An extension on Trilinear Hilbert transform

Divergence of Combinatorial Averages and the Unboundedness of the Trilinear Hilbert Transform

Abstract. We consider multilinear averages in ergodic theory and harmonic analysis and prove their divergence in some range of L spaces. This contrasts with the positive behavior exhibited by these averages in a different range, as proved in [5]. We also prove that the trilinear Hilbert transform is unbounded in a similar range of L spaces. The underlying principle behind these constructions is...

On the Bilinear Hilbert Transform

In joint work with C. Thiele, the author has shown that A. Calderón’s bilinear Hilbert transform extends to a bounded operator on certain products of L spaces. This article illustrates the method of proof by giving a complete proof of an inequality which is slightly weaker than the original conjecture of Calderón concerning this transform. 1991 Mathematics Subject Classification: 42A50

An inequality for the hilbert transform.

Comments on Hilbert Transform Based Signal Analysis

The use of the Hilbert transform for time/frequency analysis of signals is briefly considered. While the Hilbert transform is based on arbitrary continuous signals, most practical signals are digitially sampled and time-limited. To avoid aliasing in the sampling process the signals must also be bandlimited. It is argued that it is reasonable to consider such sampled signals as periodic (this is...

On the Two Dimensional Bilinear Hilbert Transform

We investigate the Bilinear Hilbert Transform in the plane and the pointwise convergence of bilinear averages in Ergodic theory, arising from Z actions. Our techniques combine novel one and a half dimensional phase-space analysis with more standard one dimensional theory.