Operator-valued martingale transform on mixed martingale Hardy spaces

Chuanzhou Zhang; Yafei Wang; Xueying Zhang

doi:10.3934/cac.2023018

Article Contents

2023, Volume 1, Issue 4: 369-378. Doi: 10.3934/cac.2023018

This issue Previous Article Reconstruction of extended regions in EIT with a generalized Robin transmission condition Next Article Approximate peak time and its application to time-domain fluorescence diffuse optical tomography

Operator-valued martingale transform on mixed martingale Hardy spaces

1.
College of Science, Wuhan University of Science and Technology, Wuhan 430065, China

^*Corresponding author: Chuanzhou Zhang
^*Corresponding author: Chuanzhou Zhang

Received: November 2023

Early access: December 8, 2023

Published online: December 8, 2023

Abstract / Introduction Full Text(HTML) Related Papers Cited by

Abstract

In this paper the operator-valued martingale transform inequalities on mixed martingale Hardy spaces are proved. Some well-known results are generalized and unified. Applications are given to classical operators such as the maximal operator and the $ p $-variation operator of vector-valued martingales, and then we can very easily obtain some new vector-valued martingale inequalities in mixed martingale Hardy spaces. These inequalities are closely related to the geometrical properties of Banach spaces.

Keywords:

Mathematics Subject Classification: Primary: 60G42, 42B30.

Citation:

FullText(HTML)

Related Papers

Cited by

References

[1]	N. Antonić and I. Ivec, On the Hörmander-Mihlin theorem for mixed-norm Lebesgue spaces, J. Math. Anal. Appl., 433 (2016), 176-199. doi: 10.1016/j.jmaa.2015.07.002.
[2]	A. Benedek and R. Panzone, The space $L^{p}$, with mixed norm, Duke Math. J., 28 (1961), 301-324. doi: 10.1215/S0012-7094-61-02828-9.
[3]	D. L. Burkholder, Martingale transform, Ann. Math. Stat., 37 (1966), 1494-1504. doi: 10.1214/aoms/1177699141.
[4]	D. L. Burkholder and R. F. Gundy, Extrapolation and interpolation of quasilinear operators on martingales, Acta Math., 124 (1970), 249-304. doi: 10.1007/BF02394573.
[5]	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.
[6]	D. L. Fernandez, Vector-valued singular integral operators on $L^{p}$ -spaces with mixed norms and applications, Pac. J. Math., 129 (1987), 257-275. doi: 10.2140/pjm.1987.129.257.
[7]	A. M. Garsia, Martingale Inequalities: Seminar Notes on Recent Progress, Math. Lecture Note Ser., W. A. Benjamin, Inc., Reading, Mass.-London-Amsterdam, 1973.
[8]	J. Hart, R. H. Torres and X. Wu, Smoothing properties of bilinear operators and Leibniz-type rules in Lebesgue and mixed Lebesgue spaces, Trans. Amer. Math. Soc., 370 (2018), 8581-8612. doi: 10.1090/tran/7312.
[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]	S. Igari, Interpolation of operators in Lebesgue spaces with mixed norm and its applications to Fourier analysis, Tohoku Math. J., 38 (1986), 469-490. doi: 10.2748/tmj/1178228457.
[12]	Y. Jiao, L. Wu and M. Popa, Operator-valued martingale transforms in rearrangement invariant spaces and applications, Sci. China Math., 56 (2013), 831-844. doi: 10.1007/s11425-013-4570-8.
[13]	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.
[14]	P. Liu, Martingale and the Geometry of Banach Spaces, Wuhan University Press, Wuhan, 1993.
[15]	P. I. Lizorkin, Multipliers of Fourier integrals and bounds of convolutions in spaces with mixed norms, Applications. Math USSR, Izv, 4 (1971), 225-255. doi: 10.1070/IM1970v004n01ABEH000901.
[16]	R. Long, Martingale spaces and inequalities, Peking University Press and Vieweg Publishing, 129 (1987), 257-275.
[17]	T. Martínez, Uniform convexity in terms of martingale H1 and BMO spaces, Houston J. Math., 27 (2001), 461-478.
[18]	T. Martínez and J. L. Torrea, Operator-valued martingale transforms, Tohoku Math. J., 52 (2000), 449-474. doi: 10.2748/tmj/1178207823.
[19]	T. Martínez and J. L. Torrea, Boundedness of vector-valued martingale transforms on extreme points and applications, J. Aust. Math. Soc., 76 (2004), 207-221. doi: 10.1017/S1446788700008909.
[20]	T. Nogayama, Mixed Morrey spaces, Positivity, 23 (2019), 961-1000. doi: 10.1007/s11117-019-00646-8.
[21]	J. L. Rubio de Francia, Martingale and integral transform of Banach space valued functions, in: Lecture Notes in Mathematics, Berlin-New York: Springer-Verlag, 1221 (1986), 195-222. doi: 10.1007/BFb0099115.
[22]	A. Stefanov and R. H. Torres, Calderon-Zygmund operators on mixed Lebesgue spaces and applications to null forms, J. Lond. Math. Soc., 70 (2004), 447-462. doi: 10.1112/S0024610704005502.
[23]	K. Szarvas and F. Weisz, Mixed martingale Hardy spaces, J. Geom. Anal., 31 (2021), 3863-3888. doi: 10.1007/s12220-020-00417-y.
[24]	R. H. Torres and E. L. Ward, Leibniz's rule, sampling and wavelets on mixed Lebesgue spaces, J. Fourier Anal. Appl., 21 (2015), 1053-1076. doi: 10.1007/s00041-015-9397-y.
[25]	L. Yu, Orlicz norm inequalities for operator-valued martingale transforms, Stat. Probab. Lett., 78 (2008), 2664-2670. doi: 10.1016/j.spl.2008.03.025.