The work is devoted to the study of boundedness properties of integration and differentiation operators of Riemann–Liouville type on the real axis and semi–axis. The operators are acting in smoothness function spaces of Besov type with Muckenhoupt weight functions (weights) of standard and local type. The problem posed is solved by decomposing elements of the function spaces with respect to spline wavelet systems, which are the main solution tools. The article presents in detail a scheme for constructing such systems. In their terms, the corresponding decomposition theorems are established in the paper. The main results of the study are conditions on weights for the fulfilment of inequalities connecting the norms of images and pre–images of Riemann–Liouville integration operators.