Let \(f\in L^p_{\text{loc}}(\Omega)\) with \(p\in [1,\infty)\). Then the mollification \(f_\varepsilon\) converges to \(f\) in \(L^p_{\text{loc}(\Omega)}\), that is for every compact subset \(U\subset \Omega\) the mollification \(f_\varepsilonl\) converges in \(L^p(U)\). [1, C.4 Theorem 6]

References Link to heading

1. L. Evans, Partial differential equations. Providence (R. I.): American mathematical society, 1998.