Weighted Norm Inequalities for Local Fractional Integrals on Gaussian Measure Spaces

Abstract

In this paper, the authors establish the weighted norm inequalities associated with the local Muckenhoupt weights for the local fractional integrals on Gaussian measure spaces. More precisely, the authors first obtain the weighted boundedness of local fractional integrals of order β from Lp(ωp) to Lq(ωq) for \(p\in (1,\infty )\) and from Lp(ωp) to \(L^{q,\infty }(\omega ^{q})\) for p = 1 under the condition of ωAp,q,a, where 1/q = 1/pβ, and then obtain the weighted boundedness of the local fractional integrals, local fractional maximal operators and local Hardy-Littlewood maximal operators on the Morrey-type spaces over Gaussian measure spaces. Moreover, the method of proving the weighted weak type endpoint estimates of local fractional integrals is new.

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Acknowledgements

The authors would like to thank Professor Liguang Liu for her careful reading and many valuable comments. The authors also sincerely express their thanks to the referee for his/her valuable remarks, which greatly improve the presentation of this article.

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Correspondence to Shengchen Mao.

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This work is supported by the National Training Program of Innovation (Grant No. 201910019171).

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Lin, H., Mao, S. Weighted Norm Inequalities for Local Fractional Integrals on Gaussian Measure Spaces. Potential Anal 53, 1377–1401 (2020). https://doi.org/10.1007/s11118-019-09810-x

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Keywords

  • Local fractional integral
  • Local fractional maximal operator
  • Gaussian measure space
  • Morrey-type space

Mathematics Subject Classification (2010)

  • Primary 42B35
  • Secondary 42B20, 42B25