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Volume 9 |Article ID 187597 | https://doi.org/10.1155/2011/187597

Haibo Lin, Eiichi Nakai, Dachun Yang, "Boundedness of Lusin-area and functions on localized Morrey-Campanato spaces over doubling metric measure spaces ", Journal of Function Spaces, vol. 9, Article ID 187597, 38 pages, 2011. https://doi.org/10.1155/2011/187597

Boundedness of Lusin-area and gλ* functions on localized Morrey-Campanato spaces over doubling metric measure spaces

Haibo Lin,1,2 Eiichi Nakai,3,4 and Dachun Yang5

1College of Science, China Agricultural University Beijing 100083, China

2School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China

3Department of Mathematics, Osaka Kyoiku University Kashiwara, Osaka 582-8582, Japan

4Department of Mathematics, Ibaraki University Mito Ibaraki 310-5812, Japan

5School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China

Academic Editor: Fernando Cobos
Received01 Jun 2009

Abstract

Let χ be a doubling metric measure space and ρ an admissible function on χ. In this paper, the authors establish some equivalent characterizations for the localized Morrey-Campanato spaces ερα,p(χ) and Morrey-Campanato-BLO spaces ε̃ρα,p(χ) when α∈(-∞,0) and p∈[1,∞). If χ has the volume regularity Property (P), the authors then establish the boundedness of the Lusin-area function, which is defined via kernels modeled on the semigroup generated by the Schrödinger operator, from ερa,p(χ) to ε̃ρa,p(χ) without invoking any regularity of considered kernels. The same is true for the gλ* function and, unlike the Lusin-area function, in this case, χ is even not necessary to have Property (P). These results are also new even for ℝd with the d-dimensional Lebesgue measure and have a wide applications.

Copyright © 2011 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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