  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.
