  In this note we introduce some nonlinear extremal nonlocal operators that
approximate the, so called, truncated Laplacians. For these operators we
construct representation formulas that lead to the construction of what, with
an abuse of notation, could be called "fundamental solutions". This, in turn,
leads to Liouville type results. The interest is double: on one hand we wish to
"understand" what is the right way to define the nonlocal version of the
truncated Laplacians, on the other, we introduce nonlocal operators whose
nonlocality is on one dimensional lines, and this dramatically changes the
prospective, as is quite clear from the results obtained that often differs
significantly with the local case or with the case where the nonlocality is
diffused. Surprisingly this is true also for operators that approximate the
Laplacian.
