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

Abstract

Let $\mathit{d}\ge 2$ and V belong to a reverse Hölder class. Let u be a strong solution to the Schrödinger equation

$$-\mathrm{\Delta }\mathit{u}+\mathit{V}\mathit{u}=\mathit{f}\text{in}{\mathbb{R}}^{\mathit{d}}.$$

For an appropriate Musielak–Orlicz function φ, we show that

$$\Vert {\mathit{D}}^2\mathit{u}{\Vert }_{{\mathit{L}}^{\mathit{\phi }(·)}({\mathbb{R}}^{\mathit{d}})}+\Vert \mathit{V}\mathit{u}{\Vert }_{{\mathit{L}}^{\mathit{\phi }(·)}({\mathbb{R}}^{\mathit{d}})}\le \mathit{C}\Vert \mathit{f}{\Vert }_{{\mathit{L}}^{\mathit{\phi }(·)}({\mathbb{R}}^{\mathit{d}})}.$$