Assume $f,g, fg\in L^1(\mathbb{R}^n)$ and $\widehat{f}\in L^1(\mathbb{R}^n)$. Prove that $\widehat{fg}=(2\pi)^{-n}\widehat{f}*\widehat{g}$

Question

To demonstrate this exercise we will use the following theorem:

Theorem: Let $f\in L^1(\mathbb{R}^n)$ and assume that $\widehat{f}\in L^1(\mathbb{R}^n)$. Then $f$ is equivalent to a continuous function. Therefore, we assume with no loss of generality that $f$ is continuous. Then for $x\in\mathbb{R}^n$, $$f(x)=(2\pi)^{-n}\displaystyle{\int_{\mathbb{R}^n}\widehat{f}(\xi)e^{ix\cdot\xi}\,d\xi}.$$

by the previous theorem and by definition of Fourier transform

\begin{align*} \widehat{fg}(\xi)&=\int_{\mathbb{R}^n}f(x)g(x)e^{-ix\cdot\xi}\,dx\\ &= \int_{\mathbb{R}^n}\left[(2\pi)^{-n}\int_{\mathbb{R}^n}\widehat{f}(\eta)e^{ix\cdot\eta}\,d\eta\right]g(x)e^{-ix\cdot\xi}\,dx\\&=(2\pi)^{-n}\int_{\mathbb{R}^n}g(x)e^{-ix\cdot\xi}\left[\int_{\mathbb{R}^n}\widehat{f}(\eta)e^{ix\cdot\eta}\,d\eta\right]\,dx\\&=(2\pi)^{-n}\int_{\mathbb{R}^n}\left[\int_{\mathbb{R}^n}\widehat{f}(\eta)g(x)e^{-ix(\xi-\eta)}\,d\eta\right]\,dx\\&=(2\pi)^{-n}\int_{\mathbb{R}^n}\left[\int_{\mathbb{R}^n}\widehat{f}(\eta)g(x)e^{-ix\cdot(\xi-\eta)}\,dx\right]\,d\eta\\&=(2\pi)^{-n}\int_{\mathbb{R}^n}\widehat{f}(\eta)\left[\int_{\mathbb{R}^n}g(x)e^{-ix\cdot(\xi-\eta)}\,dx\right]\,d\eta\\&=(2\pi)^{-n}\int_{\mathbb{R}^n}\widehat{f}(\eta)\widehat{g}(\xi-\eta)\,d\eta\\&=(2\pi)^{-n}\left(\widehat{f}*\widehat{g}\right)(\xi). \end{align*} Remark: Note that Fubini's theorem has been applied to the fifth line of the chain of equalities above, which is possible since $\widehat{f}(\eta)g(x)\in L^1(\mathbb{R}^n\times\mathbb{R}^n)$ which implies that $\displaystyle{\int_{\mathbb{R}^n\times\mathbb{R}^n}\widehat{f}g}\hspace{.2cm}$ exists. Indeed: \begin{equation} \left|\int_{\mathbb{R}^n\times\mathbb{R}^n}\widehat{f}g\right|\leq\int_{\mathbb{R}^n\times\mathbb{R}^n}|\widehat{f}g|=\|\widehat{f}g\|_{L^1}\leq\|\widehat{f}\|_{L^1}\|g\|_{L^1}<\infty\qquad (1). \end{equation} My concerns about the above proof are:

1. The chain of equalities immediately above is correct to justify that $\displaystyle{\int_{\mathbb{R}^n\times\mathbb{R}^n}\widehat{f}g}$ exists? Also I'm not sure if the last inequality in $(1)$ is true.
2. Did I apply Fubbini's theorem correctly?

Answer 1

Two observatios:

1. Since $\widehat{f}\in L_1$, the Fourier inversion theorem implies that $f\in\mathcal{C}_0(\mathbb{R}^n)$; hence $fg\in L_1$ and $\|fg\|_1\leq \|f\|_\infty\|g\|_1$. This makes line 1 in your chain of identities valid.

2. $\widehat{f}(\eta) g(x)\in L_1(\mathbb{R}^n\times\mathbb{R}^n)$ by the Fubini-Tonelli theorem, for the iterated integral $$\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}|\widehat{f}(\eta)||g(x)|\,d\eta dx=\int_{\mathbb{R}^n}|\widehat{f}(\eta)|\,d\eta\int_{\mathbb{R}^n}|g(x)|\,dx=\|\widehat{f}\|_1\|g\|_1<\infty$$ by assumption on $\widehat{f}$.