Sharp lower bounds for the volumes of nodal and positivity sets of harmonic functions
Abstract
Let B=B(p, 1)⊂ℝn be a unit ball and n≥ 3. We prove that there are positive constants c and C, depending only on n, such that every non-zero real-valued harmonic function u: 4B→ ℝ with u(p)=0 satisfies ℋn−1({u=0}∩ 2B)≥ C𝒩, and ℋn({u>0}∩ 1/2B) ≥ c(log(1+𝒩))1−n, where 𝒩 is the doubling index defined by 𝒩=log2supB|u|/sup1/2B|u|. The first estimate confirms a folklore conjecture on the nodal volume of harmonic functions, and its linear dependence on 𝒩 is optimal. The second estimate extends the planar result of Nazarov, Polterovich, and Sodin to higher dimensions, and the logarithmic order is optimal. As a further consequence of the ideas developed in the proof, we obtain an alternative proof of Nadirashvili's conjecture that does not rely on the multiscale analysis.
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BibTeX
@misc{wang2026sharp,
title = {{Sharp lower bounds for the volumes of nodal and positivity sets of harmonic functions}},
author = {Zhehui Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11444},
eprint = {2609.11444},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11444},
}
RIS
TY - UNPB
TI - Sharp lower bounds for the volumes of nodal and positivity sets of harmonic functions
AU - Zhehui Wang
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.11444
UR - https://arxiv.org/abs/2609.11444
AB - Let B=B(p, 1)⊂ℝ^n be a unit ball and n≥ 3. We prove that there are positive constants c and C, depending only on n, such that every non-zero real-valued harmonic function u: 4B→ ℝ with u(p)=0 satisfies ℋ^n−1({u=0}∩ 2B)≥ C𝒩, and ℋ^n({u>0}∩ 1/2B) ≥ c(log(1+𝒩))^1−n, where 𝒩 is the doubling index defined by 𝒩=log_2sup_B|u|/sup_1/2B|u|. The first estimate confirms a folklore conjecture on the nodal volume of harmonic functions, and its linear dependence on 𝒩 is optimal. The second estimate extends the planar result of Nazarov, Polterovich, and Sodin to higher dimensions, and the logarithmic order is optimal. As a further consequence of the ideas developed in the proof, we obtain an alternative proof of Nadirashvili's conjecture that does not rely on the multiscale analysis.
ER -