The Kato square root property for fractional elliptic operators in divergence form

We establish the Kato square root property for fractional elliptic operators L = -div^α(μ∇^α·) of order 2α∈ (0,2) on Rᵈ, built from the Riesz fractional gradient and divergence, with bounded, measurable and uniformly elliptic complex coefficients $μ$. More precisely, we show that the domain of L1/2 coincides with the distributional fractional Sobolev space Sα,2(Rᵈ) and that \|L1/2 u\|L² \|∇^αu\|L², with constants that are locally uniform in $α$. As a consequence, we recover the Kato square root property for elliptic operators -div(μ∇·) in the limit α→ 1, and we obtain continuity of the square root with respect to the parameter $α$. The proof follows the second-order approach to the Kato problem. Its two new ingredients are nonlocal off-diagonal estimates of polynomial order for the resolvents of L, and a construction of $T(b)$-test functions that circumvents the nonlocality of ∇^α by factorizing it through Riesz potentials.