On local properties of certain functions from the Hölder class (Q1317586)

Assume \(f\) is a real continuous function on an interval \(J\). Denote by \[ \omega (\delta;f) : = \sup \biggl\{ \bigl | f(x_ 1)- f(x_ 2) \bigr | : x_ 1, x_ 2 \in J,\;| x_ 1-x_ 2 | \leq \delta \biggr\},\;\delta \geq 0, \] its modulus of continuity on \(J\). If \(J : = (- \infty, \infty)\), we simply write \(\omega (\delta;f)\). Let \(I\) be a finite segment of \(J\) and \(\Omega_ f (I) : = \max_{x \in I} f(x) - \min_{x \in I} f(x)\). The author proves the following theorem. Let \(T\) be a positive number and \(\omega\) a modulus of continuity satisfying the condition \(\omega (\delta) = \omega (2T)\) for all \(\delta \geq 2T\). Then there exists a real continuous \(2T\)-periodic function \(f\) such that for any segment \(I\) the following estimates are valid: \[ \Omega_ f (I) \geq {1 \over 12} \omega \bigl( | I |; f \bigr) \] and \[ \omega \bigl( | I | \bigr) \geq \omega \bigl( | I |; f \bigr) \geq \Omega_ f (I) \geq {1 \over 48} \omega \bigl( | I | \bigr). \] The author also proves the counterpart of the above theorem in the nonperiodic case.