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


Authors: James E. Daly and Keith Phillips
Journal: Proc. Amer. Math. Soc. 106 (1989), 685-696
MSC: Primary 42B15; Secondary 42B20, 42B30
DOI: https://doi.org/10.1090/S0002-9939-1989-0957264-0
MathSciNet review: 957264
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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_m}f = {\mathcal{F}^{ - 1}}(m\mathcal{F}(f))$, is bounded on $ {H^1}({{\mathbf{R}}^2})$, then it is proved that $ {S^*}m$ has bounded variation on the circle, where the Fourier transform of $ S$ on the circle is $ \widehat{S}(n) = {( - {\text{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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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1989-0957264-0
Keywords: Singular integrals, homogeneous distributions, multipliers, Hardy spaces
Article copyright: © Copyright 1989 American Mathematical Society

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