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A new way of looking at distributional estimates; application for the bilinear Hilbert transform. (English) Zbl 1112.42006

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.

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

Citations:

Zbl 1112.42005
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