REGULARIZED FUNCTIONAL CALCULI , SEMIGROUPS , AND COSINE FUNCTIONS FOR PSEUDODIFFERENTIAL OPERATORS

Let iAj(1 ≤ j ≤ n) be generators of commuting bounded strongly continuous groups, A ≡ (A1, A2, ..., An). We show that, when f has sufficiently many polynomially bounded derivatives, then there exist k, r > 0 such that f(A) has a (1+ |A|2)−r-regularized BCk(f(Rn)) functional calculus. This immediately produces regularized semigroups and cosine functions with an explicit representation; in particular, when f(Rn) ⊆ R, then, for appropriate k, r, t → (1 − it)−ke−itf(A)(1 + |A|2)−r is a Fourier-Stieltjes transform, and when f(Rn) ⊆ [0, ∞), then t → (1+t)−ke−tf(A)(1+|A|2)−r is a Laplace-Stieltjes transform. With A ≡ i(D1, ..., Dn), f(A) is a pseudodifferential operator on Lp(Rn)(1 ≤ p < ∞) or BUC(Rn). 0. Introduction In finite dimensions, the Jordan canonical form for matrices guarantees that, although a linear operator may not be diagonalizable, which is equivalent to having a BC(C) functional calculus, it will be generalized scalar, that is, have a BC(C) functional calculus, for some k; specifically, k may be chosen to be n− 1, where n is the order of the largest Jordan block. In infinite dimensions, even a bounded linear operator on a Hilbert space may fail to be generalized scalar; consider the left shift on 2. Our favorite unbounded operators fail to be generalized scalar, on Banach spaces that are not Hilbert spaces. The operator i d dx , on L 2(R), is selfadjoint and thus has a BC(R) functional calculus. However, on L(R), p = 2, it does not have a BC(R) functional calculus, for any nonnegative integer m; that is, it is not even generalized scalar (see [2, Lemma 5.3]). 1991 Mathematics Subject Classification. Primary 47A60; secondary 47D03, 47D06, 47D09, 47F05.


Introduction
In finite dimensions, the Jordan canonical form for matrices guarantees that, although a linear operator may not be diagonalizable, which is equivalent to having a BC(C) functional calculus, it will be generalized scalar, that is, have a BC k (C) functional calculus, for some k; specifically, k may be chosen to be n − 1, where n is the order of the largest Jordan block.
In infinite dimensions, even a bounded linear operator on a Hilbert space may fail to be generalized scalar; consider the left shift on 2 .
Our favorite unbounded operators fail to be generalized scalar, on Banach spaces that are not Hilbert spaces.The operator i d dx , on L 2 (R), is selfadjoint and thus has a BC(R) functional calculus.However, on L p (R), p = 2, it does not have a BC m (R) functional calculus, for any nonnegative integer m; that is, it is not even generalized scalar (see [2,Lemma 5.3]).Differential operators in more than one dimension may be even more poorly behaved.For any n > 1, there exist constant coefficient differential operators on L p (R n ) that are not even decomposable, for any p = 2 ([1, Corollary 3.5]).
In this paper, we show that constant coefficient differential operators p(D), on L p (R n )(1 ≤ p < ∞) or BU C(R n ), have a (1 + ) −r -regularized BC k (p(R n )) functional calculus, for appropriate numbers r and k, where is the Laplacian, p is a polynomial.This means that, for any g ∈ BC k (p(R n )), g(p(D))(1 + ) −r is a bounded operator.More generally, if iA 1 , ..., iA n generate commuting bounded strongly continuous groups, A ≡ (A 1 , ..., A n ) and f has sufficiently many polynomially bounded derivatives, then f (A) has a (1 + |A| 2 ) −r -regularized BC k (f (R n )) functional calculus (Theorem 2.17).See [8] for regularized BC k (R) functional calculi for generators of polynomially bounded groups.
In Section I we give some preliminary material relating regularized functional calculi to regularized semigroups and cosine functions.Our main results are in Section II.Section III has the particular case of pseudodifferential operators on the usual function spaces See [7] for regularized functional calculi for the Schrödinger operator with potential, on such spaces.
All operators are linear, on a Banach space, X.We will write D(B) for the domain of the operator B, ρ(B) for its resolvent set, Im(B) for the image of B. We will denote by B(X) the space of all bounded operators from X into itself.Throughout this paper, C ∈ B(X) is injective, and commutes with B; that is, CB ⊆ BC.When B generates a strongly continuous semigroup, we will denote that semigroup by {e tB } t≥0 ; see [9] or [14] for material on strongly continuous semigroups and their applications.

Regularized functional calculi, regularized semigroups and regularized cosine functions
We show in this section how a regularized functional calculus produces intuitively natural constructions of regularized semigroups and regularized cosine functions.Growth estimates also follow automatically.
Definition 1.3.Suppose F is a Banach algebra of complex-valued functions, defined on a subset of the complex plane such that f 0 (z where f 1 (z) ≡ z; and (3) f 0 (B)C = C.
Remark 1.4.When F contains f 0 and g λ (z) ≡ (λ−z) −1 , for some complex λ, then (1), ( 2) and (3) of Definition 1.3 are equivalent to (1), ( 2) and (3), where (2 ) is the following: See [6] and [8] for some basic results on regularized functional calculi.Note that an I-regularized F functional calculus is a F functional calculus.Definition 1.5.A C-regularized semigroup generated by B is a strongly continuous family (2) W (t)W (s) = CW (t + s), for all s, t ≥ 0; and , with maximal domain.See [4] and the references therein, for basic material on regularized semigroups and their relationship to the abstract Cauchy problem.
A regularized cosine function deals with ill-posed second-order abstract Cauchy problems just as regularized semigroups deal with ill-posed firstorder abstract Cauchy problems.

and C((D(B)
) is dense, the same is true for all x ∈ X; that is, {W 0 (t)} t≥0 is strongly continuous.The algebraic properties of a regularized semigroup, for {W j (t)} t≥0 , follow from the definition of a C-regularized functional calculus.Thus, for j = 0, 1, 2, When the half-plane in Proposition 1.7 is replaced by the real line ([0, ∞)), we get a nice representation of the regularized semigroup, as a Fourier-Stieltjes (Laplace-Stieltjes) transform.Lemma 1.9.Suppose {W (t)} t≥0 is an exponentially bounded C-regularized semigroup generated by B. Then Proof.There exists a Banach space Z, continuously embedded between Im(C) and X, such that B| Z generates a strongly continuous semigroup, and Since the norm in Z is stronger than the norm in X, and W (t)x ∈ Z, for all x ∈ X, t ≥ 0, the result follows.
Proposition 1.10. ( Laplace-Stieltjes transform of a complex-valued measure of bounded variation. Proof.We will prove (1); it will be clear how the proof would be modified for (2).It follows from Proposition 1.7 that −iC −1 BC generates a C-regularized group {W (t)} t∈R , given by defines a bounded linear functional on C 0 (R), there exists a complex-valued measure of bounded variation, µ, such that gives us, by Lemma 1.9 and dominated convergence, for any t ≥ 0, e −its dµ(s).

Functional calculus on function spaces with polynomial growth conditions
Throughout this section, iA 1 , iA 2 , ..., iA n are generators of commuting bounded strongly continuous groups {e We will use some standard terminology.We will write Let F be the Fourier transform, F L 1 be the set of all inverse Fourier transforms of L 1 functions; that is, (2.1) Define, for f ∈ F L 1 , a bounded operator f (A) by: (2.2) We define the operator −|A| 2 as the generator of the strongly continuous semigroup {(z → e −t|z| 2 )(A)} t≥0 .

Lemma 2.3.
(a) (fg Assertions (a) and (b) are straightforward to verify, and (d) is well-known.For (c), we need the following.

Lemma 2.4 ([5, Lemma 2.2]). If A has a F functional calculus, and
Proof of Lemma 2.3(c).First, note that, since Thus we may apply Lemma 2.4 as follows. ( It is easy to check that B(l, k) is a Banach algebra, and (2) If f (t, •) is a family of functions in B(l, k) with a parameter t ≥ 0 satisfying: , where M 2 (t) ≥ 1, then there exists a constant M so that 1 and there exists M (s) ≥ 0 such that: . By Leibniz's formula, (2.9) Now we are going to follow a proof similar to the proof in [13,Lemma 2.2].By [11,Lemma 2.3], there exists a (2.10) Using Leibniz's formula, we have So, where 1 {2 m−1 ≤|x|≤2 m+1 } (x) is the characteristic function.Therefore (2.12) Using (2.12) when |α| = k and α = 0, it follows from Bernstein's theorem that g m ∈ F L 1 and: Therefore, when s > n Combining (2.10) and (2.13) concludes the proof of (1).
(2) Following exactly the same proof as in (1), replacing f (x) with f (t, x) we can show that f (t, •) ∈ F L 1 and Then Lemma 2.3 (b) concludes the proof.
Note that, by Theorem 2.7, Definition 2.15 applies to any f with [ n 2 ] + 1 polynomially bounded derivatives.
Assertion (b) follows from the fact that (1 Note that, by (b) of Lemma 2.16 and Lemma 2.3(c), the definition of Proof.According to Theorem 2.7(1), we must first show that g • f is in B(µ, k), for all g ∈ BC k (f (R n )) and there exists M ≥ 0 such that By induction on |α|, for any where A β has the form The growth conditions on D α f now imply that, for any . Theorem 2.7 and (2.18) imply that By Lemma 2.3(a), g → g(B) satisfies (1) of Definition 1.3.Suppose now that both g and gf 1 (see Definition 1.3(2)) are in BC k (f (R n )).Then for m sufficiently large, Also, from the last two lines, In fact, such a regularized semigroup can also be constructed without the polynomial growth conditions on f , using Theorem 2.1; see [4,Definition 12.10], where f (A) is defined as the generator of the regularized semigroup {(z → e tf (z) g(z))(A)} t≥0 , for appropriate g.
The proof of Corollary 2.20, with z → e tf (z) replaced by cosh(t f (z)), gives us the following.Remark 2.25.See [16] for cosine functions generated by p(A), where p is a polynomial.Finally, Theorem 2.17 and Proposition 1.10 immediately give us the following two corollaries.Corollary 2.26.Suppose f is as in Theorem 2.17 and f (R n ) ⊆ R.Then, for all s > n 2 , i(f (A)) generates a norm-continuous (1 + |A| 2 ) − µ+1 2 s -regularized group {W (t)} t≥R such that, for all x ∈ X, x * ∈ X * , the map t → (1 − it) −k W (t)x, x * is a Fourier-Stieltjes transform of a complex-valued measure of bounded variation.

Definition 1 . 1 .Definition 1 . 2 .
The complex number λ is in ρ C (B), the C-resolvent of B, if (λ − B) is injective and Im(C) ⊆ Im(λ − B).Denote by B C (X) the space of all operators G such that GC ∈ B(X), with norm