In 1966, Carleson proved the a.e. convergence of Fourier series of L2 functions. In 1973, Fefferman gave another proof. Until 2000, three years after Lacey and Thiele proved the boundedness of bilinear Hilbert transform, they just found out that the Carleson operator (whose boundedness implies a.e. convergence of Fourier series) shares the same modulation invariance with bilinear Hilbert transform, and then they gave another simplified proof for a.e. convergence of Fourier series.

 

The work of Lacey and Thiele open a road to analyze these operators which admit some modulation invariance which we cannot well control by using some powerful classic harmonic analysis tools such as the Carderon Zygmund theorem and Littlewood Paley theorem. There are many achievment done by using this tool. For example, Grafakos and Li showed the uniform bound of bilinear Hilbert transform, Tao and Muscalu and Thiele showed the bound of some muitilinear operators with singular multiplier.

 

In this talk, we will briefly explain the basic operations and estimation in time frequency analysis. Then we sketch the proof of a.e. convergence of Fourier series of l2 function and boundedness of bilinear Hilbert transform. At last, if there are still some time, we will introduce some recent works in this field.