Macroeconomics

Spectral density

Singular integral

Multiplier (economics)

Pure mathematics

Discrete mathematics

Statistics

Singularity

Economics

Computer science

Bilinear form

Embedding

Lipschitz continuity

Multilinear map

Corollary

Mathematics

Bilinear interpolation

Mathematical analysis

Artificial intelligence

Hilbert transform

Combinatorics

Subspace topology

Integral equation

Class (philosophy)

Journal of the London Mathematical Society

We establish a uniform domination of the family of trilinear multiplier forms with singularity over a one-dimensional subspace by positive sparse forms involving $L^p$-averages. This class includes the adjoint forms to the bilinear Hilbert transforms. Our result strengthens the $L^p$-boundedness proved in \cite{MTT} and entails as a corollary a rich multilinear weighted theory. In particular, we obtain $L^{q_1}(v_1) \times L^{q_2}(v_2)$-boundedness of the bilinear Hilbert transform when the weights $v_j$ belong to the class $A_{\frac{q+1}{2}}\cap RH_2$. Our proof relies on a stopping time construction based on newly developed localized outer-$L^p$ embedding theorems for the wave packet transform. In an Appendix, we show how our domination principle can be applied to recover the vector-valued bounds for the bilinear Hilbert transforms recently proved by Benea and Muscalu.

Domination of multilinear singular integrals by positive sparse forms