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.