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Hilbert transforms along curves. I: Nilpotent groups. (English) Zbl 0593.43011

In modern complex analysis there is an interesting but complicated topic to investigate the properties of general integral operators involving singular kernels supported on submanifolds of arbitrary manifolds. For references and applications see E. M. Stein [Singular integrals and differentiability properties of functions (1970; Zbl 0207.135); Invent. Math. 74, 63-83 (1983; Zbl 0522.43007)], and E. M. Stein and S. Wainger [Bull. Am. Math. Soc. 84, 1239-1295 (1978; Zbl 0393.42010)].
The main result of the paper under review is the following theorem concerning nilpotent groups. For any homogeneous nilpotent group G and any exponent \(1<p<\infty\) there exists \(C=C(p,G)<\infty\) such that \(\| M_{\gamma}f\|_{L^ p}\leq C \| f\|_{L^ p}\) for any odd homogeneous curve and \(\gamma\in G\) and for all \(f\in L^ p\). Here \[ M_{\gamma}f(x)=\sup r^{-1}\int_{| t| \leq r}| f(x\cdot \gamma (t)^{-1})| dt \] is a maximal function along \(\gamma\). Moreover for the Hilbert transform along \(\gamma\) (with some definition of principal-value of integral) \[ H_{\gamma}f(x)=principal- value\int^{\infty}_{-\infty}f(x\cdot \gamma (t)^{-1})t^{-1\quad} dt \] the inequality \(\| H_{\gamma}f\|_ p\leq C \| f\|_ p\) holds.
Reviewer: A.Venkov

MSC:

43A85 Harmonic analysis on homogeneous spaces
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
44A15 Special integral transforms (Legendre, Hilbert, etc.)
42B25 Maximal functions, Littlewood-Paley theory
31B99 Higher-dimensional potential theory
22E25 Nilpotent and solvable Lie groups
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