A Local Tb Theorem for Square Functions and Parabolic Layer Potentials

In this paper, we give a locally parabolic version of Tb theorem for a class of vector-valued operators with off-diagonal decay in L² and certain quasi-orthogonality on a subspace of L², in which the testing functions themselves are also vector-valued. As an application, we establish the boundedness of layer potentials related to parabolic operators in divergence form, defined in the upper half-space ℝ ₊ ⁿ⁺² ≔ {(x, t, λ) ∈ ℝⁿ⁺¹ × (0, ∞)}, with uniformly complex elliptic, L^∞, t, λ-independent coefficients, and satisfying the De Giorgi/Nash estimates.