One-sided fractional derivatives, fractional Laplacians, and weighted Sobolev spaces. (April 2020) - Stinga, Pablo Raúl; Vaughan, Mary
Abstract: 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 α . 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 ( − Δ ) s . These are Bourgain–Brezis–Mironescu-type characterizations for weighted Sobolev spaces. We also complement their work by studying a.e. and weighted L p limits as α, s → 0 + .