On Operators associated to the Cauchy-Riemann operator in elliptic complex numbers

The Cauchy-Kovaleskaya Theorem provides sufficient conditions for an elliptic linear equation on the plane with evolution in time to have solutions with prescribed initial value functions. That these conditions cannot be freely relaxed comes by the celebrated Lewy's example of a system with no solutions. In [3] the technique of associated operators is used to establish conditions for solvability provided the initial pair is holomorphic. This result is further generalized to the case when the initial pair is holomorphic in elliptic complex numbers in [1]. In this talk we will discuss some key aspects of this latter result and how can these be used as a tool to generalize results valid for ordinary holomorphic functions.

[1] Alayon-Solarz D., Vanegas C.J., "Operators Associated to the Cauchy-Riemann Operator in Elliptic Complex Numbers" Advances in Applied Clifford Algebras, DOI: 10.1007/s00006-011-0306-4, 2011.

[2] Lewy, H., "An example of a smooth linear partial differential equation without solution", Annals of Mathematics 66 (1): 155-158, doi:10.2307/1970121,1957.

[3] Son L. H. and Tutschke W., "First Order differential operators associated to the Cauchy-Riemann equations in the plane", Complex Variables and Elliptic Equations, Vol. 48, No. 9, pp 797-801, 2003.

This is a joint work with C.J. Vanegas