New characterizations of BLO spaces by heat semigroups and applications

Technical Summary: New Characterizations of BLO Spaces by Heat Semigroups and Applications

Problem Statement
The paper addresses the characterization of the space of functions with bounded lower oscillation (BLO) on Euclidean space $\mathbb{R}^n$. While the theory of Hardy spaces ($H^1$) and Bounded Mean Oscillation (BMO) spaces is well-established, BLO spaces—introduced by Coifman and Rochberg as a subset of BMO—have received increasing attention regarding their relationship with $A_1$ Muckenhoupt weights and singular integral operators. Existing characterizations include the classical result linking BLO to $A_1$ weights (via $e^{\epsilon f} \in A_1$) and a characterization by C. Bennett using a variant of the Hardy-Littlewood maximal function. The authors aim to provide new characterizations of BLO spaces utilizing the Gaussian heat semigroup and to explore the regularity properties of heat equation solutions with BLO boundary data.

Methodology
The primary methodology involves the use of the classical heat semigroup $W_t$ and its associated kernel $W_t(x, y) = (4\pi t)^{-n/2} e^{-|x-y|^2/4t}$. The authors employ semigroup techniques to establish equivalences between the BLO norm and quantities derived from the heat evolution of a function.

Key technical tools include:

1. Heat Semigroup Characterization: Establishing that a function $f$ belongs to BLO if and only if the oscillation of its heat evolution $W_t f$ over balls of radius $\sqrt{t}$ is uniformly bounded in $L^\infty$.
2. Weight Characterization: Utilizing the relationship between the heat semigroup and the Hardy-Littlewood maximal operator to derive a new condition involving the inequality $W_t(e^{\epsilon f}) \leq C e^{\epsilon f}$.
3. Decomposition and Kernel Estimates: In proving the boundedness of the Littlewood-Paley $g$-function, the authors decompose functions into "near" and "far" parts relative to a ball, utilizing the John-Nirenberg inequality for BMO functions and precise gradient estimates of the heat kernel.
4. Counter-examples and Stability: The paper constructs specific examples to demonstrate the non-linearity of the BLO space and proves stability under $L^\infty$ perturbations.

Key Contributions and Results

1. New Heat Semigroup Characterization (Theorem 1.2):
   The authors prove that a locally integrable function $f$ is in $BLO(\mathbb{R}^n)$ if and only if:
   $$\sup_{t>0} \left\| W_t f(\cdot) - \inf_{z \in B(\cdot, \sqrt{t})} W_t f(z) \right\|_{L^\infty(\mathbb{R}^n)} < \infty.$$
   Furthermore, the BLO norm is equivalent to this supremum. This provides a direct link between the geometric oscillation of the heat flow and the BLO property.

2. Regularity of Heat Equation Solutions (Theorems 2.1 & 2.2):
   Applying the new characterization, the paper establishes regularity estimates for solutions $u(x, t)$ to the heat equation $\partial_t u = \Delta u$ with initial data $f \in BLO(\mathbb{R}^n)$. Specifically, the solution satisfies:
   $$u(x, t) \leq \inf_{z \in B(x, \sqrt{t})} u(z, t) + C\|f\|_{BLO}.$$
   This implies a controlled local oscillation of the solution, bounded by the BLO norm of the initial data. The authors note these results extend to general parabolic equations with uniformly elliptic coefficients.

3. Heat Characterization via $A_1$ Weights (Theorem 1.5):
   The paper re-establishes the equivalence between $f \in BLO$ and $e^{\epsilon f} \in A_1$ by introducing a heat-semigroup-based condition:
   $$W_t(e^{\epsilon f})(x) \leq C e^{\epsilon f}(x) \quad \text{for all } t > 0 \text{ and a.e. } x.$$
   This offers a quantitative version of the Coifman-Rochberg theorem, providing a norm equivalence involving the infimum of constants $\epsilon$ and $C_0$ satisfying the heat inequality.

4. BMO-BLO Boundedness of the Littlewood-Paley $g$-Function (Theorem 2.4):
   The authors reprove the result that the square of the Littlewood-Paley $g$-function, $[g(f)]^2$, maps $BMO(\mathbb{R}^n)$ boundedly into $BLO(\mathbb{R}^n)$. The proof utilizes the heat semigroup characterization of BLO and kernel estimates, avoiding previous methods. Consequently, $g(f) \in BLO$ with a corresponding norm estimate.

5. Non-linearity and Stability (Section 4):
   The paper provides a concrete example ($f(x) = -\ln|x|$) to demonstrate that BLO is not a linear space (specifically, if $f \in BLO$, $-f$ may not be in BLO). However, it proves that the space is stable under $L^\infty$ perturbations: if $f \in BLO$ and $g \in L^\infty$, then $f-g \in BLO$.

Significance
The paper claims significance in providing a unified framework for analyzing BLO spaces through the lens of the heat semigroup. By characterizing BLO via the heat flow, the authors facilitate the study of regularity for diffusion equations with BLO boundary values. Additionally, the heat-semigroup approach offers an alternative, streamlined proof for the boundedness of the Littlewood-Paley $g$-function from BMO to BLO, reinforcing the utility of semigroup methods in harmonic analysis. The quantitative relationship established between the BLO norm and the $A_1$ weight constants via the heat semigroup further deepens the understanding of the structural properties of these function spaces.