Two-parameter inequality of Garcia-Rodemich-Rumsey and its application to fractional Brownian fields. (Ukrainian, English) Zbl 1164.60351

Ral’chenko, K. V.

Teor. Jmovirn. Mat. Stat. 75, 144-154 (2006); translation in Theory Probab. Math. Stat. 75, 167-178 (2007).

The author proves the following result. Let \(p\geq1, \alpha_1>p^{-1}\), \(\alpha_2>p^{-1}\). Then there exists constant \(C_{\alpha_1,\alpha_2,p}>0\) such that, for any continuous function \(f\) on \([0,T]=[0,T_1]\times[0,T_2]\) and for all \(s,t\in[0,T]\), \[ | \diamondsuit f(s,t)|^{p}\leq C_{\alpha_1,\alpha_2,p}| s_1-t_1|^{\alpha_1p-1}| s_2-t_2|^{\alpha_2p-1}\int_{[0,T]^2}{|\diamondsuit f(x,y)|^{p}\over | x_1-y_1|^{\alpha_1p+1}| x_2-y_2|^{\alpha_2p+1}}\,dx_1\,dx_2\,dy_1\,dy_2, \] where \(\diamondsuit f(s,t)=f(s_1,s_2)-f(s_1,t_2)-f(t_1,s_2)+f(t_1,t_2)\). Using this inequality, moment inequalities are obtained for fractional derivatives of fractional Brownian fields on the plane.

Reviewer: Aleksandr D. Borisenko (Kyïv)

MSC:

60G15	Gaussian processes
60G60	Random fields

Keywords:

inequality of Garcia-Rodemich-Rumsey; two-parameter processes; fractional Brownian fields; Hölder constant