R440, Astronomy-Mathematics Building, NTU

(台灣大學天文數學館 440室)

The A2 Sharp Bound for the Hilbert Transform

Chih-Chieh Hung (NCTS)

Abstract:

In Harmonic Analysis, we concerned with the boundedness of operators such as Hilbert transform, Riesz transform, or more general operators: the CZOs (Calderon-Zygmund operators). In last century, we have already known the boundedness of CZO is Lp to Lp. Next we moved to the weighted version for the same questions, which has been solved in 1970s. In particular, due to some questions arising form PDEs and other areas, we need to know the sharp boundedness. This problem has been totally solved by Tuomas Hytonen in 2010 by using Haar shift to represent any CZO and some two weight theory results, but the proof is quite complicated. To find a simpler proof, E. Sawyer et al provide another approach for Hilbert transform. In this talk, we will introduce their approach and introduce some new results. Finally, we will give some applications in PDEs.