Heat kernel bounds for the fractional Laplacian with Hardy potential in angular momentum channels

Motivated by the study of relativistic atoms, we prove sharp heat kernel bounds for the Hardy operator $(-\mathrm{\Delta }{)}^{\alpha \mathrm{/}2}-\kappa \mathrm{\mid }x{\mathrm{\mid }}^{-\alpha }$ acting on functions of the form $u(\mathrm{\mid }x\mathrm{\mid })\mathrm{\mid }x{\mathrm{\mid }}^{\mathrm{\ell }}{Y}_{\mathrm{\ell },m}(x\mathrm{/}\mathrm{\mid }x\mathrm{\mid })$ in $L^2({\mathbb{R}}^d)$, when $\alpha \in (0,2]\cap (0,d+2\mathrm{\ell })$.