Duality, BMO and Hankel operators on Bernstein spaces

Abstract

In this paper we deal with the problem of describing the dual space (Bκ1)⁎ of the Bernstein space Bκ1, that is the space of entire functions of exponential type (at most) κ>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 Hb is bounded on the Paley–Wiener space Bκ/22. We also provide a characterization of (Bκ1)⁎ as the BMO space w.r.t. the Clark measures of the inner function ei2κz 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 Pκ:L2(R)→Bκ2 induces a bounded operator from L∞(R) onto (Bκ1)⁎.Finally, we show that Bκ1 is the dual space of a suitable VMO-type space or as the space of symbols b for which the Hankel operator Hb on the Paley–Wiener space Bκ/22 is compact.