Regularity Properties and Lipschitz Spaces Adapted to High-Order Schrödinger Operators

Abstract

Let be the high-order Schrödinger operator (−Δ)2+V2, where V is a non-negative potential satisfying the reverse Hölder inequality (RHq), with q>n/2 and n≥5. In this paper, we prove that when 0<α≤2−n/q, the adapted Lipschitz spaces Λα/4L we considered are equivalent to the Lipschitz space CLα associated to the Schrödinger operator L=−Δ+V. In order to obtain this characterization, we should make use of some of the results associated to (−Δ)2. We also prove the regularity properties of fractional powers (positive and negative) of the operator ℒ, Schrödinger Riesz transforms, Bessel potentials and multipliers of the Laplace transforms type associated to the high-order Schrödinger operators.