On the Caloric Functions with BMO Traces and Their Limiting Behaviors.

Let (X , d , μ , E) be a Dirichlet metric measure space satisfying a doubling condition and supporting a scale-invariant L 2 -Poincaré inequality. Assume that L is a non-negative operator on L 2 (X) (similar to the negative Laplace operator on L 2 (R n) ) generalized by a Dirichlet form E . This paper is concerned with the boundary behavior of the caloric functions u(x, t) (i.e., the solution to the heat equation ∂ t u + L u = 0 ) on the upper half-space X × R + . We characterize all caloric functions with boundary value in bounded mean oscillation (BMO) space by means of certain Carleson measure condition. This extends the known result of Fabes–Neri [Duke Math. J., 1975, 725-734] from the Euclidean space to the metric measure space. As an application, we further consider the limiting behavior of the caloric function with the vanishing mean oscillation (CMO) trace, which is new even for the Laplace operator on Euclidean space. [ABSTRACT FROM AUTHOR]