On directional Whitney inequality

Abstract

This paper studies a new Whitney type inequality on a compact domain $\Omega \subset {\mathbb{R}}^d$ that takes the form $$\underset{Q\in {\Pi }_{r-1}^d(\mathcal{E})}{\inf }\Vert f-Q{\Vert }_p\le C(p,r,\Omega ){\omega }_{\mathcal{E}}^r(f,\mathrm{diam}(\Omega ){)}_p,r\in \mathbb{N},0<p\le \infty ,$$ where ${\omega }_{\mathcal{E}}^r(f,t{)}_p$ denotes the $r$-th order directional modulus of smoothness of $f\in L^p(\Omega )$ along a finite set of directions $\mathcal{E}\subset {\mathbb{S}}^{d-1}$ such that $\mathrm{span}(\mathcal{E})={\mathbb{R}}^d$, ${\Pi }_{r-1}^d(\mathcal{E}):=\{g\in C(\Omega ):{\omega }_{\mathcal{E}}^r(g,\mathrm{diam}(\Omega ){)}_p=0\}$. We prove that there does not exist a universal finite set of directions $\mathcal{E}$ for which this inequality holds on every convex body $\Omega \subset {\mathbb{R}}^d$, but for every connected $C^2$-domain $\Omega \subset {\mathbb{R}}^d$, one can choose $\mathcal{E}$ to be an arbitrary set of $d$ independent directions. We also study the smallest number ${\mathcal{N}}_d(\Omega )\in \mathbb{N}$ for which there exists a set of ${\mathcal{N}}_d(\Omega )$ directions $\mathcal{E}$ such that $\mathrm{span}(\mathcal{E})={\mathbb{R}}^d$ and the directional Whitney inequality holds on $\Omega$ for all $r\in \mathbb{N}$ and $p>0$. It is proved that ${\mathcal{N}}_d(\Omega )=d$ for every connected $C^2$-domain $\Omega \subset {\mathbb{R}}^d$, for $d=2$ and every planar convex body $\Omega \subset {\mathbb{R}}^2$, and for $d\ge 3$ and every almost smooth convex body $\Omega \subset {\mathbb{R}}^d$. [See the pre-print for the complete abstract - not included here due to arXiv limitations.]

the material in this article is based heavily on a part of arXiv:1910.11719