Global Hessian Estimates in Musielak–Orlicz Spaces for a Schrödinger Equation

Abstract

Let d≥2and V belong to a reverse Hölder class. Let u be a strong solution to the Schrödinger equation -Δu+Vu=f "in" R^d. For an appropriate Musielak–Orlicz function φ, we show that ‖D^2 u‖_(L^φ(·) (R^d ) )+‖Vu‖_(L^φ(·) (R^d ) )≤C‖f‖_(L^φ(·) (R^d ) )Let d≥2and V belong to a reverse Hölder class. Let u be a strong solution to the Schrödinger equation -Δu+Vu=f "in" R^d. For an appropriate Musielak–Orlicz function φ, we show that ‖D^2 u‖_(L^φ(·) (R^d ) )+‖Vu‖_(L^φ(·) (R^d ) )≤C‖f‖_(L^φ(·) (R^d ) )