Harmonic and Schr\"odinger functions of polynomial growth on gradient shrinking Ricci solitons

In this paper, we study harmonic and caloric functions of polynomial growth on a complete non-compact gradient shrinking Ricci soliton. On one hand, when the scalar curvature satisfies at least quadratic decay, we prove that the space of harmonic functions with fixed polynomial growth degree is finite dimensional. We also prove analogous results for ancient caloric functions. On the other hand, without any curvature condition, we prove sharp finite dimensional estimates for the space of Schr"odinger functions with fixed polynomial growth degree.