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

We prove bounds in the strict local L²(Rᵈ) range for trilinear Fourier multiplier forms with a d-dimensional singular subspace. Given a fixed parameter K ≥ 1, we treat multipliers with non-degenerate 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 L² range, and it implies the uniform bounds for two-dimensional bilinear Beurling transforms, which are new, in the same range.