Preservation of equivalence under the generalized upper Legendre conjugate

Ascertain whether, for arbitrary weight functions σ and τ, the generalized upper Legendre conjugate preserves equivalence relations; specifically, determine if σ ∼ σ1 and τ ∼ τ1 imply σ widehat⋆ τ ∼ σ1 widehat⋆ τ1, and whether the well-definedness condition for σ widehat⋆ τ is preserved under these equivalences without imposing additional growth hypotheses.

Background

For the generalized lower Legendre conjugate σ check⋆ τ, Lemma 3.4 (lowertransformrelationlemma) establishes that relations such as equivalence (∼) and big-O (≼) are preserved under the operation.

For the generalized upper Legendre conjugate σ widehat⋆ τ, the authors note that it is not clear whether analogous preservation properties hold in general. They develop partial results (Theorem 4.31 and Theorem 4.33) showing preservation under additional growth conditions (e.g., (ω6) or (ω1)) or specific dominance relations, but do not resolve the unrestricted case.

Thus, whether σ widehat⋆ τ behaves functorially with respect to equivalence of inputs in full generality remains an open question.

References

In general, the analogue of Lemma \ref{lowertransformrelationlemma} is not clear for the generalized upper conjugate; therefore note that the property that $\sigma\widehat{\star}\tau$ is well-defined and also conditionB are not automatically preserved under equivalences of weight functions.

conditionB:

sup⁡t>0lim sup⁡u→+∞σ(tu)τ(u)<1.\sup_{t>0}\limsup_{u\rightarrow+\infty}\frac{\sigma(tu)}{\tau(u)}<1.

— Generalized upper and lower Legendre conjugates for weight functions (2505.07497 - Schindl, 12 May 2025) in Section 4.4 (On the relations between weight functions), opening paragraph