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I have gathered that Christ's ‘method of refinements’ originated in his paper Convolution, curvature, and combinatorics: a case study to prove the sharp range of $L^p$-improving estimates for the moment curve in all dimensions. This has become a fundamental tool in proving $L^p$-improving estimates and there is an extensive literature on the subject which I have not delved into.

So, my first question is: What are the ‘refinements’ in Christ's method of refinements? In other words, what specifically are the refinements in the above paper? Are they the refined bands in Section 4.6 or the sets of Lemma 1 in Section 2.2 therein?

In his paper, Christ does not refer to any part of the method he introduces by that name. So, my second question is: Where is this nomenclature introduced?

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The "refinement" terminology in connection with Christ's 1998 paper seems to originate from Tao and Wright's 2001 article, although the earliest explicit reference to "Christ's method of refinements" I could locate is a 2005 article.

Christ himself speaks of a "refined band structure".

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