Harnack inequality for non-local Schrödinger operators

2015/07/27 by Athreya, Siva, Ramachandran, Koushik · 2 citations
#31C05 #31C35 #60J45 #60J50 #Analysis of PDEs (math.AP) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1507.07289

Abstract

Let x ∈ ℝ^d, d ≥ 3, and f: ℝ^d → ℝ be a twice differentiable function with all second partial derivatives being continuous. For 1≤ i,j ≤ d, let a_ij : ℝ^d → ℝ be a differentiable function with all partial derivatives being continuous and bounded. We shall consider the Schrödinger operator associated to Lf(x) amp;=amp; \frac12 ∑_i=1^d ∑_j=1^d (∂)/(∂ x_i) (a_ij(⋅) (∂ f)/(∂ x_j))(x) + ∫_ℝ^d∖\0\ [f(y) - f(x) ]J(x,y)dy. where J: ℝ^d × ℝ^d → ℝ is a symmetric measurable function. Let q: ℝ^d → ℝ. We specify assumptions on a,q, and J so that non-negative bounded solutions to \mathcal Lf + qf = 0 satisfy a Harnack inequality. As tools we also prove a Carleson estimate, a Uniform Boundary Harnack Principle and a 3G inequality for solutions to \mathcal Lf = 0.

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