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On the Classification of Homogeneous Multipliers Bounded on $H^1 (\mathbf{R}^2)$

James E. Daly and Keith Phillips
Proceedings of the American Mathematical Society
Vol. 106, No. 3 (Jul., 1989), pp. 685-696
DOI: 10.2307/2047423
Stable URL: http://www.jstor.org/stable/2047423
Page Count: 12
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
On the Classification of Homogeneous Multipliers Bounded on $H^1 (\mathbf{R}^2)$
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Abstract

Necessary and sufficient conditions for Calderon-Zygmund singular integral operators to be bounded operators on $H^1(\mathbf{R}^2)$ are investigated. Let $m$ be a bounded measurable function on the circle, extended to $\mathbf{R}^2$ by homogeneity $(m(rx) = m(x))$. If the Calderon-Zygmund singular integral operator $T_m$, defined by $T_mf = \mathscr{F}^{-1}(m\mathscr{F}(f))$, is bounded on $H^1(\mathbf{R}^2)$, then it is proved that $S^\ast m$ has bounded variation on the circle, where the Fourier transform of $S$ on the circle is $\widehat{S}(n) = (-\operatorname{isgn}(n))^{n + 1}$. 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 $\mathbf{R}^2$.

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