Kernel estimates for a class of fractional Kolmogorov operators

In this talk I will present some recent results obtained in [1]. We consider a measure space $(X,\mu)$ with $\sigma$-finite measure $\mu$ and a non-negative self-adjoint operator $A$ on $L^2(\mu)$. We assume that $-A$ generates a symmetric Markov semigroup on $L^2(\mu)$, namely a symmetric positivity preserving and $L^\infty$-contractive strongly continuous semigroup on $L^2(\mu)$.

We prove non-uniform bounds on the transition kernel corresponding to the Markov semigroup generated by $-A^\alpha$ for $0<\alpha<1$. The main tools are weighted Nash inequalities. Finally, we illustrate our results in concrete examples.

[1] M. Porfido, A. Rhandi, C. Tacelli: Kernel estimates for a class of fractional Kolmogorov operators, preprint (2023)