Heat Semigroup Bounds on BV(Ω) and Related Estimates

Abstract

Let be a compact Riemannian manifold with nonempty boundary. We note that if f ∈ C ∞ ( ) does not vanish identically on the boundary, then the heat semigroup e t D (with the Dirichlet boundary condition) acting on f produces a family bounded in H 1, p ( ) if and only if p = 1. This observation motivates the main result of this paper, which is that the heat semigroup is uniformly bounded on BV( ), the space of functions on with bounded variation.