An elementary proof of the L1L1 log-Sobolev inequality for Poisson point processes

In this note we provide a new proof of the L1L1 log-Sobolev inequality on the path space of Poisson point processes. Our proof is elementary in the sense that it avoids the use of the martingale representation on Poisson spaces. Moreover, the weak Poincaré inequality for the weighted Dirichlet form is presented.