Existence of the Bedrosian Identity for Singular Integral Operators

TL;DR

This paper investigates the conditions under which the Bedrosian identity holds for general singular integral operators, providing a geometric characterization of support sets and identifying those for partial Hilbert transforms.

Contribution

It extends the Bedrosian identity from the Hilbert transform to general bounded singular integral operators on R^d, with a geometric criterion for support sets.

Findings

Support sets for the Bedrosian identity are characterized geometrically.

The support sets for partial Hilbert transforms are fully determined.

The Hilbert transform satisfies the Bedrosian identity only for specific support sets.

Abstract

The Hilbert transform $H$ satisfies the Bedrosian identity $H(fg)=fHg$ whenever the supports of the Fourier transforms of $f,g\in L^2(R)$ are respectively contained in $A=[-a,b]$ and $B=R\setminus(-b,a)$, $0\le a,b\le+\infty$. Attracted by this interesting result arising from the time-frequency analysis, we investigate the existence of such an identity for a general bounded singular integral operator on $L^2(R^d)$ and for general support sets $A$ and $B$. A geometric characterization of the support sets for the existence of the Bedrosian identity is established. Moreover, the support sets for the partial Hilbert transforms are all found. In particular, for the Hilbert transform to satisfy the Bedrosian identity, the support sets must be given as above.