The maximal Hilbert transform along nonconvex curves
School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, Henan, 454003, P.R. China
Journal of Inequalities and Applications 2012, 2012:191 doi:10.1186/1029-242X-2012-191Published: 31 August 2012
The Hilbert transform along curves is defined by the principal value integral. The pointwise existence of the principal value Hilbert transform can be educed from the appropriate estimates for the corresponding maximal Hilbert transform. By using the estimates of Fourier transforms and Littlewood-Paley theory, we obtain -boundedness for the maximal Hilbert transform associated to curves , where , P is a real polynomial and γ is convex on . Then, we can conclude that the Hilbert transform along curves exists in pointwise sense.
MSC: 42B20, 42B25.