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Integral equations on compact CR manifolds
Department of Mathematics, College of Science, China Jiliang University, Hangzhou 310018, China |
$ M $ |
$ \mathcal{Y}(M) $ |
$ \begin{equation*} \Bigl| \int_M\int_M [G_\xi^\theta(\eta)]^{\frac{Q-\alpha}{Q-2}} f(\xi) g(\eta) dV_\theta(\xi) dV_\theta(\eta) \Bigr| \leq \mathcal{Y}_\alpha(M) \|f\|_{L^{\frac{2Q}{Q+\alpha}}(M)} \|g\|_{L^{\frac{2Q}{Q+\alpha}}(M)}, \end{equation*} $ |
$ G_\xi^\theta(\eta) $ |
$ \mathcal{L_\theta} = b_n\Delta_b+R $ |
$ \mathcal{Y}_\alpha(M) $ |
$ \Delta_b $ |
$ R $ |
$ f = g $ |
$ \mathcal{Y}_\alpha(M)\geq \mathcal{Y}_\alpha(\mathbb{S}^{2n+1}) $ |
$ \mathbb{C}^{n+1} $ |
$ \mathcal{Y}_\alpha(M) $ |
$ \mathcal{Y}_\alpha(M)> \mathcal{Y}_\alpha(\mathbb{S}^{2n+1}) $ |
$ \begin{equation} \varphi^{\frac{Q-\alpha}{Q+\alpha}}(\xi) = \int_M [G_\xi^\theta(\eta)]^{\frac{Q-\alpha}{Q-2}}\varphi(\eta)\ dV_\theta, \quad 0<\alpha<Q. ~~~(1) \end{equation} $ |
$ \Gamma^\alpha(M) $ |
$ \alpha = 2 $ |
References:
[1] |
A. Bahri and J.-M. Coron,
On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the togology of the domain, Comm. Pure Appl. Math., 41 (1988), 253-294.
doi: 10.1002/cpa.3160410302. |
[2] |
T. P. Branson, L. Fontana and C. Morpurgo,
Moser-Trudinger and Beckner-Onofri's inequalities on the CR sphere, Annals of Mathematics, 177 (2013), 1-52.
doi: 10.4007/annals.2013.177.1.1. |
[3] |
H. Brezis and L. Nirenberg,
Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477.
doi: 10.1002/cpa.3160360405. |
[4] |
W. Chen, C. Li and B. Ou,
Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.
doi: 10.1002/cpa.20116. |
[5] |
J.-H. Cheng, A. Malchiodi and P. Yang,
A positive mass theorem in three dimensional Cauchy-Riemann geometry, Advances in Mathematics, 308 (2017), 276-347.
doi: 10.1016/j.aim.2016.12.012. |
[6] |
W. S. Cohn and G. Lu,
Sharp constants for Moser-Trudinger inequalities on spheres in complex space $\mathbb{C}^n$, Comm. Pure Appl. Math., 57 (2004), 1458-1493.
doi: 10.1002/cpa.20043. |
[7] |
J. Dou and M. Zhu,
Nonlinear integral equations on bounded domains, J. Funct. Anal., 277 (2019), 111-134.
doi: 10.1016/j.jfa.2018.05.020. |
[8] |
S. Dragomir and G. Tomassini, Differential Geometry and Analysis on CR Manifolds, Progress in Mathematics, 246. Birkhäuser Boston, Inc., Boston, MA, 2006. |
[9] |
G. B. Folland,
A fundamental solution for a subelliptic operator, Bull. Amer. Math. Soc., 79 (1973), 373-376.
doi: 10.1090/S0002-9904-1973-13171-4. |
[10] |
G. B. Folland,
Subelliptic estimates and function spaces on nilpotent Lie groups, Arkiv för Matematik, 13 (1975), 161-207.
doi: 10.1007/BF02386204. |
[11] |
G. B. Folland and E. M. Stein,
Estimates for the $\bar{\partial}_b$ complex and analysis on the Heisenberg group, Comm. Pure Appl. Math., 27 (1974), 429-522.
doi: 10.1002/cpa.3160270403. |
[12] |
R. L. Frank and E. H. Lieb,
Sharp constants in several inequalities on the Heisenberg group, Annals of Mathematics, 176 (2012), 349-381.
doi: 10.4007/annals.2012.176.1.6. |
[13] |
N. Gamara,
The CR Yamabe conjecture the case $n = 1$, J. Eur. Math. Soc. (JEMS), 3 (2001), 105-137.
doi: 10.1007/PL00011303. |
[14] |
N. Gamara and R. Yacoub,
CR Yamabe conjecture — the conformally flat case, Pacific Journal of Mathematics, 201 (2001), 121-175.
doi: 10.2140/pjm.2001.201.121. |
[15] |
M. Gluck and M. Zhu, An extension operator on bounded domains and applications, Calc. Var. PDE, 58 (2019), 27 pp.
doi: 10.1007/s00526-019-1513-4. |
[16] |
Y. Han,
An integral type Brezis-Nirenberg problem on the Heisenberg group, J. Differential Equations, 269 (2020), 4544-4565.
doi: 10.1016/j.jde.2020.03.032. |
[17] |
Y. Han and M. Zhu,
Hardy-Littlewood-Sobolev inequalities on compact Riemannian manifolds and applications, J. Differentical Equations, 260 (2016), 1-25.
doi: 10.1016/j.jde.2015.06.032. |
[18] |
L. Hörmander,
Hypoelliptic second order differential equations, Acta Mathematica, 119 (1967), 147-171.
doi: 10.1007/BF02392081. |
[19] |
D. Jerison and J. M. Lee,
A subelliptic, nonlinear eigenvalue problem and scalar curvature on CR manifolds, Microlocal Analysis, Contemp. Math., Amer. Math. Soc., Providence, RI, 27 (1984), 57-63.
doi: 10.1090/conm/027/741039. |
[20] |
D. Jerison and J. M. Lee,
The Yamabe problem on CR manifolds, J. Differential Geom., 25 (1987), 167-197.
doi: 10.4310/jdg/1214440849. |
[21] |
D. Jerison and J. M. Lee,
Extremals for the Sobolev inequality on the Heisenberg group and the CR Yamabe problem, J. Amer. Math. Soc., 1 (1988), 1-13.
doi: 10.1090/S0894-0347-1988-0924699-9. |
[22] |
D. Jerison and J. M. Lee,
Intrinsic CR normal coordinates and the CR Yamabe problem, J. Differential Geom., 29 (1989), 303-343.
doi: 10.4310/jdg/1214442877. |
[23] |
J. M. Lee,
The Fefferman metric and pseudohermitian invariants, Trans. Amer. Math. Soc., 296 (1986), 411-429.
doi: 10.2307/2000582. |
[24] |
J. M. Lee,
Pseudo-Einstein structres on CR manifolds, Amer. J. Math., 110 (1988), 157-178.
doi: 10.2307/2374543. |
[25] |
J. M. Lee and T. H. Parker,
The Yamabe problem, Bull. Amer. Math. Soc. (N.S.), 17 (1987), 37-91.
doi: 10.1090/S0273-0979-1987-15514-5. |
[26] |
Y. Y. Li,
Remark on some conformally invariant integral equations: The method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180.
|
[27] |
S.-Y. Li, D. N. Son and X. Wang,
A new characterization of the CR sphere and the sharp eigenvalue estimate for the Kohn Laplacian, Advances in Math., 281 (2015), 1285-1305.
doi: 10.1016/j.aim.2015.06.008. |
[28] |
S.-Y. Li and X. Wang,
An Obata-type theorem in CR geometry, J. Diff. Geom., 95 (2013), 483-502.
doi: 10.4310/jdg/1381931736. |
[29] |
Y. Y. Li and M. Zhu,
Sharp Sobolev inequalities involving boundary terms, Geom. Funct. Anal., 8 (1998), 59-87.
doi: 10.1007/s000390050048. |
[30] |
Y. Li and M. Zhu,
Sharp Sobolev trace inequalities on Riemannian manifolds with boundaries, Comm. Pure Appl. Math., 50 (1997), 427-465.
doi: 10.1002/(SICI)1097-0312(199705)50:5<449::AID-CPA2>3.0.CO;2-9. |
[31] |
Y. Li and M. Zhu,
Uniqueness theorems through the method of moving spheres, Duke Math. J., 80 (1995), 383-417.
doi: 10.1215/S0012-7094-95-08016-8. |
[32] |
E. H. Lieb,
Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374.
doi: 10.2307/2007032. |
[33] |
E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, 30. Princeton University Press, Princeton, N.J. 1970. |
[34] |
X. Wang,
Some recent results in CR geometry, Tsinghua lectures in mathematics, Adv. Lect. Math. (ALM), Int. Press, Somerville, MA, 45 (2019), 469-484.
|
[35] |
X. Wang,
On a remarkable formula of Jerison and Lee in CR geometry, Math. Res. Lett., 22 (2015), 279-299.
doi: 10.4310/MRL.2015.v22.n1.a14. |
[36] |
M. Zhu,
Prescribing integral curvature equation, Differential and Integral Equations, 29 (2016), 889-904.
|
show all references
References:
[1] |
A. Bahri and J.-M. Coron,
On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the togology of the domain, Comm. Pure Appl. Math., 41 (1988), 253-294.
doi: 10.1002/cpa.3160410302. |
[2] |
T. P. Branson, L. Fontana and C. Morpurgo,
Moser-Trudinger and Beckner-Onofri's inequalities on the CR sphere, Annals of Mathematics, 177 (2013), 1-52.
doi: 10.4007/annals.2013.177.1.1. |
[3] |
H. Brezis and L. Nirenberg,
Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477.
doi: 10.1002/cpa.3160360405. |
[4] |
W. Chen, C. Li and B. Ou,
Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.
doi: 10.1002/cpa.20116. |
[5] |
J.-H. Cheng, A. Malchiodi and P. Yang,
A positive mass theorem in three dimensional Cauchy-Riemann geometry, Advances in Mathematics, 308 (2017), 276-347.
doi: 10.1016/j.aim.2016.12.012. |
[6] |
W. S. Cohn and G. Lu,
Sharp constants for Moser-Trudinger inequalities on spheres in complex space $\mathbb{C}^n$, Comm. Pure Appl. Math., 57 (2004), 1458-1493.
doi: 10.1002/cpa.20043. |
[7] |
J. Dou and M. Zhu,
Nonlinear integral equations on bounded domains, J. Funct. Anal., 277 (2019), 111-134.
doi: 10.1016/j.jfa.2018.05.020. |
[8] |
S. Dragomir and G. Tomassini, Differential Geometry and Analysis on CR Manifolds, Progress in Mathematics, 246. Birkhäuser Boston, Inc., Boston, MA, 2006. |
[9] |
G. B. Folland,
A fundamental solution for a subelliptic operator, Bull. Amer. Math. Soc., 79 (1973), 373-376.
doi: 10.1090/S0002-9904-1973-13171-4. |
[10] |
G. B. Folland,
Subelliptic estimates and function spaces on nilpotent Lie groups, Arkiv för Matematik, 13 (1975), 161-207.
doi: 10.1007/BF02386204. |
[11] |
G. B. Folland and E. M. Stein,
Estimates for the $\bar{\partial}_b$ complex and analysis on the Heisenberg group, Comm. Pure Appl. Math., 27 (1974), 429-522.
doi: 10.1002/cpa.3160270403. |
[12] |
R. L. Frank and E. H. Lieb,
Sharp constants in several inequalities on the Heisenberg group, Annals of Mathematics, 176 (2012), 349-381.
doi: 10.4007/annals.2012.176.1.6. |
[13] |
N. Gamara,
The CR Yamabe conjecture the case $n = 1$, J. Eur. Math. Soc. (JEMS), 3 (2001), 105-137.
doi: 10.1007/PL00011303. |
[14] |
N. Gamara and R. Yacoub,
CR Yamabe conjecture — the conformally flat case, Pacific Journal of Mathematics, 201 (2001), 121-175.
doi: 10.2140/pjm.2001.201.121. |
[15] |
M. Gluck and M. Zhu, An extension operator on bounded domains and applications, Calc. Var. PDE, 58 (2019), 27 pp.
doi: 10.1007/s00526-019-1513-4. |
[16] |
Y. Han,
An integral type Brezis-Nirenberg problem on the Heisenberg group, J. Differential Equations, 269 (2020), 4544-4565.
doi: 10.1016/j.jde.2020.03.032. |
[17] |
Y. Han and M. Zhu,
Hardy-Littlewood-Sobolev inequalities on compact Riemannian manifolds and applications, J. Differentical Equations, 260 (2016), 1-25.
doi: 10.1016/j.jde.2015.06.032. |
[18] |
L. Hörmander,
Hypoelliptic second order differential equations, Acta Mathematica, 119 (1967), 147-171.
doi: 10.1007/BF02392081. |
[19] |
D. Jerison and J. M. Lee,
A subelliptic, nonlinear eigenvalue problem and scalar curvature on CR manifolds, Microlocal Analysis, Contemp. Math., Amer. Math. Soc., Providence, RI, 27 (1984), 57-63.
doi: 10.1090/conm/027/741039. |
[20] |
D. Jerison and J. M. Lee,
The Yamabe problem on CR manifolds, J. Differential Geom., 25 (1987), 167-197.
doi: 10.4310/jdg/1214440849. |
[21] |
D. Jerison and J. M. Lee,
Extremals for the Sobolev inequality on the Heisenberg group and the CR Yamabe problem, J. Amer. Math. Soc., 1 (1988), 1-13.
doi: 10.1090/S0894-0347-1988-0924699-9. |
[22] |
D. Jerison and J. M. Lee,
Intrinsic CR normal coordinates and the CR Yamabe problem, J. Differential Geom., 29 (1989), 303-343.
doi: 10.4310/jdg/1214442877. |
[23] |
J. M. Lee,
The Fefferman metric and pseudohermitian invariants, Trans. Amer. Math. Soc., 296 (1986), 411-429.
doi: 10.2307/2000582. |
[24] |
J. M. Lee,
Pseudo-Einstein structres on CR manifolds, Amer. J. Math., 110 (1988), 157-178.
doi: 10.2307/2374543. |
[25] |
J. M. Lee and T. H. Parker,
The Yamabe problem, Bull. Amer. Math. Soc. (N.S.), 17 (1987), 37-91.
doi: 10.1090/S0273-0979-1987-15514-5. |
[26] |
Y. Y. Li,
Remark on some conformally invariant integral equations: The method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180.
|
[27] |
S.-Y. Li, D. N. Son and X. Wang,
A new characterization of the CR sphere and the sharp eigenvalue estimate for the Kohn Laplacian, Advances in Math., 281 (2015), 1285-1305.
doi: 10.1016/j.aim.2015.06.008. |
[28] |
S.-Y. Li and X. Wang,
An Obata-type theorem in CR geometry, J. Diff. Geom., 95 (2013), 483-502.
doi: 10.4310/jdg/1381931736. |
[29] |
Y. Y. Li and M. Zhu,
Sharp Sobolev inequalities involving boundary terms, Geom. Funct. Anal., 8 (1998), 59-87.
doi: 10.1007/s000390050048. |
[30] |
Y. Li and M. Zhu,
Sharp Sobolev trace inequalities on Riemannian manifolds with boundaries, Comm. Pure Appl. Math., 50 (1997), 427-465.
doi: 10.1002/(SICI)1097-0312(199705)50:5<449::AID-CPA2>3.0.CO;2-9. |
[31] |
Y. Li and M. Zhu,
Uniqueness theorems through the method of moving spheres, Duke Math. J., 80 (1995), 383-417.
doi: 10.1215/S0012-7094-95-08016-8. |
[32] |
E. H. Lieb,
Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374.
doi: 10.2307/2007032. |
[33] |
E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, 30. Princeton University Press, Princeton, N.J. 1970. |
[34] |
X. Wang,
Some recent results in CR geometry, Tsinghua lectures in mathematics, Adv. Lect. Math. (ALM), Int. Press, Somerville, MA, 45 (2019), 469-484.
|
[35] |
X. Wang,
On a remarkable formula of Jerison and Lee in CR geometry, Math. Res. Lett., 22 (2015), 279-299.
doi: 10.4310/MRL.2015.v22.n1.a14. |
[36] |
M. Zhu,
Prescribing integral curvature equation, Differential and Integral Equations, 29 (2016), 889-904.
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