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.

Let M be a symplectic manifold. It has a group of structure preserving automorphisms called the Symplectomorphism group. This group has a lie subgroup called the Hamiltonian group. It is known that the number of conjugacy classes of maximal tori in Ham by Symp is finite for 4 dimensional M. This result also holds in general when the torus has half the dimension of M. We hope to generalize this result to Symplectic orbifolds. This result leads to implications on conjugacy classes of maximal tori in the Contactomorphism group of a compact contact manifold.

For a field $F$ containing a primitive $p$-root of $1$, let $G$ be the Galois group of the maximal $p$-extension $F(p)$ of $F$. The group $G$ might become very fat -- i.e., the size of its open subgroups might increase arbitrarily. This does not happen (namely, the size of its open subgroups is always the same) precisely when $F$ needs to eat only the roots of $p$-power index of its elements to reach $F(p)$. In this case we may compute explicitly the structure of $G$.