On the Possibility of Mikusinski’s Operational Calculus for the Fractional Laplacian

Abstract

We try to adapt the framework of Mikusi ´nski’s operational calculus in order to apply it to differential equations involving the fractional Laplacian.We find a fundamental theorem of calculus which connects the fractional Laplacian with its inverse in general, even when no decay conditions are imposed. Ultimately, we discover that no suitable function space exists satisfying all the required conditions, and so the method of Mikusinski, in the sense that it has previously been used for fractional operators, cannot be applied to the fractional Laplacian.

Keywords: fractional Laplacian; Mikusinski’s operational calculus; fundamental theorem of fractional calculus.
2010 Mathematics Subject Classification. 44A40; 26A33