Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
block space; variable exponent analysis; Hardy-Littlewood maximal operator
block space; variable exponent analysis; Hardy-Littlewood maximal operator
Summary:
The family of block spaces with variable exponents is introduced. We obtain some fundamental properties of the family of block spaces with variable exponents. They are Banach lattices and they are generalizations of the Lebesgue spaces with variable exponents. Moreover, the block space with variable exponents is a pre-dual of the corresponding Morrey space with variable exponents. The main result of this paper is on the boundedness of the Hardy-Littlewood maximal operator on the block space with variable exponents. We find that the Hardy-Littlewood maximal operator is bounded on the block space with variable exponents whenever the Hardy-Littlewood maximal operator is bounded on the corresponding Lebesgue space with variable exponents.
References:
[1] Bennett, C., Sharpley, R.: Interpolation of Operators. Pure and Applied Mathematics vol. 129 Academic Press, Boston (1988). MR 0928802 | Zbl 0647.46057
[2] Blasco, O., Ruiz, A., Vega, L.: Non interpolation in Morrey-Campanato and block spaces. Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 28 (1999), 31-40. MR 1679077 | Zbl 0955.46013
[3] Chiarenza, F., Frasca, M.: Morrey spaces and Hardy-Littlewood maximal function. Rend. Mat. Appl., VII. Ser. 7 (1987), 273-279. MR 0985999 | Zbl 0717.42023
[4] Cruz-Uribe, D., Diening, L., Fiorenza, A.: A new proof of the boundedness of maximal operators on variable Lebesgue spaces. Boll. Unione Mat. Ital. 2 (2009), 151-173. MR 2493649 | Zbl 1207.42011
[5] Cruz-Uribe, D., Fiorenza, A., Martell, J. M., Pérez, C.: The boundedness of classical operators on variable $L^{p}$ spaces. Ann. Acad. Sci. Fenn., Math. 31 (2006), 239-264. MR 2210118 | Zbl 1100.42012
[6] Cruz-Uribe, D., Fiorenza, A., Neugebauer, C.: The maximal function on variable $L^{p}$ spaces. Ann. Acad. Sci. Fenn., Math. 28 (2003), 223-238. MR 1976842
[7] Diening, L.: Maximal functions on Musielak-Orlicz spaces and generalized Lebesgue spaces. Bull. Sci. Math. 129 (2005), 657-700. DOI 10.1016/j.bulsci.2003.10.003 | MR 2166733 | Zbl 1096.46013
[8] Diening, L.: Maximal function on generalized Lebesgue spaces $L^{p(\cdot)}$. Math. Inequal. Appl. 7 (2004), 245-253. MR 2057643
[9] Diening, L., Harjulehto, P., Hästö, P., Mizuta, Y., Shimomura, T.: Maximal functions in variable exponent spaces: limiting cases of the exponent. Ann. Acad. Sci. Fenn., Math. 34 (2009), 503-522. MR 2553809
[10] Diening, L., Harjulehto, P., Hästö, P., Růžička, M.: Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Mathematics 2017 Springer, Berlin (2011). MR 2790542 | Zbl 1222.46002
[11] Hästö, P. A.: Local-to-global results in variable exponent spaces. Math. Res. Lett. 16 (2009), 263-278. DOI 10.4310/MRL.2009.v16.n2.a5 | MR 2496743 | Zbl 1184.46033
[12] Ho, K.-P.: Atomic decompositions of Hardy-Morrey spaces with variable exponents. Ann. Acad. Sci. Fenn., Math (to appear).
[13] Ho, K.-P.: Atomic decompositions of weighted Hardy-Morrey spaces. Hokkaido Math. J. 42 (2013), 131-157. DOI 10.14492/hokmj/1362406643 | MR 3076303 | Zbl 1269.42010
[14] Ho, K.-P.: Characterizations of $BMO$ by $A_{p}$ weights and $p$-convexity. Hiroshima Math. J. 41 (2011), 153-165. DOI 10.32917/hmj/1314204559 | MR 2849152 | Zbl 1227.42024
[15] Ho, K.-P.: Generalized Boyd's indices and applications. Analysis (Munich) 32 (2012), 97-106. MR 3043715 | Zbl 1287.42014
[16] Ho, K.-P.: Littlewood-Paley spaces. Math. Scand. 108 (2011), 77-102. DOI 10.7146/math.scand.a-15161 | MR 2780808 | Zbl 1263.42021
[17] Ho, K.-P.: Vector-valued singular integral operators on Morrey type spaces and variable Triebel-Lizorkin-Morrey spaces. Ann. Acad. Sci. Fenn., Math. 37 (2012), 375-406. DOI 10.5186/aasfm.2012.3746 | MR 2987074 | Zbl 1261.42016
[18] Kokilashvili, V., Meskhi, A.: Boundedness of maximal and singular operators in Morrey spaces with variable exponent. Armen. J. Math. 1 (2008), 18-28. MR 2436241 | Zbl 1281.42012
[19] Kováčik, O., Rákosník, J.: On spaces $L^{p(x)}$ and $W^{k,p(x)}$. Czech. Math. J. 41 (1991), 592-618. MR 1134951
[20] Lerner, A. K.: On some questions related to the maximal operator on variable $L_{p}$ spaces. Trans. Am. Math. Soc. 362 (2010), 4229-4242. DOI 10.1090/S0002-9947-10-05066-X | MR 2608404 | Zbl 1208.42008
[21] Lerner, A. K.: Some remarks on the Hardy-Littlewood maximal function on variable $L_{p}$ spaces. Math. Z. 251 (2005), 509-521. DOI 10.1007/s00209-005-0818-5 | MR 2190341 | Zbl 1092.42009
[22] Meyer, Y., Taibleson, M. H., Weiss, G.: Some functional analytic properties of the space $B_{q}$ generated by blocks. Indiana Univ. Math. J. 34 (1985), 493-515. DOI 10.1512/iumj.1985.34.34028 | MR 0794574
[23] Nekvinda, A.: A note on maximal operator on $l^{\{ p_{n}\} }$ and $L^{p(x)}({\Bbb R})$. J. Funct. Spaces Appl. 5 (2007), 49-88. DOI 10.1155/2007/294367 | MR 2296013
[24] Nekvinda, A.: Hardy-Littlewood maximal operator on $L^{p(x)}({\Bbb R}^{n})$. Math. Inequal. Appl. 7 (2004), 255-265. MR 2057644
[25] Nekvinda, A.: Maximal operator on variable Lebesgue spaces for almost monotone radial exponent. J. Math. Anal. Appl. 337 (2008), 1345-1365. DOI 10.1016/j.jmaa.2007.04.047 | MR 2386383 | Zbl 1260.42010
[26] Soria, F.: Characterizations of classes of functions generated by blocks and associated Hardy spaces. Indiana Univ. Math. J. 34 (1985), 463-492. DOI 10.1512/iumj.1985.34.34027 | MR 0794573 | Zbl 0573.42015
Search
Partner of
