(y)|dy\right)^{p-1}$$

But I am having a hard time seeing which Hölder Conjugates were used.

Is the Minkowski Inequality also used here? Thank you!