Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)


For citation:

Volosivets S. S., Zaitsev N. N. Martingale Inequalities in Symmetric Spaces with Semimultiplicative Weight. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2019, vol. 19, iss. 2, pp. 126-133. DOI: 10.18500/1816-9791-2019-19-2-126-133, EDN: RCYPHX

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
28.05.2019
Full text:
(downloads: 274)
Language: 
Russian
Heading: 
Article type: 
Article
UDC: 
519.216.8
EDN: 
RCYPHX

Martingale Inequalities in Symmetric Spaces with Semimultiplicative Weight

Autors: 
Volosivets Sergei Sergeevich, Saratov State University
Zaitsev Nikolai N., Saratov State University
Abstract: 

Let (Ω, Σ, P) be a complete probability space, F = {Fn}∞ n=0 be an increasing sequence of σalgebras such that ∪∞ n=0Fn generates Σ. If f = {fn}∞ n=0 is a martingale with respect to F and En is the conditional expectation with respect to Fn, then one can introduce a maximal function M(f) = supn>0 |fn| and a square function S(f) = µP∞ i=0 |fi − fi−1| 2 ¶1/2 , f−1 = 0. In the case of uniformly integrable martingales there exists g ∈ L 1 (Ω) such that Eng = fn and we consider a sharp maximal function f ♯ = supn>0 En|g − fn−1|. The result of Burkholder – Davis – Gundy is that C1kM(f)kp 6 kS(f)kp 6 C2kM(f)k for 1 < p < ∞, where k · kp is the norm in L p (Ω) and C2 > C1 > 0. We call the inequality of type kM(f)kp 6 Ckf ♯kp, 1 < p < ∞ Fefferman – Stein inequality. It is known that Burkholder – Davis – Gundy martingale inequality is valid in rearrangement invariant Banach function spaces with non-trivial Boyd indices. We prove this inequality in a more wide class of symmetric spaces (the last notion is defined as in the famous monograph by S. G. Krein, Yu. I. Petunin and E. M. Semenov) with semimultiplicative weight. Also, the Fefferman – Stein type inequalities of sharp maximal function and sharp square functions are obtained in this class of symmetric spaces.

References: 
  1. Burkholder D. Distribution function inequalities for martingales. Ann. of Probab., 1973, vol. 1, no. 1, pp. 19–42. DOI: https://doi.org/10.1214/aop/1176997023
  2. Burkholder D., Davis B. J., Gundy R. F. Integral inequalities for convex functions of operators on martingales. Proc. Sixth Berkeley Symp. on Math. Statist. and Prob. Univ. of Calif. Press, 1972. Vol. 2, pp. 223–240.
  3. Johnson W., Schechtman G. Martingale inequalities in rearrangement invariant function space. Israel J. Math., 1988, vol. 64, no. 3, pp. 267–275. DOI: https://doi.org/10.1007/BF02882423
  4. Novikov I. Ya. Martingale inequalities in rearrangement invariant spaces. Siberian Math. J., 1993, vol. 34, no. 1, pp. 99–105. DOI: https://doi.org/10.1007/BF00971245
  5. Krein S. G., Petunin Yu. I., Semenov E. M. Interpolation of linear operators. Providence, RI, Amer. Math. Soc., 1982. 375 p. (Russ. ed.: Moscow, Nauka, 1978. 400 p.).
  6. Kikuchi M. Averaging Operators and Martingale Inequalities in Rearrangement Invariant Function Spaces. Canad. Math. Bull., 1999, vol. 42, iss. 3, pp. 321–334. DOI: https://doi.org/10.4153/CMB-1999-038-7
  7. Fefferman C., Stein E. H p spaces of several variables. Acta. Math., 1972, vol. 129, pp. 137–193. DOI: https://doi.org/10.1007/BF02392215
  8. Garsia A. M. Martingale inequalities. New York, Benjamin Inc., 1973. 184 p.
  9. Weisz F. Martingale Hardy spaces and their Applications in Fourier Analysis. Lecture Notes in Maths. Vol. 1568. Berlin, Springer-Verlag, 1994. 220 p. DOI: https://doi.org/10.1007/BFb0073448
  10. Long R. L. Rearrangement techniques in martingale setting. Illinois J. Math., 1991, vol. 35, no. 3, pp. 506–521.
  11. Ren Y. A note on some inequalities of martingale sharp functions. Math. Inequal. Appl., 2013, vol. 16, no. 1, pp. 153–157. DOI: https://doi.org/10.7153/mia-16-11
  12. Ho K. P. Martingale inequalities on rearrangement-invariant quasi-Banach function spaces. Acta Sci. Math. (Szeged), 2017, vol. 83, no. 3–4, pp. 619–627. DOI: https://doi.org/10.14232/actasm-012-817-9
  13. Hardy G. H., Littlewood J. E., Polya G. Inequalities. Cambridge, Cambridge Univ. Press, 1934. 328 p. (Russ. ed.: Мoscow, Izd-vo inostr. lit., 1948. 456 p.)
  14. Pavlov E. A. Some properties of Hardy–Littlewood operator. Math. Notes of the Academy of Sciences of the USSR, 1979, vol. 26, iss. 6, pp. 958–960. DOI: https://doi.org/10.1007/BF01142082
  15. Bagby R., Kurtz D. A rearranged good λ-inequality. Trans. Amer. Math. Soc., 1986, vol. 293, no. 1, pp. 71–81. DOI: https://doi.org/10.1090/S0002-9947-1986-0814913-7
  16. Kikuchi M. On the Davis inequality in Banach function spaces. Math. Nachrichten, 2008, vol. 281, no. 5, pp. 697–709. DOI: https://doi.org/10.1002/mana.200510635
Received: 
20.04.2018
Accepted: 
04.02.2019
Published: 
28.05.2019
Short text (in English):
(downloads: 96)