L^p-Boundedness of the Hilbert Transform

Abstract: 

Arising in 1905, the Hilbert transform is arguably one of the most interesting operators in mathematical analysis. The canonical example of a singular integral operator, it has motivated much of modern harmonic analysis and is still of great interest in itself. The LpLp-boundedness for 1<p<∞1<p<∞ of the Hilbert transform is proven using an identity attributed to Cotlar. A noncommutative analogue of the Hilbert transform preserves many of the original operator's desirable properties, including boundedness on the noncommutative Lebesgue spaces for the same choices of pp. This is seen using a remarkably similar method to that of the commutative case.