Sufficiency of limsup condition for well-definedness of the generalized upper Legendre conjugate

Determine whether the condition sup_{0 < t < t0} limsup_{u → +∞} (σ(tu)/τ(u)) ≤ 1 implies the existence, for each t ∈ (0, t0), of a constant D_t such that σ(s) − τ(s/t) ≤ D_t for all s ≥ 0, thereby ensuring that the generalized upper Legendre conjugate σ widehat⋆ τ is well-defined on (0, t0).

Background

The paper introduces the generalized upper Legendre conjugate of weight functions via (σ widehat⋆ τ)(t) := sup_{s ≥ 0} {σ(s) − τ(s/t)}. Ensuring this map is well-defined requires finiteness of the supremum, which the authors paper through limsup-type conditions.

Lemma uppertransformfinite shows that a strict limsup bound (sup_{0 < t < t0} limsup_{u → +∞} σ(tu)/τ(u) < 1) suffices to guarantee the existence of bounds σ(s) − τ(s/t) ≤ D_t for all s ≥ 0, while a weaker bound with ≤ 1 may not suffice. The general implication from the weaker bound to well-definedness is left unresolved.

The authors provide characterizations for special cases (e.g., τ = id{1/α}) and implications under additional growth conditions (such as (ω1) or (ω6)), but a general necessary-and-sufficient criterion under the weaker limsup condition remains unknown.

References

In general it is not clear if (iii)⇒(ii) in Lemma \ref{uppertransformfinite} holds; for this note that uppertransformfiniteequweak precisely means $$\forall\;t\in(0,t_0)\;\forall\;\epsilon>0\;\exists\;D_{t,\epsilon}>0\;\forall\;u\ge 0:\;\;\;\sigma(tu)\le(1+\epsilon)\tau(u)+D_{t,\epsilon},$$ but which does not imply necessarily uppetransfromwelldefequ.

uppertransformfiniteequweak:

sup⁡0<t<t0lim sup⁡u→+∞σ(tu)τ(u)≤1.\sup_{0<t<t_0}\limsup_{u\rightarrow+\infty}\frac{\sigma(tu)}{\tau(u)}\le 1.

uppetransfromwelldefequ:

∀  t∈(0,t0)  ∃  Dt>0  ∀  s≥0:      σ(s)−τ(s/t)≤Dt.\forall\;t\in(0,t_0)\;\exists\;D_t>0\;\forall\;s\ge 0:\;\;\;\sigma(s)-\tau(s/t)\le D_t.

— Generalized upper and lower Legendre conjugates for weight functions (2505.07497 - Schindl, 12 May 2025) in Note (i) after Lemma 4.5 (Lemma uppertransformfinite), Section 4.2