Hardy--Littlewood Constants Beyond Polarization

Abstract

The polynomial Hardy--Littlewood inequalities bound coefficient norms of homogeneous polynomials on ℓ_p balls by their supremum norms, with constants independent of the dimension. Their p=∞ endpoint is the Bohnenblust--Hille inequality. A longstanding obstruction to connecting their constants is the polarization loss in the usual multilinear approach. We overcome this obstruction by proving D_{m,p_m}≤(1+o(1))D_m whenever p_m/m²→∞, where D_m,p and D_m are the optimal complex polynomial Hardy--Littlewood and Bohnenblust--Hille constants. At the opposite boundary, we determine the optimal endpoint constant exactly, D_m,m=m. We then prove the full first-order asymptotic D_{m,p_m}∼ m whenever p_m≥ m and p_m−m→0, and the exact polynomial growth D_{m,p_m}=m^1+o(1) on the wider scale p_m−m=o((m/log m)^1/3).

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BibTeX RIS

BibTeX

@misc{pellegrino2026hardylittlewood,
  title = {{Hardy--Littlewood Constants Beyond Polarization}},
  author = {Daniel Pellegrino and Eduardo Teixeira},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.14909},
  eprint = {2609.14909},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.14909},
}

RIS

TY  - UNPB
TI  - Hardy--Littlewood Constants Beyond Polarization
AU  - Daniel Pellegrino
AU  - Eduardo Teixeira
PY  - 2026
DA  - 2026/10/03
PB  - arXiv
DO  - 10.48550/arxiv.2609.14909
UR  - https://arxiv.org/abs/2609.14909
AB  - The polynomial Hardy--Littlewood inequalities bound coefficient norms of homogeneous polynomials on ℓ_p balls by their supremum norms, with constants independent of the dimension. Their p=∞ endpoint is the Bohnenblust--Hille inequality. A longstanding obstruction to connecting their constants is the polarization loss in the usual multilinear approach. We overcome this obstruction by proving D_m,p_m≤(1+o(1))D_m whenever p_m/m^2→∞, where D_m,p and D_m are the optimal complex polynomial Hardy--Littlewood and Bohnenblust--Hille constants. At the opposite boundary, we determine the optimal endpoint constant exactly, D_m,m=m. We then prove the full first-order asymptotic D_m,p_m∼ m whenever p_m≥ m and p_m−m→0, and the exact polynomial growth D_m,p_m=m^1+o(1) on the wider scale p_m−m=o((m/log m)^1/3).
ER  -

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