Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 327 / 3 / c / i

Table of contents
- Solution i

Solution

The symbol class

Sym (X, R^{n}; N) = S_{1, 0}^{N}

consists of

c \in C^{\infty} (X \times R^{n})

such that, for every compact

K \subset X

and all multi-indices

α, β

,

sup_{x \in K} ∣ D_{x}^{α} D_{λ}^{β} c (x, λ) ∣ \leq C_{K, α, β} ⟨ λ ⟩^{N - ∣ β ∣} .

The defining estimate gives

D_{x}^{α} D_{λ}^{β} c_{i} \in Sym (X, R^{n}; N_{i} - ∣ β ∣) .

The Leibniz rule gives

c_{1} c_{2} \in Sym (X, R^{n}; N_{1} + N_{2}),

and the triangle inequality gives

c_{1} + c_{2} \in Sym (X, R^{n}; max {N_{1}, N_{2}}) .

These are the basic rules of symbol calculus.

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