## Taiwanese Journal of Mathematics

### MAXIMAL REGULARITY FOR INTEGRO-DIFFERENTIAL EQUATION ON PERIODIC TRIEBEL-LIZORKIN SPACES

#### Abstract

We study maximal regularity on Triebel-Lizorkin spaces $\mathrm{F} _{p,q}^s(\mathbb T, X)$ for the integro-differential equation with infinite delay: ($P_2$): $u'(t)=Au(t)+\int^{t}_{-\infty}a(t-s)Au(s)ds + f(t), \ (0\leq t \leq2\pi$) with the periodic condition $u(0)=u(2\pi)$, where $X$ is a Banach space, $a\in {\mathrm L}^1(\mathbb R_+)$ and $f$ is an $X$-valued function. Under a suitable assumption (H3) on the Laplace transform of $a$, we give a necessary and sufficient condition for ($P_2$) to have the maximal regularity property on $\mathrm{F} _{p,q}^s(\mathbb T, X)$.

#### Article information

Source
Taiwanese J. Math., Volume 12, Number 2 (2008), 281-292.

Dates
First available in Project Euclid: 20 July 2017

https://projecteuclid.org/euclid.twjm/1500574153

Digital Object Identifier
doi:10.11650/twjm/1500574153

Mathematical Reviews number (MathSciNet)
MR2402114

Zentralblatt MATH identifier
1151.45007

#### Citation

Bu, Shangquan; Fang, Yi. MAXIMAL REGULARITY FOR INTEGRO-DIFFERENTIAL EQUATION ON PERIODIC TRIEBEL-LIZORKIN SPACES. Taiwanese J. Math. 12 (2008), no. 2, 281--292. doi:10.11650/twjm/1500574153. https://projecteuclid.org/euclid.twjm/1500574153

#### References

• H. Amann, Operator-valued Fourier multipliers, vector-valued Besov spaces, and applications, Math. Nachr., 186 (1997), 5-56.
• W. Arendt and S. Bu, The operator-valued Marcinkiewicz multiplier theorem and maximal regularity, Math. Z., 240 (2002), 311-343.
• W. Arendt and S. Bu, Operator-valued Fourier multipliers on peoriodic Besov spaces and applications, Proc. Edinburgh Math. Soc., 47 (2004), 15-33.
• S. Bu and J. Kim, Operator-valued Fourier multipliers on peoriodic Triebel spaces, Acta Math. Sinica $($English Series$)$, 21(5) (2005), 1049-1056.
• J. Bergh and J. Löfström, Interpolation Spaces: an introduction, Springer-verlag, 1976.
• Ph. Clément, B. de Pagter, F. A. Sukochev and M. Witvliet, Schauder decomposition and multiplier theorems, Studia Math., 138 (2000), 135-163.
• Ph. Clément and J. Prüss, An operator-valued transference principle and maximal regularity on vector-valued $L_p$-spaces. in: Evolution Equations and Their Applications in Physics and Life Sciences, Lumer, Weis eds., Marcel Dekker (2000), 67-87.
• R. Denk, M. Hieber and J. Prüss, $R$-boundedness, Fourier Multipliers and Problems of Elliptic and Parabolic Type. Mem. Amer. Math. Soc., 166 (2003) p. 114.
• M. Girardi and L. Weis, Operator-valued Fourier multiplier theorems on $L_p(X)$ and geometry of Banach spaces, J. Funct. Analysis, 204 (2003), 320-354.
• N. J. Kalton, G. Lancien, A solution of the $L^p$-maximal regularity, Math. Z., 235 (2000), 559-568.
• V. Keyantuo and C. Lizama, Fourier multipliers and integro-differential equations in Banach spaces, J. London Math. Soc., 69(2) (2004), 737-750.
• V. Keyantuo and C. Lizama, Maximal regularity for a class of integro-differential equations with infinite delay in Banach spaces, Studia Math., 168(1) (2005), 25-50.
• H. J. Schmeisser and H. Triebel, Topics in Fourier Analysis and Function Spaces, Chichester, Wiley, 1987.
• L. Weis, Operator-valued Fourier multipliers and maximal $L_{p}$-regularity, Math. Ann., 319 (2001), 735-758.
• L. Weis, A new approach to maximal $L_{p}$-regularity, in: Evolution Equations and Their Applications in Physics and Life Sciences, Lumer, Weis eds., Marcel Dekker (2000), 195-214.