For a class of symmetric Lévy processes (Yt) t≥0 with charac- are the transition densities of (Yt) t≥0 . We estimate (from above and below) pt in terms of two metrics d ψ,t and δ ψ,t , d ψ,t controlling pt(0) and δ ψ,t the spatial decay, and we prove that the transition density π t,0 of P Xt−X 0 is controlled by δ ψ, 1 t and d ψ, 1 t now with δ ψ, 1 t controlling π t,0 (0) and d ψ, 1 t the spatial decay.