Modified Dyadic Integral and Fractional Derivative on ℝ₊

Abstract

For functions from the Lebesgue space L(ℝ₊), we introduce the modified strong dyadic integral J _α 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 J _α, α > 0. We also prove the relations D ^(α)(J _α(f)) = f and J _α(D ^(α)(f)) = f under the condition that $\smallint _{\mathbb{R}_ + } f(x)dx = 0$. We show the unboundedness of the linear operator $J_\alpha :L_{J_{_\alpha } } \to L(\mathbb{R}_ + )$, where L _{J α} is its natural domain of definition. A similar assertion is proved for the operator $D^{(\alpha )} :L_{D^{(\alpha )} } \to L(\mathbb{R}_ + )$. 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 j _α(f)(x). We prove the relations d ^(α)(J _α(f))(x) = f(x) and j _α(D ^(α)(f)) = f(x) at each dyadic Lebesgue point of the function f.