×

Atomic decomposition of Hardy spaces and characterization of \(BMO\) via Banach function spaces. (English) Zbl 1289.46049

Anal. Math. 38, No. 3, 173-185 (2012); erratum ibid. 45, No. 4, 855-856 (2019).
The author investigates what happens when replacing one of the usual \(L^p\) spaces in the definition of \(BMO\) by a Banach function space \(X\). The condition for a function to belong to \(BMO\) is \[ \left(\frac1{| B| }\int_B | f(x) - f_B| ^p \, dx\right)^{1/p} \leq C, \] with \(C\) independent of \(B\), and the space is independent of \(p\). This is replaced by saying that \(f \in BMO_X\) if \[ \frac{\| f - f_B\| _X}{\| \chi_B\| _X} \leq C \] with \(C\) independent of \(B\).
The associate space \(X^{\prime}\) plays the role for Banach function spaces that duals do for usual Banach spaces. The author also shows that, if the Hardy-Littlewood maximal function is bounded on \(X^{\prime}\), then \(BMO_X = BMO\).
The author shows that, if instead the Hardy-Littlewood maximal function is bounded on \(X\), the associated Hardy space has an atomic decomposition with \(X\) atoms, and uses that result to give conditions on a variable \(p(x)\) in order that the space \(L^{p(x)}\) gives rise to the usual \(BMO\) when \(X = L^{p(\cdot)}\).
In particular, when \(\mathrm{ess} \inf p(x) = p_{-} >1\), \(\mathrm{ess} \sup p(x) = p_{+} < \infty\), then \(L^{p(\cdot)}\) has the requisite property, and the associated Hardy space has an atomic decomposition and its dual space is the usual (non-variable) \(BMO\) space.

MSC:

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
42B35 Function spaces arising in harmonic analysis
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] H. Aoyama, Lebesgue spaces with variable exponent on a probability space, Hiroshima Math. J., 39(2009), 207–216. · Zbl 1190.46026
[2] C. Bennett and R. Sharpley, Interpolations of operators, Academic Press (New York, 1988). · Zbl 0647.46057
[3] D. Boyd Indices of function spaces and their relationship to interpolation, Canad. J. Math., 21(1969), 1245–1254. · Zbl 0184.34802 · doi:10.4153/CJM-1969-137-x
[4] D. Cruz-Uribe, L. Diening, and A. A. Fiorenza, A new proof of the boundedness of maximal operators on variable Lebesgue spaces, Boll. Unione Mat. Ital., 2(1)(2009), 151–173. · Zbl 1207.42011
[5] L. Diening, Maximal function on Orlicz-Musielak spaces and generalized Lebesgue space, Bull. Sci. Math., 129(2005), 657–700. · Zbl 1096.46013 · doi:10.1016/j.bulsci.2003.10.003
[6] L. Diening, P. Harjulehto, P. Hãstõ, Y. Mizuta, and T. Shimomura, Maximal functions in variable exponent spaces: limiting cases of the exponent, Ann. Acad. Sci. Fenn. Math., 34(2009), 503–522. · Zbl 1180.42010
[7] K.-P. Ho, Characterization of BMO in terms of rearrangement-invariant Banach function spaces, Expo. Math., 27(2009), 363–372. · Zbl 1174.42025 · doi:10.1016/j.exmath.2009.02.007
[8] K.-P. Ho, Littlewood-Paley spaces, Math. Scand., 108(2011), 77–102. · Zbl 1216.46031
[9] K.-P. Ho, Characterizations of BMO by A p weights and P-convexity, Hiroshima Math. J., 41(2011), 153–165. · Zbl 1227.42024
[10] K.-P. Ho, Generalized Boyd’s indices and applications Analysis (Munich), accepted. · Zbl 1287.42014
[11] M. Izuki, Boundedness of commutators on Herz spaces with variable exponent, Rend. Circ. Mat. Palermo (2), 59(2010), 199–213. · Zbl 1202.42029 · doi:10.1007/s12215-010-0015-1
[12] O. Kováčik and J. Rákosník, On spaces L p({\(\cdot\)}) and W k,p({\(\cdot\)}), Czechoslovak Math. J., 41(1991), 592–618.
[13] A. Lerner and S. Ombrosi, A boundedness criterion for general maximal operators, Publ. Mat., 54(2010), 53–71. · Zbl 1183.42024
[14] J. Lukeš, L. Pick, and D. Pokorný, On geometric properties of the spaces L p(x), Rev. Mat. Comput., 24(2011), 115–130. · Zbl 1223.46015 · doi:10.1007/s13163-010-0032-9
[15] E. Stein, Harmonic Analysis, Princeton University Press (Princeton, 1993). · Zbl 0821.42001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.