Multidimensional Exponential Inequalities with Weights

Abstract

We establish sufficient conditions on the weight functions u and v for the validity of the multidimensional weighted inequalitywhere 0 < p, q < ∞, Φ is a logarithmically convex function, and Tk 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.

On the equivalence of weighted inequalities for a class of operators

“…Some results of (1.1) for the multidimensional case can be found in [1,5,7,17]. In particular, in [7, theorem 3.1] it is proved that A pq δ (u) < ∞ for all δ > 1 is a necessary and sufficient condition for (1.1) to hold with 0 < p, q < ∞ and T = G φ , where φ satisfies conditions (K 1 )-(K 3 ) defined in the next section.…”