Another Esseen-type inequality for multivariate probability density functions (Q1871328)

An upper bound for the supremum of the absolute value of the difference of two multivariate probability density functions is obtained. The upper bound involves integrals of the absolute value of the difference of the corresponding characteristic functions. Results are similar to the work of \textit{N. G. Gamkrelidze} [Theory Probab. Appl. 22(1977), 877-880 (1978); translation from Teor. Veroyatn. Primen. 22, 897-900 (1977; Zbl 0395.60020)] on the Esseen's inequality for multidimensional distribution functions.