Existence of auxiliary regularly varying functions g* in boundary-index cases

Ascertain whether, for any regularly varying comparison function g with index 0 or 2 that satisfies the hypotheses in the sparse-spectrum criteria, there exists a regularly varying function g* whose reciprocal is locally integrable and which fulfills the required asymptotic identities: for Ind g = 0, ∫_t^∞ ∫_u^∞ (1/g*(s)) (ds/s) (du/u) \asymp 1/g(t); for Ind g = 2, ∫_1^t (s^2/g*(s)) (ds/s) \asymp t^2/g(t).

Background

The sparse-spectrum results (convergence class and type via the Weyl coefficient approach) require, in boundary-index cases Ind g ∈ {0,2}, the existence of an auxiliary regularly varying function g* that links certain integral transforms to 1/g or t^2/g, respectively. The existence of such a g* enables pointwise characterizations of summability of the spectrum or type conditions.

While explicit constructions of g* are available for specific Lindelöf-type g, a general existence theorem would significantly broaden the applicability of the sparse-spectrum criteria and clarify the structure of boundary-index regimes.