Consider the Fourier transform for a (complex-valued ) function $f$ defined on $\mathbb{Z}^d$. Its Fourier transform $\hat{f}$ is a function on $\mathbb{T}^d$: $$\hat{f}(p)=\sum_{x\in\mathbb{Z}^d} e^{-ip\cdot x}f(x).$$
What can be said if one wants to estimate the $L^1$ norm of $\hat{f}$ in the form of $\|\hat{f}\|_1\le \dots$?
Clearly, $\|\hat{f}\|_1 \le (2\pi)^d \|\hat{f}\|_{\infty} \le \|f\|_1$, but this estimate ignores the phases $e^{-ip\cdot x}$. Is there a way to take advantage of the cancellation due to these phases?