1 - Trace Ideal Properties of a Class of Integral Operators

Summary

We consider a particular class of integral operators $T_{\gamma,\delta}$ in $L^2(\bbR^n)$, $n \in \bbN$, $n \geq 2$, with integral kernels $T_{\gamma, \delta}(\, \cdot \,,\, \cdot \,)$ bounded (Lebesgue) a.e. by $|T_{\gamma, \delta}(x,y)| \leq C \langle x \rangle^{-\delta} |x - y|^{2 \gamma - n} \langle y \rangle^{-\delta}, \quad x, y \in \bbR^n, \; x \neq y,$ for fixed $C \in (0,\infty)$, $0 < 2 \gamma < n$, $\delta > \gamma$, and prove that$T_{\gamma, \delta} \in \cB_p\big(L^2(\bbR^n)\big) \, \text{ for } \,p > n/(2\gamma), \; p \geq 2.$ (Here $\langle x \rangle := (1+|x|^2)^{1/2}$, $x \in \bbR^n$, and $\cB_p$ abbreviates the $\ell^p$-based trace ideal.) These integral operators (and their matrix-valued analogs) naturally arise in the study of multi-dimensional Schrödinger and Dirac-type operators and we describe an application to the case of massless Dirac-type operators.