I found the paper Analytic Principle of the Large Sieve by Montgomery where he puts "linearization" as essentially "duality" of the $\ell^p$ spaces. I just thought I would add this reference for people like me who were interested in a definite place to look in the large sieve literature for related content

I think that "linearization" as Bourgain is referring to it is very closely related to what Beckenbach and Bellman call "quasilinearization" in their book Inequalities (p. 23), which is also closely related to the duality of $L^p$ spaces. The idea is to characterize $L^p$ norms as envelopes of linear quantities. For more on quasilinearization as a strategy for proving inequalities, you can look at p. 669 of the book Classical and New Inequalities in Analysis, where the starting point of the exposition is the classical Hölder and Minkowski inequalities.