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

Proposition: Classical regularity and derivative kernels

Smooth coefficient kernels, support, and continuity › Joint kernels of the observable · lines 1464–1495 · proof included or linked

X-A-conv@1#prop:regularity

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At fixed $ε$ ,

u_{1} \in C_{c}^{\infty} (M^{2}), u_{2} \in C_{c}^{\infty} (M^{3}),

with the indicated tensor components. Their supports are contained in

supp g \times supp f, supp g \times supp g \times supp f,

respectively. The coefficients are continuous polynomial functionals on the declared smooth configuration spaces.

Every joint directional derivative kernel is obtained from (eq:uone) and (eq:utwo) by slot contraction with the specified smooth directions and the finite polynomial Leibniz rule. Consequently those derivative kernels are smooth and compactly supported in the remaining slots.

Proof. Lemma lem:smoothkernel gives smoothness. The compact factors $g (y)$ , $g (y^{'})$ , and $f (z)$ , including their derivatives after integration by parts, give compact support.

Pairing a smooth compact kernel against smooth fields is a continuous multilinear map: each pairing is bounded by a finite product of compact-open seminorms of its arguments. This proves polynomial continuity. Differentiation of a finite polynomial contracts slots of its fixed kernels, proving the final assertion.

Displayed formulas in this passage

unlabeled · line 1467
$u_{1} \in C_{c}^{\infty} (M^{2}), u_{2} \in C_{c}^{\infty} (M^{3}),$
unlabeled · line 1471
$supp g \times supp f, supp g \times supp g \times supp f,$

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section: Joint kernels of the observable