Show that if $f,\hat f,g \in {L^1}(R)$ then $f.g \in {L^1}(R)$

Question

I'im wondering if the following statment is true: let $f,\hat f,g \in {L^1}(R)$, show that $f.g \in {L^1}(R)$. I proved that if $f,\hat f \in {L^1}(R)$ and $g \in {L^2}(R)$ then $f.g \in {L^1}(R)$. Is the first statment hold if $g \in {L^1}(R)$ ? thanks.

Answer 1

Since the Fourier transform maps $L^1(\mathbb R)$ into $L^\infty(\mathbb R)$ it follows that if $f,\hat f\in L^1(\mathbb R)$, then also $f,\hat f\in L^\infty(\mathbb R)$. Then the assertion follows immediately by Hölder's inequality:

$$ \|f\cdot g\|_1 \le \|f\|_\infty \cdot \|g\|_1 <\infty$$