# Applying the generalised Leibniz rule multi-index form

The multi-index Leibniz rule states $\partial^{\alpha}(fg) = \sum_{\beta \leq \alpha}{\alpha \choose \beta} (\partial^{\beta}f)(\partial^{\alpha - \beta}g)$

Where $\alpha, \beta$ are multi-indices.

I often struggle to apply this formula to problems and am looking for some guidance: For example:

1): We want to show that the space of tempered distributions is stable under multiplication by polynomials i.e. given $u \in \mathcal{S}(\mathbb{R}^{n}) \Rightarrow x_{j}u \in \mathcal{S}'(\mathbb{R}^{n})$ the space of tempered distributions.

Proof:

Letting $u , \varphi \in \mathcal{S}(\mathbb{R}^{n})$ i.e. Schwartz space and $j \in \{1,\dots,n\}.$ Then $(x_{j}u)(\varphi) = u(x_{j}\varphi).$

Since $x_{j}\varphi \in \mathcal{S}(\mathbb{R}^{n})$ we have by definition of the Schwartz space that

$$\sum_{|\alpha|, |\beta| \leq N} \sup|x^{\alpha}\partial^{\beta}(x_{j}\varphi)|$$

I don't understand this next step Applying the Lebniz rule we obtain

$$\sum_{|\alpha|, |\beta| \leq N} \sup|x^{\alpha}\partial^{\beta}(x_{j}\varphi)|\leq C_{N}' \sum_{|\alpha|,|\beta| \leq N+1}\sup|x^{\alpha}\partial^{\beta}\varphi|$$

For some $C_{N}' > 0$ and therefore $x_{j}u \in \mathcal{S}'(\mathbb{R}^{n})$

Thanks.

• Neither of your last two assertions are sentences.... – Lord Shark the Unknown Jul 9 '18 at 18:25
• @LordSharktheUnknown added more context. – VBACODER Jul 9 '18 at 18:36