On the family of singular Brascamp–Lieb inequalities with dimension datum (1,2,2,1)

Motivated by the triangular Hilbert transform, we classify a certain family of singular Brascamp–Lieb forms which we associate with the dimension datum (1,2,2,1). We determine the exact range of Lebesgue exponents, for which one has singular Brascamp–Lieb inequalities within this family. The remaining observations concern counter examples to boundedness. We compare with a counter-example showing that the triangular Hilbert form does not satisfy singular Brascamp–Lieb bounds in the endpoints.