Convolutional Approach to Fractional Calculus for Distributions of Several Variables

Abstract

Abstract We extend the semigroup {Φλ}λ∈ℂ of Gel’fand and Shilov to the semigroup {Φλ}λ∈ℂ n and the one-dimensional fractional calculus of distributions to the case of distributions on Rn with supports in the nonnegative cone ℝ ¯ + n $\overline{\mathbb{R}}_{+}^{n}$ . We use distributional mixed derivatives of an arbitrary complex order to solve Abel’s multidimensional integral equation and a distributional fractional differential equation of wave type.