New characterizations of BLO spaces by heat semigroups and applications

In this paper, we give two new characterizations of the bounded lower oscillation (BLO) space by using the Gaussian heat semigroup. Using these characterizations, we prove the boundary lower-envelope estimate of the solutions of the heat equation with BLO boundary data, a Gaussian John-Nirenberg criterion and a quantitative estimates of the BLO-norm. Also, we reprove the BMO-BLO boundedness of the Littlewood-Paley $g$-function by using the semigroup method.