Abstract: The Fourier transform, $\mathcal{F}$, enjoys certain boundedness properties as a map between various normed spaces. Typical examples are : $\|\widehat{f}\|_{L^{\infty}} \leq \|f\|_{L^1}$, $\|\widehat{f}\|_{L^2} = \|f\|_{L^2}$ (Plancherel theorem) and $\|\widehat{f}\|_{L^{p'}} \leq C \|f\|_{L^p}$ , (Hausdorff-Young inequality with $1< p \leq 2$ and $1/p + 1/p' = 1$).

This talk starts with a brief history of Fourier inequalities in weighted $L^p$ spaces and weighted Lorentz spaces. The Lorentz norm with weight $w$ is defined as $\|f\|_{\Lambda_w^p} = \|f^*\|_{L_w^p}$ where $f^*$ is decreasing rearrangement of $f$.

Then I will focus on our recent joint work with G. Sinnamon on norm inequalities for Fourier series. I will present relationship between weight functions $u(t), w(t)$ and exponents $(p,q)$ that is sufficient/necessary for $\|\widehat{f}\|_{\Lambda_u^q}\leq C\|f\|_{\Lambda_w^p}$.

An immediate consequence is some new results on boundedness of $\mathcal{F} : L_w^p (\mathbb{T}) \longrightarrow L_u^q(\mathbb{Z})$ where $u[n]$ and $w(t)$ are weight functions on $\mathbb{Z}$ and the unit circle $\mathbb{T}$ respectively.