American Institute of Mathematical Sciences

May  2021, 41(5): 2187-2204. doi: 10.3934/dcds.2020358

Integral equations on compact CR manifolds

 Department of Mathematics, College of Science, China Jiliang University, Hangzhou 310018, China

Received  September 2019 Published  October 2020

Assume that
 $M$
is a CR compact manifold without boundary and CR Yamabe invariant
 $\mathcal{Y}(M)$
is positive. Here, we devote to study a class of sharp Hardy-Littlewood-Sobolev inequality as follows
 $\begin{equation*} \Bigl| \int_M\int_M [G_\xi^\theta(\eta)]^{\frac{Q-\alpha}{Q-2}} f(\xi) g(\eta) dV_\theta(\xi) dV_\theta(\eta) \Bigr| \leq \mathcal{Y}_\alpha(M) \|f\|_{L^{\frac{2Q}{Q+\alpha}}(M)} \|g\|_{L^{\frac{2Q}{Q+\alpha}}(M)}, \end{equation*}$
where
 $G_\xi^\theta(\eta)$
is the Green function of CR conformal Laplacian
 $\mathcal{L_\theta} = b_n\Delta_b+R$
,
 $\mathcal{Y}_\alpha(M)$
is sharp constant,
 $\Delta_b$
is Sublaplacian and
 $R$
is Tanaka-Webster scalar curvature. For the diagonal case
 $f = g$
, we prove that
 $\mathcal{Y}_\alpha(M)\geq \mathcal{Y}_\alpha(\mathbb{S}^{2n+1})$
(the unit complex sphere of
 $\mathbb{C}^{n+1}$
) and
 $\mathcal{Y}_\alpha(M)$
can be attained if
 $\mathcal{Y}_\alpha(M)> \mathcal{Y}_\alpha(\mathbb{S}^{2n+1})$
. So, we got the existence of the Euler-Lagrange equations
 $$$\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 Moreover, we prove that the solution of (1) is $ \Gamma^\alpha(M) $. Particular, if $ \alpha = 2 $, the previous extremal problem is closely related to the CR Yamabe problem. Hence, we can study the CR Yamabe problem by integral equations. Citation: Yazhou Han. Integral equations on compact CR manifolds. Discrete & Continuous Dynamical Systems, 2021, 41 (5) : 2187-2204. doi: 10.3934/dcds.2020358 References:  [1] A. Bahri and J.-M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the togology of the domain, Comm. Pure Appl. Math., 41 (1988), 253-294. doi: 10.1002/cpa.3160410302. Google Scholar [2] T. P. Branson, L. Fontana and C. Morpurgo, Moser-Trudinger and Beckner-Onofri's inequalities on the CR sphere, Annals of Mathematics, 177 (2013), 1-52. doi: 10.4007/annals.2013.177.1.1. Google Scholar [3] H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477. doi: 10.1002/cpa.3160360405. Google Scholar [4] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343. doi: 10.1002/cpa.20116. Google Scholar [5] J.-H. Cheng, A. Malchiodi and P. Yang, A positive mass theorem in three dimensional Cauchy-Riemann geometry, Advances in Mathematics, 308 (2017), 276-347. doi: 10.1016/j.aim.2016.12.012. Google Scholar [6] W. S. Cohn and G. Lu, Sharp constants for Moser-Trudinger inequalities on spheres in complex space$\mathbb{C}^n$, Comm. Pure Appl. Math., 57 (2004), 1458-1493. doi: 10.1002/cpa.20043. Google Scholar [7] J. Dou and M. Zhu, Nonlinear integral equations on bounded domains, J. Funct. Anal., 277 (2019), 111-134. doi: 10.1016/j.jfa.2018.05.020. Google Scholar [8] S. Dragomir and G. Tomassini, Differential Geometry and Analysis on CR Manifolds, Progress in Mathematics, 246. Birkhäuser Boston, Inc., Boston, MA, 2006. Google Scholar [9] G. B. Folland, A fundamental solution for a subelliptic operator, Bull. Amer. Math. Soc., 79 (1973), 373-376. doi: 10.1090/S0002-9904-1973-13171-4. Google Scholar [10] G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Arkiv för Matematik, 13 (1975), 161-207. doi: 10.1007/BF02386204. Google Scholar [11] G. B. Folland and E. M. Stein, Estimates for the$\bar{\partial}_b$complex and analysis on the Heisenberg group, Comm. Pure Appl. Math., 27 (1974), 429-522. doi: 10.1002/cpa.3160270403. Google Scholar [12] R. L. Frank and E. H. Lieb, Sharp constants in several inequalities on the Heisenberg group, Annals of Mathematics, 176 (2012), 349-381. doi: 10.4007/annals.2012.176.1.6. Google Scholar [13] N. Gamara, The CR Yamabe conjecture the case$n = 1$, J. Eur. Math. Soc. (JEMS), 3 (2001), 105-137. doi: 10.1007/PL00011303. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [18] L. Hörmander, Hypoelliptic second order differential equations, Acta Mathematica, 119 (1967), 147-171. doi: 10.1007/BF02392081. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [23] J. M. Lee, The Fefferman metric and pseudohermitian invariants, Trans. Amer. Math. Soc., 296 (1986), 411-429. doi: 10.2307/2000582. Google Scholar [24] J. M. Lee, Pseudo-Einstein structres on CR manifolds, Amer. J. Math., 110 (1988), 157-178. doi: 10.2307/2374543. Google Scholar [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. Google Scholar [26] Y. Y. Li, Remark on some conformally invariant integral equations: The method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180. Google Scholar [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. Google Scholar [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. Google Scholar [29] Y. Y. Li and M. Zhu, Sharp Sobolev inequalities involving boundary terms, Geom. Funct. Anal., 8 (1998), 59-87. doi: 10.1007/s000390050048. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [33] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, 30. Princeton University Press, Princeton, N.J. 1970. Google Scholar [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. Google Scholar [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. Google Scholar [36] M. Zhu, Prescribing integral curvature equation, Differential and Integral Equations, 29 (2016), 889-904. Google Scholar 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [6] W. S. Cohn and G. Lu, Sharp constants for Moser-Trudinger inequalities on spheres in complex space$\mathbb{C}^n$, Comm. Pure Appl. Math., 57 (2004), 1458-1493. doi: 10.1002/cpa.20043. Google Scholar [7] J. Dou and M. Zhu, Nonlinear integral equations on bounded domains, J. Funct. Anal., 277 (2019), 111-134. doi: 10.1016/j.jfa.2018.05.020. Google Scholar [8] S. Dragomir and G. Tomassini, Differential Geometry and Analysis on CR Manifolds, Progress in Mathematics, 246. Birkhäuser Boston, Inc., Boston, MA, 2006. Google Scholar [9] G. B. Folland, A fundamental solution for a subelliptic operator, Bull. Amer. Math. Soc., 79 (1973), 373-376. doi: 10.1090/S0002-9904-1973-13171-4. Google Scholar [10] G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Arkiv för Matematik, 13 (1975), 161-207. doi: 10.1007/BF02386204. Google Scholar [11] G. B. Folland and E. M. Stein, Estimates for the$\bar{\partial}_b$complex and analysis on the Heisenberg group, Comm. Pure Appl. Math., 27 (1974), 429-522. doi: 10.1002/cpa.3160270403. Google Scholar [12] R. L. Frank and E. H. Lieb, Sharp constants in several inequalities on the Heisenberg group, Annals of Mathematics, 176 (2012), 349-381. doi: 10.4007/annals.2012.176.1.6. Google Scholar [13] N. Gamara, The CR Yamabe conjecture the case$n = 1$, J. Eur. Math. Soc. (JEMS), 3 (2001), 105-137. doi: 10.1007/PL00011303. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [18] L. Hörmander, Hypoelliptic second order differential equations, Acta Mathematica, 119 (1967), 147-171. doi: 10.1007/BF02392081. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [23] J. M. Lee, The Fefferman metric and pseudohermitian invariants, Trans. Amer. Math. Soc., 296 (1986), 411-429. doi: 10.2307/2000582. Google Scholar [24] J. M. Lee, Pseudo-Einstein structres on CR manifolds, Amer. J. Math., 110 (1988), 157-178. doi: 10.2307/2374543. Google Scholar [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. Google Scholar [26] Y. Y. Li, Remark on some conformally invariant integral equations: The method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180. Google Scholar [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. Google Scholar [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. Google Scholar [29] Y. Y. Li and M. Zhu, Sharp Sobolev inequalities involving boundary terms, Geom. Funct. Anal., 8 (1998), 59-87. doi: 10.1007/s000390050048. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [33] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, 30. Princeton University Press, Princeton, N.J. 1970. Google Scholar [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. Google Scholar [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. Google Scholar [36] M. Zhu, Prescribing integral curvature equation, Differential and Integral Equations, 29 (2016), 889-904. Google Scholar  [1] Andrea Cianchi, Adele Ferone. Improving sharp Sobolev type inequalities by optimal remainder gradient norms. 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