On Characterization of Poisson Integrals of Schrödinger Operators with Morrey Traces

Abstract

Let L be a Schrödinger operator of the form L = −Δ + V acting on L²(ℝⁿ) where the nonnegative potential V belongs to the reverse Hölder class B _q for some q ≥ n. In this article we will show that a function f ∈ L^2,λ(ℝⁿ), 0 < λ < n, is the trace of the solution of Lu = −u _tt + L _u = 0, u(x, 0) = f(x), where u satisfies a Carleson type condition

$$\mathop {\sup }\limits_{{x_B},{r_B}} r_B^{ - \lambda }\int_0^{{r_B}} {\int_{B\left( {{x_B},{r_B}} \right)} {t{{\left| {\nabla u\left( {x,t} \right)} \right|}^2}dxdt \leqslant C < \infty .} } $$

Its proof heavily relies on investigate the intrinsic relationship between the classical Morrey spaces and the new Campanato spaces L ^2,λ _L (ℝⁿ) associated to the operator L, i.e.

$$L_L^{2,\lambda }\left( {{\mathbb{R}^n}} \right) = {L^{2,\lambda }}\left( {{\mathbb{R}^n}} \right).$$

Conversely, this Carleson type condition characterizes all the L-harmonic functions whose traces belong to the space L^2,λ(ℝⁿ) for all 0 < λ < n.