On the Classification of Homogeneous Multipliers Bounded on 77'(r2)
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چکیده
Necessary and sufficient conditions for Calderon-Zygmund singular integral operators to be bounded operators on //' (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 = y_1C"^"(/)), is bounded on //'(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))"+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 .
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تاریخ انتشار 2010