On some issues of integration in multidimensional spaces

The paper is devoted to founding of various nontrivial estimates of the concentration function. The interest in this function is due to the fact that it is the most important tool for studying the properties of the convolutions of various probability distributions that appear in numerous applications. Some results obtained for this function in the one-dimensional case are generalized to multidimensional spaces in the presented paper. Thus, the well-known result of Enger from [1] is strengthened (see Theorem 2). In addition, it is shown that the estimate in Theorem 2 is unimprovable in the dimension of the space. The proofs of the main results are based on the use of the method of characteristic functions. The main difficulty is connected with the estimates of complex multidimensional integrals.