The ␣-times integrated semigroups, ␣ ) 0, are introduced and analyzed. It is shown that suitable differential operators generate ␣-times integrated semigroups Ž . for ␣ g 1r2, 1 . ᮊ 1997 Academic Press ␣ in Proposition 1, Corollary 1, Theorem 2, and Corollary 2 the properties of the generator A of an ␣-times integrated semigroup S and apply the ␣ 790

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We give necessary and sufficient conditions for the continuity of the Hilbert transform on complex quasi-Hilbert spaces, i.e. on complex, reflexive, strictly convex Banach spaces with Gâteauxdifferentiable norm and with generalized inner product.

We obtain a formula of decomposition for Φ(A) = A R n S(f (x))ϕ(x) dx + R n ϕ(x) dx using the method of stationary phase. Here (S(t)) t∈R is once integrated, exponentially bounded group of operators in a Banach space X, with generator A, which satisfies the condition: For every x ∈ X there exists δ = δ(x) > 0 such that S(t)x t 1/2+δ → 0 as t → 0. The function ϕ(x) is infinitely differentiable, defined on R n , with values in X, with a compact support supp ϕ, the function f (x) is infinitely differentiable, defined on R n , with values in R, and f (x) on supp ϕ has exactly one nondegenerate stationary point x 0 .