Finite Hardy Phase Theory for Power Means: An exact analytic shooting family through the Carleman parameter
Abstract
For the power mean Pt, t<1, let ΛN(t)=supxk>0 (∑n=1N Pt(x1,…,xn))/(∑n=1N xn) be its finite Hardy constant. Negative powers, the geometric mean, and the usual positive-exponent Hardy inequality correspond respectively to t<0, t=0, and 0<t<1. With the analytic parameter q=t/(1−t)∈(−1,∞), and writing λN(q)=ΛN(q/(1+q)), we derive one exact scalar shooting map for all three regimes; its apparent singularity at q=0 is removable and gives the finite Carleman problem exactly. The map has a common terminal condition and its limiting vector field has a quadratic critical bottleneck at y*(q)=1+q, Λ*(q)=(1+q)(1+q)/q. We prove that there is a unique real-analytic phase κ(q) for which the finite-defect expansion has zero cubic coefficient and, uniformly for q in compact subsets of (−1,∞), for every fixed L≥2, Λ*(q)−λN(q) =∑j=2LAj(q)/((log N+κ(q))j) +O ((log N)−L−1). In this phase-normalized scale, A2=2π2(1+q)Λ*, A3=0, A4=−π2/3(2q2+5q+5)A2. Thus the classical Carleman and positive-Hardy leading corrections are projections of a single analytic family. At the Carleman parameter we obtain A2(0)=2eπ2 and A4(0)=−(10/3)eπ4; the latter is also recovered by a direct local residue calculation. The phase itself contains the global discrete defect. After phase normalization, the first coefficient involving a nonconstant discrete response is A5, at order (log N+κ(q))−5.
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BibTeX
@misc{huang2026finite,
title = {{Finite Hardy Phase Theory for Power Means: An exact analytic shooting family through the Carleman parameter}},
author = {Tianchi Huang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10598},
eprint = {2609.10598},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10598},
}
RIS
TY - UNPB
TI - Finite Hardy Phase Theory for Power Means: An exact analytic shooting family through the Carleman parameter
AU - Tianchi Huang
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.10598
UR - https://arxiv.org/abs/2609.10598
AB - For the power mean P_t, t<1, let Λ_N(t)=sup_x_k>0 (∑_n=1^N P_t(x_1,…,x_n))/(∑_n=1^N x_n) be its finite Hardy constant. Negative powers, the geometric mean, and the usual positive-exponent Hardy inequality correspond respectively to t<0, t=0, and 0<t<1. With the analytic parameter q=t/(1−t)∈(−1,∞), and writing λ_N(q)=Λ_N(q/(1+q)), we derive one exact scalar shooting map for all three regimes; its apparent singularity at q=0 is removable and gives the finite Carleman problem exactly. The map has a common terminal condition and its limiting vector field has a quadratic critical bottleneck at y_*(q)=1+q, Λ_*(q)=(1+q)^(1+q)/q. We prove that there is a unique real-analytic phase κ(q) for which the finite-defect expansion has zero cubic coefficient and, uniformly for q in compact subsets of (−1,∞), for every fixed L≥2, Λ_*(q)−λ_N(q) =∑_j=2^LA_j(q)/((log N+κ(q))^j) +O ((log N)^−L−1). In this phase-normalized scale, A_2=2π^2(1+q)Λ_*, A_3=0, A_4=−π^2/3(2q^2+5q+5)A_2. Thus the classical Carleman and positive-Hardy leading corrections are projections of a single analytic family. At the Carleman parameter we obtain A_2(0)=2eπ^2 and A_4(0)=−(10/3)eπ^4; the latter is also recovered by a direct local residue calculation. The phase itself contains the global discrete defect. After phase normalization, the first coefficient involving a nonconstant discrete response is A_5, at order (log N+κ(q))^−5.
ER -