Sharp estimate of norms in \(L_ p\) on certain classes of conjugate functions (Q2639289)

Given the modulus of continuity \(\omega\), the author considers the class \(W^ rH_{\omega}\) of \(2\pi\)-periodic r-smooth functions f whose modulus of continuity satisfies \(\omega (f^{(r)},t)\leq \omega (t),\) \(\tilde W^ rH_{\omega}\) denotes the class of trigonometrically conjugate functions. In terms of \[ {\tilde \Omega}_ r(\omega;t)=^{def}\sup_{\tilde f\in \tilde W^ rH_{\omega}}\max_{x}| \tilde f(x+t)-\tilde f(x)|, \] the author gives exact estimates for \(L^ p\)-norms of functions from \(\tilde W^ rH_{\omega}\). (The explicit expression for \({\tilde \Omega}{}_ r(\omega;t)\) was obtained in previous articles.)