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$L^2$ Boundedness of Hilbert Transforms along Variable Flat Curves (1808.01447v3)

Published 4 Aug 2018 in math.CA

Abstract: In this paper, the $L2$ boundedness of the Hilbert transform along variable flat curve $(t,P(x_1)\gamma(t))$ $$H_{P,\gamma}f(x_1,x_2):=\mathrm{p.\,v.}\int_{-\infty}{\infty}f(x_1-t,x_2-P(x_1)\gamma(t))\,\frac{\textrm{d}t}{t},\quad \forall\, (x_1,x_2)\in\mathbb{R}2,$$ is studied, where $P$ is a real polynomial on $\mathbb{R}$. A new sufficient condition on the curve $\gamma$ is introduced.

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