On the classification of homogeneous multipliers bounded on 𝐻¹(𝑅²)

Daly, James E.; Phillips, Keith

doi:10.1090/S0002-9939-1989-0957264-0

On the classification of homogeneous multipliers bounded on $H^ 1(\textbf {R}^ 2)$
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Proc. Amer. Math. Soc. 106 (1989), 685-696 Request permission

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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