a) $ |h(x)| = \left|\int{f(x-y)g(y)dy}\right| \le \int{|f(x-y)g(y)|dy} \le \left(\int{|f(x-y)|^p dy}\right)^{1/p}\left(\int{|g(y)|^q dy}\right)^{1/q} $ by Holder's Inequality.
Let $ z = y - x $. Then $ \left(\int{|f(x-y)|^p dy}\right)^{1/p} \left(\int{|g(y)|^q}\right)^{1/q} = \left(\int{|f(-z)|^p dz}\right)^{1/p} ||g||_q = ||f||_p ||g||_q $ because $ \int{|f(-x)|} = \int{|f(x)|} $.
Since $ f \in L^p $ and $ g \in L^q $ we get $ |h(x)| \le ||f||_p ||g||_q < \infty $
b)
