Bernstein Inequality for Multivariate Functions with Smooth Fourier Images

Let K be a compact set in ℝⁿ with (O)-property and let 1 ≤ p ≤ ∞. Then there exists a constant C_K < ∞ independent of f and α and such that

$$\begin{array}{cc}{\Vert {D}^{\alpha }f\Vert }_{p}\le {C}_{K}\underset{\xi \in K}{\mathrm{sup}}\left|{\xi }^{\alpha }\right|{\Vert f\Vert }_{{\mathcal{H}}_{p,K,3}}& \begin{array}{cc}\mathrm{for all}& \begin{array}{cc}\alpha \in {\mathbb{Z}}_{+}^{n}& \begin{array}{cc}\mathrm{and}& f\in {\mathcal{H}}_{p,K,3}\end{array}\end{array},\end{array}\end{array}$$

where ${\mathcal{H}}_{p,K,3}=\left\{f\in {L}^{p}\left({\mathbb{R}}^{n}\right):\mathrm{supp }\widehat{f}\subset K, {D}^{\left(\mathrm{3,3},\dots ,3\right)}\widehat{f}\in C\left({\mathbb{R}}^{n}\right)\right\}$, ${\Vert f\Vert }_{{\mathcal{H}}_{p,K,3}}={\Vert {D}^{\left(\mathrm{3,3},\dots ,3\right)}\widehat{f}\Vert }_{\infty },$ 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

$$\begin{array}{cc}\underset{\mathrm{x}\in K}{\mathrm{sup}}\left|{\mathrm{x}}^{\alpha +{e}_{j}}\right|\ge C\underset{\mathrm{x}\in K}{\mathrm{sup}}\left|{\mathrm{x}}^{\alpha }\right|& \begin{array}{cc}\mathrm{for all}& \begin{array}{cc}\alpha \in {\mathbb{Z}}_{+}^{n}& \begin{array}{cc}\mathrm{and}& j=\mathrm{1,2},\dots ,n.\end{array}\end{array}\end{array}\end{array}$$