Preprint

Inserted: 31 aug 2026
Last Updated: 31 aug 2026

Pages: 17
Year: 2026

Abstract:

Let $0<s<1$ and $A$ be a constant symmetric positive definite matrix. Given the quadratic form \[ Q_A(u)=\int_{\mathbb{R}^N} A\nabla^s u(x)\cdot \nabla^s u(x)\,dx, \] where $\nabla^s$ denotes the Riesz fractional gradient, we establish necessary and sufficient conditions such that $Q_A$ defines a Dirichlet form. In strike contrast with the local case $s=1$, positive definiteness of a constant anisotropic matrix is not sufficient for the Markov property when $0<s<1$. We also describe the associated symmetric Markov semigroup and its $L^p$ realizations, prove the existence of a smooth nonnegative heat kernel and establish a bounded $H^\infty$ functional calculus.

Keywords: analytic semigroups, Dirichlet forms, Harmonic Analysis, Riesz fractional gradient, Strongly continuous contraction semigroups, $H^\infty$ calculus

Download:

• Semigroups_generated_by_fractional_divergence_form_operators.pdf