This monograph covers the existing results regarding Green's functions for differential
equations with involutions (DEI).The first part of the book is devoted to the study of the most
useful aspects of involutions from an analytical point of view and the associated algebras of
differential operators. The work combines the state of the art regarding the existence and
uniqueness results for DEI and new theorems describing how to obtain Green's functions proving
that the theory can be extended to operators (not necessarily involutions) of a similar nature
such as the Hilbert transform or projections due to their analogous algebraic properties.
Obtaining a Green's function for these operators leads to new results on the qualitative
properties of the solutions in particular maximum and antimaximum principles.
