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For $K \in C^{\infty}([0,1]\times[0,1])$ and $f \in L^{1}[0,1]$ is

$Tf(x) = \int_0^1K(x,y)f(y) \, dy \in C^{\infty}[0,1]$.

I can show that $Tf(x)$ does indeed exist but I don't know how to show that it is infinitely differentiable.

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Notice that for every $n\geq0$ there exists $M_n>0$ such that $$\left|\frac{\partial^nK}{\partial x^n}(x,y)\right|<M_n,\quad \forall\,(x,y)\in [0,1]\times [0,1]$$ (if this is not clear, see here). Then, $$\left|\frac{\partial^nK}{\partial x^n}(x,y)f(y)\right|\leq M_n\,f(y)\in L^1([0,1]),$$ so you can differentiate under the integral sign any number $n$ of times (if this is not clear, see the accepted answer here).

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