Quantitative Korovkin theorems for sublinear, monotone and strongly translatable operators in $L^p([0,1]),1\le p\le +\mathrm{\infty }$

By extending the classical quantitative approximation results for positive and linear operators in $L^p([0,1]),1\le p\le +\mathrm{\infty }$ of Berens and DeVore in 1978 and of Swetits and Wood in 1983 to the more general case of sublinear, monotone and strongly translatable operators, in this paper we obtain quantitative estimates in terms of the second order and third order moduli of smoothness, in Korovkin type theorems. Applications to concrete examples are included and an open question concerning interpolation theory for sublinear, monotone and strongly translatable operators is raised.