Hilbert-type inequalities with non-homogeneous kernel: another view

We will survey some recent results on the theory of fractional Orlicz-Sobolev spaces with a special focus on their role in Sobolev type embeddings with an optimal Orlicz target. We will also mention related Hardy type inequalities, criteria for compact embeddings, and limits of these spaces when the smoothness parameter tends to either of its natural endpoints. A novel approach to the so-called Hilbert's double series theorem with non-homogeneous kernel and several generalizations are considered. The applications concern among others the multiple discrete Hilbert inequality, the Mordell-Tornhemi-Witten and Matsumoto, and Tsumura Zeta functions integral expressions. The main tools are the Cahen's Laplace integral formula for Dirichlet series developed for the generalized (a,λ)-series theory and the Euler-Maclaurin summation formula.