$$\mu^2(M_fg(-i\nabla))\prec\prec 532 \mu^2(f\otimes g),$$

in the sense of Hardy-Littlewood majorisation, where $\mu(M_fg(-i\nabla))$ denotes the sequence of singular values of the operator $M_fg(-i\nabla)$ and $\mu(f\otimes g)$ denotes the decreasing rearrangement of $f\otimes g$. As a corollary of this estimate and the Lorentz-Shimogaki interpolation theorem we obtain Cwikel estimates in any arbitrary symmetric space $E$, which is an interpolation space for the Banach couple $(L_2,L_\infty)$, that is

if $f\otimes g\in E(\mathbb{R}^{2d})$, then $M_f g(-i\nabla)\in E(L_2(\mathbb{R}^d))$ and

$$\|M_f g(-i\nabla)\|_E\leq {\rm const}(E)\, \|f\otimes g\|_E.$$

Furthermore, we show that our assumption on the symmetric function space $E$ is optimal, that is, if we omit the assumption that $E$ is an $(L_2,L_\infty)$-interpolation space, then the corresponding Cwikel estimates fail.

We also provide a version of these estimates for a class of interpolation spaces in the pair $(L_q, L_2)$ with $0<q<2.$

Joint work with G. Levitina and D. Zanin.