On the Π-operator in Clifford analysis

The Π‐operator on some conformally flat manifolds and hyperbolic half space

“…As special cases of this general Hilbert space approach, one has the L 2 isometry of the usual Π‐operator in one complex variable and the Π‐operator in described in other studies 3,4,6 and elsewhere. In the remaining sections of this paper, we show how this Hilbert space technique can be readily applied to a number of basic examples, and the operator D is invertible over the Hilbert spaces that we use.…”

“…Kähler studied Beltrami equations in the case of quaternions in his study, 5 which gave an overview of possible generalizations of complex Beltrami equation and their properties in the quaternionic case. In their study, 6 Blaya et al studied the Π‐operator in Clifford analysis by using two orthogonal bases of a Euclidean space, which allows to find the expression of the jump of the generalized Π‐operator across the boundary of the domain. The case of the Π‐operator and the Beltrami equation on the unit sphere has also been discussed in Cheng et al 7 with most useful properties inherited from the complex Π‐operator.…”

∂¯‐problem for a second‐order elliptic system in Clifford analysis

“…Basic properties of structural sets can be found in earlier studies [15][16][17][18] and the references therein. The flexibility introduced by structural sets ensembles makes it possible to seek new perspectives in investigations concerning the mapping properties of multidimensional Ahlfors-Beurling transforms and additive decompositions of harmonic functions [16,[19][20][21][22][23][24][25][26]. This permits us to consider a much wider class of second-order partial differential equation from two given structural sets 𝜈 and 𝜗, that is, the general sandwich equation…”

On a Generalized Lamé-Navier system in $\mathbb{R}^3$

“…In the state of equilibrium the three-dimensional displacement vector u should satisfy the Lamé-Navier system µ△ u + (µ + λ)grad(div u) = 0, (1) at any point within a homogeneous isotropic linear elastic body without volume forces. The quantities µ > 0 and λ > − 2 3 µ are called the Lamé constants.…”

“…As pointed out in [11], the class of ψ-hyperholomorphic functions is wider than the one we get by rotations from the class of monogenic functions. The flexibility introduced by the structural sets allows us to look for new perspectives in several lines of research concerning the mapping properties of a related Π-operator, geometric conformal mappings and additive decompositions of harmonic functions [1,2,5,7,8,10,12,16].…”