The spectral theory on the S-spectrum was introduced to give an appropriate mathematical setting to quaternionic quantum mechanics, but it was soon realized that there were different applications of this theory, for example, to fractional heat diffusion and to the spectral theory for the Dirac operator on manifolds. In this seminal paper we introduce the harmonic functional calculus based on the S-spectrum and on an integral representation of axially harmonic functions. This calculus can be seen as a bridge between harmonic analysis and the spectral theory. The resolvent operator of the harmonic functional calculus is the commutative version of the pseudo S-resolvent operator. This new calculus also appears, in a natural way, in the product rule for the F-functional calculus.

On a smooth domain Ω ⊂⊂ C n , we consider the Dirichlet problem for the complex Monge-Ampère equation ((dd c u) n = f dV, u| bΩ ≡ ϕ). We state the Hölder regularity of the solution u when the boundary value ϕ is Hölder continuous and the density f is only

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Let eℓ, for ℓ = 1,2,3, be orthogonal unit vectors in R3 and let normalΩ⊂R3 be a bounded open set with smooth boundary ∂Ω. Denoting by x_ a point in Ω, the heat equation, for nonhomogeneous materials, is obtained replacing the Fourier law, given by the following: T=a(xtrue_)∂xe1+b(xtrue_)∂ye2+c(xtrue_)∂ze3, into the conservation of energy law, here a, b, c:normalΩ→double-struckR are given functions. With the S‐spectrum approach to fractional diffusion processes we determine, in a suitable way, the fractional powers of T. Then, roughly speaking, we replace the fractional powers of T into the conservation of energy law to obtain the fractional evolution equation. This method is important for nonhomogeneous materials where the Fourier law is not simply the negative gradient. In this paper, we determine under which conditions on the coefficients a, b, c:normalΩ→double-struckR the fractional powers of T exist in the sense of the S‐spectrum approach. More in general, this theory allows to compute the fractional powers of vector operators that arise in different fields of science and technology. This paper is devoted to researchers working in fractional diffusion and fractional evolution problems, partial differential equations, and noncommutative operator theory.

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