One-Sided and Parabolic BLO Spaces with Time Lag and Their Applications to Muckenhoupt $A_1$ Weights and Doubly Nonlinear Parabolic Equations

In this article, we first introduce the one-sided BLO space ${\mathrm{B}\mathrm{L}\mathrm{O}}^{+}(\mathbb{R})$ and characterize it, respectively, in terms of the one-sided Muckenhoupt class $A_1^{+}(\mathbb{R})$ and the one-sided John–Nirenberg inequality. Using these, we establish the Coifman–Rochberg type decomposition of ${\mathrm{B}\mathrm{L}\mathrm{O}}^{+}(\mathbb{R})$ functions and show that ${\mathrm{B}\mathrm{L}\mathrm{O}}^{+}(\mathbb{R})$ is independent of the distance between the two intervals, which further induces the characterization of this space in terms of the one-sided BMO space ${\mathrm{B}\mathrm{M}\mathrm{O}}^{+}(\mathbb{R})$ (the Bennett type lemma). As applications, we prove that any ${\mathrm{B}\mathrm{M}\mathrm{O}}^{+}(\mathbb{R})$ function can split into the sum of two ${\mathrm{B}\mathrm{L}\mathrm{O}}^{+}(\mathbb{R})$ functions and we provide an explicit description of the distance from ${\mathrm{B}\mathrm{L}\mathrm{O}}^{+}(\mathbb{R})$ functions to $L^{\mathrm{\infty }}(\mathbb{R})$. Finally, as a higher-dimensional analogue we introduce the parabolic BLO space ${\mathrm{P}\mathrm{B}\mathrm{L}\mathrm{O}}_{\gamma }^{-}({\mathbb{R}}^{n+1})$ with time lag, and we extend all the above one-dimensional results to ${\mathrm{P}\mathrm{B}\mathrm{L}\mathrm{O}}_{\gamma }^{-}({\mathbb{R}}^{n+1})$; furthermore, as applications, we not only establish the relationships between ${\mathrm{P}\mathrm{B}\mathrm{L}\mathrm{O}}_{\gamma }^{-}({\mathbb{R}}^{n+1})$ and the solutions of doubly nonlinear parabolic equations, but also provide a necessary condition for the negative logarithm of the parabolic distance function to belong to ${\mathrm{P}\mathrm{B}\mathrm{L}\mathrm{O}}_{\gamma }^{-}({\mathbb{R}}^{n+1})$ in terms of the weak porosity of the set.