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We prove the existence of free objects in certain subcategories of Banach lattices, including  p -convex Banach lattices, Banach lattices with upper  p -estimates, and AM-spaces. From this we immediately deduce that projectively universal objects exist in each of these subcategories, extending results of Leung, Li, Oikhberg and Tursi ( Israel J. Math.  2019). In the  p -convex and AM-space cases, we are able to explicitly identify the norms of the free Banach lattices, and we conclude by investigating the structure of these norms in connection with nonlinear  p -summing maps.

Free Banach lattices under convexity conditions