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 $\mathrm{BLO}^+(\mathbb{R})$ 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 $\mathrm{BLO}^+(\mathbb{R})$ functions and show that $\mathrm{BLO}^+(\mathbb{R})$ 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 $\mathrm{BMO}^+(\mathbb{R})$ (the Bennett type lemma). As applications, we prove that any $\mathrm{BMO}^+(\mathbb{R})$ function can split into the sum of two $\mathrm{BLO}^+(\mathbb{R})$ functions and we provide an explicit description of the distance from $\mathrm{BLO}^+(\mathbb{R})$ functions to $L^\infty(\mathbb{R})$. Finally, as a higher-dimensional analogue we introduce the parabolic BLO space $\mathrm{PBLO}_γ^-(\mathbb{R}^{n+1})$ with time lag, and we extend all the above one-dimensional results to $\mathrm{PBLO}_γ^-(\mathbb{R}^{n+1})$; furthermore, as applications, we not only establish the relationships between $\mathrm{PBLO}_γ^-(\mathbb{R}^{n+1})$ 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 $\mathrm{PBLO}_γ^-(\mathbb{R}^{n+1})$ in terms of the weak porosity of the set.

💡 Research Summary

This paper develops a systematic theory of one‑sided and parabolic BLO (bounded lower oscillation) spaces and demonstrates their deep connections with one‑sided Muckenhoupt weights, maximal operators, and doubly nonlinear parabolic equations.

Motivation and background.
The classical BMO space (functions of bounded mean oscillation) is intimately linked with the Muckenhoupt A_q classes: for q>1, BMO consists of logarithms of A_q weights, while the endpoint q=1 fails. Coifman and Rochberg introduced BLO (bounded lower oscillation) to capture the missing endpoint: BLO = {λ ln ω : ω∈A₁, λ≥0}. Recent work on one‑sided analysis (Sawyer, Martín‑Reyes, etc.) has defined one‑sided A⁺_q and BMO⁺ spaces, but the endpoint q=1 still lacks a satisfactory description.

One‑sided BLO⁺(ℝ).
The authors define BLO⁺(ℝ) by measuring the difference between the average of a function over a left interval I⁻ and its essential infimum over a right interval I⁺, allowing an arbitrary separation between the intervals. The main results are:

1. Theorem 2.4 – Every BLO⁺ function is exactly a non‑negative multiple of the logarithm of a one‑sided A⁺₁ weight, i.e. BLO⁺(ℝ)= {λ ln ω : ω∈A⁺₁(ℝ), λ≥0}. This answers the open question of what replaces BMO⁺ in the endpoint case.

2. Theorem 2.7 – A one‑sided John–Nirenberg inequality holds: the distribution of the lower oscillation decays exponentially, with constants independent of the distance between I⁻ and I⁺.

3. Theorem 2.8 – The definition of BLO⁺ does not depend on the distance between the two intervals; any pair of intervals with the same lengths yields the same norm.

Bennett‑type characterization and maximal operators.
Using the uncentered one‑sided natural maximal operator N⁻, the authors prove (Proposition 3.1) that N⁻ maps BMO⁺ boundedly into BLO⁺. Consequently (Theorem 3.3) BLO⁺ can be described as the image of BMO⁺ under N⁻ modulo bounded functions – a Bennett‑type lemma for the one‑sided setting.

Coifman–Rochberg decomposition.
Theorem 4.2 provides a decomposition of any BLO⁺ function f as
 f = λ ln ω + g,
where ω∈A⁺₁ and g∈L^∞. As corollaries:

• Every BMO⁺ function splits into the sum of two BLO⁺ functions (Corollary 4.3).
• The distance from a BLO⁺ function to L^∞ is given explicitly by the infimum of the associated λ (Corollary 4.4).

Parabolic extension PBLO⁻_γ(ℝⁿ⁺¹).
The paper then lifts the theory to the parabolic setting with a time lag γ∈(0,½]. A parabolic rectangle R⁻_γ consists of a spatial cube and a time interval shifted backward by γ. The space PBLO⁻_γ is defined analogously to BLO⁺, measuring lower oscillation between a forward and a backward rectangle.

Key results:

• Theorem 5.3 – PBLO⁻_γ consists precisely of λ ln ω with ω in the parabolic one‑sided A⁺₁(γ) class (weights satisfying a one‑sided parabolic Muckenhoupt condition).
• Theorem 5.6 – Parabolic John–Nirenberg inequality holds for PBLO⁻_γ.
• Theorem 5.8 – PBLO⁻_γ is independent of both the time lag γ and the spatial separation of the rectangles.

The authors establish the boundedness of the parabolic natural maximal operator N⁻_γ from PBMO⁻_γ to PBLO⁻_γ (Proposition 5.9) and obtain a Bennett‑type characterization (Theorem 5.12).

Coifman–Rochberg decomposition in the parabolic case.
Theorem 5.14 gives f∈PBLO⁻_γ as λ ln ω + g with ω∈A⁺₁(γ) and g∈L^∞. Consequently, any PBMO⁻_γ function can be written as the sum of two PBLO⁻_γ functions (Corollary 5.15), and the distance to L^∞ is quantified (Corollary 5.16).

Applications to doubly nonlinear parabolic equations.
The authors consider the doubly nonlinear equation
 ∂ₜ(|u|^{p−2}u) – div(|∇u|^{p−2}∇u) = 0, p∈(1,∞).
They show (Proposition 5.17 and Corollaries 5.18‑5.19) that if u is a weak solution, then the logarithm of its positive part belongs to PBLO⁻_γ, providing a new regularity tool: the lower oscillation of log u is uniformly bounded in space‑time, which yields Harnack‑type estimates and continuity results.

Geometric condition via weak porosity.
Finally, Theorem 5.23 establishes that for a set E⊂ℝⁿ⁺¹, the function –log d_p(·,E) (parabolic distance) can belong to PBLO⁻_γ only if E is weakly porous in the parabolic sense (γ‑FIT weak porosity). This links geometric measure properties of sets with membership in a function space governing PDE regularity.

Overall significance.
The paper fills a long‑standing gap in the endpoint theory of one‑sided BMO by introducing BLO⁺, provides a complete suite of characterizations (weight, John–Nirenberg, maximal operator, Bennett lemma), and extends all these ideas to the parabolic setting with time lag. The results not only deepen the harmonic analysis of one‑sided operators but also supply powerful tools for the study of nonlinear parabolic equations and the geometry of porous sets. The work is likely to influence future research on anisotropic function spaces, weighted estimates for directional operators, and regularity theory for degenerate/singular evolution equations.