In this paper, the $$L^2$$ L 2 boundedness of the Hilbert transform along variable flat curve $$(t,P(x_1)\gamma (t))$$ ( t , P ( x 1 ) γ ( t ) ) $$\begin{aligned} H_{P,\gamma }f(x_1,x_2):=\mathrm {p.\,v.}\int _{-\infty }^{\infty }f(x_1-t,x_2-P(x_1)\gamma (t))\,\frac{\text {d}t}{t},\quad \forall \, (x_1,x_2)\in {\mathbb {R}}^2, \end{aligned}$$ H P , γ f ( x 1 , x 2 ) : = p . v . ∫ - ∞ ∞ f ( x 1 - t , x 2 - P ( x 1 ) γ ( t ) ) d t t , ∀ ( x 1 , x 2 ) ∈ R 2 , is studied, where P is a real polynomial on $${\mathbb {R}}$$ R . A new sufficient condition on the curve $$\gamma $$ γ is introduced.

中文翻译：

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$$L^2$$ 沿可变平坦曲线的希尔伯特变换的有界性

在本文中，希尔伯特变换沿可变平坦曲线 $$(t,P(x_1)\gamma (t))$$ ( t , P ( x 1 ) γ ( t ) ) $$\begin{aligned} H_{P,\gamma }f(x_1,x_2):=\mathrm {p.\,v.}\int _{-\infty }^{\infty }f( x_1-t,x_2-P(x_1)\gamma (t))\,\frac{\text {d}t}{t},\quad \forall \, (x_1,x_2)\in {\mathbb {R }}^2, \end{aligned}$$ HP , γ f ( x 1 , x 2 ) : = p 。诉。∫ - ∞ ∞ f ( x 1 - t , x 2 - P ( x 1 ) γ ( t ) ) dtt , ∀ ( x 1 , x 2 ) ∈ R 2 ，被研究，其中P是$$上的实多项式{\mathbb {R}}$$ R . 引入了曲线 $$\gamma $$ γ 上的一个新的充分条件。