Modified Dyadic Integral and Fractional Derivative on $\mathbb R_+$

For functions from the Lebesgue space $L(\mathbb R_+)$, we introduce the modified strong dyadic integral $J_\alpha$ and the fractional derivative $D^{(\alpha)}$ of order $\alpha>0$. We establish criteria for their existence for a given function $f\in L(\mathbb R_+)$. We find a countable set of eigenfunctions of the operators $D^{(\alpha)}$ and $J_\alpha$, $\alpha>0$. We also prove the relations $D^{(\alpha)}(J_\alpha(f))=f$ and $J_\alpha(D^{(\alpha)}(f))=f$ under the condition that $\int_{\mathbb R_+}f(x)\,dx=0$. We show the unboundedness of the linear operator $J_\alpha\colon L_{J_\alpha}\to L(\mathbb R_+)$, where $L_{J_\alpha}$ is its natural domain of definition. A similar assertion is proved for the operator $D^{(\alpha)}\colon L_{D^{(\alpha)}}\to L(\mathbb R_+)$. Moreover, for a function $f\in L(\mathbb R_+)$ and a given point $x\in\mathbb R_+$, we introduce the modified dyadic derivative $d^{(\alpha)}(f)(x)$ and the modified dyadic integral $j_\alpha(f)(x)$. We prove the relations$d^{(\alpha)}(J_\alpha(f))(x)=f(x)$ and $j_\alpha(D^{(\alpha)}(f))=f(x)$ at each dyadic Lebesgue point of the function $f$.