On Characterization of Poisson Integrals of Schrödinger Operators with Morrey Traces

On Characterization of Poisson Integrals of Schrödinger Operators with Morrey Traces Let L be a Schrödinger operator of the form L = −Δ + V acting on L 2(ℝ n ) 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(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 $$\mathop {\sup }\limits_{{x_B},{r_B}} r_B^{ - \lambda }\int_0^{{r_B}} {\int_{B\left( {{x_B},{r_B}} \right)} {t{{\left| {\nabla u\left( {x,t} \right)} \right|}^2}dxdt \leqslant C < \infty .} } $$ sup x B , r B r B − λ ∫ 0 r B ∫ B ( x B , r B ) t | ∇ u ( x , t ) | 2 d x d t ≤ C < ∞ . 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. $$L_L^{2,\lambda }\left( {{\mathbb{R}^n}} \right) = {L^{2,\lambda }}\left( {{\mathbb{R}^n}} \right).$$ L L 2 , λ ( ℝ n ) = L 2 , λ ( ℝ n ) . Conversely, this Carleson type condition characterizes all the L-harmonic functions whose traces belong to the space L 2,λ(ℝ n ) for all 0 < λ < n. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematica Sinica, English Series Springer Journals

On Characterization of Poisson Integrals of Schrödinger Operators with Morrey Traces

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Publisher
Institute of Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society
Copyright
Copyright © 2018 by 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
Subject
Mathematics; Mathematics, general
ISSN
1439-8516
eISSN
1439-7617
D.O.I.
10.1007/s10114-018-7368-3
Publisher site
See Article on Publisher Site

Abstract

Let L be a Schrödinger operator of the form L = −Δ + V acting on L 2(ℝ n ) 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(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 $$\mathop {\sup }\limits_{{x_B},{r_B}} r_B^{ - \lambda }\int_0^{{r_B}} {\int_{B\left( {{x_B},{r_B}} \right)} {t{{\left| {\nabla u\left( {x,t} \right)} \right|}^2}dxdt \leqslant C < \infty .} } $$ sup x B , r B r B − λ ∫ 0 r B ∫ B ( x B , r B ) t | ∇ u ( x , t ) | 2 d x d t ≤ C < ∞ . 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. $$L_L^{2,\lambda }\left( {{\mathbb{R}^n}} \right) = {L^{2,\lambda }}\left( {{\mathbb{R}^n}} \right).$$ L L 2 , λ ( ℝ n ) = L 2 , λ ( ℝ n ) . Conversely, this Carleson type condition characterizes all the L-harmonic functions whose traces belong to the space L 2,λ(ℝ n ) for all 0 < λ < n.

Journal

Acta Mathematica Sinica, English SeriesSpringer Journals

Published: Mar 15, 2018

References

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