A modified strong dyadic integral and derivative

For a function $f\in L(\mathbb R_+)$ its modified strong dyadic integral $J(f)$ and the modified strong dyadic derivative $D(f)$ are defined. A criterion for the existence of a modified strong dyadic integral for an integrable function is proved, and the equalities $J(D(f))=f$ and $D(J(f))=f$ are established under the assumption that $\displaystyle\int_{\mathbb R_+}f(x)\,dx=0$. A countable system of eigenfunctions of the operators $D$ and $J$ is found. The linear span $L$ of this set is shown to be dense in the dyadic Hardy space $H(\mathbb R_+)$, and the linear operator $\widetilde J\colon L\to L(\mathbb R_+)$, $\widetilde J(f)=J(f)^\sim$, is proved to be bounded. Hence this operator can be uniquely continuously extended to $H(\mathbb R_+)$ and the resulting linear operator $\widetilde J\colon H(\mathbb R_+)\to L(\mathbb R_+)$ is bounded.