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