, Volume 39, Issue 1, pp 69-98,
Open Access This content is freely available online to anyone, anywhere at any time.
Date: 09 Nov 2012

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.

The research was supported in part by NCN grant no. 2011/03/B/ST1/00423.