Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 327 / 2 / ii

Table of contents
- Solution ii

Solution

The multivariable Leibniz rule writes each derivative of

a_{1} a_{2}

as a finite sum of products

(D_{x}^{α_{1}} D_{θ}^{β_{1}} a_{1}) (D_{x}^{α_{2}} D_{θ}^{β_{2}} a_{2}), β_{1} + β_{2} = β .

Each term is bounded by a constant times

(1 + ∣ θ ∣)^{N_{1} - ∣ β_{1} ∣} (1 + ∣ θ ∣)^{N_{2} - ∣ β_{2} ∣} = (1 + ∣ θ ∣)^{N_{1} + N_{2} - ∣ β ∣} .

Hence

a_{1} a_{2} \in Sym (X, R^{k}; N_{1} + N_{2}) .

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