Bernstein inequality for multivariate functions with smooth Fourier images

Abstract

UDC 517.5

Let $K$ be a compact set in ${\Bbb R}^n$ with $(O)$-property and let $1\leq p\leq \infty$. Then there exists a constant $C_K< \infty$ independent of $f$ and $\alpha$ such that $$\|D^{\alpha } f \|_p \leq C_K \sup\limits_{\xi \in K } |\xi ^{\alpha} |\, \|f\|_{\mathcal{H}_{p,K,3}}$$  for all $\alpha \in \mathbb{Z}_+^n$ and $f\in \mathcal{H}_{p,K,3},$ where $\mathcal{H}_{p,K,3}=\big\{f \in L^p({\Bbb R}^n)\colon {\rm supp\,} \widehat f \subset K ,D^{(3,3,\ldots,3)} \widehat{f} \in C({\Bbb R}^n) \big\},$ $\|f\|_{\mathcal{H}_{p,K,3}} = \big\|D^{(3,3,\ldots,3)} \widehat{f}\,\big\|_\infty$, and $\widehat{f}$ is the Fourier transform of $f$. Note that $K$ is said to have the $(O)$-property if there exists a constant $C>0$ such that  $$\sup\limits_{{\bf x} \in K} |{\bf x}^{\alpha + e_j} | \geq C\sup\limits_{{\bf x} \in K } |{\bf x}^{\alpha} |$$  for all $\alpha \in \mathbb{Z}_+^n$ and $j=1,2, \ldots ,n$.