Let $f: \mathbb{R}^n \rightarrow \mathbb{C}$ be in $L^2\cap L^1,$ then the Fourier transform is in $L^2 \cap L^\infty.$

Does this imply that we can take common norms in the sense that we can estimate

$$\sup_{x_i \in \mathbb{R}} \sqrt{\int_{\mathbb{R}^{n-1}} \left\lvert \hat{f}(x_1,...,x_i ,...,x_n) \right\rvert^2 dx_1...dx_{i-1}dx_{i+1}...dx_{n}}$$ or alternatively $$ \sqrt{\int_{\mathbb{R}^{n-1}}\sup_{x_i \in \mathbb{R}} \left\lvert \hat{f}(x_1,...,x_i ,...,x_n) \right\rvert^2 dx_1...dx_{i-1}dx_{i+1}...dx_{n}}?$$

Maybe this can even be bounded by mixed norms

$$\int_{\mathbb{R}} \sqrt{\int_{\mathbb{R}^{n-1}} \left\lvert f(x_1,...,x_i ,...,x_n) \right\rvert^2 dx_1...dx_{i-1}dx_{i+1}...dx_{n}}dx_i$$ or

$$ \sqrt{\int_{\mathbb{R}^{n-1}}\left(\int_{\mathbb{R}} \left\lvert f(x_1,...,x_i ,...,x_n) \right\rvert dx_i\right)^2 dx_1...dx_{i-1}dx_{i+1}...dx_{n}}?$$

However, I do not see yet whether this is true or not.