On the domains of first order differential operators on the Sierpiński gasket

Abstract: We study the first order structure of $L^2$-differential one-forms on the Sierpi\'{n}ski gasket. We consider piecewise energy finite functions related to one-forms, normal parts, and show self-similarity properties of one-forms as in the case of energy finite functions on the Sierpi\'{n}ski gasket. We introduce first order differential operators, taking functions into functions, that can be understood as total derivatives with respect to some reference function or form. The main result is a pointwise representation result for certain elements in the domain of said first order differential operator by a ratio of normal derivatives. At last we provide three classes of examples in the domain of said first order differential operator with different types of discontinuities.