Space of ancient caloric functions on some manifolds beyond volume doubling
Fanghua Lin, Hongbing Qiu, Jun Sun, Qi S. Zhang
TL;DR
This work investigates spaces of harmonic and caloric functions on complete noncompact manifolds under a relaxed geometric framework given by Condition $(BA)$, which replaces volume doubling with a polynomial-volume bound and a weighted heat-kernel upper bound. It proves that ancient caloric functions with polynomial growth are time polynomials of degree bounded by $k$, where $k$ is the least integer greater than $2m+\frac{5}{2}\alpha+\frac{d}{2}$, and that finiteness of the caloric-growth space $f K^d$ is equivalent to the finiteness of the corresponding harmonic-growth space. A key tool is a modified mean value inequality with a polynomial weight, derived from weighted heat-kernel estimates and used to obtain time-derivative bounds and global integral controls. The paper also establishes sharpness results, including a non-doubling counterexample, and proves finite-dimensionality results for manifolds with finitely many Euclidean ends or in exterior-domain settings, thereby extending classical CM97/Li97 results beyond volume doubling and highlighting the role of end-geometry in harmonic analysis on manifolds.
Abstract
Under a condition that breaks the volume doubling barrier, we obtain a time polynomial structure result on the space of ancient caloric functions with polynomial growth on manifolds. As a byproduct, it is shown that the finiteness result for the space of harmonic functions with polynomial growth on manifolds in \cite{CM97} and \cite{Li97} are essentially sharp, except for the multi-end cases, addressing an issue raised in \cite{CM98} and removing all {\it local} topological or geometric conditions on the manifold with respect to a reference point.