Trace ideal criteria for Hankel operators, and applications to Besov spaces (Q794253)

Let \(H^ 2=H^ 2({\mathbb{R}})\) denote the usual Hardy space on the real line and let P denote the orthogonal projection of \(L^ 2\) onto \(H^ 2\). Let b be a function on \({\mathbb{R}}\), and consider the Hankel operator \(H_ b\) defined by \(H_ bf=(I-P)bPf.\) \textit{V. V. Peller} [Mat. Sb., Nov. Ser. 113(155), 538-581 (1980; Zbl 0458.47022)] has given a characterization of those functions b such that \(H_ b \in {\mathcal S}^ p\), \(1\leq p<\infty\), i.e., such that \(H_ b\) is compact and \((H^*_ bH_ b)^{{1\over2}}\) has p-summable eigenvalues. An alternate proof has been given by \textit{R. R. Coifman} and \textit{R. Rochberg} [Astérisque 77, 67-151 (1980; Zbl 0472.46040)] for \(p=1\) and by \textit{R. Rochberg} [Indiana Univ. Math. J. 31, 913-925 (1982; Zbl 0514.47020)] for \(1\leq p<\infty\). The paper under review extends this characterization to \(0<p<1\). The proof uses the results of Coifman and Rochberg to almost diagonalize \(H_ b\) well enough to estimate b in terms of the \({\mathcal S}^ p\) norm of \(H_ b\). Professor Peller [Prepr. LOMI E-6-82, Leningrad (1982)] has given another proof of this by a different method. Applications are given similar to those of Peller and Rochberg.