A description of Hankel operators of class $\mathfrak S_p$ for $p>0$, an investigation of the rate of rational approximation, and other applications

The main result is the following description of Hankel operators in the Schatten-von Neumann class $\mathfrak{S}_p$ when $0<p<1$:
$$ \Gamma_\varphi\in\mathfrak S_p\Leftrightarrow\varphi\in B_p^{1/p}, $$
where $\Gamma_\varphi$ is the Hankel operator with symbol $\varphi$, and $B_p^{1/p}$ is the Besov class. This result extends results obtained earlier for $1\leqslant p<+\infty$ by the author to the case $ 0<p<1$. Also described are the Hankel operators in the Schatten–Lorentz classes $\mathfrak S_{pq}$, $0<p<+\infty$, $ 0<q\leqslant\infty$.
Precise descriptions of classes of functions defined in terms of rational approximation in the bounded mean oscillation norm are given as an application, along with a complete investigation of the case where the decrease is of power order, and some precise results on rational approximation in the $L^\infty$-norm. Certain other applications are also considered.
Bibliography: 57 titles.