Abstract
In this paper, we consider continuous parameter martingale Hardy–Lorentz spaces and describe their real interpolation spaces when we apply function parameter to Hardy–Lorentz and BMO spaces. Some new interpolation theorems concerning continuous parameter Hardy–Lorentz spaces are formulated. The results generalize some fundamental interpolation theorems in continuous parameter martingale Hardy spaces.
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C. Bennett and R. Sharpley, Interpolation of Operators, Pure Appl. Math., Vol. 129, Academic Press, Inc. (Boston, MA, 1988).
C. Dellacherie and P. A. Meyer, Probabilities and Potential. B, North-Holland Math. Stud. 72, North-Holland (Amsterdam, 1982).
C. Fefferman and E. M. Stein, H p spaces of several variables, Acta Math., 129 (1972), 137–194.
C. Fefferman, N. M. Riviere and Y. Sagher, Interpolation between H p spaces: the real method, Trans. Amer. Math. Soc., 191 (1974), 75–81.
A. M. Garsia, Martingale Inequalities: Seminar Notes on Recent Progress, Mathematics Lecture Notes Series, W. A. Benjamin, Inc. (Reading, Mass.–London–Amsterdam, 1973).
R. Hanks, Interpolation by the real method between BMO, Lα (0 < α < ∞) and Hα (0 < α < ∞), Indiana Univ. Math. J., 26 (1977), 679–689.
H. P. Heinig, Interpolation of quasi-normed spaces involving weights, in: Seminar on Harmonic Analysis (Montreal, Que., 1980), Amer. Math. Soc. (Providence, RI, 1981), pp. 245–267.
S. Janson and P. W. Jones, Interpolation between Hp spaces: the complex method, J. Funct. Anal., 48 (1982), 58–80.
Y. Jiao, L. Peng and P. Liu, Atomic decompositions of Lorentz martingale spaces and applications, J. Funct. Spaces Appl., 7 (2009), 153–166.
R. L. Long, Martingale Spaces and Inequalities, Peking University Press (Beijing, 1993).
Milman, On the interpolation of martingale L p spaces, Indiana Univ. Math. J., 30 (1981), 313–318.
M. Mohsenipour, Burkholder–Gundy–Davis’ inequalities on weighted Lorentz martingale spaces, Func. Anal.-TMA., 2 (2016), 52–55.
M. Mohsenipour and Gh. Sadeghi, Atomic decomposition of martingale weighted Lorentz spaces with two-parameter and applications, Rocky Mountain J. Math., 47 (2017), 927–947.
M. Mohsenipour and Gh. Sadeghi, Atomic decompositions of martingale Hardy–Lorentz spaces and interpolation, Filomat, 31 (2017), 5921–5929.
A. Osekowski, Weighted maximal inequalities for martingales, Tohoku Math. J., 65 (2013), 75–91.
L. E. Persson, Interpolation with a parameter function, Math. Scand., 59 (1986), 199–222.
M. Pratelli, Sur certains espaces de martingales localement de carré intégrable, in: Séminaire de Probabilitiés (Univ. Strasbourg, année universitaire 1974/1975), Lecture Notes in Math., Vol. 511, Springer (Berlin, 1976).
N. M. Riviere and Y. Sagher, Interpolation between L ∞ and H 1: the real method, J. Funct. Anal., 14 (1973), 401–409.
F. Weisz, Martingale BMO spaces with continuous time, Anal. Math., 22 (1996), 65–79.
F. Weisz, Martingale Hardy spaces with continuous time, in: Probability Theory and Applications, Math. Appl., Vol. 80, Kluwer Acad. Publ. (Dordrecht, 1992), pp. 47–75.
F. Weisz, Interpolation between continuous parameter martingale spaces: The real method, Acta Math. Hungar., 68 (1995), 37–54.
F. Weisz, Martingale Hardy Spaces and Their Applications in Fourier Analysis, Lecture Notes in Math., Vol. 1568, Springer-Verlag (Berlin, 1994).
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Mohsenipour, M., Sadeghi, G. Interpolation Between Continuous Parameter Martingale Hardy–Lorentz and BMO Spaces. Anal Math 45, 375–389 (2019). https://doi.org/10.1007/s10476-018-0716-9
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DOI: https://doi.org/10.1007/s10476-018-0716-9