Weighted L^p-representations of Banach function spaces

Abstract

This thesis studies which conditions on the norm of a Banach function space force it to be a weighted L^p-space. We start from a result we call the seed theorem: if X is a Banach function space over a σ-finite measure space whose norm satisfies ∥f+g∥_X = ∥f∥_X + ∥g∥_X for all f,g ∈ X_+, then X = L^1(Ω,w dμ) isometrically for some weight w > 0. We then relate the seed theorem to three more general statements: an isometric L^p(w dμ)-representation under disjoint p-additivity, a theorem about weighted L^p-inequalities controlling a family of maps and finally an L^p(w dμ)-equivalence for (p,b)-convexity and (p,d)-concavity. Recovering the seed theorem from these general statements exposes an asymmetry: once the general theorem is known, deriving the specific case is very easy. Finding the general theorem from the specific case is harder, since the general theorem may be formulated in a different mathematical language.

To exploit this asymmetry, we introduce a protocol that uses large language models to generate specific cases of general theorem statements and add them as corollary notes. We evaluate the protocol on the three general statements above, using the seed theorem as the target specific case. In the second iteration, the protocol produces corollary notes that come very close to the seed theorem for two of the three general statements. In a controlled search experiment, adding these corollary notes moves both corresponding general statements into the top ten results for every query about the seed theorem. Without these corollary notes, neither appears in the top ten.