Modified Dyadic Integral and Fractional Derivative on ℝ+

For functions from the Lebesgue space L(ℝ+), we introduce the modified strong dyadic integral Jα and the fractional derivative D(α) of order α > 0. We establish criteria for their existence for a given function f ∈ L(ℝ+). We find a countable set of eigenfunctions of the operators D(α) and Jα, α > 0. We also prove the relations D(α)(Jα(f)) = f and Jα(D(α)(f)) = f under the condition that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\smallint _{\mathbb{R}_ + } f(x)dx = 0$$ \end{document}. We show the unboundedness of the linear operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$J_\alpha :L_{J_{_\alpha } } \to L(\mathbb{R}_ + )$$ \end{document}, where LJα is its natural domain of definition. A similar assertion is proved for the operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$D^{(\alpha )} :L_{D^{(\alpha )} } \to L(\mathbb{R}_ + )$$ \end{document}. Moreover, for a function f ∈ L(ℝ+) and a given point x ∈ ℝ+, we introduce the modified dyadic derivative d(α)(f)(x) and the modified dyadic integral jα(f)(x). We prove the relations d(α)(Jα(f))(x) = f(x) and jα(D(α)(f)) = f(x) at each dyadic Lebesgue point of the function f.