An improvement of Bernstein’s inequality for functions in Orlicz spaces with smooth Fourier image

Abstract

Let $\mathrm{\Phi }:[0,+\infty )\to [0,+\infty ]$ be an arbitrary Young function and $K$ be a compact set in ${ℝ}^n$ having (O)-property. Then there exists a constant independent of $f$ such that

$$\parallel D^{\alpha }f{\parallel }_{(\mathrm{\Phi })}\le C_{K,\mathrm{\Phi }}\underset{z\in K}{\text{sup}}\left|z^{\alpha }\right|\parallel f{\parallel }_{{\mathcal{ℒ}}_{K,3}}$$

for all $\alpha \in {\mathbb{Z}}_{+}^n$ and $f\in {\mathcal{ℒ}}_{K,3}$, where ${\mathcal{ℒ}}_{K,3}=\{f\in {\mathcal{𝒮}}^{\prime }({ℝ}^n):\text{supp}\widehat{f}\subset K,D^{(3,3,\dots ,3)}\widehat{f}\in C({ℝ}^n)\}$, $\parallel f{\parallel }_{{\mathcal{ℒ}}_{K,3}}=\parallel D^{(3,3,\dots ,3)}\widehat{f}{\parallel }_{\infty }$, $\widehat{f}$ is the Fourier transform of $f$ and $\parallel \cdot {\parallel }_{(\mathrm{\Phi })}$ is the Luxemburg norm. As an application, a new Paley–Wiener theorem is given.