New characterizations of BLO spaces by heat semigroups and applications

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.

💡 Research Summary

The paper investigates the bounded lower oscillation (BLO) space, a non‑linear subspace of BMO, by exploiting the Gaussian heat semigroup. After recalling the classical definition of BLO as the supremum over balls of the average excess over the essential infimum, the authors introduce two novel characterizations.

The first (Theorem 1.2) states that a locally integrable function f belongs to BLO if and only if the quantity
  sup_{t>0} ‖W_t f(·) − inf_{z∈B(·,√t)} W_t f(z)‖_{L^∞}
is finite, where W_t is the heat semigroup with kernel (4πt)^{-n/2} e^{-|x−y|²/(4t)}. Moreover, this supremum is comparable to the usual BLO norm. The proof uses the gradient estimate |∇_x W_t(x,y)| ≤ C t^{-1/2} W_t(x,y) and the mean‑value theorem to control the difference W_t f(x)−W_t f(z) for points within distance √t. Conversely, the approximation‑of‑the‑identity property of W_t as t→0 yields the BLO bound from the assumed supremum.

The second characterization (Theorem 1.5) links BLO to Muckenhoupt A₁ weights. It shows the equivalence of three statements: (i) f∈BLO; (ii) there exists ε>0 such that e^{εf}∈A₁; (iii) there are ε>0 and C>0 such that for all t>0, W_t(e^{εf})(x) ≤ C e^{εf}(x) almost everywhere. Thus the heat semigroup preserves the A₁ condition, providing a new weight‑theoretic perspective on BLO.

Using these characterizations, the authors obtain two applications. First, they consider the heat equation ∂t u = Δu with initial data f∈BLO. The solution u(x,t)=W_t f(x) satisfies the regularity estimate
  u(x,t) ≤ inf{z∈B(x,√t)} u(z,t) + C‖f‖{BLO},
and consequently the oscillation of u over any ball of radius √t is bounded by C‖f‖{BLO}. This demonstrates that BLO initial data control the local behavior of the heat flow.

Second, they re‑prove the boundedness of the Littlewood‑Paley g‑function from BMO to BLO. Defining
  g(f)(x)=\big(∫₀^∞|t∂_tW_t f(x)|² dt/t\big)^{1/2},
they show that