Take the Schwartz space $\mathcal{S}(\mathbb{R}^{n})$ with family of norms: $$\|f\|_{\alpha,\beta} = \sup_{x\in \mathbb{R}^{n}}|(\partial^{\alpha}f)(x)x^{\beta}|.$$ Now consider the following mapping: $$\mathcal{S}(\mathbb{R}^{n}) \ni f \mapsto g(x) =\int_{\mathbb{R}^{n}}\varphi(y)f(x+y)dy$$ Does the following inequality: $$\|g\|_{\alpha,\beta} \le \|f\|_{\alpha,\beta}\int_{\mathbb{R}^{n}}|\varphi(y)|dy$$ hold?
I sketched some calculations: $$\bigg{|}\bigg{(}\partial^{\alpha}\int \varphi(y)f(x+y)dy\bigg{)}x^{\beta}\bigg{|} = \bigg{|}\bigg{(}\int \varphi(y)(\partial^{\alpha}f)(x+y)dy\bigg{)}x^{\beta}\bigg{|} \le \int |\varphi(y)||x^{\beta}(\partial^{\alpha}f)(x+y)|dy \le \int |\varphi(y)|\sup_{x\in \mathbb{R}^{n}}|x^{\beta}(\partial^{\alpha}f)(x)|dy = \|f\|_{\alpha,\beta}\int|\varphi(y)|dy$$ but I am not sure if the steps are right.
Thanks!