Ultracontractivity of heat semigroups with non-local Robin boundary conditions via two-sided bounds for a positive eigenfunction

We study heat semigroups on bounded Lipschitz domains $\mathrm{\Omega }\subset {\mathbb{R}}^d$ with dimension $d>2$ under non-local Robin boundary conditions. The boundary operator $B\in \mathcal{L}({\mathrm{L}}^2(\mathrm{\partial }\mathrm{\Omega }))$ is allowed to destroy the positivity of the semigroup. We assume that there exists a positive operator $C\in \mathcal{L}({\mathrm{L}}^2(\mathrm{\partial }\mathrm{\Omega }))$ such that $$\mathrm{\mid }Bu\mathrm{\mid }\le C\mathrm{\mid }u\mathrm{\mid }\text{for every }u\in {\mathrm{L}}^2(\mathrm{\partial }\mathrm{\Omega }),C\mathbf{1}\in {\mathrm{L}}^{\mathrm{\infty }}(\mathrm{\partial }\mathrm{\Omega })\mathrm{.}$$ Under this assumption, we prove that the semigroup generated by the corresponding uniformly elliptic operator is ultracontractive. More precisely, its norm from ${\mathrm{L}}^2(\mathrm{\Omega })$ to ${\mathrm{L}}^{\mathrm{\infty }}(\mathrm{\Omega })$ has short-time order $t^{-d\mathrm{/}4}$. The main step is the construction of a positive eigenfunction $\phi$ of a dominating elliptic operator such that $$0<\delta \le \phi \le M$$ almost everywhere in $\mathrm{\Omega }$. The upper bound is obtained by power truncations and an iteration of Sobolev exponents. The lower bound follows by comparison with the Neumann semigroup. The two-sided estimate gives an ${\mathrm{L}}^{\mathrm{\infty }}$-bound for the comparison semigroup. Nash’s inequality and duality then yield ultracontractivity.