Double Hilbert Transforms

The Hilbert transform is a classical example of a singular integral operator. It is weak-type 1-1 and Lp bounded for all finite p >1. The Hilbert transform can be thought of as a convolution operator whose kernel is a tempered distribution and this has led to the study of weak-type 1-1 and Lp bounds of convolution operators that are more general than the Hilbert transform. It is known that convolution operators whose kernel satisfy certain smoothness condition and cancellation condition obey bounds similar to that of Hilbert transform. However, operators whose kernel are supported along curves or surfaces are hard to study for such type of bounds. In the recent past a lot of work has been published along this line and the problems considered in this book are motivated from some of these recent results. The book mainly discusses the Lp bounds of double Hilbert transforms in one and three dimensions that are associated to the polynomials in two real variables. It is fascinating to realize how some of the monomials of a given polynomial play a dominating role in deciding the boundedness of the associated double Hilbert transform.