Weak interaction condition [wI] for log-convex weight sequences

Determine whether every log-convex weight sequence M = (M_α)_{α ∈ N^n} satisfies the weak interaction condition [wI] as defined for weight sequences in Section 2.3, namely: there exist constants C, H > 0 such that for all multi-indices α, β ∈ N^n one has M_α ≤ C H^{|β|} M_{α+β}. This seeks to clarify whether the weaker condition [wI] holds universally for log-convex weight sequences, given that the stronger condition [I] is known to fail for some log-convex sequences.

Background

The paper studies inclusion relations between Gelfand-Shilov type spaces built from weight sequence systems and weight function systems. Central to the analysis are structural conditions on weight sequences, notably [I] and the weaker [wI], which govern how indices combine under addition and control growth.

Remark 2.10 explains that while isotropic log-convex weight sequences satisfy strong multiplicative inequalities implying [I], there exist general log-convex weight sequences that do not satisfy [I]. The authors explicitly state that it is unknown whether all log-convex weight sequences satisfy the weaker condition [wI]. Resolving this would impact the generality of their inclusion characterizations, since several main results assume [wI] on the underlying sequence systems.