Well-defined ness of defining Fourier transform of functions in $L^p$ ,for $1<p<2$, by decomposing as a sum of $L^1$ and$L^2$ functions

Question

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 $f \in L^p$ , and let $g\in L^1 , h\in L^2$ be such that $f=g+h$ . Let $f_1 =f.1_{\{|f(x)|\le1\}}\in L^2$ and $f_2=f.1_{\{|f(x)|\ge1\}}\in L^1 $

Then is it true that $\hat g + \hat h=\widehat f_1 + \widehat f_2$ ?

Answer 1

First, a comment for anyone who thinks that $\hat g+\hat h=\hat f_1+\hat f_2$ is trivial by linearity of the Fourier transform, since $g+h=f_1+f_2$:

No. The problem is that at this point we don't know of a single vector space $V$ such that $g,h,f_1,f_2\in V$ and also such that the Fourier transform is defined (and linear) on $V$. (For example, we need to do this exercise first in order to define the Fourier transform on $L^p$.)

Hint: $h-f_1=f_2-g\in L^1\cap L^2$. The definition of the Fourier transform on $L^2$ shows that it agrees with the previously defined $L^1$ transform on $L^2\cap L^1$...