Notes on BLO spaces with parabolic structure

For the natural parabolic maximal operators $\mathfrak M^{\pm}_{\gamma}$ with time lag $\gamma\in(0,1)$, the author first establishes its boundedness from parabolic $\mathrm{BMO}$ space to $\mathrm{BLO}$ space with parabolic structure, and then proves that $f$ belongs to the $\mathrm{BLO}$ space with parabolic structure if and only if $f={\mathfrak M}^{\pm}_{\gamma}g+h$ for some $g$ in the parabolic $\mathrm{BMO}$ space, and some bounded function $h$. These characterizations generalize the classical decomposition for $\mathrm {BLO}$ functions to the parabolic setting.