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## Membership of distributions of polynomials in the Nikolskii–Besov class

The main result of this paper asserts that the distribution density of any non-constant polynomial *f*(ξ1,ξ2,...) of degree *d* in independent standard Gaussian random variables ξ1(possibly, in infinitely many variables) always belongs to the Nikol’skii–Besov space *B*1/*d* (R1) of fractional order 1/*d* (see the definition below) depending only on the degree of the polynomial. A natural analog of this assertion is obtained for the density of the joint distribution of *k*polynomials of degree *d*, also with a fractional order that is independent of the number of variables, but depends only on the degree *d* and the number of polynomials. We also give a new simple sufficient condition for a measure on R*k* to possess a density in the Nikol’skii–Besov class *B*α(R)*k*. This result is applied for obtaining an upper bound on the total variation distance between two probability measures on R*k* via the Kantorovich distance between them and a certain Nikol’skii–Besov norm of their difference. Applications are given to estimates of distributions of polynomials in Gaussian random variables.