Adapt the Fourier-transform proof of the Nash inequality to the weighted half-line setting

Determine how to adapt the standard Fourier-transform proof of the Nash inequality to a half-space domain with a weighted $L^1$ term, in the setting required for the convergence analysis of the shape defect function for the FKPP model with nonlinear advection.

Background

The paper uses a weighted Nash inequality on the positive half-line to obtain decay estimates for the shape defect function associated with the nonlinear-advection FKPP equation. The inequality involves a weighted $L^1$ norm of the form $\int_0^\infty |\varphi(x)|x^\beta\,dx$, rather than the standard unweighted whole-line norm.

The authors note that the classical proof of the Nash inequality is based on the Fourier transform, but the combination of the half-line domain and the weighted $L^1$ term prevents an immediate adaptation. They therefore develop a real-space proof instead. Constructing a suitable Fourier-analytic proof for this weighted half-line setting remains an explicit unresolved methodological question.