A Lorentz Endpoint and Quantitative Obstruction for a Cowling-Price Moment Reduction
Abstract
Let Mr,p(d)(g)=∫ℝd|x|r|g(x)|p dx, 1<p<∞. Write p'=p/(p−1) and ωd=|B(0,1)|. A direct step in the Wigderson--Wigderson moment method asks for a single-function estimate lVert grVert1≤ KlVert grVertq 1−pη (Mr,p(d)(g))η, η=(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≤ |x|≤ R the endpoint loss is exactly (dωdlog R)1/p'. The critical line nevertheless has two sharp Lorentz endpoints. The strong moment controls L1,p, lVert grVertL1,p≤ ωd1/p'Md(p−1),p(d)(g)1/p, and strengthening the weighted norm from Lp to Lp,1 restores the strong target, lVert grVert1≤ ωd1/p' lVert |x|d/p'grVertLp,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.
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BibTeX
@misc{menezes2026lorentz,
title = {{A Lorentz Endpoint and Quantitative Obstruction for a Cowling-Price Moment Reduction}},
author = {Dean Menezes},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22345},
eprint = {2609.22345},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22345},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Lorentz Endpoint and Quantitative Obstruction for a Cowling-Price Moment Reduction
AU - Dean Menezes
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.22345
UR - https://arxiv.org/abs/2609.22345
AB - Let M_r,p^(d)(g)=∫_ℝ^d|x|^r|g(x)|^p dx, 1<p<∞. Write p'=p/(p−1) and ω_d=|B(0,1)|. A direct step in the Wigderson--Wigderson moment method asks for a single-function estimate lVert grVert_1≤ KlVert grVert_q^ 1−pη (M_r,p^(d)(g))^η, η=(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≤ |x|≤ R the endpoint loss is exactly (dω_dlog R)^1/p'. The critical line nevertheless has two sharp Lorentz endpoints. The strong moment controls L^1,p, lVert grVert_L^1,p≤ 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 grVert_1≤ ω_d^1/p' lVert |x|^d/p'grVert_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.
ER -