Uniform bounds for bilinear symbols with linear K-quasiconformally embedded singularity

Abstract

We prove bounds in the strict local L 2 ( R d ) range for trilinear Fourier multiplier forms with a d -dimensional singular subspace. Given a fixed parameter K = 1 , we treat multipliers with nondegenerate singularity that are push-forwards by K -quasiconformal matrices of suitable symbols. As particular applications, our result recovers the uniform bounds for the one-dimensional bilinear Hilbert transforms in the strict local L 2 range, and it implies the uniform bounds for two-dimensional bilinear Beurling transforms, which are new, in the same range.

The authors were funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under project numbers 390685813 (EXC 2047: Hausdorff Center for Mathematics) and 211504053 (CRC 1060: Mathematics of Emergent Effects). Fraccaroli was supported by the Basque Government through the BERC 2022-2025 program and by the Ministry of Science and Innovation: BCAM Severo Ochoa accreditation CEX2021-001142-S / MICIN / AEI / 10.13039/501100011033. Saari was supported by Generalitat de Catalunya through the grant 2021-SGR-00087 and by the Spanish State Research Agency MCIN/AEI/10.13039/501100011033, Next Generation EU and by ERDF “A way of making Europe” through the grant RYC2021-032950-I, the grant PID2021-123903NB-I00 and the Severo Ochoa and Maria de Maeztu Program for Centers and Units of Excellence in R&D, grant number CEX2020-001084-M.

Peer Reviewed

Postprint (published version)