$L^p$ Spaces, Young's Theorem, Convolutions, and Minkowski's Inequality

Question

I need to show \begin{align} \|f*g\|_p \le \|f\|_p\|g\|_1 \end{align} By using the generalized Minkowski inequality instead of just Young's Theorem. I have spent a lot of time, but I keep hitting a dead end. Thanks a million in advance!

Answer 1

$$\Big(\int |(f*g)(x)|^p\ dx\Big)^\frac{1}{p} = \Big(\int \Big|\int f(x - y)g(y)dy\Big|^p\ dx\Big)^\frac{1}{p} \le \int \Big(\int |f(x - y)g(y)|^pdx\Big)^{\frac{1}{p}}dy = \int \Big(\int |f(x - y)|^p dx\Big)^{\frac{1}{p}}|g(y)|dy = \|f\|_p\|g\|_1.$$

Here the only inequality sign is given exactly by the Minkowski inequality (just to emphasize that there is nothing hidden somewhere :D )