Manuscript Em01-A@1 · Em01-A.tex · proposition · 880b67af

Proposition: Noninjective witness contraction

Witness data versus complete distributions › Partial contraction loses information · lines 897–936 · proof included or linked

Em01-A@1#prop:partial-contraction

Back to manuscript

Let $M_{i} = R^{d_{i}}$ , $d_{i} \geq 1$ , and let $w_{1}, w_{2}$ be fixed smooth scalar functions on $M_{1}, M_{2}$ . For smooth compact kernels $T$ on $M_{1} \times M_{2} \times M_{3}$ , define the witness response

W (T) (φ) = ⟨ T, w_{1} \otimes w_{2} \otimes φ ⟩, φ \in C_{c}^{\infty} (M_{3}) .

Then $W$ is not injective. Its kernel contains a nonzero smooth compact kernel.

Proof. Choose a nonzero $v_{1} \in C_{c}^{\infty} (M_{1})$ with

\int_{M_{1}} v_{1} w_{1} = 0.

Such a function exists: the displayed integral is one linear functional on an infinite-dimensional test space. Explicitly, two linearly independent test functions yield a nonzero linear combination annihilating it.

Choose nonzero $v_{2} \in C_{c}^{\infty} (M_{2})$ and $v_{3} \in C_{c}^{\infty} (M_{3})$ , and set

B = v_{1} \otimes v_{2} \otimes v_{3} .

Then $B \neq = 0$ , but

W (B) (φ) = (\int v_{1} w_{1}) (\int v_{2} w_{2}) (\int v_{3} φ) = 0

for every $φ$ . All these kernels are smooth and compact.

Displayed formulas in this passage

unlabeled · line 903
$W (T) (φ) = ⟨ T, w_{1} \otimes w_{2} \otimes φ ⟩, φ \in C_{c}^{\infty} (M_{3}) .$
unlabeled · line 914
$\int_{M_{1}} v_{1} w_{1} = 0.$
unlabeled · line 924
$B = v_{1} \otimes v_{2} \otimes v_{3} .$
unlabeled · line 928
$W (B) (φ) = (\int v_{1} w_{1}) (\int v_{2} w_{2}) (\int v_{3} φ) = 0$

Proposition: Noninjective witness contraction

Displayed formulas in this passage

Nearby in this section