Manuscript XX-D@1 · XX-D.tex · section · c83a526d

The Hessian: microcausality, noncompactness, and internal contraction

The Hessian: microcausality, noncompactness, and internal contraction · lines 884–918

XX-D@1#eq:b

Put

q (z) = q_{loc} (z) - 2 \calRz \cdot \nabla ℓ z .

Let $b (z, w)$ be its symmetric polarization:

b (z, w) = \frac{1}{2} (q (z + w) - q (z) - q (w)) .

Thus

b (z, w) = b_{loc} (z, w) - \calRz \cdot \nabla ℓ w - \calRw \cdot \nabla ℓ z .

Define $K_{f}$ by

⟨ K_{f}, z \otimes w ⟩ = \int f b (z, w) d V_{g ˉ} .

Then

F_{f}^{(2)} = s^{2} K_{f}

with respect to the free first jet $z$ .

All component kernels may be written relative to coordinate measures by absorbing the smooth $a^{3}$ factors. This does not change wavefront sets or scaling degrees.

Displayed formulas in this passage

XX-D@1#eq:b · line 897
$b (z, w) = b_{loc} (z, w) - \calRz \cdot \nabla ℓ w - \calRw \cdot \nabla ℓ z .$
unlabeled · line 888
$q (z) = q_{loc} (z) - 2 \calRz \cdot \nabla ℓ z .$
unlabeled · line 893
$b (z, w) = \frac{1}{2} (q (z + w) - q (z) - q (w)) .$
unlabeled · line 904
$⟨ K_{f}, z \otimes w ⟩ = \int f b (z, w) d V_{g ˉ} .$
unlabeled · line 909
$F_{f}^{(2)} = s^{2} K_{f}$