Gradient Estimates of q-Harmonic Functions of Fractional Schrödinger Operator

Abstract

We study gradient estimates of q-harmonic functions u of the fractional Schrödinger operator Δ^α/2 + q, α ∈ (0, 1] in bounded domains D ⊂ ℝ^d. For nonnegative u we show that if q is Hölder continuous of order η > 1 − α then $\nabla u(x)$ exists for any x ∈ D and $|\nabla u(x)| \le c u(x)/ ({\rm dist}(x,\partial D) \wedge 1)$. The exponent 1 − α is critical i.e. when q is only 1 − α Hölder continuous $\nabla u(x)$ may not exist. The above gradient estimates are well known for α ∈ (1, 2] under the assumption that q belongs to the Kato class $\mathcal{J}^{\alpha - 1}$. The case α ∈ (0, 1] is different. To obtain results for α ∈ (0, 1] we use probabilistic methods. As a corollary, we obtain for α ∈ (0, 1) that a weak solution of Δ^α/2 u + q u = 0 is in fact a strong solution.