Abstract: We study the averaging operators associated with curved models of the triangular Hilbert transform. Our main result provides purely geometric criteria on curves that guarantee smoothing inequalities for the associated averaging operators. Connections to o-minimal theory are also explored. In particular, uniformity of the estimates is derived via tools from o-minimal theory. Such uniformity enables various applications in bilinear singular integral theory, maximal operator theory, and pattern finding. As one specific example, we prove the boundedness of the curved models of the triangular Hilbert transform for a wide class of curves.

 

The talk is based on joint work in preparation with Fred Yu-Hsiang Lin.

 

Organizer: Chun-Yen Shen (NTU)