American Institute of Mathematical Sciences

Advanced
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  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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

C. Fefferman, Characterizations of bounded mean oscillation, Bull. Am. Math. Soc., 77 (1971), 587–588. doi: 10.1090/S0002-9904-1971-12763-5.  Google Scholar

[6]

C. Fefferman and E. M. Stein, $H^p$ spaces of several variables, Acta Math., 129 (1972), 137-194.  doi: 10.1007/BF02392215.  Google Scholar

[7]

A. M. Garsia, Martingale Inequalities. Seminar Notes on Recent Progress, Math. Lecture Note. Benjamin, New York, 1973.  Google Scholar

[8]

C. Herz, $H_p$-spaces of martingales, $0 < p \leq 1$, Z. Wahrscheinlichkeitstheorie Verw. Geb., 28 (1974), 189-205.  doi: 10.1007/BF00533241.  Google Scholar

[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.  Google Scholar

[10]

K. P. Ho, Mixed norm Lebesgue spaces with variable exponents and applications, Riv. Mat. Univ. Parma (N.S.), 9 (2018), 21–44.  Google Scholar

[11]

L. Hörmander, Estimates for translation invariant operators in $L^p$ spaces, Acta Math., 104 (1960), 93–140. doi: 10.1007/BF02547187.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

L. Huang and D. Yang, On function spaces with mixed norms-a survey, arXiv: 1908.03291.  Google Scholar

[17]

Y. Jiao, F. Weisz, L. Wu and D. Zhou, Dual spaces for variable martingale Lorentz-Hardy spaces, preprint.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

R. Long, Martingale Spaces and Inequalities, Peking University Press and Vieweg Publishing, 1993. doi: 10.1007/978-3-322-99266-6.  Google Scholar

[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.  Google Scholar

[28]

K. Szarvas and F. Weisz, Mixed martingale Hardy spaces., J. Geom. Anal., (2020), 26pp. doi: 10.1007/s12220-020-00417-y.  Google Scholar

[29]

F. Weisz, Martingale Hardy spaces for $0 < p \leq 1$, Probab. Th. Rel. Fields, 84 (1990), 361-376.  doi: 10.1007/BF01197890.  Google Scholar

[30]

F. Weisz, Martingale Hardy Spaces and their Applications in Fourier Analysis, Springer, Berlin, 1994. doi: 10.1007/BFb0073448.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

A. Benedek and R. Panzone, The spaces $L^p$, with mixed norm, Duke Math. J., 28 (1961), 301–324.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

C. Fefferman, Characterizations of bounded mean oscillation, Bull. Am. Math. Soc., 77 (1971), 587–588. doi: 10.1090/S0002-9904-1971-12763-5.  Google Scholar

[6]

C. Fefferman and E. M. Stein, $H^p$ spaces of several variables, Acta Math., 129 (1972), 137-194.  doi: 10.1007/BF02392215.  Google Scholar

[7]

A. M. Garsia, Martingale Inequalities. Seminar Notes on Recent Progress, Math. Lecture Note. Benjamin, New York, 1973.  Google Scholar

[8]

C. Herz, $H_p$-spaces of martingales, $0 < p \leq 1$, Z. Wahrscheinlichkeitstheorie Verw. Geb., 28 (1974), 189-205.  doi: 10.1007/BF00533241.  Google Scholar

[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.  Google Scholar

[10]

K. P. Ho, Mixed norm Lebesgue spaces with variable exponents and applications, Riv. Mat. Univ. Parma (N.S.), 9 (2018), 21–44.  Google Scholar

[11]

L. Hörmander, Estimates for translation invariant operators in $L^p$ spaces, Acta Math., 104 (1960), 93–140. doi: 10.1007/BF02547187.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

L. Huang and D. Yang, On function spaces with mixed norms-a survey, arXiv: 1908.03291.  Google Scholar

[17]

Y. Jiao, F. Weisz, L. Wu and D. Zhou, Dual spaces for variable martingale Lorentz-Hardy spaces, preprint.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

R. Long, Martingale Spaces and Inequalities, Peking University Press and Vieweg Publishing, 1993. doi: 10.1007/978-3-322-99266-6.  Google Scholar

[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.  Google Scholar

[28]

K. Szarvas and F. Weisz, Mixed martingale Hardy spaces., J. Geom. Anal., (2020), 26pp. doi: 10.1007/s12220-020-00417-y.  Google Scholar

[29]

F. Weisz, Martingale Hardy spaces for $0 < p \leq 1$, Probab. Th. Rel. Fields, 84 (1990), 361-376.  doi: 10.1007/BF01197890.  Google Scholar

[30]

F. Weisz, Martingale Hardy Spaces and their Applications in Fourier Analysis, Springer, Berlin, 1994. doi: 10.1007/BFb0073448.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Alexandre B. Simas, Fábio J. Valentim. $W$-Sobolev spaces: Higher order and regularity. Communications on Pure & Applied Analysis, 2015, 14 (2) : 597-607. doi: 10.3934/cpaa.2015.14.597
[2]

Alexander A. Davydov, Massimo Giulietti, Stefano Marcugini, Fernanda Pambianco. Linear nonbinary covering codes and saturating sets in projective spaces. Advances in Mathematics of Communications, 2011, 5 (1) : 119-147. doi: 10.3934/amc.2011.5.119
[3]

Xue-Ping Luo, Yi-Bin Xiao, Wei Li. Strict feasibility of variational inclusion problems in reflexive Banach spaces. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2495-2502. doi: 10.3934/jimo.2019065
[4]

Ademir Fernando Pazoto, Lionel Rosier. Uniform stabilization in weighted Sobolev spaces for the KdV equation posed on the half-line. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1511-1535. doi: 10.3934/dcdsb.2010.14.1511
[5]

Lucas C. F. Ferreira, Jhean E. Pérez-López, Élder J. Villamizar-Roa. On the product in Besov-Lorentz-Morrey spaces and existence of solutions for the stationary Boussinesq equations. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2423-2439. doi: 10.3934/cpaa.2018115
[6]

Demetres D. Kouvatsos, Jumma S. Alanazi, Kevin Smith. A unified ME algorithm for arbitrary open QNMs with mixed blocking mechanisms. Numerical Algebra, Control & Optimization, 2011, 1 (4) : 781-816. doi: 10.3934/naco.2011.1.781
[7]

Carlos Fresneda-Portillo, Sergey E. Mikhailov. Analysis of Boundary-Domain Integral Equations to the mixed BVP for a compressible stokes system with variable viscosity. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3059-3088. doi: 10.3934/cpaa.2019137
[8]

Yimin Zhang, Youjun Wang, Yaotian Shen. Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1037-1054. doi: 10.3934/cpaa.2011.10.1037
[9]

Z. Reichstein and B. Youssin. Parusinski's "Key Lemma" via algebraic geometry. Electronic Research Announcements, 1999, 5: 136-145.

[10]

Ronald E. Mickens. Positivity preserving discrete model for the coupled ODE's modeling glycolysis. Conference Publications, 2003, 2003 (Special) : 623-629. doi: 10.3934/proc.2003.2003.623
[11]

Bernold Fiedler, Carlos Rocha, Matthias Wolfrum. Sturm global attractors for $S^1$-equivariant parabolic equations. Networks & Heterogeneous Media, 2012, 7 (4) : 617-659. doi: 10.3934/nhm.2012.7.617
[12]

Hakan Özadam, Ferruh Özbudak. A note on negacyclic and cyclic codes of length $p^s$ over a finite field of characteristic $p$. Advances in Mathematics of Communications, 2009, 3 (3) : 265-271. doi: 10.3934/amc.2009.3.265
[13]

Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (48)
  • HTML views (72)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences