Eigenvalue decay of operators on harmonic function spaces

Let $Ω$ be an open set in $\R^d$ $(d > 1)$ and $h(Ω)$ the Fréchet space of harmonic functions on $Ω$. Given a bounded linear operator $L :h(Ω)\to h(Ω)$, we show that its eigenvalues $λ_n$, arranged in decreasing order and counting multiplicities, satisfy $|λ_n|\leq K\exp(-cn^{1/(d-1)})$, where $K$ and $c$ are two explicitly computable positive constants.

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