Rough averages of triangular Hilbert transforms

2026/07/22 by Fred Yu-Hsiang Lin, Lenka Slavíková
#math.CA

paper · pdf

Abstract

We study a bilinear singular integral operator obtained by taking rough averages of certain directional variants of the triangular Hilbert transform. This operator can be interpreted as the twisted paraproduct with a rough homogeneous kernel. Under a balance condition on the L^q integrability of the kernel on the unit sphere, we establish a range of L^p₁ × L^p₂ → L^p bounds for this operator. Our results are optimal within the known boundedness range for the twisted paraproduct with a smooth kernel.

Citations

• Smoothing inequalities for corner-type bilinear averages: geometric characterization and applications

Related

• Quantitative weighted estimates for rough homogeneous singular integrals
• An alternate approach to bilinear rough singular integrals
• Sparse bounds for maximal rough singular integrals via the Fourier transform
• Improved estimates for bilinear rough singular integrals
• A study guide for "Trilinear smoothing inequalities and a variant of the triangular Hilbert transform"