end point estimates for the bilinear circular averaging operator

We prove that the bilinear circular averaging operator is bounded from $L^1(\mathbb{R})\times L^1(\mathbb{R})$ to $L^{1\mathrm{/}2,\mathrm{\infty }}(\mathbb{R})$. 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 $L^{1\mathrm{/}2}(\mathbb{R})$. The weak estimate is proved by a dyadic decomposition of the inputs and a summation over interval pairs near each quadratic fold.