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

We prove bounds in the strict local $L^2({\mathbb{R}}^d)$ range for trilinear Fourier multiplier forms with a $d$-dimensional singular subspace. Given a fixed parameter $K\ge 1$, 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 $L^2$ range, and it implies the uniform bounds for two-dimensional bilinear Beurling transforms, which are new, in the same range.