January  2019, 18(1): 435-453. doi: 10.3934/cpaa.2019022

A note on commutators of the fractional sub-Laplacian on Carnot groups

Department of mathematics and natural sciences, American University of Ras Al Khaimah, PO Box 10021, Ras Al Khaimah, UAE

* Corresponding author

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 $α ∈ (0, Q)$.

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
References:
[1]

A. Bonfiglioli, A. Lanconelli and F. Uguzzoni, Stratified Lie Groups and Potential Theory for Their Sub-Laplacians, Springer Monographs in Mathematics. Springer, Berlin, 2007. Google Scholar

[2]

G. Citti, Semilinear Dirichlet problem involving critical exponent for the Kohn Laplacian, Annali di Matematica pura ed applicata (IV), CLXIX (1995), 375-392. doi: 10.1007/BF01759361. Google Scholar

[3]

S. Chanillo, A note on commutators, Indiana Univ. Math. J., 31 (1982), 7-16. doi: 10.1512/iumj.1982.31.31002. Google Scholar

[4]

F. Ferrari and B. Franchi, Harnack inequality for fractional sub-Laplacians in Carnot groups, Math. Z., 279 (2015), 435-458. doi: 10.1007/s00209-014-1376-5. Google Scholar

[5]

F. Da Lio and T. Riviére, Sub-criticality of non-local Schrödinger systems with antisymmetric potentials and applications to half-harmonic maps, Advances in Mathematics, 3 (2011), 1300-1348. doi: 10.1016/j.aim.2011.03.011. Google Scholar

[6]

F. Da Lio and T. Riviére, Three-term commutator estimates and the regularity of 1/2-harmonic maps into spheres, Analysis and PDE, 1 (2011), 149-190. doi: 10.2140/apde.2011.4.149. Google Scholar

[7]

Y. Fang and M. del Mar González, Asymptotic behaviour of Palais-Smale sequences associated with fractional Yamabe type equations, Pacific Journal of Mathematics, 278 (2015), 369-405. doi: 10.2140/pjm.2015.278.369. Google Scholar

[8]

G. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat., 13 (1975), 161-207. doi: 10.1007/BF02386204. Google Scholar

[9]

G. Folland, A fundamental solution for a subelliptic operator, Bull. Amer. Math. Soc., 79 (1973), 373-376. doi: 10.1090/S0002-9904-1973-13171-4. Google Scholar

[10]

G. Folland and E. Stein, Hardy Spaces on Homogeneous Groups, Mathematical Notes, 28. Princeton University Press, N. J.; University of Tokyo Press, Tokyo, 1982. Google Scholar

[11]

R. L. FrankM. del Mar GonzálezD. D. Monticelli and J. Tan, An extension problem for the CR fractional Laplacian, Adv. Math., 270 (2015), 97-137. doi: 10.1016/j.aim.2014.09.026. Google Scholar

[12]

A. Gover and C. R. Graham, CR invariant powers of the sub-Laplacian, J. Reine Angew. Math., 583 (2005), 1-27. doi: 10.1515/crll.2005.2005.583.1. Google Scholar

[13]

C. R. GrahamR. JenneL. J. Mason and G. J. Sparling, Conformally invariant powers of the Laplacian I. Existence, J. Lond. Math. Soc., 46 (1992), 557-565. doi: 10.1112/jlms/s2-46.3.557. Google Scholar

[14]

C. E. KenigG. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46 (1993), 527-620. doi: 10.1002/cpa.3160460405. Google Scholar

[15]

E. Lenzmann and A. Schikorra, Sharp commutator estimates via harmonic extensions, preprint arXiv: math/1609.08547.Google Scholar

[16]

L. Roncala and S. Thangavelub, Hardy's inequality for fractional powers of the sublaplacian on the Heisenberg group, Advances in Mathematics, 302 (2016), 106-158. doi: 10.1016/j.aim.2016.07.010. Google Scholar

[17]

A. Schikorra, $\varepsilon$-regularity for systems involving non-local, antisymmetric operators, Calc. Var. Partial Differential Equations, 54 (2015), 3531. doi: 10.1007/s00526-015-0913-3. Google Scholar

[18]

S. Thangavelu, Harmonic Analysis on the Heisenberg Group, Progress in Mathematics 159. Birkhauser, Boston, MA, 1998. doi: 10.1007/978-1-4612-1772-5. Google Scholar

show all references

References:
[1]

A. Bonfiglioli, A. Lanconelli and F. Uguzzoni, Stratified Lie Groups and Potential Theory for Their Sub-Laplacians, Springer Monographs in Mathematics. Springer, Berlin, 2007. Google Scholar

[2]

G. Citti, Semilinear Dirichlet problem involving critical exponent for the Kohn Laplacian, Annali di Matematica pura ed applicata (IV), CLXIX (1995), 375-392. doi: 10.1007/BF01759361. Google Scholar

[3]

S. Chanillo, A note on commutators, Indiana Univ. Math. J., 31 (1982), 7-16. doi: 10.1512/iumj.1982.31.31002. Google Scholar

[4]

F. Ferrari and B. Franchi, Harnack inequality for fractional sub-Laplacians in Carnot groups, Math. Z., 279 (2015), 435-458. doi: 10.1007/s00209-014-1376-5. Google Scholar

[5]

F. Da Lio and T. Riviére, Sub-criticality of non-local Schrödinger systems with antisymmetric potentials and applications to half-harmonic maps, Advances in Mathematics, 3 (2011), 1300-1348. doi: 10.1016/j.aim.2011.03.011. Google Scholar

[6]

F. Da Lio and T. Riviére, Three-term commutator estimates and the regularity of 1/2-harmonic maps into spheres, Analysis and PDE, 1 (2011), 149-190. doi: 10.2140/apde.2011.4.149. Google Scholar

[7]

Y. Fang and M. del Mar González, Asymptotic behaviour of Palais-Smale sequences associated with fractional Yamabe type equations, Pacific Journal of Mathematics, 278 (2015), 369-405. doi: 10.2140/pjm.2015.278.369. Google Scholar

[8]

G. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat., 13 (1975), 161-207. doi: 10.1007/BF02386204. Google Scholar

[9]

G. Folland, A fundamental solution for a subelliptic operator, Bull. Amer. Math. Soc., 79 (1973), 373-376. doi: 10.1090/S0002-9904-1973-13171-4. Google Scholar

[10]

G. Folland and E. Stein, Hardy Spaces on Homogeneous Groups, Mathematical Notes, 28. Princeton University Press, N. J.; University of Tokyo Press, Tokyo, 1982. Google Scholar

[11]

R. L. FrankM. del Mar GonzálezD. D. Monticelli and J. Tan, An extension problem for the CR fractional Laplacian, Adv. Math., 270 (2015), 97-137. doi: 10.1016/j.aim.2014.09.026. Google Scholar

[12]

A. Gover and C. R. Graham, CR invariant powers of the sub-Laplacian, J. Reine Angew. Math., 583 (2005), 1-27. doi: 10.1515/crll.2005.2005.583.1. Google Scholar

[13]

C. R. GrahamR. JenneL. J. Mason and G. J. Sparling, Conformally invariant powers of the Laplacian I. Existence, J. Lond. Math. Soc., 46 (1992), 557-565. doi: 10.1112/jlms/s2-46.3.557. Google Scholar

[14]

C. E. KenigG. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46 (1993), 527-620. doi: 10.1002/cpa.3160460405. Google Scholar

[15]

E. Lenzmann and A. Schikorra, Sharp commutator estimates via harmonic extensions, preprint arXiv: math/1609.08547.Google Scholar

[16]

L. Roncala and S. Thangavelub, Hardy's inequality for fractional powers of the sublaplacian on the Heisenberg group, Advances in Mathematics, 302 (2016), 106-158. doi: 10.1016/j.aim.2016.07.010. Google Scholar

[17]

A. Schikorra, $\varepsilon$-regularity for systems involving non-local, antisymmetric operators, Calc. Var. Partial Differential Equations, 54 (2015), 3531. doi: 10.1007/s00526-015-0913-3. Google Scholar

[18]

S. Thangavelu, Harmonic Analysis on the Heisenberg Group, Progress in Mathematics 159. Birkhauser, Boston, MA, 1998. doi: 10.1007/978-1-4612-1772-5. Google Scholar

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