论文标题
关于定向惠特尼不平等
On directional Whitney inequality
论文作者
论文摘要
本文研究了紧凑型域$ω\ subset {\ mathbb {r}}^d $上的一种新的惠特尼类型不等式，该{\ mathbb {r}}^d $采用表格$ \ inf_ {q \inπ_{r-n-c}^d（r-1}^d（{\ mathcal {\ mathcal {e}}}}}}}}}} \ f-q \ | q \ | | | | | _p c（c）东$ r $ -th沿l^p（ω）$的平滑度的订购方向模量，沿一组有限的方向$ {\ Mathcal {e}}} \ subset {\ mathbb {s}}^{d-1 {d-1}} $跨度}（{\ Mathcal {e}}）= {\ Mathbb {r}}^d $，$π_{r-1}^d（{\ Mathcal {e}}}）：= \ {g \ {g \ in c（g \） （ω））_ p = 0 \} $。 我们证明，不存在一套通用有限的方向集$ {\ Mathcal {e}} $，此不平等在每个凸面上都存在$ω\ subset {\ Mathbb {r}}^d $ $ {\ Mathcal {e}} $是任意集合的$ D $独立说明。 我们还研究了最小数字$ {\ MATHCAL {n}} _ d（ω）\ in {\ Mathbb {n}} $，其中存在一组$ {\ Mathcal {n}} _ d（ω）$ { SPAN}（{\ Mathcal {e}}）= {\ Mathbb {r}}^d $，定向惠特尼不平等在{\ mathbb {n}}} $和$ p> 0 $中的所有$ r \ in Doriestal Whitney Inquartion in $ω$上。事实证明，每个连接的$ C^2 $ - domain $ω\ subset {\ Mathbb {r}}^d $，$ {\ Mathcal {n}} _ d（ω）= D $，对于$ d = 2 $ = 2 $，每个平面盒$ d = $ d = $ d = $ d $ $ d = 2 $ and $ d = $ gemset gebb gebb，ge 3 $和d $光滑的凸面$ω\ subset {\ mathbb {r}}^d $。 [有关完整摘要，请参见预印 - 由于arxiv的限制，此处未包含。]
This paper studies a new Whitney type inequality on a compact domain $Ω\subset {\mathbb{R}}^d$ that takes the form $$\inf_{Q\in Π_{r-1}^d({\mathcal{E}})} \|f-Q\|_p \leq C(p,r,Ω) ω_{\mathcal{E}}^r(f,{\rm diam}(Ω))_p,\ \ r\in {\mathbb{N}},\ \ 0<p\leq \infty,$$ where $ω_{\mathcal{E}}^r(f, t)_p$ denotes the $r$-th order directional modulus of smoothness of $f\in L^p(Ω)$ along a finite set of directions ${\mathcal{E}}\subset {\mathbb{S}^{d-1}}$ such that ${\rm span}({\mathcal{E}})={\mathbb{R}}^d$, $Π_{r-1}^d({\mathcal{E}}):=\{g\in C(Ω):\ ω^r_{\mathcal{E}} (g, {\rm diam} (Ω))_p=0\}$. We prove that there does not exist a universal finite set of directions ${\mathcal{E}}$ for which this inequality holds on every convex body $Ω\subset {\mathbb{R}}^d$, but for every connected $C^2$-domain $Ω\subset {\mathbb{R}}^d$, one can choose ${\mathcal{E}}$ to be an arbitrary set of $d$ independent directions. We also study the smallest number ${\mathcal{N}}_d(Ω)\in{\mathbb{N}}$ for which there exists a set of ${\mathcal{N}}_d(Ω)$ directions ${\mathcal{E}}$ such that ${\rm span}({\mathcal{E}})={\mathbb{R}}^d$ and the directional Whitney inequality holds on $Ω$ for all $r\in{\mathbb{N}}$ and $p>0$. It is proved that ${\mathcal{N}}_d(Ω)=d$ for every connected $C^2$-domain $Ω\subset {\mathbb{R}}^d$, for $d=2$ and every planar convex body $Ω\subset {\mathbb{R}}^2$, and for $d\ge 3$ and every almost smooth convex body $Ω\subset {\mathbb{R}}^d$. [See the pre-print for the complete abstract - not included here due to arXiv limitations.]
