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Dual spaces of mixed-norm martingale Hardy spaces
Department of Numerical Analysis, Eötvös L. University, H-1117 Budapest, Pázmány P. sétány 1/C., Hungary |
In this paper, we generalize the Doob's maximal inequality for mixed-norm $ L_{\vec{p}} $ spaces. We consider martingale Hardy spaces defined with the help of mixed $ L_{{\vec{p}}} $-norm. A new atomic decomposition is given for these spaces via simple atoms. The dual spaces of the mixed-norm martingale Hardy spaces is given as the mixed-norm $ BMO_{\vec{r}}(\vec{\alpha}) $ spaces. This implies the John-Nirenberg inequality $ BMO_{1}(\vec{\alpha}) \sim BMO_{\vec{r}}(\vec{\alpha}) $ for $ 1<\vec{r}<\infty $. These results generalize the well known classical results for constant $ p $ and $ r $.
References:
[1] |
A. Benedek and R. Panzone, The spaces $L^p$, with mixed norm, Duke Math. J., 28 (1961), 301–324. |
[2] |
W. Chen, K. P. Ho, Y. Jiao and D. Zhou., Weighted mixed-norm inequality on Doob's maximal operator and John-Nirenberg inequalities in Banach function spaces, Acta Math. Hung., 157 (2019), 408–433.
doi: 10.1007/s10474-018-0889-5. |
[3] |
G. Cleanthous and A. G. Georgiadis., Mixed-norm $\alpha$-modulation spaces, T. Am. Math. Soc., 373 (2020), 3323–3356.
doi: 10.1090/tran/8023. |
[4] |
G. Cleanthous, A. G. Georgiadis and M. Nielsen., Anisotropic mixed-norm Hardy spaces, J. Geom. Anal., 27 (2017), 2758–2787.
doi: 10.1007/s12220-017-9781-8. |
[5] |
C. Fefferman, Characterizations of bounded mean oscillation, Bull. Am. Math. Soc., 77 (1971), 587–588.
doi: 10.1090/S0002-9904-1971-12763-5. |
[6] |
C. Fefferman and E. M. Stein,
$H^p$ spaces of several variables, Acta Math., 129 (1972), 137-194.
doi: 10.1007/BF02392215. |
[7] |
A. M. Garsia, Martingale Inequalities. Seminar Notes on Recent Progress, Math. Lecture Note. Benjamin, New York, 1973. |
[8] |
C. Herz,
$H_p$-spaces of martingales, $0 < p \leq 1$, Z. Wahrscheinlichkeitstheorie Verw. Geb., 28 (1974), 189-205.
doi: 10.1007/BF00533241. |
[9] |
K. P, Ho, Strong maximal operator on mixed-norm spaces, Ann. Univ. Ferrara, Sez. VII, Sci. Mat., 62 (2016), 275–291.
doi: 10.1007/s11565-016-0245-z. |
[10] |
K. P. Ho, Mixed norm Lebesgue spaces with variable exponents and applications, Riv. Mat. Univ. Parma (N.S.), 9 (2018), 21–44. |
[11] |
L. Hörmander, Estimates for translation invariant operators in $L^p$ spaces, Acta Math., 104 (1960), 93–140.
doi: 10.1007/BF02547187. |
[12] |
L. Huang, J. Liu, D. Yang and W. Yuan,
Atomic and Littlewood-Paley characterizations of anisotropic mixed-norm Hardy spaces and their applications, J. Geom. Anal., 29 (2019), 1991-2067.
doi: 10.1007/s12220-018-0070-y. |
[13] |
L. Huang, J. Liu, D. Yang and W. Yuan, Dual spaces of anisotropic mixed-norm Hardy spaces, Proc. Amer. Math. Soc., 147 (2019), 1201–1215.
doi: 10.1090/proc/14348. |
[14] |
L. Huang, J. Liu, D. Yang and W. Yuan, Identification of anisotropic mixed-norm Hardy spaces and certain homogeneous Triebel-Lizorkin spaces, J. Approx. Theory, 258 (2020), 105459.
doi: 10.1016/j.jat.2020.105459. |
[15] |
L. Huang, J. Liu, D. Yang and W. Yuan, Real-variable characterizations of new anisotropic mixed-norm hardy spaces, Commun. Pure Appl. Anal., 19 (2020), 3033–3082.
doi: 10.3934/cpaa.2020132. |
[16] |
L. Huang and D. Yang, On function spaces with mixed norms-a survey, arXiv: 1908.03291. |
[17] |
Y. Jiao, F. Weisz, L. Wu and D. Zhou, Dual spaces for variable martingale Lorentz-Hardy spaces, preprint. |
[18] |
Y. Jiao, F. Weisz, L. Wu and D. Zhou,
Variable martingale Hardy spaces and their applications in Fourier analysis, Dissertationes Math., 550 (2020), 1-67.
doi: 10.4064/dm807-12-2019. |
[19] |
Y. Jiao, L. Wu, A. Yang and R. Yi,
The predual and John-Nirenberg inequalities on generalized BMO martingale space, T. Am. Math. Soc., 369 (2017), 537-553.
doi: 10.1090/tran/6657. |
[20] |
Y. Jiao, G. Xie and D. Zhou,
Dual spaces and John-Nirenberg inequalities of martingale Hardy-Lorentz-Karamata spaces, Quart. J. Math., 66 (2015), 605-623.
doi: 10.1093/qmath/hav003. |
[21] |
Y. Jiao, D. Zhou, Z. Hao and W. Chen,
Martingale Hardy spaces with variable exponents, Banach J. Math, 10 (2016), 750-770.
doi: 10.1215/17358787-3649326. |
[22] |
Y. Jiao, Y. Zuo, D. Zhou and L. Wu,
Variable Hardy-Lorentz spaces $H^{p(\cdot), q}(\mathbb R^n)$, Math. Nachr., 292 (2019), 309-349.
doi: 10.1002/mana.201700331. |
[23] |
F. John and L. Nirenberg, On functions of bounded mean oscillation, Commun. Pure Appl. Math., 14 (1961), 415–426.
doi: 10.1002/cpa.3160140317. |
[24] |
J. Liu, F. Weisz, D. Yang and W. Yuan,
Variable anisotropic Hardy spaces and their applications, Taiwanese J. Math., 22 (2018), 1173-1216.
doi: 10.11650/tjm/171101. |
[25] |
J. Liu, F. Weisz, D. Yang and W. Yuan,
Littlewood-Paley and finite atomic characterizations of anisotropic variable Hardy-Lorentz spaces and their applications, J. Fourier Anal. Appl., 25 (2019), 874-922.
doi: 10.1007/s00041-018-9609-3. |
[26] |
R. Long, Martingale Spaces and Inequalities, Peking University Press and Vieweg Publishing, 1993.
doi: 10.1007/978-3-322-99266-6. |
[27] |
E. Nakai and Y. Sawano, Hardy spaces with variable exponents and generalized Campanato spaces, J. Funct. Anal., 262 (2012), 3665–3748.
doi: 10.1016/j.jfa.2012.01.004. |
[28] |
K. Szarvas and F. Weisz, Mixed martingale Hardy spaces., J. Geom. Anal., (2020), 26pp.
doi: 10.1007/s12220-020-00417-y. |
[29] |
F. Weisz,
Martingale Hardy spaces for $0 < p \leq 1$, Probab. Th. Rel. Fields, 84 (1990), 361-376.
doi: 10.1007/BF01197890. |
[30] |
F. Weisz, Martingale Hardy Spaces and their Applications in Fourier Analysis, Springer, Berlin, 1994.
doi: 10.1007/BFb0073448. |
[31] |
G. Xie, Y. Jiao and D. Yang, Martingale Musielak-Orlicz Hardy spaces, Sci. China, Math., 62 (2019), 1567–1584.
doi: 10.1007/s11425-017-9237-3. |
[32] |
G. Xie, F. Weisz, D. Yang and Y. Jiao,
New martingale inequalities and applications to Fourier analysis, Nonlinear Anal., 182 (2019), 143-192.
doi: 10.1016/j.na.2018.12.011. |
[33] |
G. Xie and D. Yang, Atomic characterizations of weak martingale Musielak-Orlicz Hardy spaces and their applications, Banach J. Math. Anal., 13 (2019), 884–917.
doi: 10.1215/17358787-2018-0050. |
[34] |
X. Yan, D. Yang, W. Yuan and C. Zhuo,
Variable weak Hardy spaces and their applications, J. Funct. Anal., 271 (2016), 2822-2887.
doi: 10.1016/j.jfa.2016.07.006. |
[35] |
D. Yang, Y. Liang and L. D. Ky, Real-Variable Theory of Musielak-Orlicz Hardy Spaces, Springer, 2017.
doi: 10.1007/978-3-319-54361-1. |
show all references
References:
[1] |
A. Benedek and R. Panzone, The spaces $L^p$, with mixed norm, Duke Math. J., 28 (1961), 301–324. |
[2] |
W. Chen, K. P. Ho, Y. Jiao and D. Zhou., Weighted mixed-norm inequality on Doob's maximal operator and John-Nirenberg inequalities in Banach function spaces, Acta Math. Hung., 157 (2019), 408–433.
doi: 10.1007/s10474-018-0889-5. |
[3] |
G. Cleanthous and A. G. Georgiadis., Mixed-norm $\alpha$-modulation spaces, T. Am. Math. Soc., 373 (2020), 3323–3356.
doi: 10.1090/tran/8023. |
[4] |
G. Cleanthous, A. G. Georgiadis and M. Nielsen., Anisotropic mixed-norm Hardy spaces, J. Geom. Anal., 27 (2017), 2758–2787.
doi: 10.1007/s12220-017-9781-8. |
[5] |
C. Fefferman, Characterizations of bounded mean oscillation, Bull. Am. Math. Soc., 77 (1971), 587–588.
doi: 10.1090/S0002-9904-1971-12763-5. |
[6] |
C. Fefferman and E. M. Stein,
$H^p$ spaces of several variables, Acta Math., 129 (1972), 137-194.
doi: 10.1007/BF02392215. |
[7] |
A. M. Garsia, Martingale Inequalities. Seminar Notes on Recent Progress, Math. Lecture Note. Benjamin, New York, 1973. |
[8] |
C. Herz,
$H_p$-spaces of martingales, $0 < p \leq 1$, Z. Wahrscheinlichkeitstheorie Verw. Geb., 28 (1974), 189-205.
doi: 10.1007/BF00533241. |
[9] |
K. P, Ho, Strong maximal operator on mixed-norm spaces, Ann. Univ. Ferrara, Sez. VII, Sci. Mat., 62 (2016), 275–291.
doi: 10.1007/s11565-016-0245-z. |
[10] |
K. P. Ho, Mixed norm Lebesgue spaces with variable exponents and applications, Riv. Mat. Univ. Parma (N.S.), 9 (2018), 21–44. |
[11] |
L. Hörmander, Estimates for translation invariant operators in $L^p$ spaces, Acta Math., 104 (1960), 93–140.
doi: 10.1007/BF02547187. |
[12] |
L. Huang, J. Liu, D. Yang and W. Yuan,
Atomic and Littlewood-Paley characterizations of anisotropic mixed-norm Hardy spaces and their applications, J. Geom. Anal., 29 (2019), 1991-2067.
doi: 10.1007/s12220-018-0070-y. |
[13] |
L. Huang, J. Liu, D. Yang and W. Yuan, Dual spaces of anisotropic mixed-norm Hardy spaces, Proc. Amer. Math. Soc., 147 (2019), 1201–1215.
doi: 10.1090/proc/14348. |
[14] |
L. Huang, J. Liu, D. Yang and W. Yuan, Identification of anisotropic mixed-norm Hardy spaces and certain homogeneous Triebel-Lizorkin spaces, J. Approx. Theory, 258 (2020), 105459.
doi: 10.1016/j.jat.2020.105459. |
[15] |
L. Huang, J. Liu, D. Yang and W. Yuan, Real-variable characterizations of new anisotropic mixed-norm hardy spaces, Commun. Pure Appl. Anal., 19 (2020), 3033–3082.
doi: 10.3934/cpaa.2020132. |
[16] |
L. Huang and D. Yang, On function spaces with mixed norms-a survey, arXiv: 1908.03291. |
[17] |
Y. Jiao, F. Weisz, L. Wu and D. Zhou, Dual spaces for variable martingale Lorentz-Hardy spaces, preprint. |
[18] |
Y. Jiao, F. Weisz, L. Wu and D. Zhou,
Variable martingale Hardy spaces and their applications in Fourier analysis, Dissertationes Math., 550 (2020), 1-67.
doi: 10.4064/dm807-12-2019. |
[19] |
Y. Jiao, L. Wu, A. Yang and R. Yi,
The predual and John-Nirenberg inequalities on generalized BMO martingale space, T. Am. Math. Soc., 369 (2017), 537-553.
doi: 10.1090/tran/6657. |
[20] |
Y. Jiao, G. Xie and D. Zhou,
Dual spaces and John-Nirenberg inequalities of martingale Hardy-Lorentz-Karamata spaces, Quart. J. Math., 66 (2015), 605-623.
doi: 10.1093/qmath/hav003. |
[21] |
Y. Jiao, D. Zhou, Z. Hao and W. Chen,
Martingale Hardy spaces with variable exponents, Banach J. Math, 10 (2016), 750-770.
doi: 10.1215/17358787-3649326. |
[22] |
Y. Jiao, Y. Zuo, D. Zhou and L. Wu,
Variable Hardy-Lorentz spaces $H^{p(\cdot), q}(\mathbb R^n)$, Math. Nachr., 292 (2019), 309-349.
doi: 10.1002/mana.201700331. |
[23] |
F. John and L. Nirenberg, On functions of bounded mean oscillation, Commun. Pure Appl. Math., 14 (1961), 415–426.
doi: 10.1002/cpa.3160140317. |
[24] |
J. Liu, F. Weisz, D. Yang and W. Yuan,
Variable anisotropic Hardy spaces and their applications, Taiwanese J. Math., 22 (2018), 1173-1216.
doi: 10.11650/tjm/171101. |
[25] |
J. Liu, F. Weisz, D. Yang and W. Yuan,
Littlewood-Paley and finite atomic characterizations of anisotropic variable Hardy-Lorentz spaces and their applications, J. Fourier Anal. Appl., 25 (2019), 874-922.
doi: 10.1007/s00041-018-9609-3. |
[26] |
R. Long, Martingale Spaces and Inequalities, Peking University Press and Vieweg Publishing, 1993.
doi: 10.1007/978-3-322-99266-6. |
[27] |
E. Nakai and Y. Sawano, Hardy spaces with variable exponents and generalized Campanato spaces, J. Funct. Anal., 262 (2012), 3665–3748.
doi: 10.1016/j.jfa.2012.01.004. |
[28] |
K. Szarvas and F. Weisz, Mixed martingale Hardy spaces., J. Geom. Anal., (2020), 26pp.
doi: 10.1007/s12220-020-00417-y. |
[29] |
F. Weisz,
Martingale Hardy spaces for $0 < p \leq 1$, Probab. Th. Rel. Fields, 84 (1990), 361-376.
doi: 10.1007/BF01197890. |
[30] |
F. Weisz, Martingale Hardy Spaces and their Applications in Fourier Analysis, Springer, Berlin, 1994.
doi: 10.1007/BFb0073448. |
[31] |
G. Xie, Y. Jiao and D. Yang, Martingale Musielak-Orlicz Hardy spaces, Sci. China, Math., 62 (2019), 1567–1584.
doi: 10.1007/s11425-017-9237-3. |
[32] |
G. Xie, F. Weisz, D. Yang and Y. Jiao,
New martingale inequalities and applications to Fourier analysis, Nonlinear Anal., 182 (2019), 143-192.
doi: 10.1016/j.na.2018.12.011. |
[33] |
G. Xie and D. Yang, Atomic characterizations of weak martingale Musielak-Orlicz Hardy spaces and their applications, Banach J. Math. Anal., 13 (2019), 884–917.
doi: 10.1215/17358787-2018-0050. |
[34] |
X. Yan, D. Yang, W. Yuan and C. Zhuo,
Variable weak Hardy spaces and their applications, J. Funct. Anal., 271 (2016), 2822-2887.
doi: 10.1016/j.jfa.2016.07.006. |
[35] |
D. Yang, Y. Liang and L. D. Ky, Real-Variable Theory of Musielak-Orlicz Hardy Spaces, Springer, 2017.
doi: 10.1007/978-3-319-54361-1. |
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