Fractional Modified Dyadic Integral and Derivative on $\mathbb{R}_+$

For functions in the Lebesgue space $L(\mathbb{R}_+)$, a modified strong dyadic integral $J_\alpha$ and a modified strong dyadic derivative $D^{(\alpha)}$ of fractional order $\alpha>0$ are introduced. For a given function $f\in L(\mathbb{R}_+)$, criteria for the existence of these integrals and derivatives are obtained. A countable set of eigenfunctions for the operators $J_\alpha$ and $D^{(\alpha)}$ is indicated. The formulas $D^{(\alpha)}(J_\alpha(f))=f$ and $J_\alpha(D^{(\alpha)}(f))=f$ are proved for each $\alpha>0$ under the condition that $\int_{\mathbb{R}_+} f(x)\,dx=0$. We prove that the linear operator $J_\alpha\colon L_{J_\alpha}\to L(\mathbb{R}_+)$ is unbounded, where $L_{J_\alpha}$ is the natural domain of $J_\alpha$. A similar statement for the operator $D^{(\alpha)}\colon L_{D^{(\alpha)}}\to L(\mathbb{R}_+)$ is proved. A modified dyadic derivative $d^{(\alpha)}(f)(x)$ and a modified dyadic integral $j_\alpha(f)(x)$ are also defined for a function $f\in L(\mathbb{R}_+)$ and a given point $x\in\mathbb{R}_+$. The formulas $d^{(\alpha)}(J_\alpha(f))(x)=f(x)$ and $j_\alpha(D^{(\alpha)}(f))=f(x)$ are shown to be valid at each dyadic Lebesgue point $x\in\mathbb{R}_+$ of $f$.