Gradient Estimates of q-Harmonic Functions of Fractional Schr?dinger Operator

摘要

We study gradient estimates of q-harmonic functions u of the fractional Schr?dinger operator Δ~(α/2) + q, α ∈ (0, 1] in bounded domains D ? ?~d. For nonnegative u we show that if q is H?lder continuous of order η > 1 - α then ?u(x) exists for any x ∈ D and {pipe}?u(x){pipe} ≤ cu(x)/(dist(x, ?D) ∧ 1). The exponent 1 - α is critical i.e. when q is only 1 - α H?lder continuous ?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 J~(α-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.