Bilinear rough singular integrals under a fractional geometric condition

Abstract

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 \[ \sup_{ξ\in \mathbb{S}^{1}}\int_{\mathbb{S}^{1}} \frac{|Ω(θ)|}{|θ\cdot ξ|^{a}} \, dσ(θ) < \infty, \qquad \frac12 < a < 1. \] 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 $L(\log L{)}^{\hat{I}}\pm ({\mathbb{S}}^1)$ ($\hat{I}\pm >1$). Our proof avoids traditional wavelet decompositions of the multiplier, instead using local Fourier series expansions of the input functions.

19 pages