One-sided fractional derivatives, fractional Laplacians, and weighted Sobolev spaces. (April 2020)

• Record Type: Journal Article
• Authors: Stinga, Pablo Raúl; Vaughan, Mary
• Is Part Of: Nonlinear analysis. Volume 193(2020)
• Publication Date: 2020-04
• Subjects: primary 26A24 26A33 46E35 -- secondary 35R11 42B25 46E30; Marchaud fractional derivative -- One-sided weights -- Weighted Sobolev space -- Maximal operator -- Fractional Laplacian; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2019.04.004
• Languages: English
• ISSNs: 0362-546X

Abstract

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 + .