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February  2021, 20(2): 681-695. doi: 10.3934/cpaa.2020285

Dual spaces of mixed-norm martingale Hardy spaces

Ferenc Weisz

Department of Numerical Analysis, Eötvös L. University, H-1117 Budapest, Pázmány P. sétány 1/C., Hungary

Received  May 2020 Revised  October 2020 Published  February 2021 Early access  December 2020

Fund Project: This research was supported by the Hungarian National Research, Development and Innovation Office-NKFIH, KH130426

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 $.

Keywords: Mixed Lebesgue spaces, mixed martingale Hardy spaces, atomic decompositions, Doob's inequality, $ BMO $ spaces, John-Nirenberg inequality.
Mathematics Subject Classification: Primary: 42B35; Secondary: 42B30, 60G42, 42B25, 46E30.
Citation: Ferenc Weisz. Dual spaces of mixed-norm martingale Hardy spaces. Communications on Pure and Applied Analysis, 2021, 20 (2) : 681-695. doi: 10.3934/cpaa.2020285
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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