I believe that the statement below is a standard fact but I haven't figured out yet:
Suppose $f\in L^{1}(\mathbb{R}^{n})$ has integrable partial derivatives of order $n+1$ and $D^{\alpha}f\in L^{1}(\mathbb{R}^{n})$ for all multiindex with $|\alpha|\le n+1$ (here $D^{\alpha}$ is the mixed partial derivative). Prove that the Fourier transform $\hat{f}$ is in $L^{1}(\mathbb{R}^{n})$.
Does anyone know the proof or a reference for this statement?