On Modified Strong Dyadic Integral and Derivative

Abstract

For functions f ∈ L(R +), we define a modified strong dyadic integral J(f) ∈ L(R +) and a modified strong dyadic derivative D(f) ∈ L(R +). We establish a necessary and sufficient condition for the existence of the modified strong dyadic integral J(f). Under the condition $$\smallint _{R_ + }$$ f(x)dx = 0, we prove the equalities J(D(f)) = f and D(J(f)) = f. We find a countable set of eigenfunctions of the operators J and D. We prove that the linear span L of this set is dense in the dyadic Hardy space H(R +). For the functions f ∈ H(R +), we define a modified uniform dyadic integral J(f) ∈ L ∞(R +).