The Bilinear Hilbert Transform and Sub-bilinear Maximal Function Along Curves

Multi-linear operators play an important role in analysis due to their multiple connections with and applications to other mathematical areas such as ergodic theory, elliptic regularity, and other problems in partial differential equations.

Within the area of multi-linear operators, powerful methods were developed originating from the problem of the almost everywhere convergence of Fourier series. Indeed, in their work, Carleson and Fefferman lay the foundation of time-frequency analysis. By further refining their methods, M. Lacey and C. Thiele proved the boundedness of the classical bilinear Hilbert transform for a suitable range of Hölder indices.

In this thesis, we consider the general boundedness properties of the bilinear Hilbert transform and the sub-bilinear maximal function along a suitable family of curves.

In the first part of our work, we present a short proof of the maximal boundedness range for the sub-bilinear maximal function along non-flat curves, giving a unified treatment of both the singular and the maximal operators.

In the second part, we discuss the boundedness of these operators along hybrid curves. This work aims to present a unified perspective that treats the case obtained by joining the zero-curvature features of the operators along flat curves with the non-zero curvature features along non-flat curves.