We establish sufficient conditions on the weight functions $u$ and $v$ for the validity of the multidimensional weighted inequality

$${{\left( \int_{E}{\Phi {{\left( {{T}_{k}}f\left( x \right) \right)}^{q}}u\left( x \right)dx} \right)}^{1/q}}\,\le \,C{{\left( \int_{E}{\Phi {{\left( f\left( x \right) \right)}^{p}}v\left( x \right)dx} \right)}^{1/p}},$$

where $0\,<\,p,\,q\,<\,\infty $, $\Phi $ is a logarithmically convex function, and ${{T}_{k}}$ is an integral operator over star-shaped regions. The condition is also necessary for the exponential integral inequality. Moreover, the estimation of $C$ is given and we apply the obtained results to generalize some multidimensional Levin–Cochran-Lee type inequalities.

In this paper, we apply the technique of decoupling to obtain some exponential inequalities for semi-bounded martingale, which extend the results of de la Peña, Ann. probab.27 (1999) 537–564.

We establish new exponential inequalities for partial sums of random fields. Next, using classicalchaining arguments, we give sufficient conditions for partial sum processes indexed by large classes ofsets to converge to a set-indexed Brownian motion. For stationary fields of bounded random variables, thecondition is expressed in terms of a series of conditional expectations. For non-uniform ϕ-mixingrandom fields, we require both finite fourth moments and an algebraic decay of the mixing coefficients.