Slice-wise integration by parts before the time s #
On every time slice τ < s the backward heat kernel is smooth, so a spatial derivative can be
moved across the integral slice by slice and then integrated in time. This gives the three
integrations by parts in the cut-off heat representation of the interior estimates in the proof
of lem:aniso:annulus.
The half-space {τ < s} as a product set, and Lebesgue measure on it as a product measure.
Integration by parts in a spatial direction on the half-space {τ < s}, performed slice by
slice: on each slice the first factor is smooth with compact support inside an open set where the
second factor is smooth.
Integrability on the half-space of a function that vanishes off a closed window, is continuous on the window, and is dominated there by an integrable function.
The spatial partial of a field smooth on an open set is continuous there.
The cut-off heat weight is smooth before the time s.
On a slice before s, the heat weight is smooth, compactly supported, and supported in the
window slice.
The backward window is compact.
The translated heat kernel restricted to the strip 0 < s - τ ≤ ρ² is integrable.
The main-term integration by parts: before s, ∫ Ψ ∂ⱼY = -∫ ∂ⱼΨ Y for Y smooth on an
open set containing the window.
In the window before s, the products of the heat kernel (or its spatial derivative) with
the partials of the cutoff are bounded by C ρ⁻⁵.
Before s and off the window, the commutator kernel vanishes.
The commutator kernel is bounded by C ρ⁻⁵ in the window before s.
The divergence-term integration by parts: before s,
∫ (∑ₗ ∂ₗUₗ) K₀ = -∑ₗ ∫ Uₗ ∂ₗK₀ for U smooth on an open set containing the window.