On Bernstein type theorems concerning the growth of derivatives of entire functions

Abstract:

A subspace $X$ of $L^1_{loc}(\mathbb {R})$ which is invariant under all left translation operators $T_t,$ $t\in \mathbb {R},$ is called admissible if $X$ is a Banach space satisfying the following properties: (i) If $\Vert f_n\Vert _X\to 0,$ then there exists a subsequence $(n_k)$ such that $f_{n_k}(s)\to 0$ almost everywhere. (ii) The group ${\mathcal T}_X:=\{T_t\vert _X: t\in \mathbb {R}\}$ is a bounded strongly continuous group. In this case, let \[ C_X:=\sup \{\Vert T_t\Vert _X: t\in \mathbb {R}\}.\] Typical admissible spaces are $C_0(\mathbb {R}),$ $BUC(\mathbb {R})$ and all spaces $L^p(\mathbb {R})$ for $1\leq p<\infty .$ More generally, all of the Peetre interpolation spaces of two admissible spaces $X_1,X_2$ are also admissible. A function $g\in L^1_{loc}(\mathbb {R})$ is called subexponential if for every $\delta >0,$ $e^{-\delta \vert t\vert } g(t) \in L^1(\mathbb {R}).$ With these definitions our main result goes as follows:

Theorem. If $g$ is an entire function of exponential type $\tau$ such that its restriction to the real axis, denoted by $g_\mathbb {R}$, is subexponential and belongs to some admissible space $X,$ then the derivative $g^\prime _\mathbb {R}$ is also in $X.$ Moreover, $\Vert \alpha g_\mathbb {R}+g^\prime _\mathbb {R}\Vert _X\leq (\alpha ^2+\tau ^2)^{1/2} \cdot C_X\cdot \Vert g_\mathbb {R}\Vert _X$ for each real $\alpha .$

This result yields as consequences and in a systematic way many new and old Bernstein type inequalities.