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Sharp Sobolev space estimates for solutions of neutral difference-differential equations with arbitrary index are obtained without the assumption that the roots of the characteristic quasipolynomial are separated. The proof is based on the fact that the system of divided differences of the exponential solutions forms a Riesz basis. Moreover, it is proved that, under more general conditions, the system of exponential solutions is minimal and complete. §0. Introduction In the present paper, we study the initial problem for the neutral difference-differential equation a0mu (t) + · · ·+ akmu(t− hk) + · · ·+ anmu(t− hn) = 0, t > 0, with several delays and with the initial condition u|(−hn,0) = g. Here, the terms with integrals and the terms with lower derivatives are omitted. The solutions are studied in the scale of Sobolev spacesH for s ≥ m. The basic assumption is that a0m 6= 0 and anm 6= 0. Under this assumption, the zeros {λq} of the characteristic quasipolynomial L(λ) lie in the strip κ− ≤ Re z ≤ κ+, where κ+ := sup Reλq and κ− := inf Reλq. Using some properties of the family of exponential solutions, we prove the following sharp estimates for the solutions: ‖u‖Hs(T−h,T ) ≤ d(T + 1)M−1eκ+T ‖g‖Hs(−h,0), T ≥ 0, (0.1) ‖u‖Hs(T−h,T ) ≥ c(g)eκ−T , c > 0, (0.2) where the constant M is determined by the function L(λ). We note that the estimates obtained are valid also for all s ≥ m such that s is not a half-integer even if the zeros of L(z) are not separated. Under the separation assumption, the above estimates were obtained for all integral s in [7], [8]; for the nonintegral s, they were announced in [9], [27]. Estimates similar to (0.1) but with κ+ replaced by κ+ + e, e > 0, have long been known; see, e.g., [10], [14], [15]. Our estimate (0.1) is sharp in the following sense: the constants κ± cannot be replaced by κ+ − e and κ− + e, respectively, with any e > 0. Moreover, the exponent M also cannot be reduced to fit for all g. The proof is based on the properties of the family V of exponential solutions of the problem (in the case where the quasipolynomial L(λ) has only simple roots λq, this family 2000 Mathematics Subject Classification. Primary 34K40, 42B30, 46E35.

Sobolev space estimates for solutions of equations with delay, and the basis of divided differences