Integral operators defined “up to a polynomial”

Serena Dipierro, Aleksandr Dzhugan, Enrico Valdinoci

Research output: Contribution to journalArticlepeer-review

Abstract

We introduce a suitable notion of integral operators (comprising the fractional Laplacian as a particular case) acting on functions with minimal requirements at infinity. For these functions, the classical definition would lead to divergent expressions, thus we replace it with an appropriate framework obtained by a cut-off procedure. The notion obtained in this way quotients out the polynomials which produce the divergent pattern once the cut-off is removed. We also present results of stability under the appropriate notion of convergence and compatibility results between polynomials of different orders. Additionally, we address the solvability of the Dirichlet problem. The theory is developed in general in the pointwise sense. A viscosity counterpart is also presented under the additional assumption that the interaction kernel has a sign, in conformity with the maximum principle structure.

Original languageEnglish
Pages (from-to)60-108
Number of pages49
JournalFractional Calculus and Applied Analysis
Volume25
Issue number1
DOIs
Publication statusPublished - Feb 2022

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