Lp-Lq estimates for electromagnetic Helmholtz equation. Singular potentials

Abstract

In space dimension n ≥ 3, we consider the electromagnetic Schrödinger Hamiltonian H = (∇ − iA(x))2 + V and the corresponding Helmholtz equation (∇ − iA(x))2u + (1 ± i )u + V (x)u = f in Rn, where the magnetic and electric potentials are allowed to have singularities at the origin and decay at infinity. We extend the well known Lp–Lq estimates for the solution of the free Helmholtz equation to the case when the electromagnetic Hamiltonian H is considered. This work extends the results that appear in [7].