Cwikel type estimate as a consequence of certain properties of the heat kernel

Abstract:

Estimates for the singular values of the operator $\mathbb {T}_{fg}:=f(H)g(x)$ are investigated for suitable functions $f(\lambda )$, $\lambda \in \mathbb {R}$, $g(x)$, $x\in \mathbb {R}^{d}$, and a selfadjoint operator $H$ in $L_{2}(\mathbb {R}^{d})$. It is assumed that the kernel of the semigroup $e^{-tH}$ satisfies special conditions. Power-like estimates for the singular values of the operator $\mathbb {T}_{fg}$ are obtained, in particular, in the case where $\mathbb {T}_{fg}\in \mathfrak {S}_{2}$. Conditions for the operator $\mathbb {T}_{fg}$ to belong to the trace class are established. Neither any smoothness conditions for the kernel of the operator $\mathbb {T}_{fg}$, nor any knowledge of the (partial) diagonalization of the operator $H$ are required. The results admit further refinement under additional conditions imposed on the generalized eigenfunctions of the operator $H$.