分野別セミナー

Abstract:
The celebrated Yau’s $L^p$-Liouville theorem says that a complete Riemannian manifold does not admit any non-trivial $L^p$ harmonic function for $p>1$. In general, the case of $p=2$ is related with the essential self-adjointness of the Laplacian, and hence, we observe the $L^2$-Liouville property fails for certain incomplete manifolds. For the case p=1, the situation drastically changes even for complete manifolds, and in this talk, inspired by the observation above, we will learn how to construct integrable harmonic functions under certain potential analytical conditions of the manifold.

Based on a joint work with A. Grigoryan and M. Murata.