Hostname: page-component-5d84bcc8dc-vpzxz Total loading time: 0 Render date: 2026-09-07T04:41:53.575Z Has data issue: false hasContentIssue false
  • English
  • Français

Characterizations of Besov-Type and Triebel–Lizorkin–Type Spaces via Averages on Balls

Published online by Cambridge University Press:  20 November 2018

Ciqiang Zhuo ,
Winfried Sickel ,
Dachun Yang  and
Wen Yuan
Ciqiang Zhuo
Affiliation:
Key Laboratory of High Performance Computing and Stochastic Information Processing (HPCSIP) (Ministry of Education of China), College of Mathematics and Computer Science, Hunan Normal University, Changsha, Hunan 410081, P. R. China. e-mail: cqzhuo@mail.bnu.edu.cn
Winfried Sickel
Affiliation:
Mathematisches Institut, Friedrich-Schiller-Universität Jena, Jena 07743, Germany. e-mail: winfried.sickel@uni-jena.de
Dachun Yang
Affiliation:
School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, P. R. China. e-mail: dcyang@bnu.edu.cn wenyuan@bnu.edu.cn
Wen Yuan
Affiliation:
School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, P. R. China. e-mail: dcyang@bnu.edu.cn wenyuan@bnu.edu.cn
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

Let $\ell \in \mathbb{N}$ and $\alpha \in (0,2\ell )$ . In this article, the authors establish equivalent characterizations of Besov-type spaces, Triebel–Lizorkin-type spaces, and Besov–Morrey spaces via the sequence ${{\{f-{{B}_{\ell ,{{2}^{-k}}}}f\}}_{k}}$ consisting of the difference between $f$ and the ball average ${{B}_{\ell ,{{2}^{-k}}}}f$ . These results lead to the introduction of Besov-type spaces, Triebel–Lizorkin-type spaces, and Besov–Morrey spaceswith any positive smoothness order onmetricmeasure spaces. As special cases, the authors obtain a new characterization of Morrey–Sobolev spaces and ${{\text{Q}}_{\alpha }}$ spaces with $\alpha \in (0,1)$ , which are of independent interest.

Keywords

42B2546E3542B35Besov spaceTriebel–Lizorkin spaceball averageCalderón reproducing formula

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 60 , Issue 3 , 01 September 2017 , pp. 655 - 672
Copyright
Copyright © Canadian Mathematical Society 2017

References

[1] Adams, D. R. and J. Xiao, Morrey spaces in harmonic analysis. Ark. Mat. 50(2012), 201–230. http://dx.doi.Org/1 0.1007/s11512-010-0134-0 Google Scholar
[2] Alabern, R., Mateu, J., and Verdera, J., A new characterization ofSobolev spaces on R”. Math. Ann. 354(2012), 589–626. http://dx.doi.Org/1 0.1007/s00208-011 -0738-0 Google Scholar
[3] Chang, D.-C., Liu, J., Yang, D., and Yuan, W., Littlewood-Paley characterizations of Hajlasz-Sobolev and Triebel-Lizorkin spaces via averages on balls. Potential Anal. (2016). http://dx.doi.Org/10.1007/s11118-016-9579-5 Google Scholar
[4] Chiarenza, F. and Frasca, M., Morrey spaces and Hardy-Littlewood maximal function, Rend. Math. 7 (1987), 273–279. Google Scholar
[5] Dafni, G. and Xiao, J., Some new tent spaces and duality theorems for fractional Carleson measures and Qa (R“). J. Funct. Anal. 208(2004), 377–422. http://dx.doi.Org/10.1016/SOO22-1236(3)00181-2 Google Scholar
[6] Dafni, G. and Xiao, J., The dyadic structure and atomic decomposition ofQ spaces in several real variables. Tohoku Math. J. (2) 57(2005), 119–145. http://dx.doi.Org/!0.2748/tmj/111 3234836 Google Scholar
[7] Dai, F., Gogatishvili, A., Yang, D., and Yuan, W., Characterizations ofSobolev spaces via averages on balls. Nonlinear Anal. 128(2015), 86–99. http://dx.doi.Org/10.1016/j.na.2O15.07.024 Google Scholar
[8] Dai, F., Gogatishvili, A., Yang, D., and Yuan, W., Characterizations ofBesov and Triebel-Lizorkin spaces via averages on balls. J. Math. Anal. Appl. 433(2016), 1350–1368. http://dx.doi.Org/10.1016/j.jmaa.2015.08.054 Google Scholar
[9] Dai, F., Liu, J., Yang, D., and Yuan, W., Littlewood-Paley characterizations of fractional Sobolev spaces via averages on balls. arxiv:1 511.07598 Google Scholar
[10] DeVore, R. A. and Lorentz, G. G., Constructive approximation. Grundlehren der Mathematischen Wissenschaften, 303, Springer-Verlag, Berlin, 1993. http://dx.doi.Org/10.1007/978-3-662-02888-9 Google Scholar
[11] El Baraka, A., An embedding theorem for Campanato spaces. Electron. J. Differential Equations 66(2002), 1–17. Google Scholar
[12] El Baraka, A., Function spaces ofBMO and Campanato type. In: Proceedings of the 2002 Fez Conference on Partial Differential Equations, Electron. J. Differ. Equ. Conf., 9, Southwest Texas State Univ., San Marcos, TX, 2002, pp. 109–115.Google Scholar
[13] El Baraka, A., Littlewood-Paley characterization for Campanato spaces. J. Funct. Spaces Appl. 4(2006), 193–220. http://dx.doi.Org/10.1155/2006/921520 Google Scholar
[14] Essen, M., Janson, S., Peng, L., and Xiao, J., Q spaces of several real variables. Indiana Univ. Math. J. 49(2000), 575–615. http://dx.doi.Org/10.1512/iumj.2000.49.1732 Google Scholar
[15] Fefferman, C. and Stein, E. M., Some maximal inequalities. Amer. J. Math. 93(1971), 107–115. http://dx.doi.Org/1 0.2307/2373450 Google Scholar
[16] Frazier, M. and Jawerth, B., Decomposition ofBesov spaces. Indiana Univ. Math. J. 34(1985), 777–799. http://dx.doi.Org/10.1512/iumj.1 985.34.34041 Google Scholar
[17] Frazier, M. and Jawerth, B., A discrete transform and decompositions of distribution spaces. J. Funct. Anal. 93(1990), 34–170. http://dx.doi.Org/10.101 6/0022-1236(90)90137-A Google Scholar
[18] Frazier, M., Jawerth, B., and Weiss, G., Littlewood-Paley theory and the study of function spaces. CBMS Regional Conference Series in Mathematics, 79, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1991. http://dx.doi.Org/10.1090/cbms/079 Google Scholar
[19] Gogatishvili, A., Koskela, P., and Zhou, Y., Characterizations ofBesov and Triebel-Lizorkin spaces on metric measure spaces. Forum Math. 25(2013), 787–819. doil0.1515/form.2011.135 Google Scholar
[20] Grafakos, L., Classical Fourier analysis. Third ed., Graduate Texts in Mathematics, 249, Springer, New York, 2014. http://dx.doi.Org/10.1007/978-1-4939-1194-3 Google Scholar
[21] Koskela, P., Yang, D., and Zhou, Y., Pointwise characterizations ofBesov and Triebel-Lizorkin spaces and quasiconformal mappings. Adv. Math. 226(2011), 3579–3621. http://dx.doi.Org/1 0.101 6/j.aim.2O10.10.020 Google Scholar
[22] Kozono, H. and Yamazaki, M., Semilinear heat equations and the Navier-Stokes equation with distributions in new function spaces as initial data. Comm. Partial Differential Equations 19(1994), 959–1014. http://dx.doi.Org/10.1080/03605309408821042 Google Scholar
[23] Sawano, Y. and Tanaka, H., Decompositions of Besov-Morrey spaces and Triebel-Lizorkin-Morrey spaces. Math. Z. 257(2007), 871–905. http://dx.doi.Org/10.1007/s00209-007-0150-3 Google Scholar
[24] Sawano, Y., Yang, D., and Yuan, W., New applications of Besov-type and Triebel-Lizorkin-type spaces. J. Math. Anal. Appl. 363(2010), 73–85. http://dx.doi.Org/10.1016/j.jmaa.2OO9.O8.OO2 Google Scholar
[25] Sickel, W., Smoothness spaces related to Morrey spaces-a survey. I. Eurasian Math. J. 3(2012), 110–149. Google Scholar
[26] Stein, E. M., Harmonic analysis: Real-variable methods, orthogonality, and oscillatory integrals. Princeton Mathematical Series, 43, Princeton University Press, Princeton, NJ, 1993. Google Scholar
[27] Tang, L. and Xu, J., Some properties of Morrey type Besov-Triebel spaces. Math. Nachr. 278(2005), 904–917. http://dx.doi.Org/10.1002/mana.200310281 Google Scholar
[28] Triebel, H., Theory of function spaces., Monographs in Mathematics, 78, Birkhauser Verlag, Basel, 1983. http://dx.doi.Org/10.1007/978-3-0346-0416-1 Google Scholar
[29] Triebel, H., Hybrid function spaces, heat and Navier-Stokes equations. EMS Tracts in Mathematics, 24, European Mathematical Society (EMS), Zurich, 2014.Google Scholar
[30] Triebel, H., Tempered homogeneous function spaces. EMS Series of Lectures in Mathematics, European Mathematical Society (EMS), Zurich, 2015. http://dx.doi.Org/10.4171/1 55 Google Scholar
[31] Yang, D. and Yuan, W., A new class of function spaces connecting Triebel-Lizorkin spaces and Q spaces. J. Funct. Anal. 255(2008), 2760–2809. http://dx.doi.Org/10.1016/j.jfa.2008.09.005 Google Scholar
[32] Yang, D. and Yuan, W., New Besov-type spaces and Triebel-Lizorkin-type spaces including Q spaces. Math. Z. 265(2010), 451–480. http://dx.doi.Org/10.1007/s00209-009-0524-9 Google Scholar
[33] Yang, D. and Yuan, W., Relations among Besov-type spaces, Triebel-Lizorkin-type spaces and generalized Carleson measure spaces. Appl. Anal. 92(2013), 549–561. http://dx.doi.Org/! 0.1080/00036811.2011.629610 Google Scholar
[34] Yang, D. and Yuan, W., Pointwise characterizations ofBesov and Triebel-Lizorkin spaces in terms of averages on balls. Trans. Amer. Math. Soc. (2016). http://dx.doi.Org/10.1090/tran/6871 Google Scholar
[35] Yang, D., Yuan, W., and Zhou, Y., A new characterization of Triebel-Lizorkin spaces on W. Publ. Mat. 57(2013), 57–82. http://dx.doi.Org/10.5565/PUBLMAT_57113_O2 Google Scholar
[36] Yuan, W., Sickel, W., and Yang, D., Morrey and Campanato meet Besov, Lizorkin and Triebel. Lecture Notes in Mathematics, 2005, Springer-Verlag, Berlin, 2010. http://dx.doi.Org/10.1007/978-3-642-14606-0 Google Scholar
[37] Yuan, W., Sickel, W., and Yang, D., On the coincidence of certain approaches to smoothness spaces related to Morrey spaces. Math. Nachr. 286(2013), 1571–1584.Google Scholar