PreprintCC0arXiv
Sharp estimates for imaginary powers of discrete Laplacians
Abstract
Let Δd be the nonnegative discrete Laplacian on ℤd. We prove that, for 0<p≤1, |Δdiγ|HpΔd→ HpΔd ≃ (1+|γ|)d(1/p−1/2), and, for 1<p<∞, |Δdiγ|ℓp→ℓp ≃ (1+|γ|)d|1/p−1/2|. The estimates are sharp with respect to the exponent of 1+|γ|. We also explain why the general spectral multiplier theorem and the usual Calderón--Zygmund approach do not give the sharp power.
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BibTeX
@misc{bui2026sharp,
title = {{Sharp estimates for imaginary powers of discrete Laplacians}},
author = {The Anh Bui and The Quan Bui and Thanh Tinh Cao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27385},
eprint = {2610.27385},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27385},
note = {CC0},
}
RIS
TY - UNPB
TI - Sharp estimates for imaginary powers of discrete Laplacians
AU - The Anh Bui
AU - The Quan Bui
AU - Thanh Tinh Cao
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.27385
UR - https://arxiv.org/abs/2610.27385
AB - Let Δ_d be the nonnegative discrete Laplacian on ℤ^d. We prove that, for 0<p≤1, |Δ_d^iγ|_H^p_Δ_d→ H^p_Δ_d ≃ (1+|γ|)^d(1/p−1/2), and, for 1<p<∞, |Δ_d^iγ|_ℓ^p→ℓ^p ≃ (1+|γ|)^d|1/p−1/2|. The estimates are sharp with respect to the exponent of 1+|γ|. We also explain why the general spectral multiplier theorem and the usual Calderón--Zygmund approach do not give the sharp power.
ER -