Controllability for a class of semilinear fractional evolution systems via resolvent operators
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January 2019, 18(1): 435-453. doi: 10.3934/cpaa.2019022
A note on commutators of the fractional sub-Laplacian on Carnot groups
Ali Maalaoui ,
Department of mathematics and natural sciences, American University of Ras Al Khaimah, PO Box 10021, Ras Al Khaimah, UAE
The author is supported by the Seed Grant of AURAK, No.: AAS/001/18, Critical Problems in the Sub-Elliptic Setting
Received February 2018 Revised April 2018 Published August 2018
In this manuscript, we provide a point-wise estimate for the 3-commutators involving fractional powers of the sub-Laplacian on Carnot groups of homogeneous dimension $Q$. This can be seen as a fractional Leibniz rule in the sub-elliptic setting. As a corollary of the point-wise estimate, we provide an $(L^{p}, L^{q})\to L^{r}$ estimate for the commutator, provided that $\frac{1}{r} = \frac{1}{p}+\frac{1}{q}-\frac{α}{Q}$ for $α ∈ (0, Q)$.
Keywords: Fractional sub-Laplacian, commutator, Carnot groups.
Mathematics Subject Classification: Primary: 35R11, 35R03; Secondary: 35H20.
Citation: Ali Maalaoui. A note on commutators of the fractional sub-Laplacian on Carnot groups. Communications on Pure & Applied Analysis, 2019, 18 (1) : 435-453. doi: 10.3934/cpaa.2019022
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Ali Maalaoui