We find two-sided estimates for the best uniform approximations of classes convolutions 2????-periodic functions from a unit ball space Lp,1 ? p < ?; with fixed kernels such that moduli their Fourier coefficients satisfy condition $$ \sum \limits_{k=n+1}^{\infty}\psi (k)<\psi (n) : In case (k)=o(1)\psi ; obtained become asymptotic equalities.