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Fundamental Solutions of the Logarithmic Laplacian: An Approach via the Division Problem
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abstract
Existence of the fundamental solution of the logarithmic Laplacian (in dimensions $d \geq 3$) was established by Huyuan Chen and Laurent V\'eron (2024). In this note, we present an alternative approach, based on a modification on the classical division problem. This is inspired by the theory of fundamental solutions by Malgrange and Ehrenpreis. Moreover, we give a variant of the Liouville theorem for the logarithmic Laplacian and give some further clarification regarding a conjecture posed by Chen and V\'eron regarding the behavior of solutions in dimensions 1 and 2.
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Extension theorems for logarithmic Schr\"odinger and discrete Laplacian operators
Logarithmic operators log L_V and log(−Δ_d) are realized as boundary values of solutions to suitable extension problems, in a more involved way than the fractional Laplacian case.
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