On estimates of the $L^p$-norms of derivatives in spaces of entire functions

In the present work, weighted $L^p$-norms of derivatives are studied in the spaces of entire functions $\mathcal H^p(E)$ generalizing the de Branges spaces. A description of the spaces $\mathcal H^p(E)$ such that the differentiation operator $\mathcal D\colon F\mapsto F'$ is bounded in $\mathcal H^p(E)$ is obtained in terms of the generating entire function $E$ of the Hermite–Biehler class. It is shown that for a broad class of the spaces $\mathcal H^p(E)$ the boundedness criterion is given by the condition $E'/E\in L^\infty(\mathbb R)$. In the general case a necessary and sufficient condition is found in terms of a certain embedding theorem for the space $\mathcal H^p(E)$; moreover, the boundedness of the operator $\mathcal D$ depends essentially on the exponential $p$. Also we obtain a number of conditions sufficient for the compactness of the differentiation operator in $\mathcal H^p(E)$.