Abstract
In this paper we present anew approach based on the heat equation and extension problems to some intertwining formulas arising in conformal CR geometry.
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A. V. Balakrishnan, An operational calculus for infinitesimal generators of semigroups, Trans. Amer. Math. Soc. 91 (1959), 330–353.
H. Baum and A. Juhl, Conformal Differential Geometry, Birkhäuser, Basel, 2010.
T. P. Branson, Sharp inequalities, the functional determinant, and the complementary series, Trans. Amer. Math. Soc. 347 (1995), 3671–3742.
T. P. Branson, Spectral theory of invariant operators, sharp inequalities, and representation theory, Rend. Circ. Mat. Palermo (2) Suppl. 46 (1997), 29–54.
T. P. Branson, L. Fontana and C. Morpurgo, Moser-Trudinger and Beckner-Onofri’s inequalities on the CR sphere, Ann. of Math. (2) 177 (2013), 1–52.
L. Caffarelli and L. Silvestre, An extension problem related, to the fractional Laplacian, Comm. Partial Differential Equations 32 (2007), 1245–1260.
S.-Y. A. Chang and M. d. M. González, Fractional Laplacian in conformal geometry, Adv. Math. 226 (2011), 1410–1432.
W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math. 59 (2006), 330–343.
L. Corwin and F. P. Greenleaf, Representations of Nilpotent Lie Groups and their Applications, Part I: Basic Theory and Examples, Cambridge University Press, Cambridge, 1990.
M. Cowling, Unitary and uniformly bounded representations of some simple Lie groups, in Harmonic Analysis and Group Representations, Liguori Editore, Naples, 1982, pp. 49–128.
M. Cowling, A. H. Dooley, A. Korányi and F. Ricci, H-type groups and Iwasawa decompositions, Adv. Math. 87 (1991), 1–41.
M. Cowling and U. Haagerup, Completely bounded multipliers of the Fourier algebra of a simple Lie group of real rank one, Invent. Math. 96 (1989), 507–549.
F. Ferrari and B. Franchi, Harnack inequality for fractional sub-Laplacians in Carnot groups, Math. Z. 279 (2015), 435–458.
G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat. 13 (1975), 161–207.
R. L. Frank and E. H. Lieb, Sharp constants in several inequalities on the Heisenberg group, Ann. of Math. (2) 176 (2012), 349–381.
R. L. Frank, M. d. M. González, D. Monticelli and J. Tan, An extension problem for the CR fractional Laplacian, Adv. Math. 270 (2015), 97–137.
N. Garofalo and D. Vassilev, Regularity near the characteristic set in the non-linear Dirichlet problem and conformal geometry of sub-Laplacians on Carnot groups, Math. Ann. 318 (2000), 453–516.
N. Garofalo and D. Vassilev, Symmetry properties of positive entire solutions of Yamabe-type equations on groups of Heisenberg type, Duke Math. J. 106 (2001), 411–448.
N. Garofalo, Some properties of sub-Laplaceans, in Proceedings of the International Conference “Two nonlinear days in Urbino 2017”, Texas State University, Department of Mathematics, San Marcos, TX, 2018, pp. 103–131.
N. Garofalo, Fractional thoughts, in New Developments in the Analysis of Nonlocal Operators, American Mathematical Society, Providence, RI, 2019, pp. 1–135.
N. Garofalo and G. Tralli, Feeling the heat in a group of Heisenberg type, Adv. Math. 381 (2021), 107635.
N. Garofalo and G. Tralli, Heat kernels for a class of hybrid evolution equations, Potential Anal., to appear, DOI: https://doi.org/10.1007/s11118-022-10003-2.
M. d. M. González, Recent progress on the fractional Laplacian in conformal geometry, in Recent Developments in Nonlocal Theory, De Gruyter, Berlin, 2018, pp. 236–273.
C. Guidi, A. Maalaoui and V. Martino, Palais-Smale sequences for the fractional CR Yamabe functional and multiplicity results, Calc. Var. Partial Differential Equations 57 (2018), Article no. 152.
L. Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147–171.
D. Jerison and J. M. Lee, The Yamabe problem on CR manifolds, J. Differential Geom. 25 (1987), 167–197.
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.
A. Kaplan, Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms, Trans. Amer. Math. Soc. 258 (1980), 147–153.
A. W. Knapp and E. M. Stein, Intertwining operators for semisimple groups, Ann. of Math. (2) 93 (1971), 489–578.
A. Kristály, Nodal solutions for the fractional Yamabe problem on Heisenberg groups, Proc. Roy. Soc. Edinburgh Sect. A 150 (2020), 771–788.
E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math. (2) 118 (1983), 349–374.
M. Riesz, Intégrales de Riemann-Liouville et potentiels, Acta Sci. Math. Szeged, 9 (1938), 1–42.
L. Roncal and S. Thangavelu, Hardy’s inequality for fractional powers of the sublaplacian on the Heisenberg group, Adv. Math. 302 (2016), 106–158.
L. Roncal and S. Thangavelu, An extension problem and trace Hardy inequality for the sublaplacian on H-type groups, Int. Math. Res. Not. IMRN 2020 (2020), 4238–4294.
E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, NJ, 1970.
N. Th. Varopoulos, L. Saloff-Coste and T. Coulhon, Analysis and Geometry on Groups, Cambridge University Press, Cambridge, 1992.
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Both authors are supported in part by a Progetto SID: “Non-local Sobolev and isoperimetric inequalities”, University of Padova, 2019.
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Garofalo, N., Tralli, G. A heat equation approach to intertwining. JAMA 149, 113–134 (2023). https://doi.org/10.1007/s11854-022-0246-z
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DOI: https://doi.org/10.1007/s11854-022-0246-z