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 $L^{\mathrm{\infty }}$-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)