Manuscript X-A-conv@1 · X-A-conv.tex · section · c76fffa3

Joint kernels of the observable

Smooth coefficient kernels, support, and continuity › Joint kernels of the observable · lines 1416–1463

X-A-conv@1#eq:uone, X-A-conv@1#eq:Pkernel, X-A-conv@1#eq:Qkernel, X-A-conv@1#eq:utwoformula, X-A-conv@1#eq:utwo

Integration by parts in the scalar argument gives

O^{(1)} [h, ϕ] = \int u_{1, A} (y, z) h_{A} (y) ϕ (z) d^{4} y d^{4} z, u_{1, A} = \partial_{z^{μ}} (f (z) Λ_{A}^{μ} (z, y)) .

For the quadratic metric part define

P_{A B}^{μ} (z; y, y^{'}) = Q_{A B}^{μν} (z; y, y^{'}) = \frac{1}{2} (Λ_{A}^{ν} (z, y) \partial_{z^{ν}} Λ_{B}^{μ} (z, y^{'}) + Λ_{B}^{ν} (z, y^{'}) \partial_{z^{ν}} Λ_{A}^{μ} (z, y)) - Ω_{A B}^{μ} (z; y, y^{'}), \frac{1}{4} (Λ_{A}^{μ} (z, y) Λ_{B}^{ν} (z, y^{'}) + Λ_{B}^{μ} (z, y^{'}) Λ_{A}^{ν} (z, y)) .

Then

O^{(2)} [h, k, ϕ] = u_{2, A B} = \int u_{1, A} (y, z) k_{A} (y) ϕ (z) d^{4} y d^{4} z + \int u_{2, A B} (y, y^{'}, z) h_{A} (y) h_{B} (y^{'}) ϕ (z) d^{4} y d^{4} y^{'} d^{4} z, - \partial_{z^{μ}} (f P_{A B}^{μ}) + \partial_{z^{μ}} \partial_{z^{ν}} (f Q_{A B}^{μν}) .

The kernels are symmetric in their two metric slots.

Displayed formulas in this passage

X-A-conv@1#eq:uone · line 1419
$O^{(1)} [h, ϕ] = \int u_{1, A} (y, z) h_{A} (y) ϕ (z) d^{4} y d^{4} z, u_{1, A} = \partial_{z^{μ}} (f (z) Λ_{A}^{μ} (z, y)) .$
X-A-conv@1#eq:Pkernel · line 1429
$P_{A B}^{μ} (z; y, y^{'}) = Q_{A B}^{μν} (z; y, y^{'}) = \frac{1}{2} (Λ_{A}^{ν} (z, y) \partial_{z^{ν}} Λ_{B}^{μ} (z, y^{'}) + Λ_{B}^{ν} (z, y^{'}) \partial_{z^{ν}} Λ_{A}^{μ} (z, y)) - Ω_{A B}^{μ} (z; y, y^{'}), \frac{1}{4} (Λ_{A}^{μ} (z, y) Λ_{B}^{ν} (z, y^{'}) + Λ_{B}^{μ} (z, y^{'}) Λ_{A}^{ν} (z, y)) .$
X-A-conv@1#eq:Qkernel · line 1429
$P_{A B}^{μ} (z; y, y^{'}) = Q_{A B}^{μν} (z; y, y^{'}) = \frac{1}{2} (Λ_{A}^{ν} (z, y) \partial_{z^{ν}} Λ_{B}^{μ} (z, y^{'}) + Λ_{B}^{ν} (z, y^{'}) \partial_{z^{ν}} Λ_{A}^{μ} (z, y)) - Ω_{A B}^{μ} (z; y, y^{'}), \frac{1}{4} (Λ_{A}^{μ} (z, y) Λ_{B}^{ν} (z, y^{'}) + Λ_{B}^{μ} (z, y^{'}) Λ_{A}^{ν} (z, y)) .$
X-A-conv@1#eq:utwoformula · line 1447
$O^{(2)} [h, k, ϕ] = u_{2, A B} = \int u_{1, A} (y, z) k_{A} (y) ϕ (z) d^{4} y d^{4} z + \int u_{2, A B} (y, y^{'}, z) h_{A} (y) h_{B} (y^{'}) ϕ (z) d^{4} y d^{4} y^{'} d^{4} z, - \partial_{z^{μ}} (f P_{A B}^{μ}) + \partial_{z^{μ}} \partial_{z^{ν}} (f Q_{A B}^{μν}) .$
X-A-conv@1#eq:utwo · line 1447
$O^{(2)} [h, k, ϕ] = u_{2, A B} = \int u_{1, A} (y, z) k_{A} (y) ϕ (z) d^{4} y d^{4} z + \int u_{2, A B} (y, y^{'}, z) h_{A} (y) h_{B} (y^{'}) ϕ (z) d^{4} y d^{4} y^{'} d^{4} z, - \partial_{z^{μ}} (f P_{A B}^{μ}) + \partial_{z^{μ}} \partial_{z^{ν}} (f Q_{A B}^{μν}) .$

Nearby in this section

proposition: Proposition: Classical regularity and derivative kernels