Duality, BMO and Hankel operators on Bernstein spaces

Abstract

In this paper we deal with the problem of describing the dual space ${\left(B_{\kappa }^1\right)}^{⁎}$ of the Bernstein space $B_{\kappa }^1$, 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 characterizations, showing that such dual space can be described as a quotient of the space of entire functions of exponential type κ whose restriction to the real line are in a suitable BMO-type space, or as the space of symbols b for which the Hankel operator $H_b$ is bounded on the Paley–Wiener space $B_{\kappa /2}^2$. We also provide a characterization of ${\left(B_{\kappa }^1\right)}^{⁎}$ as the BMO space w.r.t. the Clark measures of the inner function $e^{i2\kappa z}$ 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 $P_{\kappa }:L^2\left(R\right)\to B_{\kappa }^2$ induces a bounded operator from $L^{\infty }\left(R\right)$ onto ${\left(B_{\kappa }^1\right)}^{⁎}$.

Finally, we show that $B_{\kappa }^1$ is the dual space of a suitable VMO-type space or as the space of symbols b for which the Hankel operator $H_b$ on the Paley–Wiener space $B_{\kappa /2}^2$ is compact.