2012/09/26 by Tadeusz Kulczycki, Kulczycki, Tadeusz
Mathematics · #Advanced Harmonic Analysis Research #Analysis of PDEs (math.AP) #Bounded function #Combinatorics #Exponent #FOS: Mathematics #Hölder condition #Mathematical analysis #Mathematical physics #Mathematics #Nabla symbol #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #Omega #Operator (biology) #Order (exchange) #Physics #Probability (math.PR) #Quantum mechanics #math.AP #math.PR
paper · pdf · doi:10.48550/arxiv.1209.5904
published in arXiv (Cornell University) (Cornell University)
arxiv created 2012/09/26 · openalex publication_date 2012/09/26 · arxiv updated 2012/09/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study gradient estimates of q-harmonic functions u of the fractional\nSchr "odinger operator \Δ\α/2 + q, \α \∈ (0,1] in bounded\ndomains D \⊂ Rd. For nonnegative u we show that if q is H "older\ncontinuous of order \η > 1 - \α then \∇ u(x) exists for any x\n\∈ D and |\∇ u(x)| \≤ c u(x)/ ( dist(x,\∂ D) wedge 1). The\nexponent 1 - \α is critical i.e. when q is only 1 - \α H "older\ncontinuous \∇ u(x) may not exist. The above gradient estimates are well\nknown for \α \∈ (1,2] under the assumption that q belongs to the Kato\nclass calJ\α - 1. The case \α \∈ (0,1] is different. To obtain\nresults for \α \∈ (0,1] we use probabilistic methods. As a corollary, we\nobtain for \α \∈ (0,1) that a weak solution of \Δ\α/2u + q u\n= 0 is in fact a strong solution.\n