Real powers of the Japanese bracket act on Schwartz space

Put $q(\xi )=1+\mathrm{\mid }\xi {\mathrm{\mid }}^2$. Induction on $\mathrm{\mid }\alpha \mathrm{\mid }$, differentiating either the polynomial factor or $q^{s\mathrm{/}2-j}$, expresses each derivative as a finite sum ${\mathrm{\partial }}^{\alpha }{w}_s(\xi )={\sum }_j{P}_{\alpha ,j}(\xi )q(\xi {)}^{s\mathrm{/}2-j},$ where every $P_{\alpha ,j}$ is a polynomial of degree at most $\mathrm{\mid }\alpha \mathrm{\mid }$. Since $q=⟨\xi {⟩}^2$, each term is bounded by a constant times $⟨\xi {⟩}^{s+\mathrm{\mid }\alpha \mathrm{\mid }}$, and hence by $C_{\alpha ,s}⟨\xi {⟩}^{\max (0,s+\mathrm{\mid }\alpha \mathrm{\mid })}$. Enlarging the exponent to an integer gives a polynomial-growth bound for this derivative.

The same induction with $-s$ gives a polynomial-growth bound for every derivative of $w_{-s}$.

The bounds in steps 1.1 and 2.1 meet the hypotheses of [F2], so multiplication by either weight is continuous on Schwartz space. Pointwise $w_s{w}_{-s}=1$, so both compositions on $\mathcal{S}$ are the identity.

For $T\in {\mathcal{S}}^{\mathrm{\prime }}$, transposition defines $⟨w_{\pm s}T,\phi ⟩=⟨T,w_{\pm s}\phi ⟩$. By [F2] these maps are continuous for the weak and strong dual topologies; their compositions evaluate $T$ on $w_s{w}_{-s}\phi =\phi$, so they are inverse on ${\mathcal{S}}^{\mathrm{\prime }}$.