Chin. Ann. Math. 30B(1), 2009, 39–50 DOI: 10.1007/s11401-007-0553-9 Chinese Annals of Mathematics, Series B c  The Editorial Office of CAM and Springer-Verlag Berlin Heidelberg 2009 Maximal B-Regular Integro-Differential Equation Veli SHAKHMUROV ∗ Rishad SHAHMUROV ∗∗ Abstract By using Fourier multiplier theorems, the maximal B-regularity of ordinary integro-differential operator equations is investigated. It is shown that the corresponding differential operator is positive and satisfies coercive estimate. Moreover, these results are used to establish maximal regularity for infinite systems of integro-differential equations. Keywords Banach-valued Besov spaces, Operator-valued multipliers, Boundary value problems, Integro-differential equations 2000 MR Subject Classification 34G10, 35J25, 35J70 1 Introduction, Notations and Background In recent years, the maximal regularity of differential operator equations has been studied extensively, e.g. in [1–4, 7–9]. Moreover, integro-differential equations (IDEs) have been studied, e.g. in [6, 10–12] and the reference therein. However, the integro-differential operator equation (IDOE) is a relatively less investigated subject. The main aim of present paper is to establish the maximal regularity of convolution differential operator equation Lu = l  k=0 a k ∗ d k u d x k + A ∗ u = f (x) in E-valued Besov spaces, where E is an arbitrary Banach space, A = A(x) is a possible unbounded operator in E, a k = a k (x) are complex-valued functions. Particularly, we prove that the differential operator generated by this equation is a generator of analytic semigroup. Let x =(x 1 ,x 2 , ··· ,x n ) ∈ Ω ⊂ R n . L p (Ω; E) denotes the space of all strongly measurable E-valued functions that are defined on the measurable subset Ω ⊂ R n with the norm ‖f ‖ Lp(Ω;E) =   ‖f (x)‖ p E d x  1 p , 1 ≤ p ≤∞, ‖f ‖ L∞(Ω;E) = ess sup x∈Ω [‖f (x)‖ E ]. Let S = S(R n ; E) denote a Schwartz class, i.e., a space of E-valued rapidly decreasing smooth functions on R n and S ′ (R n ; E) denotes the space of E-valued tempered distributions. Let α =(α 1 ,α 2 , ··· ,α n ), where α i are integers. An E-valued generalized function D α f is called Manuscript received December 25, 2007. Revised April 21, 2008. Published online January 9, 2009. * Department of Electronics Engineering and Communication, Okan University, Istanbul 34959, Turkey. E-mail: veli.sahmurov@okan.edu.tr ** Vocational High School, Okan University, Istanbul 34959, Turkey. E-mail: shahmurov@hotmail.com