January  2021, 20(1): 17-54. doi: 10.3934/cpaa.2020255

A note on Riemann-Liouville fractional Sobolev spaces

1. 

Dipartimento di Matematica e Fisica, Università del Salento, Via Per Arnesano, 73100 Lecce, Italy

2. 

Fachbereich Mathematik, Universität Hamburg, Bundesstraße 55, 20146 Hamburg, Germany

* Corresponding author

Received  June 2020 Revised  July 2020 Published  October 2020

Taking inspiration from a recent paper by Bergounioux et al., we study the Riemann-Liouville fractional Sobolev space $ W^{s, p}_{RL, a+}(I) $, for $ I = (a, b) $ for some $ a, b \in \mathbb{R}, a < b $, $ s \in (0, 1) $ and $ p \in [1, \infty] $; that is, the space of functions $ u \in L^{p}(I) $ such that the left Riemann-Liouville $ (1 - s) $-fractional integral $ I_{a+}^{1 - s}[u] $ belongs to $ W^{1, p}(I) $. We prove that the space of functions of bounded variation $ BV(I) $ and the fractional Sobolev space $ W^{s, 1}(I) $ continuously embed into $ W^{s, 1}_{RL, a+}(I) $. In addition, we define the space of functions with left Riemann-Liouville $ s $-fractional bounded variation, $ BV^{s}_{RL,a+}(I) $, as the set of functions $ u \in L^{1}(I) $ such that $ I^{1 - s}_{a+}[u] \in BV(I) $, and we analyze some fine properties of these functions. Finally, we prove some fractional Sobolev-type embedding results and we analyze the case of higher order Riemann-Liouville fractional derivatives.

Citation: Alessandro Carbotti, Giovanni E. Comi. A note on Riemann-Liouville fractional Sobolev spaces. Communications on Pure & Applied Analysis, 2021, 20 (1) : 17-54. doi: 10.3934/cpaa.2020255
References:
[1] Robert A. Adams and John J. F. Fournier, Sobolev Spaces., Elsevier/Academic Press, Amsterdam, 2003.   Google Scholar
[2]

Mark AllenLuis Caffarelli and Alexis Vasseur, A parabolic problem with a fractional time derivative., Arch. Ration. Mech. Anal., 221 (2016), 603-630.  doi: 10.1007/s00205-016-0969-z.  Google Scholar

[3]

Ricardo AlmeidaNuno R. O. Bastos and M. Teresa T. Monteiro, Modeling some real phenomena by fractional differential equations., Math. Methods Appl. Sci., 39 (2016), 4846-4855.  doi: 10.1002/mma.3818.  Google Scholar

[4] Luigi AmbrosioNicola Fusco and Diego Pallara, Functions of Bounded Variation and Free Discontinuity Problems., The Clarendon Press, Oxford University Press, New York, 2000.   Google Scholar
[5]

George A. Anastassiou, Fractional Differentiation Inequalities., Springer, Dordrecht, 2009. doi: 10.1007/978-0-387-98128-4.  Google Scholar

[6]

Emil Artin, The Gamma Function., Holt, Rinehart and Winston, New York-Toronto-London, 1964.  Google Scholar

[7]

Maïtine BergouniouxAntonio LeaciGiacomo Nardi and Franco Tomarelli, Fractional Sobolev spaces and functions of bounded variation of one variable., Fract. Calc. Appl. Anal., 20 (2017), 936-962.  doi: 10.1515/fca-2017-0049.  Google Scholar

[8]

Loïc Bourdin and Dariusz Idczak, A fractional fundamental lemma and a fractional integration by parts formula——Applications to critical points of Bolza functionals and to linear boundary value problems., Adv. Differ. Equ., 20 (2015), 213-232.   Google Scholar

[9]

Michele Caputo, Linear models of dissipation whose $Q$ is almost frequency independent. II., Fract. Calc. Appl. Anal., 11 (2008), 4-14.   Google Scholar

[10]

Alessandro Carbotti, Serena Dipierro and Enrico Valdinoci, Local density of Caputo-stationary functions of any order., Complex Var. and Ellipti Equ., 65 (2018), 1115-1138. doi: 10.1080/17476933.2018.1544631.  Google Scholar

[11]

Alessandro Carbotti, Serena Dipierro and Enrico Valdinoci, Local Density Of Solutions To Fractional Equations., De Gruyter, 2019. Google Scholar

[12]

Michele CarrieroAntonio Leaci and Franco Tomarelli, Second order variational problems with free discontinuity and free gradient discontinuity., Calculus of variations: topics from the mathematical heritage of E. De Giorgi, Quad. Mat., 14 (2004), 135-186.   Google Scholar

[13]

Giovanni E. Comi and Giorgio Stefani, A distributional approach to fractional Sobolev spaces and fractional variation: existence of blow-up., J. Funct. Anal., 277 (2019), 3373-3435.  doi: 10.1016/j.jfa.2019.03.011.  Google Scholar

[14]

Giovanni E. Comi and Giorgio Stefani, A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics I., arXiv: 1910.13419. Google Scholar

[15]

Françoise Demengel, Fonctions à hessien borné., Ann. Inst. Fourier (Grenoble), 34 (1984), 155-190.   Google Scholar

[16]

Eleonora Di NezzaGiampiero Palatucci and Enrico 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

[17]

Mario Di PaolaFrancesco Paolo Pinnola and Massimiliano Zingales, Fractional differential equations and related exact mechanical models., Comput. Math. Appl., 66 (2013), 608-620.  doi: 10.1016/j.camwa.2013.03.012.  Google Scholar

[18]

Serena Dipierro and Enrico Valdinoci, A Simple Mathematical Model Inspired by the Purkinje Cells: From Delayed Travelling Waves to Fractional Diffusion., Bull. Math. Biol., 80 (2018), 1849-1870.  doi: 10.1007/s11538-018-0437-z.  Google Scholar

[19]

Bartłomiej Dyda, A fractional order Hardy inequality., Illinois J. Math., 48 (2004), 575-588.   Google Scholar

[20] Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties of functions., CRC Press, Boca Raton, FL, 2015.   Google Scholar
[21]

Fausto Ferrari, Weyl and Marchaud Derivatives: A Forgotten History., Mathematics, 6 (2018) doi: 10.3390/math6010006.  Google Scholar

[22]

Loukas Grafakos, Classical Fourier Analysis., Springer, New York, 2014. doi: 10.1007/978-1-4939-1194-3.  Google Scholar

[23]

Loukas Grafakos, Modern Fourier Analysis., Springer, New York, 2014. doi: 10.1007/978-1-4939-1230-8.  Google Scholar

[24]

G. H. Hardy and J. E. Littlewood, Some properties of fractional integrals. I., Math. Z., 27 (1928), 565-606.  doi: 10.1007/BF01171116.  Google Scholar

[25]

Dariusz Idczak and Stanisław Walczak, Fractional Sobolev spaces via Riemann-Liouville derivatives., J. Funct. Spaces Appl., 2013 (2013), 15pp. doi: 10.1155/2013/128043.  Google Scholar

[26]

Gottfried Wilhelm Leibniz, Letter from Hanover, Germany, to G. F. A. L'Hôpital, September 30; 1695., Mathematische Schriften, 2 (1849), 301-302.   Google Scholar

[27]

Luca Lombardini, Minimization problems involving nonlocal functionals: nonlocal minimal surfaces and a free boundary problem., arXiv: 1811.09746. Google Scholar

[28]

Alessandra Lunardi, Interpolation Theory., Edizioni della Normale, Pisa, 2018. doi: 10.1007/978-88-7642-638-4.  Google Scholar

[29]

Mironescu Petru and Sickel Winfried, A Sobolev non embedding., Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 26 (2015), 291-298.  doi: 10.4171/RLM/707.  Google Scholar

[30]

Stefan G. Samko, Anatoly A. Kilbas and Oleg I. Marichev, Fractional Integrals and Derivatives., Gordon and Breach Science Publishers, Yverdon, 1993.  Google Scholar

[31]

Armin SchikorraTien-Tsan Shieh and Daniel Spector, $L^p$ theory for fractional gradient PDE with $VMO$ coefficients., Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 26 (2015), 433-443.  doi: 10.4171/RLM/714.  Google Scholar

[32]

Tien-Tsan Shieh and Daniel E. Spector, On a new class of fractional partial differential equations., Adv. Calc. Var., 8 (2015), 321-336.  doi: 10.1515/acv-2014-0009.  Google Scholar

[33]

Tien-Tsan Shieh and Daniel E. Spector, On a new class of fractional partial differential equations Ⅱ., Adv. Calc. Var., 11 (2018), 289-307.  doi: 10.1515/acv-2016-0056.  Google Scholar

[34]

Miroslav Šilhavý, Fractional vector analysis based on invariance requirements (Critique of coordinate approaches)., Continuum Mech. Therm., 32 (2019), 207-228.  doi: 10.1007/s00161-019-00797-9.  Google Scholar

[35] Elias M. Stein, Singular integrals and differentiability properties of functions., Princeton University Press, Princeton, N. J., 1970.   Google Scholar

show all references

References:
[1] Robert A. Adams and John J. F. Fournier, Sobolev Spaces., Elsevier/Academic Press, Amsterdam, 2003.   Google Scholar
[2]

Mark AllenLuis Caffarelli and Alexis Vasseur, A parabolic problem with a fractional time derivative., Arch. Ration. Mech. Anal., 221 (2016), 603-630.  doi: 10.1007/s00205-016-0969-z.  Google Scholar

[3]

Ricardo AlmeidaNuno R. O. Bastos and M. Teresa T. Monteiro, Modeling some real phenomena by fractional differential equations., Math. Methods Appl. Sci., 39 (2016), 4846-4855.  doi: 10.1002/mma.3818.  Google Scholar

[4] Luigi AmbrosioNicola Fusco and Diego Pallara, Functions of Bounded Variation and Free Discontinuity Problems., The Clarendon Press, Oxford University Press, New York, 2000.   Google Scholar
[5]

George A. Anastassiou, Fractional Differentiation Inequalities., Springer, Dordrecht, 2009. doi: 10.1007/978-0-387-98128-4.  Google Scholar

[6]

Emil Artin, The Gamma Function., Holt, Rinehart and Winston, New York-Toronto-London, 1964.  Google Scholar

[7]

Maïtine BergouniouxAntonio LeaciGiacomo Nardi and Franco Tomarelli, Fractional Sobolev spaces and functions of bounded variation of one variable., Fract. Calc. Appl. Anal., 20 (2017), 936-962.  doi: 10.1515/fca-2017-0049.  Google Scholar

[8]

Loïc Bourdin and Dariusz Idczak, A fractional fundamental lemma and a fractional integration by parts formula——Applications to critical points of Bolza functionals and to linear boundary value problems., Adv. Differ. Equ., 20 (2015), 213-232.   Google Scholar

[9]

Michele Caputo, Linear models of dissipation whose $Q$ is almost frequency independent. II., Fract. Calc. Appl. Anal., 11 (2008), 4-14.   Google Scholar

[10]

Alessandro Carbotti, Serena Dipierro and Enrico Valdinoci, Local density of Caputo-stationary functions of any order., Complex Var. and Ellipti Equ., 65 (2018), 1115-1138. doi: 10.1080/17476933.2018.1544631.  Google Scholar

[11]

Alessandro Carbotti, Serena Dipierro and Enrico Valdinoci, Local Density Of Solutions To Fractional Equations., De Gruyter, 2019. Google Scholar

[12]

Michele CarrieroAntonio Leaci and Franco Tomarelli, Second order variational problems with free discontinuity and free gradient discontinuity., Calculus of variations: topics from the mathematical heritage of E. De Giorgi, Quad. Mat., 14 (2004), 135-186.   Google Scholar

[13]

Giovanni E. Comi and Giorgio Stefani, A distributional approach to fractional Sobolev spaces and fractional variation: existence of blow-up., J. Funct. Anal., 277 (2019), 3373-3435.  doi: 10.1016/j.jfa.2019.03.011.  Google Scholar

[14]

Giovanni E. Comi and Giorgio Stefani, A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics I., arXiv: 1910.13419. Google Scholar

[15]

Françoise Demengel, Fonctions à hessien borné., Ann. Inst. Fourier (Grenoble), 34 (1984), 155-190.   Google Scholar

[16]

Eleonora Di NezzaGiampiero Palatucci and Enrico 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

[17]

Mario Di PaolaFrancesco Paolo Pinnola and Massimiliano Zingales, Fractional differential equations and related exact mechanical models., Comput. Math. Appl., 66 (2013), 608-620.  doi: 10.1016/j.camwa.2013.03.012.  Google Scholar

[18]

Serena Dipierro and Enrico Valdinoci, A Simple Mathematical Model Inspired by the Purkinje Cells: From Delayed Travelling Waves to Fractional Diffusion., Bull. Math. Biol., 80 (2018), 1849-1870.  doi: 10.1007/s11538-018-0437-z.  Google Scholar

[19]

Bartłomiej Dyda, A fractional order Hardy inequality., Illinois J. Math., 48 (2004), 575-588.   Google Scholar

[20] Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties of functions., CRC Press, Boca Raton, FL, 2015.   Google Scholar
[21]

Fausto Ferrari, Weyl and Marchaud Derivatives: A Forgotten History., Mathematics, 6 (2018) doi: 10.3390/math6010006.  Google Scholar

[22]

Loukas Grafakos, Classical Fourier Analysis., Springer, New York, 2014. doi: 10.1007/978-1-4939-1194-3.  Google Scholar

[23]

Loukas Grafakos, Modern Fourier Analysis., Springer, New York, 2014. doi: 10.1007/978-1-4939-1230-8.  Google Scholar

[24]

G. H. Hardy and J. E. Littlewood, Some properties of fractional integrals. I., Math. Z., 27 (1928), 565-606.  doi: 10.1007/BF01171116.  Google Scholar

[25]

Dariusz Idczak and Stanisław Walczak, Fractional Sobolev spaces via Riemann-Liouville derivatives., J. Funct. Spaces Appl., 2013 (2013), 15pp. doi: 10.1155/2013/128043.  Google Scholar

[26]

Gottfried Wilhelm Leibniz, Letter from Hanover, Germany, to G. F. A. L'Hôpital, September 30; 1695., Mathematische Schriften, 2 (1849), 301-302.   Google Scholar

[27]

Luca Lombardini, Minimization problems involving nonlocal functionals: nonlocal minimal surfaces and a free boundary problem., arXiv: 1811.09746. Google Scholar

[28]

Alessandra Lunardi, Interpolation Theory., Edizioni della Normale, Pisa, 2018. doi: 10.1007/978-88-7642-638-4.  Google Scholar

[29]

Mironescu Petru and Sickel Winfried, A Sobolev non embedding., Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 26 (2015), 291-298.  doi: 10.4171/RLM/707.  Google Scholar

[30]

Stefan G. Samko, Anatoly A. Kilbas and Oleg I. Marichev, Fractional Integrals and Derivatives., Gordon and Breach Science Publishers, Yverdon, 1993.  Google Scholar

[31]

Armin SchikorraTien-Tsan Shieh and Daniel Spector, $L^p$ theory for fractional gradient PDE with $VMO$ coefficients., Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 26 (2015), 433-443.  doi: 10.4171/RLM/714.  Google Scholar

[32]

Tien-Tsan Shieh and Daniel E. Spector, On a new class of fractional partial differential equations., Adv. Calc. Var., 8 (2015), 321-336.  doi: 10.1515/acv-2014-0009.  Google Scholar

[33]

Tien-Tsan Shieh and Daniel E. Spector, On a new class of fractional partial differential equations Ⅱ., Adv. Calc. Var., 11 (2018), 289-307.  doi: 10.1515/acv-2016-0056.  Google Scholar

[34]

Miroslav Šilhavý, Fractional vector analysis based on invariance requirements (Critique of coordinate approaches)., Continuum Mech. Therm., 32 (2019), 207-228.  doi: 10.1007/s00161-019-00797-9.  Google Scholar

[35] Elias M. Stein, Singular integrals and differentiability properties of functions., Princeton University Press, Princeton, N. J., 1970.   Google Scholar
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