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Boundedness of nonregular bilinear pseudodifferential operators on variable Besov and Triebel--Lizorkin spaces
Abstract
In this article, we introduce two bilinear, non-regular symbol classes\linebreak BSℬmδ(r,μ,v;N,λ) and BSℱmδ(r,μ,v;N,λ), where m∈ℝ, 0≤δ<1, 0<μ,v≤∞, 1≤λ≤2, r>n/μ, and N>2n/λ, with μ<∞ for the latter class. For μ≥1, (1−δ)r≥ n/μ, and sufficiently large N, we establish boundedness of the associated bilinear pseudodifferential operators on Besov and Triebel--Lizorkin spaces with variable smoothness, integrability and summation exponents. To the best of the authors' knowledge, these operator estimates are new even on the classical Besov and Triebel--Lizorkin spaces with constant exponents. As an application, we establish the multiplication algebra property for both variable exponent spaces.
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BibTeX
@misc{bai2026boundedness,
title = {{Boundedness of nonregular bilinear pseudodifferential operators on variable Besov and Triebel--Lizorkin spaces}},
author = {Tengfei Bai and Jingshi Xu and Jinlai Zhu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25234},
eprint = {2610.25234},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25234},
}
RIS
TY - UNPB
TI - Boundedness of nonregular bilinear pseudodifferential operators on variable Besov and Triebel--Lizorkin spaces
AU - Tengfei Bai
AU - Jingshi Xu
AU - Jinlai Zhu
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.25234
UR - https://arxiv.org/abs/2610.25234
AB - In this article, we introduce two bilinear, non-regular symbol classes\linebreak BSℬ^m_δ(r,μ,v;N,λ) and BSℱ^m_δ(r,μ,v;N,λ), where m∈ℝ, 0≤δ<1, 0<μ,v≤∞, 1≤λ≤2, r>n/μ, and N>2n/λ, with μ<∞ for the latter class. For μ≥1, (1−δ)r≥ n/μ, and sufficiently large N, we establish boundedness of the associated bilinear pseudodifferential operators on Besov and Triebel--Lizorkin spaces with variable smoothness, integrability and summation exponents. To the best of the authors' knowledge, these operator estimates are new even on the classical Besov and Triebel--Lizorkin spaces with constant exponents. As an application, we establish the multiplication algebra property for both variable exponent spaces.
ER -