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Extension of Campanato–Sobolev type spaces associated with Schrödinger operators

  • Jizheng Huang
  • Pengtao LiEmail author
  • Yu Liu
Original Paper

Abstract

Let \(L=-\varDelta +V\) be a Schrödinger operator acting on \(L^2({\mathbb {R}}^{d})\), where V belongs to the reverse Hölder class \(B_q\) for some \(q\ge d\). For \(\alpha , \beta \in [0,1)\), let \(\varLambda _{\alpha ,\beta }^L({\mathbb {R}}^d)\) be the Campanato–Sobolev space associated with L. Via the Poisson semigroup \(\{e^{-t\sqrt{L}}\}_{t\ge 0}\), we extend \(\varLambda _{\alpha ,\beta }^L({\mathbb {R}}^d)\) to \({\mathcal {T}}^{\alpha ,\beta }_{L}({\mathbb {R}}^{d+1}_{+})\) which is defined as the set of all distributional solutions u of \(-u_{tt}+Lu=0\) on the upper half space \({\mathbb {R}}_+^{d+1}\) satisfying
$$\begin{aligned} \sup _{(x_0,r)\in {\mathbb {R}}_+^{d+1}}r^{-(2\alpha +d)}\int _{B(x_0,r)}\int _0^r|\nabla _{x,t}u(x,t)|^2t^{1-2\beta }dtdx<\infty . \end{aligned}$$

Keywords

Companato–Sobolev spaces Schrödinger operator Carleson measure Poisson integral 

Mathematics Subject Classification

42B35 35J10 42B25 

Notes

Acknowledgements

J.Z. Huang was supported by the National Natural Science Foundation of China under Grants (No. 11471018) and the Beijing Natural Science Foundation under Grant (No. 1142005). P.T. Li was supported by the National Natural Science Foundation of China under Grants (No. 11871293); Shandong Natural Science Foundation of China (Nos. ZR2017JL008, ZR2016AM05); University Science and Technology Projects of Shandong Province (No. J15LI15). Y. Liu was supported by the National Natural Science Foundation of China under Grants (No. 11671031).

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Copyright information

© Tusi Mathematical Research Group (TMRG) 2019

Authors and Affiliations

  1. 1.School of ScienceBeijing University of Posts and TelecommunicationsBeijingChina
  2. 2.School of Mathematics and StatisticsQingdao UniversityQingdaoChina
  3. 3.School of Mathematics and PhysicsUniversity of Science and Technology BeijingBeijingChina

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