Geodesic
The modified multiplicative integral and derivative of arbitrary order on the semiaxis
Izvestiya. Mathematics , Tome 70 (2006) no. 2, pp. 211-231.

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We consider the modified strong dyadic integral and derivative in $L_q({\mathbb R}_+)$, $1\le q\le 2$. We establish conditions for their existence, study how the behaviour of the structural characteristics of a function is related to that of its derivative (integral), and prove an embedding theorem of Hardy–Littlewood–Sobolev type. 
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S. S. Volosivets. The modified multiplicative integral and derivative of arbitrary order on the semiaxis. Izvestiya. Mathematics , Tome 70 (2006) no. 2, pp. 211-231. http://geodesic.mathdoc.fr/item/IM2_2006_70_2_a0/

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