Acta Mathematica Sinica, English Series

, Volume 34, Issue 4, pp 787–800 | Cite as

On Characterization of Poisson Integrals of Schrödinger Operators with Morrey Traces

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Abstract

Let L be a Schrödinger operator of the form L = −Δ + V acting on L2(ℝ n ) where the nonnegative potential V belongs to the reverse Hölder class B q for some qn. In this article we will show that a function fL2,λ(ℝ n ), 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 L 2,λ (ℝ n ) 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 L2,λ(ℝ n ) for all 0 < λ < n.

Keywords

Schrödinger operators Dirichlet problem Morrey spaces Campanato spaces Poisson semigroup 

MR(2010) Subject Classification

42B37 42B35 47B38 

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Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsSun Yat-sen UniversityGuangzhouP. R. China

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