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Necessary and sufficient conditions for Calderon-Zygmund singular integral operators to be bounded operators on H1 (R2) are investigated. Let m be a bounded measurable function on the circle, extended to R2 by homogeneity (m(rx) = m(x)) . If the Calderon-Zygmund singular integral operator Tm, defined by Tmf = F-I(mF(f)), is bounded on HI(R2), then it is proved that S* m has bounded variation on the circle, where the Fourier transform of S on the circle is S(n) = (isgn(n))n+1 . This implies that m must have an absolutely convergent Fourier series on the circle, and other relations on the Fourier series of m. Partial converses are also given. The problems are formulated in terms of distributions on the circle and on R2.

On the classification of homogeneous multipliers bounded on $H\sp 1({\bf R}\sp 2)$