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Schwarz-contractivity of the Poisson operator
Abstract
We say that the Poisson operator Pr is Schwarz-contractive from a space X to a space Y if |Pr|X→ Y≤ r holds for all 0<r<1. The classical Schwarz lemma is the case H∞0→ H∞, the subscript indicating zero mean. We are concerned with the non-holomorphic case L∞0 → Lp. For real functions Schwarz-contractivity holds up to the sharp exponent pℝ=4.109…. For complex functions we prove it for p≤ 3 and conjecture that the critical exponent is $4$.
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BibTeX
@misc{kovalev2026schwarzcontractivity,
title = {{Schwarz-contractivity of the Poisson operator}},
author = {Leonid V. Kovalev},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28938},
eprint = {2609.28938},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28938},
}
RIS
TY - UNPB
TI - Schwarz-contractivity of the Poisson operator
AU - Leonid V. Kovalev
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.28938
UR - https://arxiv.org/abs/2609.28938
AB - We say that the Poisson operator P_r is Schwarz-contractive from a space X to a space Y if |P_r|_X→ Y≤ r holds for all 0<r<1. The classical Schwarz lemma is the case H^∞_0→ H^∞, the subscript indicating zero mean. We are concerned with the non-holomorphic case L^∞_0 → L^p. For real functions Schwarz-contractivity holds up to the sharp exponent p_ℝ=4.109…. For complex functions we prove it for p≤ 3 and conjecture that the critical exponent is $4$.
ER -