Inequalities for $\Lambda$-derivatives of functions defined on a metric space and some of their applications

We introduce a concept of a $\Lambda$-derivative operator, which is a certain generalization of hypersingular integral operators, which in turn are used in the definitions of the Marchaud and the Riesz fractional derivatives. We study the problem of sharp Kolmogorov-type inequalities in spaces of continuous bounded functions equipped with a seminorm defined by a certain modulus of continuity for such kind of operators. We also consider an application of the obtained Kolmogorov-type inequality to the Stechkin problem about the best approximation of an operator by bounded ones.