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Euclidean and block-permutation pullbacks preserve Schwartz space

Statement

For every integer $n \geq 1$ , every Euclidean isometry $g (x) = R x + a$ of $R^{2}$ , and every permutation $π \in S_{n}$ , the pullbacks $T_{g, n} f (x_{1}, \dots, x_{n}) = f (g^{- 1} x_{1}, \dots, g^{- 1} x_{n}), T_{π, n} f (x_{1}, \dots, x_{n}) = f (x_{π^{- 1} (1)}, \dots, x_{π^{- 1} (n)})$ are continuous linear automorphisms of $S ((R^{2})^{n})$ .

Statement references

Schwartz space and its seminorms

Facts & Assumptions

Given: $n \geq 1$ , $g (x) = R x + a$ with $R \in O (2)$ and $a \in R^{2}$ , a permutation $π \in S_{n}$ , and $f \in S ((R^{2})^{n})$ .

[F1]

The Schwartz space is defined by the seminorms $p_{α β} (f) = \sup_{x} ∣ x^{α} \partial^{β} f (x) ∣$ for all multi-indices $α, β$ (Schwartz space and its seminorms).

[A1]

The pullbacks use the block-diagonal affine map $(x_{1}, \dots, x_{n}) \mapsto (g^{- 1} x_{1}, \dots, g^{- 1} x_{n})$ and the linear block permutation $(x_{1}, \dots, x_{n}) \mapsto (x_{π^{- 1} (1)}, \dots, x_{π^{- 1} (n)})$ .

Proof

1.1givenF1A1algebra

In dimension $2 n$ , each pullback is $f \circ L$ for an affine map $L (x) = A x + b$ whose linear part $A$ is invertible. By the chain rule, every derivative $\partial^{β} (f \circ L)$ is a finite linear combination of $(\partial^{γ} f) \circ L$ . Since $x = A^{- 1} (y - b)$ , each monomial $x^{α}$ is a finite linear combination of monomials $y^{δ}$ with $y = L (x)$ . Therefore each seminorm $p_{α β} (f \circ L)$ is bounded by a finite linear combination of seminorms $p_{δ γ} (f)$ from [F1]. This proves that the pullback maps Schwartz space to itself and is continuous in the stated seminorm topology.

2.1givenA1step 1.1algebra∎

The inverse pullbacks are $T_{g^{- 1}, n}$ and $T_{π^{- 1}, n}$ , respectively; direct substitution gives $T_{g, n} T_{g^{- 1}, n} = T_{g^{- 1}, n} T_{g, n} = i d$ and the same identities for $π$ . Hence both maps are linear automorphisms.

Euclidean and block-permutation pullbacks preserve Schwartz space

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources