Eigenvalue decay of operators on harmonic function spaces

Let $\mathrm{\Omega }$ be an open set in ${\mathbb{R}}^d$ $(d>1)$ and $h(\mathrm{\Omega })$ the Fr'echet space of harmonic functions on $\mathrm{\Omega }$. Given a bounded linear operator $L:h(\mathrm{\Omega })\to h(\mathrm{\Omega })$, we show that its eigenvalues ${\lambda }_n$, arranged in decreasing order and counting multiplicities, satisfy $\mathrm{\mid }{\lambda }_n\mathrm{\mid }\le K\exp (-c{n}^{1\mathrm{/}(d-1)})$, where $K$ and $c$ are two explicitly computable positive constants.