An estimate for an integral modulus of smoothness (Q1311157)

In this short note, the author proves an equivalence theorem for moduli of smoothness of \(2\pi\)-periodic \(L_p\)-integrable functions \(f\). Let \(\Delta_h f(x)= f(x+ h)- f(x)\), \(0< p< \infty\) and \(\delta> 0\). Then there exists a positive constant \(C_p\) only depending on \(p\) such that \[ \sup_{0\leq h\leq \delta} |\Delta_h f|_p\leq C_p \Biggl({1\over \delta} \int^\delta_0 |\Delta_h f|^p_p dh\Biggr)^{{1\over p}}. \]