# American Institute of Mathematical Sciences

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  January 2021 Early access  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 and 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. [2] Mark Allen, Luis 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. [3] Ricardo Almeida, Nuno 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. [4] Luigi Ambrosio, Nicola Fusco and Diego Pallara, Functions of Bounded Variation and Free Discontinuity Problems., The Clarendon Press, Oxford University Press, New York, 2000. [5] George A. Anastassiou, Fractional Differentiation Inequalities., Springer, Dordrecht, 2009. doi: 10.1007/978-0-387-98128-4. [6] Emil Artin, The Gamma Function., Holt, Rinehart and Winston, New York-Toronto-London, 1964. [7] Maïtine Bergounioux, Antonio Leaci, Giacomo 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. [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. [9] Michele Caputo, Linear models of dissipation whose $Q$ is almost frequency independent. II., Fract. Calc. Appl. Anal., 11 (2008), 4-14. [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. [11] Alessandro Carbotti, Serena Dipierro and Enrico Valdinoci, Local Density Of Solutions To Fractional Equations., De Gruyter, 2019. [12] Michele Carriero, Antonio 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. [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. [14] Giovanni E. Comi and Giorgio Stefani, A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics I., arXiv: 1910.13419. [15] Françoise Demengel, Fonctions à hessien borné., Ann. Inst. Fourier (Grenoble), 34 (1984), 155-190. [16] Eleonora Di Nezza, Giampiero 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. [17] Mario Di Paola, Francesco 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. [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. [19] Bartłomiej Dyda, A fractional order Hardy inequality., Illinois J. Math., 48 (2004), 575-588. [20] Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties of functions., CRC Press, Boca Raton, FL, 2015. [21] Fausto Ferrari, Weyl and Marchaud Derivatives: A Forgotten History., Mathematics, 6 (2018) doi: 10.3390/math6010006. [22] Loukas Grafakos, Classical Fourier Analysis., Springer, New York, 2014. doi: 10.1007/978-1-4939-1194-3. [23] Loukas Grafakos, Modern Fourier Analysis., Springer, New York, 2014. doi: 10.1007/978-1-4939-1230-8. [24] G. H. Hardy and J. E. Littlewood, Some properties of fractional integrals. I., Math. Z., 27 (1928), 565-606.  doi: 10.1007/BF01171116. [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. [26] Gottfried Wilhelm Leibniz, Letter from Hanover, Germany, to G. F. A. L'Hôpital, September 30; 1695., Mathematische Schriften, 2 (1849), 301-302. [27] Luca Lombardini, Minimization problems involving nonlocal functionals: nonlocal minimal surfaces and a free boundary problem., arXiv: 1811.09746. [28] Alessandra Lunardi, Interpolation Theory., Edizioni della Normale, Pisa, 2018. doi: 10.1007/978-88-7642-638-4. [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. [30] Stefan G. Samko, Anatoly A. Kilbas and Oleg I. Marichev, Fractional Integrals and Derivatives., Gordon and Breach Science Publishers, Yverdon, 1993. [31] Armin Schikorra, Tien-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. [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. [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. [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. [35] Elias M. Stein, Singular integrals and differentiability properties of functions., Princeton University Press, Princeton, N. J., 1970.

show all references

##### References:
 [1] Robert A. Adams and John J. F. Fournier, Sobolev Spaces., Elsevier/Academic Press, Amsterdam, 2003. [2] Mark Allen, Luis 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. [3] Ricardo Almeida, Nuno 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. [4] Luigi Ambrosio, Nicola Fusco and Diego Pallara, Functions of Bounded Variation and Free Discontinuity Problems., The Clarendon Press, Oxford University Press, New York, 2000. [5] George A. Anastassiou, Fractional Differentiation Inequalities., Springer, Dordrecht, 2009. doi: 10.1007/978-0-387-98128-4. [6] Emil Artin, The Gamma Function., Holt, Rinehart and Winston, New York-Toronto-London, 1964. [7] Maïtine Bergounioux, Antonio Leaci, Giacomo 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. [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. [9] Michele Caputo, Linear models of dissipation whose $Q$ is almost frequency independent. II., Fract. Calc. Appl. Anal., 11 (2008), 4-14. [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. [11] Alessandro Carbotti, Serena Dipierro and Enrico Valdinoci, Local Density Of Solutions To Fractional Equations., De Gruyter, 2019. [12] Michele Carriero, Antonio 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. [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. [14] Giovanni E. Comi and Giorgio Stefani, A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics I., arXiv: 1910.13419. [15] Françoise Demengel, Fonctions à hessien borné., Ann. Inst. Fourier (Grenoble), 34 (1984), 155-190. [16] Eleonora Di Nezza, Giampiero 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. [17] Mario Di Paola, Francesco 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. [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. [19] Bartłomiej Dyda, A fractional order Hardy inequality., Illinois J. Math., 48 (2004), 575-588. [20] Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties of functions., CRC Press, Boca Raton, FL, 2015. [21] Fausto Ferrari, Weyl and Marchaud Derivatives: A Forgotten History., Mathematics, 6 (2018) doi: 10.3390/math6010006. [22] Loukas Grafakos, Classical Fourier Analysis., Springer, New York, 2014. doi: 10.1007/978-1-4939-1194-3. [23] Loukas Grafakos, Modern Fourier Analysis., Springer, New York, 2014. doi: 10.1007/978-1-4939-1230-8. [24] G. H. Hardy and J. E. Littlewood, Some properties of fractional integrals. I., Math. Z., 27 (1928), 565-606.  doi: 10.1007/BF01171116. [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. [26] Gottfried Wilhelm Leibniz, Letter from Hanover, Germany, to G. F. A. L'Hôpital, September 30; 1695., Mathematische Schriften, 2 (1849), 301-302. [27] Luca Lombardini, Minimization problems involving nonlocal functionals: nonlocal minimal surfaces and a free boundary problem., arXiv: 1811.09746. [28] Alessandra Lunardi, Interpolation Theory., Edizioni della Normale, Pisa, 2018. doi: 10.1007/978-88-7642-638-4. [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. [30] Stefan G. Samko, Anatoly A. Kilbas and Oleg I. Marichev, Fractional Integrals and Derivatives., Gordon and Breach Science Publishers, Yverdon, 1993. [31] Armin Schikorra, Tien-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. [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. [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. [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. [35] Elias M. Stein, Singular integrals and differentiability properties of functions., Princeton University Press, Princeton, N. J., 1970.
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