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Extension problem of fractional laplacian and Fourier transform of $L^1_\text{loc}$ function?

I have understand the proof oh the lemma 4.1.9 "SOME NONLOCAL OPERATORS AND EFFECTS DUE TO NONLOCALITY" by C.Bocur, there is a link. In this paper, we define, for $u\in L^1_\text{loc}(\...