Uniform bounds for bilinear symbols with linearK-quasiconformally embedded singularity

We prove bounds in the strict local L 2 ( 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.1. Introduction 2293 2. Outline of the proof 2297 3. Proof of Proposition 2.3: paraproduct 2301 4. Proof of Proposition 2.7: tree estimate 2304 5. Proof of Proposition 2.8: tree selection 2307 6. Proof of Proposition 2.2: tensorized model form 2319 7. Proof of Theorem 1.1