Consider a closed smooth hyperbolic surface ${\Sigma = \Gamma \backslash \mathbb{H}}$. Let ${k(x,y)}$ be a continuous function which depends only on the hyperbolic distance between ${x,y \in \mathbb{H}}$, and has some ``nice'' decay properties. Using ${k(x,y)}$, we construct a trace-class integral operator ${T_k}$ on ${L^2 (\Sigma)}$. The trace of ${T_k}$ is computed in two different ways using the Lidski's trace formula. The resulting Selberg's trace formula gives a relation between the length of closed geodesics and the eigenvalues of the hyperbolic Laplacian on $\Sigma$.

This talk is on the mapping properties of the Fourier transform between Banach function spaces. These are generalizations of Hausdorff-Young and Pitt's inequalities.

We provide several relations between weight functions, that guarantee the boundedness of the Fourier series coefficients, viewed as a map between weighted Lorentz spaces. As a useful machinery, we briefly introduce the quasi concave functions and generalize a number of known inequalities. Finally, we apply our results to Fourier inequalities in weighted Lebesgue spaces and Lorentz-Zygmund spaces