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

Existence of the Bedrosian Identity for Singular Integral Operators