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In this note we introduce some nonlinear extremal nonlocal operators thatapproximate the, so called, truncated Laplacians. For these operators weconstruct representation formulas that lead to the construction of what, withan 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 thetruncated Laplacians, on the other, we introduce nonlocal operators whosenonlocality is on one dimensional lines, and this dramatically changes theprospective, as is quite clear from the results obtained that often differssignificantly with the local case or with the case where the nonlocality isdiffused. Surprisingly this is true also for operators that approximate theLaplacian.

Fractional truncated Laplacians: representation formula, fundamental solutions and applications