论文标题
$ l^p（\ mathbb {r}^2）$ - 希尔伯特（Hilbert
$L^p(\mathbb{R}^2)$-boundedness of Hilbert Transforms and Maximal Functions along Plane Curves with Two-variable Coefficients
论文作者
论文摘要
在本文中，对于一般平面曲线$γ$满足了一些合适的平滑度和曲率条件，我们获得了单个环形$ l^p（\ mathbb {r}^2）$ - 希尔伯特的界限 - 希尔伯特（Hilbert $ l^p（\ mathbb {r}^2）$ - 相应最大函数的界限$ m^\ infty_ {u，γ} $，其中$ p> 2 $和$ u $是可测量的函数。 $ p $上的范围很清晰。此外，在$ 1 <p \ leq 2 $的情况下，在其他条件下，用$γ（2 \ varepsilon_0）\ varepsilon_0）\ leq 1/4 \ | u \ | $ h^{\ varepsilon_0} _ {u，γ} $和$ m^{\ varepsilon_0} _ {u，γ} $。
In this paper, for general plane curves $γ$ satisfying some suitable smoothness and curvature conditions, we obtain the single annulus $L^p(\mathbb{R}^2)$-boundedness of the Hilbert transforms $H^\infty_{U,γ}$ along the variable plane curves $(t,U(x_1, x_2)γ(t))$ and the $L^p(\mathbb{R}^2)$-boundedness of the corresponding maximal functions $M^\infty_{U,γ}$, where $p>2$ and $U$ is a measurable function. The range on $p$ is sharp. Furthermore, for $1<p\leq 2$, under the additional conditions that $U$ is Lipschitz and making a $\varepsilon_0$-truncation with $γ(2 \varepsilon_0)\leq 1/4\|U\|_{\textrm{Lip}}$, we also obtain similar boundedness for these two operators $H^{\varepsilon_0}_{U,γ}$ and $M^{\varepsilon_0}_{U,γ}$.
