Bilyk, Dmitriy; Grafakos, Loukas A new way of looking at distributional estimates; application for the bilinear Hilbert transform. (English) Zbl 1112.42006 Collect. Math. 2006, Spec. Iss., 141-169 (2006). This paper provides expository background for the restricted weak-type estimates that the authors obtained for the bilinear Hilbert transform in [J. Geom. Anal. 16, No. 4, 563–584 (2006; Zbl 1112.42005)]. In particular, a general distributional inequality along the lines of good-lambda inequalities, but exploiting characteristic functions, is established and used as a vehicle for refining the restricted weak-type \(L^1\) inequality for the Hilbert transform \(H\) in the sense of showing exponential decay of \(| \{x:H(\chi_F)(x)>\lambda\}| \) as \(\lambda\to\infty\). The rest of the paper outlines the methods used in the work cited above. Reviewer: Joseph Lakey (Las Cruces) Cited in 1 ReviewCited in 4 Documents MSC: 42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type 46B70 Interpolation between normed linear spaces 44A15 Special integral transforms (Legendre, Hilbert, etc.) 46F99 Distributions, generalized functions, distribution spaces Keywords:bilinear Hilbert transform; distributional inequality; restricted weak-type inequality Citations:Zbl 1112.42005 PDFBibTeX XMLCite \textit{D. Bilyk} and \textit{L. Grafakos}, Collect. Math. 2006, 141--169 (2006; Zbl 1112.42006) Full Text: EuDML