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Riesz transforms and related singular integrals. (English) Zbl 0847.42015

The paper under review gives important results for many of the most important singular integral operators. For example, the \(L^p\)-norm of the scalar Riesz transform is explicitly calculated \((|R_\ell|_{L^p}= \text{tan } {\pi\over 2\pi}\) if \(1< p\leq 2\) and \(= \text{cot } {\pi\over 2\pi}\) for \(2\leq p< \infty\)) or estimates for polynomials in the Riesz transform are obtained. Moreover, the complex Riesz transform is handled and its \(L^p\)-norm as well as the \(L^p\)-norms of its iterates are calculated. The results do have of course many applications in getting estimates for derivatives of functions. It should be noted that the results for the complex Riesz transform imply many results for operators like the Beurling-Ahlfors transform and related operators.
Reviewer: N.Jacob (Erlangen)

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
47G10 Integral operators
45P05 Integral operators
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