On trilinear singular Brascamp–Lieb integrals
Lars Becker, Polona Durcik, Fred Yu‐Hsiang LinAbstract
We classify trilinear singular Brascamp–Lieb forms, extending a result of Demeter and Thiele. We use results from the representation theory of finite‐dimensional algebras, namely the classification of indecomposable representations of the four subspace quiver. Our classification lays out a roadmap for proving bounds for all higher‐dimensional bilinear Hilbert transforms. As a further step toward this goal, we prove new bounds for a general class of forms that arises as a natural next candidate from our classification. Moreover, we prove conditional bounds for forms associated with mutually related representations. For this purpose, we develop a method of rotations to decompose any homogeneous ‐dimensional singular integral kernel into ‐dimensional kernels on hyperplanes.