The modulus of smoothness
\omega (f, \delta)_{\varphi, p} = \mathrm {sup}_{0 < h ≤ \delta} \| \Delta^2_{h, \varphi} \|_{L^p}
has arisen during the investigation of positive operators of the Kantorovich type. Here we show that
\omega_{\varphi, p}
resembles the ordinary case
\varphi = 1
and we give the characterization of those functions
f
for which
\omega (f, \delta)_{\varphi, p} = O (\delta^2)
. The results obtained have applications to positive operators.