Convexity and concavity in Banach lattices

These notes are a detailed, self-contained introductory course on convexity and concavity in Banach lattices, suitable for both experts and beginners. We revisit, from a modern perspective, the classical notions of $(p,q)$-convexity, $(p,q)$-concavity and upper and lower $p$-estimates, and the main relations between these properties, integrating more recent developments in the area. We explain in full detail the $p$-convexification and $p$-concavification techniques. We also provide a comprehensive exposition of the main factorization results for $(p,q)$-convex and $(p,q)$-concave operators, including well-known results from Krivine, Maurey–Nikishin, Pietsch and Pisier, and their applications to the renorming and the representation of convex and concave Banach lattices.