Analysis of nonlocal equations: One-sided weighted fractional Sobolev spaces and Harnack inequality for fractional nondivergence form elliptic equations

Abstract

<p>This dissertation deals with the following two projects.</p> <p>First, we characterize one-sided weighted Sobolev spaces W^{1,p}(R,ω), where ω is a one-sided Sawyer weight, in terms of a.e. and weighted L^p limits as α → 1− of Marchaud fractional derivatives of order 0 < α < 1. These are Bourgain–Brezis–Mironescu-type characterizations for one-sided weighted Sobolev spaces. Similar results for weighted Sobolev spaces W^{2,p}(R^n,ν), where ν is an A_p-Muckenhoupt weight, are proved in terms of limits as s → 1− of fractional Laplacians (−∆)^s. We also additionally study the a.e. and weighted L^p limits as α, s → 0+.</p> <p>Second, we define fractional powers of nondivergence form elliptic operators (−aij(x)∂ij)^s for 0 < s < 1 with Hölder coefficients and characterize a Poisson problem driven by (−aij(x)∂ij)^s with a local degenerate extension problem. An interior Harnack inequality for nonnegative solutions to such an extension equation with bounded, measurable coefficients is proved. This in turn implies the interior Harnack inequality for the fractional problem.</p>