New characterizations of BLO spaces by heat semigroups and applications

TL;DR

The paper develops two heat semigroup-based characterizations of BLO spaces on $\mathbb{R}^n$, linking BLO to Gaussian averaging via $W_t$ and to exponential integrability through $A_1$ weights. It proves a sharp oscillation-based BLO criterion $\sup_{t>0}\|W_t f(\cdot)-\inf_{z\in B(\cdot,\sqrt t)}W_t f(z)\|_{L^\infty}<\infty$, and leverages this to obtain a regularity estimate for the heat equation with BLO boundary data and a parabolic oscillation bound. The semigroup approach yields a semigroup-based proof that the Littlewood-Paley $g$-function maps $BMO$ into $BLO$, with $[g(f)]^2\in BLO$ and $\|[g(f)]^2\|_{BLO} \le C\|f\|_{BMO}^2$. A complementary colorable result shows the BLO condition is equivalent to an exponential A1 bound via $e^{\varepsilon f}$, and a quantitative relation $N(f)$ ties BLO to exponential integrability with tight upper and lower bounds. The work also addresses BLO nonlinearity and stability under $L^\infty$ perturbations, and extends the framework to parabolic equations with general elliptic coefficients.

Abstract

In this paper, we give two new characterizations of the bounded lower oscillation(BLO) space by using the Gaussian heat semigroup. By the new characterizations, we prove the regularity property of the solutions to the heat equation with BLO boundary value. Also, we reprove the BMO-BLO boundedness of the Littlewood-Paley $g$-function by using the semigroup method.

Theorems & Definitions (23)

• Definition 1.1
• Theorem 1.2
• Definition 1.3
• Theorem 1.4
• Theorem 1.5: Heat Characterization via $A_1$ Weights
• proof : Proof of Theorem \ref{['Thm:heatC']}
• Theorem 2.1
• proof
• Theorem 2.2
• proof