Endpoint Criteria for One-Dimensional Bilinear Rough Singular Integrals

Abstract

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 $\hat{I}\text{c◯}\in L^1({\mathbb{S}}^1)$, the finite-part angular multiplier associated with $T_{\hat{I}}\text{c◯}$ has bounded variation if and only if the antipodal even part of $\hat{I}\text{c◯}$ 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 $\hat{I}\text{c◯}\in L\log L({\mathbb{S}}^1)$, then $T_{\hat{I}}\text{c◯}$ 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 $\hat{I}\text{c◯}\in {\mathcal{K}}_{1/2,{\hat{I}}^2}({\mathbb{S}}^1)$, provided that ${\hat{I}}^2>\frac{3}{2}\max \{p_1,p_1^{\prime },p_2,p_2^{\prime }\}-1.$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.

40 pages