Sharp diagonal weight exponent in the unresolved region

Determine the largest admissible diagonal weight exponent t for which the dimension-free weighted coefficient inequality \[ \left(\sum_{|\alpha|=m}|a_\alpha u^\alpha|^q\right)^{1/q}\le C\|u\|_t^m\|P\|_p \] holds for every m-homogeneous polynomial P on \ell_p^n and every u\in\ell_t^n, in the parameter range 2<p<m, 2<q<\infty, and D=m-(\lceil p\rceil-1)<G=2m/p-1.

Background

The paper studies dimension-free weighted Hardy–Littlewood inequalities for homogeneous polynomial coefficients. In the range 2<p<m and 2<q<\infty with D<G, the authors derive a necessary condition from the growth exponent of the unweighted coefficient embedding: any admissible exponent t must satisfy 1/t\ge\beta_{m,p}(q)/m, where \beta_{m,p}(q) is determined in Theorem C.

The authors also prove a sufficient estimate using quadratic-to-maximum interpolation, with 1/t=(2/q)(1/p-1/(2m)). They explicitly state that these bounds do not determine the optimal exponent throughout this parameter range. The unresolved problem is therefore to identify the exact largest admissible t, equivalently to close the gap between the necessary and sufficient conditions.