Bounds on short cylinders and uniqueness in Cauchy problem for degenerate Kolmogorov equations

We consider the Cauchy problem for degenerate Kolmogorov equations in the form ∂ t u = ∑ i , j = 1 m a i , j ( x , t ) ∂ x i x j u + ∑ j = 1 m a j ( x , t ) ∂ x j u + ∑ i , j = 1 N b i , j x i ∂ x j u , ( x , t ) ∈ R N × ] 0 , T [ , 1 ⩽ m ⩽ N , as well as in its divergence form. We prove that, if | u ( x , t ) | ⩽ M exp ( a ( t − β + | x | 2 ) ) , for some positive constants a, M and β ∈ ] 0 , 1 [ and u ( ⋅ , 0 ) ≡ 0 , then u ≡ 0 . The proof of the main result is based on some previous uniqueness result and on the application of some “estimates in short cylinders”, previously used by Ferretti in the study of uniformly parabolic operators.