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

Integral equations on compact CR manifolds

Yazhou Han

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
$ \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} $
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.
Keywords: Integral equations, CR Yamabe problem, sharp Hardy-Littlewood-Sobolev inequalities, blowup analysis, CR manifold.
Mathematics Subject Classification: Primary: 45G05; Secondary: 32V20.
Citation: Yazhou Han. Integral equations on compact CR manifolds. Discrete & Continuous Dynamical Systems - A, 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

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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

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S. Dragomir and G. Tomassini, Differential Geometry and Analysis on CR Manifolds, Progress in Mathematics, 246. Birkhäuser Boston, Inc., Boston, MA, 2006.  Google Scholar

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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

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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

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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

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