Atomic decomposition of predictable martingale Hardy space with variable exponents

Hao, Zhiwei

Czechoslovak Mathematical Journal, Tome 65 (2015) no. 4, pp. 1033-1045

Abstract (VO)

This paper is mainly devoted to establishing an atomic decomposition of a predictable martingale Hardy space with variable exponents defined on probability spaces. More precisely, let $(\Omega , \mathcal {F}, \mathbb {P})$ be a probability space and $p(\cdot )\colon \Omega \rightarrow (0,\infty )$ be a $\mathcal {F}$-measurable function such that $0\inf \nolimits _{x\in \Omega }p(x)\leq \sup \nolimits _{x\in \Omega }p(x)\infty $. It is proved that a predictable martingale Hardy space $\mathcal P_{p(\cdot )}$ has an atomic decomposition by some key observations and new techniques. As an application, we obtain the boundedness of fractional integrals on the predictable martingale Hardy space with variable exponents when the stochastic basis is regular.

Détail

DOI : 10.1007/s10587-015-0226-x

Classification : 60G42, 60G46
Keywords: variable exponent; atomic decomposition; martingale Hardy space; fractional integral

Bibliographie

[1] Chao, J.-A., Ombe, H.: Commutators on dyadic martingales. Proc. Japan Acad., Ser. A 61 (1985), 35-38. | MR | Zbl

[2] Cheung, K. L., Ho, K.-P.: Boundedness of Hardy-Littlewood maximal operator on block spaces with variable exponent. Czech. Math. J. 64 (2014), 159-171. | DOI | MR

[3] Cruz-Uribe, D. V., Fiorenza, A.: Variable Lebesgue Spaces: Foundations and Harmonic Analysis. Applied and Numerical Harmonic Analysis Birkhäuser, New York (2013). | MR | Zbl

[4] 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 | Zbl

[5] Cruz-Uribe, D., Wang, L.-A. D.: Variable Hardy spaces. Indiana Univ. Math. J. 63 (2014), 447-493. | DOI | MR | Zbl

[6] Diening, L.: Maximal function on generalized Lebesgue spaces {$L^{p(\cdot)}$}. Math. Inequal. Appl. 7 (2004), 245-253. | MR | Zbl

[7] Diening, L., H{ä}stö, P., Roudenko, S.: Function spaces of variable smoothness and integrability. J. Funct. Anal. 256 (2009), 1731-1768. | DOI | MR | Zbl

[8] Fan, X., Zhao, D.: On the spaces {$L^{p(x)}(\Omega)$} and {$W^{m,p(x)}(\Omega)$}. J. Math. Anal. Appl. 263 (2001), 424-446. | MR | Zbl

[9] Hao, Z., Jiao, Y.: Fractional integral on martingale Hardy spaces with variable exponents. Fract. Calc. Appl. Anal. 18 (2015), 1128-1145. | MR

[10] Ho, K.-P.: John-Nirenberg inequalities on Lebesgue spaces with variable exponents. Taiwanese J. Math. 18 (2014), 1107-1118. | DOI | MR

[11] Jiao, Y., Peng, L., Liu, P.: Atomic decompositions of Lorentz martingale spaces and applications. J. Funct. Spaces Appl. 7 (2009), 153-166. | DOI | MR

[12] Jiao, Y., Wu, L., Yang, A., Yi, R.: The predual and John-Nirenberg inequalities on generalized BMO martingale spaces. (to appear) in Trans. Amer. Math. Soc (2014), arXiv:1408.4641v1 [math.FA], 20 Aug. 2014. | MR

[13] Jiao, Y., Xie, G., Zhou, D.: Dual spaces and John-Nirenberg inequalities of martingale Hardy-Lorentz-Karamata spaces. Q. J. Math. 66 (2015), 605-623. | DOI | MR | Zbl

[14] Kováčik, O., Rákosník, J.: On spaces {$L^{p(x)}$} and {$W^{k,p(x)}$}. Czech. Math. J. 41 (1991), 592-618. | MR

[15] Liu, P., Hou, Y.: Atomic decompositions of Banach-space-valued martingales. Sci. China, Ser. A 42 (1999), 38-47. | DOI | MR | Zbl

[16] Miyamoto, T., Nakai, E., Sadasue, G.: Martingale Orlicz-Hardy spaces. Math. Nachr. 285 (2012), 670-686. | DOI | MR | Zbl

[17] Nakai, E., Sadasue, G.: Martingale Morrey-Campanato spaces and fractional integrals. J. Funct. Spaces Appl. 2012 (2012), Article ID 673929, 29 pages. | MR | Zbl

[18] Nakai, E., Sawano, Y.: Hardy spaces with variable exponents and generalized Campanato spaces. J. Funct. Anal. 262 (2012), 3665-3748. | DOI | MR | Zbl

[19] Ohno, T., Shimomura, T.: Sobolev embeddings for Riesz potentials of functions in grand Morrey spaces of variable exponents over non-doubling measure spaces. Czech. Math. J. 64 (2014), 209-228. | DOI | MR

[20] Orlicz, W.: Über konjugierte Exponentenfolgen. Stud. Math. 3 German (1931), 200-211. | DOI | Zbl

[21] Sadasue, G.: Fractional integrals on martingale Hardy spaces for $0. Mem. Osaka Kyoiku Univ., Ser. III, Nat. Sci. Appl. Sci. 60 (2011), 1-7. MR 2963747

[22] Sawano, Y.: Atomic decompositions of Hardy spaces with variable exponents and its application to bounded linear operators. Integral Equations Oper. Theory 77 (2013), 123-148. | DOI | MR | Zbl

[23] Weisz, F.: Martingale Hardy Spaces and Their Applications in Fourier Analysis. Lecture Notes in Mathematics 1568 Springer, Berlin (1994). | DOI | MR | Zbl

[24] Wu, L., Hao, Z., Jiao, Y.: John-Nirenberg inequalities with variable exponents on probability spaces. Tokyo J. Math. 38 (2) (2015). | MR

[25] Yi, R., Wu, L., Jiao, Y.: New John-Nirenberg inequalities for martingales. Stat. Probab. Lett. 86 (2014), 68-73. | DOI | MR | Zbl