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Let $G$ be a linear Lie group, say $\mathrm{SL}(2,\mathbb{R})$, and $\pi:G\to\mathrm{GL}(H)$ a continuous representation of $G$ in a Banach space $H$. Let $\pi^1:\mathcal{C}_c(G)\to\mathrm{End}(H)$ be defined by $$\pi^1(\varphi)v:=\int_G \varphi(x)\pi(x)v\,\mathrm{d}x.$$ I want to show that $\pi^1(\varphi)v$ is a smooth vector for any $\varphi\in\mathcal{C}_c^\infty(G), v\in H$. This essentially boils down to proving $f\ast g\in\mathcal{C}^\infty(G)$ for $f\in\mathcal{L}^1_\mathrm{loc}(G), g\in\mathcal{C}_c^\infty(G).$ For $G=\mathbb{R}$, the integrand in the expression $$(f\ast g)(y)=\int_G f(x)g(x^{-1}y)\,\mathrm{d}x$$ is a differentiable function of $y$ (of course, $x^{-1}y$ here is to be understood as $y-x$ since $G=\mathbb{R}$), its derivative being bounded by an integrable function of $x$. This boundedness is uniform with respect to $y$ if $y$ is restricted to take values in a bounded interval. Differentiability hence smoothness of $f\ast g$ then follows by the mean value and dominated convergence theorems. I try to follow the same procedure for $G=\mathrm{SL}(2,\mathbb{R})$, pulling back $f\ast g$ to local coordinates, but I cannot even see why a (partial) derivative should be bounded by an integrable function—the computations are too cumbersome. In his book on $\mathrm{SL}(2,\mathbb{R})$, Lang is quite terse at this point (p. 93, proof of Lemma 3). Could you give an idea on how to proceed, or a reference?

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