Folland's real analysis text spends a lot of time on various basic $L^p$ inequalities including $L^p$ norms of convolutions. The material on convolutions is (I believe) in Chapter 8 which is titled ``Elements of Fourier Analysis," and the basic $L^p$ stuff is in Chapter 6. Incidentally, this is definitely not a research level question, and in fact, I would say it's a standard exercise (using HölderMinkowskiFubini) in a first course in measure theory.
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