This isn't supposed to be hard, but I just can't seem to be able to write this down. My Radon transform is defined as follows: Let $f\in C_c^{\infty}(\mathbb{R}^2)$, then $$Rf(s,\omega)=\int_{\mathbb{R}}f(s\omega+t\omega^{\perp})\,dt,\quad s\in\mathbb{R}, \omega\in S^1,$$ where $\omega^{\perp}=(-\omega_2,\omega_1)$. I am supposed to show that if $f\in C_c^{\infty}(\mathbb{R}^2)$ then $Rf\in C^{\infty}(\mathbb{R}\times S^1)$. I have already used Dominated convergence to show that $Rf$ is continuous and I know that the derivatives can be shown to be continuous the same way. My problem is to show that the derivatives exist. I know this is a silly question, but I am just stuck. I am trying to show the existence using difference quotients and Dominated convergence, but I don't know what should be the dominating function here.
EDIT:
I used Leibniz integration rule for the above mentioned problem.
But a related problem is this: If $f$ is a Schwartz function then I need to show that $Rf\in C^{\infty}(\mathbb{R}\times S^1)$. For this I have shown that $Rf$ is continuous but I don't know how to prove the existence of derivatives. Again I am trying to use difference quotients. My definition for the Schwartz function is that $f$ is a Schwartz function if $f\in C^{\infty}(\mathbb{R}^2)$ and $x^a\partial^bf\in L^{\infty}(\mathbb{R}^2)$ for all multi-indices $a$ and $b$.