On some properties of fractional dyadic derivative and integral

Summary

For the fractional dyadic derivative and integral, the following analogues of two theorems of Lebesgue  are proved: the theorem on differentiation of the indefinite Lebesgue integral of  an integrable function at its Lebesgue points, and the theorem on reconstruction of an absolutely  continuous function by means of its derivative. Dyadic fractional analogues of the formula of integration by  parts are also obtained. In addition, some theorems are proved on dyadic fractional differentiation and  integration of a Lebesgue integral depending on a parameter.  Most of the results are new even for dyadic derivatives and integrals of natural order.