Abstract
We prove that the bilinear Hilbert transforms and maximal functions along certain general plane curves are bounded from \(L^2({{\mathbb {R}}})\times L^2({{\mathbb {R}}})\) to \(L^1({{\mathbb {R}}})\).
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Notes
In the problems considered in this paper, we can always remove the constant \(a\) from the definition of \(\Gamma \) by a translation argument, hence there is no need to specify the dependence of \(\Gamma \) on \(a\) and we will always let \(a=0\).
The condition (2.2) implies that there exist constants \(K_1, K_2>0\) such that
$$\begin{aligned} \big |\gamma '(\epsilon )\big |\le K_1|\epsilon |^{c_1}\quad \mathrm{for}\,\, 0<|\epsilon |<c_0 \end{aligned}$$or
$$\begin{aligned} \big |\gamma '(\epsilon )\big |\ge K_2|\epsilon |^{-c_1}\quad \mathrm{for}\,\, 0<|\epsilon |<c_0. \end{aligned}$$See also Lie [13, p. 4] Observation (6) and (7).
\(B(t, r)\) denotes the interval \((t-r, t+r)\).
We actually do not need the condition (2.4) for this proposition.
In this part we mainly follow the argument contained in Section 6 of the preprint arXiv:0805.0107 and make necessary modifications in order to adapt it to the current case.
In this part we mainly generalize the argument contained in Sections 8, 10, and 11 of the preprint arXiv:0805.0107. In particular we apply Lemma 3.3 with a carefully-chosen function set \(U(\mathbf I )\).
We do not need the condition (2.4) for this proposition.
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Acknowledgments
We would like to express our sincere gratitude to Xiaochun Li for valuable advice and helpful discussion.
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Guo, J., Xiao, L. Bilinear Hilbert Transforms Associated with Plane Curves. J Geom Anal 26, 967–995 (2016). https://doi.org/10.1007/s12220-015-9580-z
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DOI: https://doi.org/10.1007/s12220-015-9580-z