2023/05/11 by De Martino, Antonino, Diki, Kamal, Adán, Ali Guzmán · 1 citation
#Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.2305.06998
The Fueter-Sce theorem provides a procedure to obtain axially monogenic functions, which are in the kernel of generalized Cauchy-Riemann operator in ℝn+1. This result is obtained by using two operators. The first one is the slice operator, which extends holomorphic functions of one complex variable to slice monogenic functions in ℝn+1. The second one is a suitable power of the Laplace operator in n+1 variables. Another way to get axially monogenic functions is the generalized Cauchy-Kovalevskaya (CK) extension. This characterizes axial monogenic functions by their restriction to the real line. In this paper, using the connection between the Fueter-Sce map and the generalized CK-extension, we explicitly compute the actions Δℝn+1(n-1)/(2) xk, where x ∈ ℝn+1. The expressions obtained is related to a well-known class of Clifford-Appell polynomials. These are the building blocks to write a Taylor series for axially monogenic functions. Moreover, we focus on some elementary axially monogenic functions, where the action of the Fueter-Sce map and the generalized CK-extension coincide. In order to get algebraic relations between the elementary functions, as in complex analysis, we define a new product between axially monogenic functions. By using the connections between the Fueter-Sce map and the generalized CK extension we characterize the range and the kernel of the Fueter-Sce map. Furthermore, we focus on studying the Clifford-Appell-Fock space and the Clifford-Appell-Hardy space. Finally, using the polyanalytic Fueter-Sce theorems we obtain a new family of polyanalytic monogenic polynomials, which extends to higher dimensions the Clifford-Appell polynomials.