doi: 10.3934/cpaa.2022118
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The Riemann-Liouville fractional integral in Bochner-Lebesgue spaces I

1. 

Department of Mathematics, Federal University of Santa Catarina, R. Eng. Agronômico Andrei Cristian Ferreira, Florianópolis SC, Brazil

2. 

Department of Mathematics, Federal University of Espírito Santo, Av. Fernando Ferrari, 514, Vitória - ES, Brazil

*Corresponding author: Renato Fehlberg Júnior

Received  February 2022 Revised  May 2022 Early access July 2022

In this paper we study the Riemann-Liouville fractional integral of order $ \alpha>0 $ as a linear operator from $ L^p(I,X) $ into itself, when $ 1\leq p\leq \infty $, $ I=[t_0,t_1] $ (or $ I=[t_0,\infty) $) and $ X $ is a Banach space. In particular, when $ I=[t_0,t_1] $, we obtain necessary and sufficient conditions to ensure its compactness. We also prove that Riemann-Liouville fractional integral defines a $ C_0- $semigroup but does not defines a uniformly continuous semigroup. We close this study by presenting lower and higher bounds to the norm of this operator.

Citation: Paulo Mendes de Carvalho Neto, Renato Fehlberg Júnior. The Riemann-Liouville fractional integral in Bochner-Lebesgue spaces I. Communications on Pure and Applied Analysis, doi: 10.3934/cpaa.2022118
References:
[1]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York, 1972.

[2]

M. AllenL. Caffarelli and A. 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]

W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-Valued Laplace Transforms and Cauchy Problems, Second Edition, Birkhäuser, Basel, 2011. doi: 10.1007/978-3-0348-0087-7.

[4]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011.

[5]

P. M. Carvalho-Neto and R. Fehlberg Júnior, On the fractional version of Leibniz rule, Math. Nachr., 293 (2020), 670-700.  doi: 10.1002/mana.201900097.

[6]

L. Chen, Nonlinear stochastic time-fractional diffusion equations on $\mathbb{R}$: moments, Hölder regularity and intermittency, Trans. Amer. Math. Soc., 369 (2017), 8497-8535.  doi: 10.1090/tran/6951.

[7]

D. DierJ. KemppainenJ. Siljander and R. Zacher, On the parabolic Harnack inequality for non-local diffusion equations, Math. Z., 295 (2020), 1751-1769.  doi: 10.1007/s00209-019-02421-7.

[8]

H. Dong and D. Kim, $L_p$-estimates for time fractional parabolic equations with coefficients measurable in time, Adv. Math., 345 (2019), 289-345.  doi: 10.1016/j.aim.2019.01.016.

[9]

H. Dong and D. Kim, $L_p$-estimates for time fractional parabolic equations in divergence form with measurable coefficients, J. Funct. Anal., 278 (2020). doi: 10.1016/j. jfa. 2019.108338.

[10]

G. B. Folland, Real Analysis: Modern Techniques and Their Applications, John Wiley & Sons, 1999.

[11]

J. Diestel and J. J. Uhl Jr., Vector Measures, Mathematical Surveys and Monographs, Am. Math. Soc., Vol 15, 1977.

[12]

P. N. DowlingZ. Hu and D. Mupasiri, Complex convexity in Lebesgue-Bochner function spaces, Trans. Amer. Math. Soc., 348 (1996), 127-139.  doi: 10.1090/S0002-9947-96-01508-5.

[13]

Y. Giga and T. Namba, Well-posedness of Hamilton-Jacobi equations with Caputo's time fractional derivative, Commun. Partial Differ. Equ., 42 (2017), 1088-1120.  doi: 10.1080/03605302.2017.1324880.

[14]

G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, Cambridge Mathematical Library, 1952.

[15]

E. Hille and R. S. Phillips, Functional Analysis and Semi-groups, Publications Amer. Mathematical Soc., Colloquium Publications, Vol. 31, 1996.

[16]

E. W. Hobson, On the second mean-value theorem of the integral calculus, Proc. London Math. Soc., S2-7 (1909), 14-23.  doi: 10.1112/plms/s2-7.1.14.

[17]

Y. KianZ. LiY. Liu and M. Yamamoto, The uniqueness of inverse problems for a fractional equation with a single measurement, Math. Ann., 380 (2021), 1465-1495.  doi: 10.1007/s00208-020-02027-z.

[18]

I. KimK. -H. Kim and S. Lim, An Lq(Lp)-theory for the time fractional evolution equations with variable coefficients, Adv. Math., 306 (2017), 123-176.  doi: 10.1016/j.aim.2016.08.046.

[19]

G. W. Leibniz, Letter from Hanover, Germany, to G. F. A. L'Hopital, September 30, 1695, Math. Schr., 2 (1849), 301-302. 

[20]

E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. Math., 118 (1983), 349-374.  doi: 10.2307/2007032.

[21]

C. MartinezM. Sanz and D. Martinez, About fractional integrals in the space of locally integrable functions, J. Math. Anal. Appl., 167 (1992), 111-122.  doi: 10.1016/0022-247X(92)90239-A.

[22]

J. Mikusiński, The Bochner Integral, Mathematische Reihe, Birkhäuser Basel, 1978.

[23]

T. Namba and P. Rybka, On viscosity solutions of space-fractional diffusion equations of Caputo type, SIAM J. Math. Anal., 52 (2020), 653-681.  doi: 10.1137/19M1259316.

[24]

A. Pazy, Semigroups of Linear Operators and Applications to PDEs, Springer, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[25]

J. Pettis, On integration in vector spaces, Trans. Amer. Math. Soc., 44 (1938), 277-304.  doi: 10.2307/1989973.

[26]

R. Ponce, On the well-posedness of degenerate fractional differential equations in vector valued function spaces, Israel J. Math., 219 (2017), 727-755.  doi: 10.1007/s11856-017-1496-9.

[27]

B. Ross, A brief history and exposition of the fundamental theory of fractional calculus, Fract. Calc. Appl. Anal., 457 (1975), 1-36. 

[28]

B. Ross, The development of fractional calculus 1695-1900, Hist. Math., 4 (1977), 75-89.  doi: 10.1016/0315-0860(77)90039-8.

[29]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon & Breach Sci. Publishers, Yverdon, 1993.

[30]

L. Schwartz, Théorie des distributions à valeurs vectorielles, I, Ann. de l'Institut Fourier, 7 (1957), 1-141. 

[31]

R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Mathematical Surveys and Monographs, AMS, 1997. doi: 10.1090/surv/049.

[32]

J. Simon, Compact sets in the space $L^p(0, T;B)$, Ann. di Mat. Pura ed Appl., 146 (1986), 65-96.  doi: 10.1007/BF01762360.

[33]

M. A. Smith, Rotundity and extremity in $l^p(X_i)$ and $L^p(u, X)$, Contemp. Math., 52 (1986), 143-162.  doi: 10.1090/conm/052/840706.

[34]

L. L. Vrabie, $C_0$-Semigroups and Applications, Elsevier, Amsterdam, 2003.

[35]

A. H. Zemanian, Realizability Theory for Continuous Linear Systems, Elsevier, 1973.

show all references

References:
[1]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York, 1972.

[2]

M. AllenL. Caffarelli and A. 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]

W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-Valued Laplace Transforms and Cauchy Problems, Second Edition, Birkhäuser, Basel, 2011. doi: 10.1007/978-3-0348-0087-7.

[4]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011.

[5]

P. M. Carvalho-Neto and R. Fehlberg Júnior, On the fractional version of Leibniz rule, Math. Nachr., 293 (2020), 670-700.  doi: 10.1002/mana.201900097.

[6]

L. Chen, Nonlinear stochastic time-fractional diffusion equations on $\mathbb{R}$: moments, Hölder regularity and intermittency, Trans. Amer. Math. Soc., 369 (2017), 8497-8535.  doi: 10.1090/tran/6951.

[7]

D. DierJ. KemppainenJ. Siljander and R. Zacher, On the parabolic Harnack inequality for non-local diffusion equations, Math. Z., 295 (2020), 1751-1769.  doi: 10.1007/s00209-019-02421-7.

[8]

H. Dong and D. Kim, $L_p$-estimates for time fractional parabolic equations with coefficients measurable in time, Adv. Math., 345 (2019), 289-345.  doi: 10.1016/j.aim.2019.01.016.

[9]

H. Dong and D. Kim, $L_p$-estimates for time fractional parabolic equations in divergence form with measurable coefficients, J. Funct. Anal., 278 (2020). doi: 10.1016/j. jfa. 2019.108338.

[10]

G. B. Folland, Real Analysis: Modern Techniques and Their Applications, John Wiley & Sons, 1999.

[11]

J. Diestel and J. J. Uhl Jr., Vector Measures, Mathematical Surveys and Monographs, Am. Math. Soc., Vol 15, 1977.

[12]

P. N. DowlingZ. Hu and D. Mupasiri, Complex convexity in Lebesgue-Bochner function spaces, Trans. Amer. Math. Soc., 348 (1996), 127-139.  doi: 10.1090/S0002-9947-96-01508-5.

[13]

Y. Giga and T. Namba, Well-posedness of Hamilton-Jacobi equations with Caputo's time fractional derivative, Commun. Partial Differ. Equ., 42 (2017), 1088-1120.  doi: 10.1080/03605302.2017.1324880.

[14]

G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, Cambridge Mathematical Library, 1952.

[15]

E. Hille and R. S. Phillips, Functional Analysis and Semi-groups, Publications Amer. Mathematical Soc., Colloquium Publications, Vol. 31, 1996.

[16]

E. W. Hobson, On the second mean-value theorem of the integral calculus, Proc. London Math. Soc., S2-7 (1909), 14-23.  doi: 10.1112/plms/s2-7.1.14.

[17]

Y. KianZ. LiY. Liu and M. Yamamoto, The uniqueness of inverse problems for a fractional equation with a single measurement, Math. Ann., 380 (2021), 1465-1495.  doi: 10.1007/s00208-020-02027-z.

[18]

I. KimK. -H. Kim and S. Lim, An Lq(Lp)-theory for the time fractional evolution equations with variable coefficients, Adv. Math., 306 (2017), 123-176.  doi: 10.1016/j.aim.2016.08.046.

[19]

G. W. Leibniz, Letter from Hanover, Germany, to G. F. A. L'Hopital, September 30, 1695, Math. Schr., 2 (1849), 301-302. 

[20]

E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. Math., 118 (1983), 349-374.  doi: 10.2307/2007032.

[21]

C. MartinezM. Sanz and D. Martinez, About fractional integrals in the space of locally integrable functions, J. Math. Anal. Appl., 167 (1992), 111-122.  doi: 10.1016/0022-247X(92)90239-A.

[22]

J. Mikusiński, The Bochner Integral, Mathematische Reihe, Birkhäuser Basel, 1978.

[23]

T. Namba and P. Rybka, On viscosity solutions of space-fractional diffusion equations of Caputo type, SIAM J. Math. Anal., 52 (2020), 653-681.  doi: 10.1137/19M1259316.

[24]

A. Pazy, Semigroups of Linear Operators and Applications to PDEs, Springer, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[25]

J. Pettis, On integration in vector spaces, Trans. Amer. Math. Soc., 44 (1938), 277-304.  doi: 10.2307/1989973.

[26]

R. Ponce, On the well-posedness of degenerate fractional differential equations in vector valued function spaces, Israel J. Math., 219 (2017), 727-755.  doi: 10.1007/s11856-017-1496-9.

[27]

B. Ross, A brief history and exposition of the fundamental theory of fractional calculus, Fract. Calc. Appl. Anal., 457 (1975), 1-36. 

[28]

B. Ross, The development of fractional calculus 1695-1900, Hist. Math., 4 (1977), 75-89.  doi: 10.1016/0315-0860(77)90039-8.

[29]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon & Breach Sci. Publishers, Yverdon, 1993.

[30]

L. Schwartz, Théorie des distributions à valeurs vectorielles, I, Ann. de l'Institut Fourier, 7 (1957), 1-141. 

[31]

R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Mathematical Surveys and Monographs, AMS, 1997. doi: 10.1090/surv/049.

[32]

J. Simon, Compact sets in the space $L^p(0, T;B)$, Ann. di Mat. Pura ed Appl., 146 (1986), 65-96.  doi: 10.1007/BF01762360.

[33]

M. A. Smith, Rotundity and extremity in $l^p(X_i)$ and $L^p(u, X)$, Contemp. Math., 52 (1986), 143-162.  doi: 10.1090/conm/052/840706.

[34]

L. L. Vrabie, $C_0$-Semigroups and Applications, Elsevier, Amsterdam, 2003.

[35]

A. H. Zemanian, Realizability Theory for Continuous Linear Systems, Elsevier, 1973.

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