Bilinear rough singular integrals under a fractional geometric condition

We establish the Banach-range boundedness of bilinear rough singular integral operators, together with their maximal and maximally truncated forms, under the fractional geometric condition on the mean-zero angular kernel $$\underset{\xi \in {\mathbb{S}}^1}{\sup }{\int }_{{\mathbb{S}}^1}\frac{\mathrm{\mid }\mathrm{\Omega }(\theta )\mathrm{\mid }}{\mathrm{\mid }\theta \cdot \xi {\mathrm{\mid }}^a}d\sigma (\theta )<\mathrm{\infty },\frac{1}{2}<a<1.$$ This condition imposes integrability strictly weaker than the $L^q({\mathbb{S}}^1)(q>1)$ constraints considered by Grafakos, He, Honzík (Adv. Math., 2018), Dosidis and Slavíková (Math. Ann., 2024), while defining a class of functions that is neither contained in nor contains the classical Orlicz space $L(\log L{)}^{\alpha }({\mathbb{S}}^1)$ ($\alpha >1$). Our proof avoids traditional wavelet decompositions of the multiplier, instead using local Fourier series expansions of the input functions.