The integral modulus of continuity of the Dirichlet kernel and the conjugate Dirichlet kernel

Asymptotic estimates for the integral modulus of continuity of order s of the Dirichlet kernel and the conjugate Dirichlet kernel are obtained. For example, if k/2, then _s (D _k ,)=2 ^s +1/²sin ^s k/2 log(1+k/s)+O(2 ^s sin ^s k/2)holds uniformly with respect to all the parameters.Translated from Matematicheskie Zametki, Vol. 54, No. 3, pp. 98–105, September, 1993.