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  May 2021 Early access  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.
Citation: Yazhou Han. Integral equations on compact CR manifolds. Discrete and 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.

[2]

T. P. BransonL. 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. ChenC. 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. ChengA. 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. LiD. 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. BransonL. 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. ChenC. 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. ChengA. 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. LiD. 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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