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June  2018, 11(3): 477-491. doi: 10.3934/dcdss.2018026

Fractional Laplacians, perimeters and heat semigroups in Carnot groups

Fausto Ferrari 1, , Michele Miranda Jr 2, , Diego Pallara 3, , Andrea Pinamonti 4, and  Yannick Sire 5,,

1. 

Dip. di Matematica, Università di Bologna, Piazza di Porta San Donato, 5, 40126, Bologna, Italy
2. 

Dip. di Matematica e Informatica, Università di Ferrara, via Machiavelli 30, 44121 Ferrara, Italy
3. 

Dip. di Matematica e Fisica "Ennio De Giorgi", Università del Salento, P.O.B. 193, and INFN, 73100 Lecce, Italy
4. 

Dip. di Matematica, Università di Trento, via Sommarive, 14, 38123 Povo (TN), Italy
5. 

Department of Mathematics, Johns Hopkins University, 404 Krieger Hall, 3400 N. Charles Street, Baltimore, MD 21218, USA

* Corresponding author.

Received  May 2017 Revised  August 2017 Published  October 2017

We define and study the fractional Laplacian and the fractional perimeter of a set in Carnot groups and we compare the perimeter with the asymptotic behaviour of the fractional heat semigroup.

Keywords: Fractional Laplacians, fractional perimeter, heat semi-group.
Mathematics Subject Classification: Primary: 35C15, 35J15; Secondary: 35K08.
Citation: Fausto Ferrari, Michele Miranda Jr, Diego Pallara, Andrea Pinamonti, Yannick Sire. Fractional Laplacians, perimeters and heat semigroups in Carnot groups. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 477-491. doi: 10.3934/dcdss.2018026
References:
[1]

L. Ambrosio, M. Miranda Jr and D. Pallara, Some fine properties of $BV$ functions on Wiener spaces, Anal. Geom. Metr. Spaces, 3 (2015), 212-230.  doi: 10.1515/agms-2015-0013.  Google Scholar

[2]

L. Angiuli, M. Miranda Jr, D. Pallara and F. Paronetto, $BV$ functions and parabolic initial boundary value problems on domains, Ann. Mat. Pura Appl.(4), 188 (2009), 297-311.  doi: 10.1007/s10231-008-0076-3.  Google Scholar

[3]

R. M. Blumenthal and R. K. Getoor, Some theorems on stable processes, Trans. Amer. Math. Soc., 95 (1960), 263-273.  doi: 10.1090/S0002-9947-1960-0119247-6.  Google Scholar

[4]

item BLUbook (MR2363343) A. Bonfiglioli, E. Lanconelli and F. Uguzzoni, Stratified Lie Groups and Potential Theory for Their Sub-Laplacians, Springer, 2007. Google Scholar

[5]

N. Bouleau and F. Hirsch, Dirichlet Forms and Analysis on Wiener Spaces, vol 14 of De Gruyter studies in Mathematics, De Gruyter, 1991. doi: 10.1515/9783110858389.  Google Scholar

[6]

J. Bourgain, H. Brezis and P. Mironescu, Another look at Sobolev spaces, in: J. L. Menaldi, E. Rofman and A. Sulem (Eds), in: J. L. Menaldi, E. Rofman and A. Sulem (Eds), Optimal Control and Partial Differential Equations (volume in Honour of A. Bensoussan’s 60th birthday), IOS Press, (2001), 439-455.   Google Scholar

[7]

M. Bramanti, M. Miranda Jr and D. Pallara, Two characterization of $BV$ functions on Carnot groups via the heat semigroup, Int. Math. Res. Not, 17 (2012), 3845-3876.  doi: 10.1093/imrn/rnr170.  Google Scholar

[8]

C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, UMI Lectute Notes 20, Springer, 2016. doi: 10.1007/978-3-319-28739-3.  Google Scholar

[9]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.  Google Scholar

[10]

L. Capogna, D. Danielli, S. D. Pauls and J. T. Tyson, An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem, Birkhäuser, 2007.  Google Scholar

[11]

Z.-Q. Chen and T. Kumagai, Heat kernel estimates for jump processes of mixed types on metric measure spaces, Probab. Theory Related Fields, 140 (2008), 277-317.  doi: 10.1007/s00440-007-0070-5.  Google Scholar

[12]

W. Cygan and T. Grzywny, Heat content for convolution semigroups, J. Math. Anal. Appl, 446 (2017), 1393-1414.  doi: 10.1016/j.jmaa.2016.09.051.  Google Scholar

[13]

J. Davila, On an open question about functions of bounded variation, Calc. Var. Partial Differential Equations, 15 (2001), 519-527.  doi: 10.1007/s005260100135.  Google Scholar

[14]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136) (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[15]

F. Ferrari and B. Franchi, Harnack inequality for fractional sub-Laplacians in Carnot groups, Math. Z., 279 (2015), 435-458.  doi: 10.1007/s00209-014-1376-5.  Google Scholar

[16]

G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat., 13 (1975), 161-207.  doi: 10.1007/BF02386204.  Google Scholar

[17]

G. B. Folland and E. Stein, Hardy Spaces on Homogeneous Groups, Princeton University Press, 1982.  Google Scholar

[18]

M. Kwaśnicki, Ten equivalent definitions of the fractional Laplace operator, Frac. Calc. Appl. Anal., 20 (2017), 17-51.  doi: 10.1515/fca-2017-0002.  Google Scholar

[19]

M. Ludwig, Anisotropic fractional perimeters, J. Diff. Geom., 96 (2014), 77-93.  doi: 10.4310/jdg/1391192693.  Google Scholar

[20]

M. Ludwig, Anisotropic fractional Sobolev norms, Adv. Math., 252 (2014), 150-157.  doi: 10.1016/j.aim.2013.10.024.  Google Scholar

[21]

M. Miranda Jr, D. Pallara, F. Paronetto and M. Preunkert, Short-time heat flow and functions of bounded variation in $\mathbf{R}^{n}$, Ann. Fac. Sci. Toulouse Math., 16 (2007), 125-145.  doi: 10.5802/afst.1142.  Google Scholar

[22]

H. M. Nguyen, A. Pinamonti, M. Squassina and E. Vecchi, New characterization of magnetic Sobolev spaces, arXiv:1703. 09801. Google Scholar

[23]

A. Pinamonti, M. Squassina and E. Vecchi, Magnetic BV functions and the Bourgain-Brezis-Mironescu formula Adv. Calc. Var. (2017). doi: 10.1515/acv-2017-0019.  Google Scholar

[24]

A. Pinamonti, M. Squassina and E. Vecchi, The Maz'ya-Shaposhnikova limit in the magnetic setting, J. Math. Anal. Appl., 449 (2017), 1152-1159.  doi: 10.1016/j.jmaa.2016.12.065.  Google Scholar

[25]

A. Ponce, A new approach to Sobolev spaces and connections to $Γ$-convergence, Calc. Var. Partial Differential Equations, 19 (2004), 229-255.  doi: 10.1007/s00526-003-0195-z.  Google Scholar

[26]

K. Saka, Besov spaces and Sobolev spaces on a nilpotent Lie group, Tohoku Math. Journ., 31 (1979), 383-437.  doi: 10.2748/tmj/1178229728.  Google Scholar

[27]

A. Sánchez-Calle, Fundamental solutions and geometry of the sum of squares of vector fields, Invent. Math., 78 (1984), 143-160.  doi: 10.1007/BF01388721.  Google Scholar

[28]

item St (MR1232192) E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals, Princeton University Press, 1993. Google Scholar

[29]

P. R. Stinga and J. L. Torrea, Extension problem and Harnack's inequality for some fractional operators, Comm. Partial Differential Equations, 35 (2010), 2092-2122.  doi: 10.1080/03605301003735680.  Google Scholar

[30]

K. Yosida, Functional Analysis Springer, Sixth Ed. , 1980.  Google Scholar

show all references

References:
[1]

L. Ambrosio, M. Miranda Jr and D. Pallara, Some fine properties of $BV$ functions on Wiener spaces, Anal. Geom. Metr. Spaces, 3 (2015), 212-230.  doi: 10.1515/agms-2015-0013.  Google Scholar

[2]

L. Angiuli, M. Miranda Jr, D. Pallara and F. Paronetto, $BV$ functions and parabolic initial boundary value problems on domains, Ann. Mat. Pura Appl.(4), 188 (2009), 297-311.  doi: 10.1007/s10231-008-0076-3.  Google Scholar

[3]

R. M. Blumenthal and R. K. Getoor, Some theorems on stable processes, Trans. Amer. Math. Soc., 95 (1960), 263-273.  doi: 10.1090/S0002-9947-1960-0119247-6.  Google Scholar

[4]

item BLUbook (MR2363343) A. Bonfiglioli, E. Lanconelli and F. Uguzzoni, Stratified Lie Groups and Potential Theory for Their Sub-Laplacians, Springer, 2007. Google Scholar

[5]

N. Bouleau and F. Hirsch, Dirichlet Forms and Analysis on Wiener Spaces, vol 14 of De Gruyter studies in Mathematics, De Gruyter, 1991. doi: 10.1515/9783110858389.  Google Scholar

[6]

J. Bourgain, H. Brezis and P. Mironescu, Another look at Sobolev spaces, in: J. L. Menaldi, E. Rofman and A. Sulem (Eds), in: J. L. Menaldi, E. Rofman and A. Sulem (Eds), Optimal Control and Partial Differential Equations (volume in Honour of A. Bensoussan’s 60th birthday), IOS Press, (2001), 439-455.   Google Scholar

[7]

M. Bramanti, M. Miranda Jr and D. Pallara, Two characterization of $BV$ functions on Carnot groups via the heat semigroup, Int. Math. Res. Not, 17 (2012), 3845-3876.  doi: 10.1093/imrn/rnr170.  Google Scholar

[8]

C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, UMI Lectute Notes 20, Springer, 2016. doi: 10.1007/978-3-319-28739-3.  Google Scholar

[9]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.  Google Scholar

[10]

L. Capogna, D. Danielli, S. D. Pauls and J. T. Tyson, An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem, Birkhäuser, 2007.  Google Scholar

[11]

Z.-Q. Chen and T. Kumagai, Heat kernel estimates for jump processes of mixed types on metric measure spaces, Probab. Theory Related Fields, 140 (2008), 277-317.  doi: 10.1007/s00440-007-0070-5.  Google Scholar

[12]

W. Cygan and T. Grzywny, Heat content for convolution semigroups, J. Math. Anal. Appl, 446 (2017), 1393-1414.  doi: 10.1016/j.jmaa.2016.09.051.  Google Scholar

[13]

J. Davila, On an open question about functions of bounded variation, Calc. Var. Partial Differential Equations, 15 (2001), 519-527.  doi: 10.1007/s005260100135.  Google Scholar

[14]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136) (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[15]

F. Ferrari and B. Franchi, Harnack inequality for fractional sub-Laplacians in Carnot groups, Math. Z., 279 (2015), 435-458.  doi: 10.1007/s00209-014-1376-5.  Google Scholar

[16]

G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat., 13 (1975), 161-207.  doi: 10.1007/BF02386204.  Google Scholar

[17]

G. B. Folland and E. Stein, Hardy Spaces on Homogeneous Groups, Princeton University Press, 1982.  Google Scholar

[18]

M. Kwaśnicki, Ten equivalent definitions of the fractional Laplace operator, Frac. Calc. Appl. Anal., 20 (2017), 17-51.  doi: 10.1515/fca-2017-0002.  Google Scholar

[19]

M. Ludwig, Anisotropic fractional perimeters, J. Diff. Geom., 96 (2014), 77-93.  doi: 10.4310/jdg/1391192693.  Google Scholar

[20]

M. Ludwig, Anisotropic fractional Sobolev norms, Adv. Math., 252 (2014), 150-157.  doi: 10.1016/j.aim.2013.10.024.  Google Scholar

[21]

M. Miranda Jr, D. Pallara, F. Paronetto and M. Preunkert, Short-time heat flow and functions of bounded variation in $\mathbf{R}^{n}$, Ann. Fac. Sci. Toulouse Math., 16 (2007), 125-145.  doi: 10.5802/afst.1142.  Google Scholar

[22]

H. M. Nguyen, A. Pinamonti, M. Squassina and E. Vecchi, New characterization of magnetic Sobolev spaces, arXiv:1703. 09801. Google Scholar

[23]

A. Pinamonti, M. Squassina and E. Vecchi, Magnetic BV functions and the Bourgain-Brezis-Mironescu formula Adv. Calc. Var. (2017). doi: 10.1515/acv-2017-0019.  Google Scholar

[24]

A. Pinamonti, M. Squassina and E. Vecchi, The Maz'ya-Shaposhnikova limit in the magnetic setting, J. Math. Anal. Appl., 449 (2017), 1152-1159.  doi: 10.1016/j.jmaa.2016.12.065.  Google Scholar

[25]

A. Ponce, A new approach to Sobolev spaces and connections to $Γ$-convergence, Calc. Var. Partial Differential Equations, 19 (2004), 229-255.  doi: 10.1007/s00526-003-0195-z.  Google Scholar

[26]

K. Saka, Besov spaces and Sobolev spaces on a nilpotent Lie group, Tohoku Math. Journ., 31 (1979), 383-437.  doi: 10.2748/tmj/1178229728.  Google Scholar

[27]

A. Sánchez-Calle, Fundamental solutions and geometry of the sum of squares of vector fields, Invent. Math., 78 (1984), 143-160.  doi: 10.1007/BF01388721.  Google Scholar

[28]

item St (MR1232192) E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals, Princeton University Press, 1993. Google Scholar

[29]

P. R. Stinga and J. L. Torrea, Extension problem and Harnack's inequality for some fractional operators, Comm. Partial Differential Equations, 35 (2010), 2092-2122.  doi: 10.1080/03605301003735680.  Google Scholar

[30]

K. Yosida, Functional Analysis Springer, Sixth Ed. , 1980.  Google Scholar

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