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Mathematics. Mechanics. Informatics

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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

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Martingale Inequalities in Symmetric Spaces with Semimultiplicative Weight

Volosivets Sergei Sergeevich, Saratov State University
Zaitsev Nikolai N., Saratov State University

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.

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