Startseite On relations between weak and strong type inequalities for modified maximal operators on non-doubling metric measure spaces
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On relations between weak and strong type inequalities for modified maximal operators on non-doubling metric measure spaces

  • Dariusz Kosz ORCID logo EMAIL logo
Veröffentlicht/Copyright: 8. März 2019

Abstract

In this article, we investigate a special class of non-doubling metric measure spaces in order to describe the possible configurations of Pk,sc, Pk,s, Pk,wc and Pk,w, the sets of all p[1,] for which the weak and strong type (p,p) inequalities hold for the centered and non-centered modified Hardy–Littlewood maximal operators Mkc and Mk, k1. For any fixed k we describe the necessary conditions that Pk,sc, Pk,s, Pk,wc and Pk,w must satisfy in general and illustrate each admissible configuration with a properly chosen non-doubling metric measure space. We also give some partial results related to an analogous problem stated for varying k.

MSC 2010: 42B25; 46E30

Communicated by Christopher D. Sogge


Funding source: Narodowe Centrum Nauki

Award Identifier / Grant number: 2016/21/N/ST1/01496

Funding statement: Research was supported by the National Science Centre of Poland (no. 2016/21/N/ST1/01496).

Acknowledgements

I would like to express my deep gratitude to my supervisor Professor Krzysztof Stempak for his suggestion to study the problem discussed in this article. I thank him for all the remarks made during the preparation of the manuscript. I am also indebted to the referee for a very thorough reading of the article and for constructive comments and hints resulting in an improvement of the presentation.

References

[1] E. DiBenedetto, Real Analysis, Birkhäuser Adv. Texts Basler Lehrbücher, Birkhäuser, Boston, 2002. 10.1007/978-1-4612-0117-5Suche in Google Scholar

[2] D. Kosz, On relations between weak and restricted weak type inequalities for maximal operators on non-doubling metric measure spaces, Studia Math. 241 (2018), 57–70. 10.4064/sm8724-5-2017Suche in Google Scholar

[3] D. Kosz, On relations between weak and strong type inequalities for maximal operators on non-doubling metric measure spaces, Publ. Mat. 62 (2018), no. 1, 37–54. 10.5565/PUBLMAT6211802Suche in Google Scholar

[4] F. Nazarov, S. Treil and A. Volberg, Weak type estimates and Cotlar inequalities for Calderón–Zygmund operators on nonhomogeneous spaces, Int. Math. Res. Not. IMRN 1998 (1998), no. 9, 463–487. 10.1155/S1073792898000312Suche in Google Scholar

[5] Y. Sawano, Sharp estimates of the modified Hardy–Littlewood maximal operator on the nonhomogeneous space via covering lemmas, Hokkaido Math. J. 34 (2005), 435–458. 10.14492/hokmj/1285766231Suche in Google Scholar

[6] Y. Sawano and T. Shimomura, Sobolev embeddings for Riesz potentials of functions in non-doubling Morrey spaces of variable exponents, Collect. Math. 64 (2013), no. 3, 313–350. 10.1007/s13348-013-0082-7Suche in Google Scholar

[7] K. Stempak, Modified Hardy–Littlewood maximal operators on nondoubling metric measure spaces, Ann. Acad. Sci. Fenn. Math. 40 (2015), 443–448. 10.5186/aasfm.2015.4024Suche in Google Scholar

[8] K. Stempak, Examples of metric measure spaces related to modified Hardy–Littlewood maximal operators, Ann. Acad. Sci. Fenn. Math. 41 (2016), 313–314. 10.5186/aasfm.2016.4119Suche in Google Scholar

[9] Y. Terasawa, Outer measures and weak type (1,1) estimates of Hardy–Littlewood maximal operators, J. Inequal. Appl. 2006 (2006), Article ID 15063. 10.1155/JIA/2006/15063Suche in Google Scholar

Received: 2018-05-21
Revised: 2018-01-10
Published Online: 2019-03-08
Published in Print: 2019-05-01

© 2019 Walter de Gruyter GmbH, Berlin/Boston

Heruntergeladen am 21.9.2025 von https://www.degruyterbrill.com/document/doi/10.1515/forum-2018-0126/html
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