Abstract. We investigate fractional powers of the generalized gradient operator
with non-constant coefficients, formulated in the Clifford algebra setting.
The vector nature of this operator makes classical complex spectral theory
inadequate, and the analysis must instead be carried out within the
S-spectrum framework.
A fundamental obstruction arises from the fact that fractional powers
are not defined on the negative real line, which seriously complicates
their construction in this setting.
Extending earlier work on bisectorial vector operators, we show that previously
established weak solutions are in fact strong solutions, and we prove
injectivity of the gradient operator.
We then introduce a novel approach that rigorously circumvents the
negative real line obstruction, yielding a precise definition of the
fractional powers together with their main operator-theoretic properties.
As an application, we derive a non-local Fourier diffusion law governed
by these fractional operators.

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This initiative is part of the "PhD Lectures" activity of the project
"Departments of Excellence 2023-2027" of the Department of Mathematics of Politecnico di Milano.
This activity consists of seminars open to PhD students, followed by meetings with the speaker to discuss and go into detail on the topics presented at the talk.