Endpoint Criteria for One-Dimensional Bilinear Rough Singular Integrals

We prove endpoint theorems for one-dimensional bilinear rough singular integrals. Our starting point is a sharp structural characterization of the associated angular multiplier. For every mean-zero $\Omega\in L^1(\mathbb{S}^1)$, the finite-part angular multiplier associated with $T_\Omega$ has bounded variation if and only if the antipodal even part of $\Omega$ belongs to $H^1(\mathbb{S}^1)$. This characterization identifies the precise rotational regularity required in the one-dimensional bilinear setting. It also yields a Stieltjes decomposition compatible with uniform estimates for the bilinear Hilbert transform. We then establish two boundedness criteria under critical kernel assumptions. First, if $\Omega\in L\log L(\mathbb{S}^1)$, then $T_\Omega$ is bounded from $L^{p_1}(\mathbb{R})\times L^{p_2}(\mathbb{R})\text{to} L^p(\mathbb{R})$ whenever $1<p_1,p_2,p<\infty$ and $\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}.$ Moreover, the logarithmic exponent $1$ is optimal within the scale $L(\log L)^A$. Second, at the critical directional index, the same boundedness holds for $\Omega\in\mathcal{K}_{1/2,\beta}(\mathbb{S}^1)$, provided that $\beta>\frac{3}{2}\max\bigl\{p_1,p_1',p_2,p_2'\bigr\}-1.$The two critical kernel classes are incomparable. The $L\log L$ result is obtained by reducing the multiplier to a finite-part angular profile of bounded variation. The directional result follows from endpoint Fourier decay, product wavelet decompositions, and interpolation.