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end point estimates for the bilinear circular averaging operator
Abstract
We prove that the bilinear circular averaging operator is bounded from L1(ℝ) × L1(ℝ) to L1/2,∞(ℝ). The corresponding strong estimate fails, even when the inputs are nonnegative, smooth, and compactly supported. We also construct nonnegative f,g ∈ L1(ℝ) for which the average does not belong to L1/2(ℝ). The weak estimate is proved by a dyadic decomposition of the inputs and a summation over interval pairs near each quadratic fold.
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BibTeX
@misc{wei2026point,
title = {{end point estimates for the bilinear circular averaging operator}},
author = {Xuyi Wei},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24465},
eprint = {2610.24465},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24465},
}
RIS
TY - UNPB
TI - end point estimates for the bilinear circular averaging operator
AU - Xuyi Wei
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2610.24465
UR - https://arxiv.org/abs/2610.24465
AB - We prove that the bilinear circular averaging operator is bounded from L^1(ℝ) × L^1(ℝ) to L^1/2,∞(ℝ). The corresponding strong estimate fails, even when the inputs are nonnegative, smooth, and compactly supported. We also construct nonnegative f,g ∈ L^1(ℝ) for which the average does not belong to L^1/2(ℝ). The weak estimate is proved by a dyadic decomposition of the inputs and a summation over interval pairs near each quadratic fold.
ER -