Year: 2026
Type: preprint
Abstract: Let $$ M_{r,p}^{(d)}(g)=\int_{\mathbb R^d}|x|^r|g(x)|^p\,dx, \qquad 1d(p-1)$. We then quantify the transition: the best constant blows up like $[r-d(p-1)]^{-1/p'}$ at the critical line, while on the annulus $1\le |x|\le R$ the endpoint loss is exactly $(dω_d\log R)^{1/p'}$. The critical line nevertheless has two sharp Lorentz endpoints. The strong moment controls $L^{1,p}$, $$ \lVert g\rVert_{L^{1,p}}\le ω_d^{1/p'}M_{d(p-1),p}^{(d)}(g)^{1/p}, $$ and strengthening the weighted norm from $L^p$ to $L^{p,1}$ restores the strong target, $$ \lVert g\rVert_1\le ω_d^{1/p'} \lVert |x|^{d/p'}g\rVert_{L^{p,1}}. $$ More generally, every power moment has a sharp target Lorentz space, and its secondary index is optimal. These results identify the endpoint lost by the direct same-$q$ reduction. They do not obstruct auxiliary-norm versions of the Wigderson framework, which reach the Cowling--Price range by a different route.
Source: arXiv (Cornell University)
Author: Menezes, Dean
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214056282