论文标题
双线性最大功能与退化表面相关
Bilinear maximal functions associated with degenerate surfaces
论文作者
论文摘要
我们研究$ l^{p} \ times l^{q} \ rightarrow l^{r} $ - （sub）双线性最大功能与退化超曲面相关的（sub）双线性最大函数。首先，我们获得了指数范围$ p，q，r $（除某些边界线案例除外）双线性最大功能的最大界限。 \ Mathbb {r}^{n}：| y |^{l_ {1}}+| z | |^{l_ {2}} = 1 \ big \} $，$（l_ {1}，l_ {2}，l_ {2}） 我们的结果表明，与（sub）线性最大算子与HyperSurfaces相关的（sub）线性最大算子的$ l^p $结合性相反，非变化的高斯曲率还不够好，以表征最好的最大界限。其次，我们考虑与$ \ Mathbb r^2 $中有限类型曲线关联的双线性最大函数，并获得最大结合的完整表征。我们还证明了上述结果的多线性概括。
We study $L^{p}\times L^{q}\rightarrow L^{r}$-boundedness of (sub)bilinear maximal functions associated with degenerate hypersurfaces. First, we obtain the maximal bound on the sharp range of exponents $p,q,r$ (except some border line cases) for the bilinear maximal functions given by the model surface $\big\{(y,z)\in\mathbb{R}^{n}\times \mathbb{R}^{n}:|y|^{l_{1}}+|z|^{l_{2}}=1\big\}$, $(l_{1},l_{2})\in [1,\infty)^2$, $n\ge 2$. Our result manifests that nonvanishing Gaussian curvature is not good enough, in contrast with $L^p$-boundedness of the (sub)linear maximal operator associated to hypersurfaces, to characterize the best possible maximal boundedness. Secondly, we consider the bilinear maximal function associated to the finite type curve in $\mathbb R^2$ and obtain a complete characterization of the maximal bound. We also prove multilinear generalizations of the aforementioned results.
