Duality, $BMO$ and Hankel operators on Bernstein spaces

In this paper we deal with the problem of describing the dual space $(B^1_\kappa)^*$ of the Bernstein space $B^1_\kappa$, that is the space of entire functions of exponential type at most $\kappa>0$ whose restriction to the real line is Lebesgue integrable. We provide several characterisations, showing that such dual space can be described as a quotient of the space of entire functions of exponential type $\kappa$ whose restrictions to the real line is Lebesgue integrable. We provide several characterisations, showing that such dual space can be described as a quotient of the space of entire functions of exponential type $\kappa$ whose restrictions to the real line is in a suitable $BMO$-type space, or as the space of symbols $b$ for which the Hankel operatorc $H_b$ is bounded on the Paley-Wiener space $B^2_{\kappa/2}$. We also provide a characterisation of $(B^1_\kappa)^*$ as the $BMO$ space w.r.t. the Clark measure of the inner function $e^{i\kappa z}$ on the upper half-plane, in analogy with the known description of the dual of backward-shift invariant $1$-spaces on the torus. Furthermore, we show that the orthogonal projection $P_\kappa\ : L^2(R)\to B^2_\kappa$ induces a bounded operator from $L^\infty(R)$ onto $(B^1_\kappa)^*$. Finally, we show that $B^1_\kappa$ is the dual space of the suitable $VMO$-type space or as the space of symbols $b$ for which the Hankel opertor $H_b$ on the Paley-Wiener space $B^2_{k/2}$ is compact.