Year: 2026
Type: preprint
Abstract: We prove that the bilinear circular averaging operator is bounded from $L^1(\mathbb{R}) \times L^1(\mathbb{R})$ to $L^{1/2,\infty}(\mathbb{R})$. The corresponding strong estimate fails, even when the inputs are nonnegative, smooth, and compactly supported. We also construct nonnegative $f,g \in L^1(\mathbb{R})$ for which the average does not belong to $L^{1/2}(\mathbb{R})$. The weak estimate is proved by a dyadic decomposition of the inputs and a summation over interval pairs near each quadratic fold.
Source: arXiv (Cornell University)
Author: Wei, Xuyi
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223032216