Martingale inequalities in symmetric spaces with semimultiplicative weight

S. S. Volosivets ; N. N. Zaitsev

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 19 (2019) no. 2, pp. 126-133

Résumé

Let $(\Omega,\Sigma,P)$ be a complete probability space, $\mathcal F=\{\mathcal F_n\}^\infty_{n=0}$ be an increasing sequence of $\sigma$-algebras such that $\cup^\infty_{n=0}\mathcal F_n$ generates $\Sigma$. If $f=\{f_n\}^\infty_{n=0}$ is a martingale with respect to $\mathcal F$ and $\mathbb E_n$ is the conditional expectation with respect to $\mathcal F_n$, then one can introduce a maximal function $M(f)=\sup_{n\geq 0}|f_n|$ and a square function $S(f)=\left(\sum\limits^\infty_{i=0}|f_i-f_{i-1}|^2\right)^{1/2}$, $f_{-1}=0$. In the case of uniformly integrable martingales there exists $g\in L^1(\Omega)$ such that $\mathbb E_ng=f_n$ and we consider a sharp maximal function $f^\sharp=\sup_{n\geq 0}\mathbb E_n|g-f_{n-1}|$. The result of Burkholder – Davis – Gundy is that $C_1\|M(f)\|_p\leq \|S(f)\|_p\leq C_2\|M(f)\|$ for $1<p<\infty$, where $\|\cdot\|_p$ is the norm in $L^p(\Omega)$ and $C_2>C_1>0$. We call the inequality of type $\|M(f)\|_p\leq C\|f^\sharp\|_p$, $1<p<\infty$ 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.

Bibliographie

[1] Burkholder D., “Distribution function inequalities for martingales”, Ann. of Probab., 1:1 (1973), 19–42 | DOI | MR | Zbl

[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., v. 2, Univ. of Calif. Press, 1972, 223–240 | MR | Zbl

[3] Johnson W., Schechtman G., “Martingale inequalities in rearrangement invariant function space”, Israel J. Math., 64:3 (1988), 267–275 | DOI | MR

[4] Novikov I. Ya., “Martingale inequalities in rearrangement invariant spaces”, Siberian Math. J., 34:1 (1993), 99–105 | DOI | MR | Zbl

[5] Krein S. G., Petunin Yu. I., Semenov E. M., Interpolation of linear operators, Amer. Math. Soc., Providence, RI, 1982, 375 pp. | MR | MR

[6] Kikuchi M., “Averaging Operators and Martingale Inequalities in Rearrangement Invariant Function Spaces”, Canad. Math. Bull., 42:3 (1999), 321–334 | DOI | MR | Zbl

[7] Fefferman C., Stein E., “$H^p$ spaces of several variables”, Acta. Math., 129 (1972), 137–193 | DOI | MR | Zbl

[8] Garsia A. M., Martingale inequalities, Benjamin Inc., N. Y., 1973, 184 pp. | MR | Zbl

[9] Weisz F., Martingale Hardy spaces and their Applications in Fourier Analysis, Lecture Notes in Math., 1568, Springer-Verlag, Berlin, 1994, 220 pp. | DOI | MR | Zbl

[10] Long R. L., “Rearrangement techniques in martingale setting”, Illinois J. Math., 35:3 (1991), 506–521 | DOI | MR | Zbl

[11] Ren Y., “A note on some inequalities of martingale sharp functions”, Math. Inequal. Appl., 16:1 (2013), 153–157 | DOI | MR | Zbl

[12] Ho K. P., “Martingale inequalities on rearrangement-invariant quasi-Banach function spaces”, Acta Sci. Math. (Szeged), 83:3–4 (2017), 619–627 | DOI | MR | Zbl

[13] Hardy G. H., Littlewood J. E., Polya G., Inequalities, Cambridge Univ. Press, Cambridge, 1934, 328 pp. | MR

[14] Pavlov E. A., “Some properties of Hardy – Littlewood operator”, Math. Notes of the Academy of Sciences of the USSR, 26:6 (1979), 958–960 | DOI | MR

[15] Bagby R., Kurtz D., “A rearranged good $\lambda$-inequality”, Trans. Amer. Math. Soc., 293:1 (1986), 71–81 | DOI | MR | Zbl

[16] Kikuchi M., “On the Davis inequality in Banach function spaces”, Math. Nachrichten, 281:5 (2008), 697–709 | DOI | MR | Zbl