\(L^ p\) estimates for maximal functions and Hilbert transforms along flat convex curves in \(R^ 2\). (English) Zbl 0588.44007

Carlsson, Hasse; Christ, Michael; Cordoba, Antonio; Duoandikoetxea, Javier; Rubio de Francia, Jose L.; Vance, James; Wainger, Stephen; Weinberg, David

Bull. Am. Math. Soc., New Ser. 14, 263-267 (1986).

Various interesting results concerning \(L^ p\) estimates for maximal functions and Hilbert transforms along convex curves in \(R^ 2\) are established. The main result of the paper is contained in the following Theorem: Let \(\Gamma,\Gamma_ e,\Gamma_ o\) satisfy the ”doubling property”. Then \(\| M_{\Gamma}f\|_ p\leq C\| f\|_ p\) for \(1<p\leq \infty\), and \(\| H_{\Gamma e}f\|_ p+\| H_{\Gamma_ 0}f\|_ p\leq C\| f\|_ p\) for \(1<p<\infty\) (various symbols used here are explained fully in the paper). More precisely, the latter assertion is that the operators \(H_{\Gamma}\), initially defined only for test functions, extend to bounded operators on \(L^ p\).

Reviewer: S.P.Goyal

MSC:

44A15

Special integral transforms (Legendre, Hilbert, etc.)

Keywords:

Paley-Littlewood decomposition; maximal functions; Hilbert transforms

References:

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