A Lorentz Endpoint and Quantitative Obstruction for a Cowling-Price Moment Reduction

Abstract

Let $$ M_{r,p}^{(d)}(g)=\int_{\mathbb R^d}|x|^r|g(x)|^p\,dx, \qquad 1<p<\infty. $$ Write $p'=p/(p-1)$ and $\omega_d=|B(0,1)|$. A direct step in the Wigderson--Wigderson moment method asks for a single-function estimate $$ \lVert g\rVert_1\le K\lVert g\rVert_q^{\,1-p\eta} \bigl(M_{r,p}^{(d)}(g)\bigr)^\eta, \qquad \eta=\frac{d(1-1/q)}{r+d-dp/q}. $$ We give a self-contained proof that, in its scale-invariant regime, this estimate holds exactly when $r>d(p-1)$. 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 $(d\omega_d\log R)^{1/p'}$. The critical line nevertheless has two sharp Lorentz endpoints. The strong moment controls $L^{1,p}$, $$ \lVert g\rVert_{L^{1,p}}\le \omega_d^{1/p'}M_{d(p-1),p}^{(d)}(g)^{1/p}, $$ and strengthening the weighted norm from $L^p$ to $L^{p,1}$ restores the strong target, $$ \lVert g\rVert_1\le \omega_d^{1/p'} \lVert |x|^{d/p'}g\rVert_{L^{p,1}}. $$ 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.