A new metric characteristic 0(5) of rectifiable Jordan curves is introduced. We will find an estimate of the type of the Zygmund estimate for an arbitrary rectifiable closed Jordan curve in its terms. It is shown that the Plemel'-Privalov theorem on the invariance of HSlder's spaces is true for the class of curves satisfying the condition 0(6)~ 6, which is much wider than the class of piecewise smooth curves (the presence of cusps is admissible). The Bari-Stechkin theorem on the necessary conditions of action of a singular operator in the spaces Hr is generalized. It is shown that this theorem is valid for every curve which has a continuous tangent at least at one point and 0(6)~ 5.1. Let ~/be a rectifiable closed Jordan curve, t = t(s), 0 -< s _< l, be the equation of the curve in arcual coordinates.Let us consider the singular integral t, ~ (~)t d~If 7 is the unit circle, then the singular integral~(t) reduces to the singular integralFor the case of a continuous density U(~) in the result of the investigations by A. Zygmund, N. K. Bari, and S. B. Stechkin the specific problems in which direct and inverse estimates of the singular integral U are found in terms of the moduli of continuity of various orders by means of the same characteristic of density U are complicated.A direct estimate of the operator U for the modulus of continuity of the first order has been obtained in [1]. The Plemel'-Privalov theorem on the behavior of a singular integral in the HSlder scale of the spaces H a precedes this estimate.The following inequality holds for k-curves, i.e., for curves satisfying the condition: (~ k ~ i) s (t, t) ~ k I t --t i, where s(t, ~') is not greater than the lengths of the arcs subtended by the points t,T (7 (see Azerbaijan State University.

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