论文标题
$ l^p $最大界限和索博莱夫的两参数的规律性
$L^p$ maximal bound and Sobolev regularity of two-parameter averages over tori
论文作者
论文摘要
我们研究了由Tori $ \ Mathbb {T} _t} _t _t _t^{s}的两个参数家族，研究由平均运算符$ f \ to \ Mathcal {a} _t^s f $定义的最大函数的$ l^p $有界性。 （t+s \cosθ）\ coscct，\，（t+s \cosθ）\ sinϕ，\，s \sinθ）：θ，ϕ \ in [0,2π）\} $ in [0,2π）\} $ in $ c_0t> s> s> 0 $ for Some $ c_0 f而言（0,1）in（0,1）$。我们证明，当$ l^p $时，仅当$ p> 2 $时，相关的（两参数）最大功能在$ l^p $上有限。我们还获得了$ l^p $ - $ l^q $估计，用于本地最大运营商的额定范围，$ p，q $。此外，证明了尖锐的平滑估计值，包括操作员的急剧局部平滑估计值$ f \ to \ MATHCAL A_T^S F $和$ f \ to \ Mathcal A_T^{C_0T} F $。为了目的，我们利用波尔加恩（Demeter）对圆锥体和guth-wang-wang-zhang的脱钩不平等，对$ 2 $ dimensional Wave Operator的本地平滑估计值。
We investigate $L^p$ boundedness of the maximal function defined by the averaging operator $f\to \mathcal{A}_t^s f$ over the two-parameter family of tori $\mathbb{T}_t^{s}:=\{ ( (t+s\cosθ)\cosϕ,\,(t+s\cosθ)\sinϕ,\, s\sinθ): θ, ϕ\in [0,2π) \}$ with $c_0t>s>0$ for some $c_0\in (0,1)$. We prove that the associated (two-parameter) maximal function is bounded on $L^p$ if and only if $p>2$. We also obtain $L^p$--$L^q$ estimates for the local maximal operator on a sharp range of $p,q$. Furthermore, the sharp smoothing estimates are proved including the sharp local smoothing estimates for the operators $f\to \mathcal A_t^s f$ and $f\to \mathcal A_t^{c_0t} f$. For the purpose, we make use of Bourgain--Demeter's decoupling inequality for the cone and Guth--Wang--Zhang's local smoothing estimates for the $2$ dimensional wave operator.
