Geodesic
The differentiation operator on subsets of the field of $p$-adic numbers
Izvestiya. Mathematics , Tome 41 (1993) no. 2, pp. 289-305.

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A study is made of the spectral properties of the operator $D^\alpha$ (the $p$-adic analogue of the operator of fractional differentiation) in the space $L_2(G)$, where $G\subset Q_p$ is an open subset. 
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A. N. Kochubei. The differentiation operator on subsets of the field of $p$-adic numbers. Izvestiya. Mathematics , Tome 41 (1993) no. 2, pp. 289-305. http://geodesic.mathdoc.fr/item/IM2_1993_41_2_a5/

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