Showing that $f \in L^p$ if both $f$ and its Fourier coefficients are in $L^1$

Question

How do I go about proving that $f \in L^p(\mathbb{R})$ for $1\leq p \leq \infty $ if $f, \hat{f} \in L^1(\mathbb{R})$ ? I feel like I will have to somehow use the Fourier inversion theorem with some other inequality like Holder's inequality but I am stuck. Any help is highly appreciated.

Answer 1

Since $\hat{f}\in L^1$ then $f$ is uniformly continuous. So you can set $$f = f_++f_- \qquad\qquad f_+(x) = f(x) 1_{|f(x)| \ge 1} \qquad\qquad f_-(x) = f(x) 1_{|f(x)| < 1}$$ and $f_+$ is compactly supported.

Finally since $f \in L^1$ $$\|f\|_{L^p}^p \le \|f_-\|_{L^1}^p+\|f_+\|_{L^p}^p < \infty$$