Solomyak-type eigenvalue estimates for the Birman-Schwinger operator

We revise the Cwikel-type estimate for the singular values of the operator $(1-\Delta_{\mathbb{T}^d})^{-d/4}M_f(1-\Delta_{\mathbb{T}^d})^{-d/4}$ on the torus $\mathbb{T}^d$, for the ideal $\mathcal{L}_{1,\infty}$ and $f\in L\log L(\mathbb{T}^d)$ (the Orlicz space), which was established by Solomyak in even dimensions, and we extend it to odd dimensions. We show that this result does not literally extend to Laplacians on $\mathbb{R}^d$, neither for Orlicz spaces on $\mathbb{R}^d$, nor for any symmetric function space on $\mathbb{R}^d$. Nevertheless, we obtain a new positive result on (symmetrized) Solomyak-type estimates for Laplacians on $\mathbb{R}^d$ for an arbitrary positive integer $d$ and $f$ in $L\log L(\mathbb{R}^d)$. The last result reveals the conformal invariance of Solomyak-type estimates.
Bibliography: 44 titles.