A hierarchical structure within Harmonic Analysis

In this talk we will discuss a natural hierarchical structure that governs a vast teritory within the classical harmonic analysis area: (1) non-zero curvature problems: this usually involves the study of objects that lack (generalized) modulation invariance; promi- nent examples within this class are the “curved” Carleson oper- ator and the linear and bilinear Hilbert transforms along “non- flat” curves. (2) zero-curvature problems: this focuses on objects that, on top of the standard dilation and translation symmetries, also ex- hibit a (generalized) modulation invariance; prominent exam- ples within this class are the classical Carleson operator and the Bilinear Hilbert transform. (3) hybrid problems: this refers to the study of objects that share both zero and non-zero curvature features; prominent examples within this class are the Polynomial Carleson operator and the newly introduced Bilinear Hilbert–Carleson operator as well as the “hybrid” Trilinear Hilbert transform. In the first part of the talk we will elaborate on the main concepts and problems, with an emphasis on an intuitive and at the same time panoramic view of the subject. The second part of the talk will be centered around some very recent (joint) contributions of the speaker within the realm of the hybrid prob- lem category. In this context, we will discuss the key role played by the LGC-methodology in the resolution of some difficult questions regard- ing the behavior of suitable quadrilinear forms that serve as models for the celebrated open question on the boundedness of the Trilinear Hilbert transform.