Let $1<p<2$ . Let us denote$L^p(\mathbb R^n)$ by $L^p$ in short and let $1/p+1/p'=1$ . Let $f \in L^p$ , let $g_n$ be a sequence in $C_c(\mathbb R^n)$ converging to $f$ in $L^p$ . Let $g$ be the $L^{p'}$ limit of $\hat g_n$ in $L^{p'}$ . Let $f_1 =f.1_{\{|f(x)|\le1\}}\in L^2$ and $f_2=f.1_{\{|f(x)|\ge1\}}\in L^1$
Then how to show that $g=\hat f_1+\hat f_2$ ?
I can show that it is enough to check for one sequence in $C_c$ converging to $f$ in $L^p$ i.e. that if two sequences in $C_c$ converges to $f$ in $L^p$ , then their Fourier transforms converges to the same function in $L^{p'}$. But I don't know if it is helpful.
Please help.