A New Characterization of the Domains of Integral Powers of the Self-Adjoint Friedrichs-Legendre Operator

2026/07/22 by Lance Littlejohn, Richard Wellman, Quinn Wicks
Mathematics · #math.FA

paper · pdf

Abstract

Let A be the self-adjoint operator in L²(-1,1), generated by the second-order classical Legendre differential equation% ℓ\lbrack y](t)=-( (1-t²)y^′(t)) ^′+ky(t)=λy(t) (t∈(-1,1)), which has the Legendre polynomials \P_m\_m=0^∞ as a complete sequence of eigenfunctions; here k is a fixed, non-negative real number. This is the Friedrichs extension of the minimal operator associated with ℓ[⋅] in L²(-1,1). For each n ∈ ℕ, we show that D(Aⁿ) is characterized by one integrability condition instead of 2n boundary conditions as dictated by the classical Glazman-Krein-Naimark theory. We also prove that if f\inD(Aⁿ) then f⁽ⁿ⁾∈ L²(-1,1). This smoothness result extends known results when n=1 and n=2. Furthermore, this result is optimal in the sense that there exists g\inD(Aⁿ) with g⁽ⁿ⁺¹⁾∉ L²(-1,1).

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