This paper is concerned with evolution equations of fractional order Dαu(t)=Au(t); u(0)=u0, u′(0)=0, where A is a differential operator corresponding to a coercive polynomial taking values in a sector of angle less than π and 1<α<2. We show that such equations are well posed in the sense that there always exists an α-times resolvent family for the operator A.

1. Introduction

It is well known that the abstract Cauchy problem of first order
u′(t)=Au(t),t>0;u(0)=x
is well posed if and only if A is the generator of a C0-semigroup. However, many partial differential operators (PDOs) such as the Schrödinger operator iΔ on Lp(ℝn) (p≠2) cannot generate C0-semigroups. It was Kellermann and Hieber [1] who first showed that some elliptic differential operators on some function spaces generate integrated semigroups, and their results are improved and developed in [2, 3]. Because of the limitations of integrated semigroups, the results in [1–3] are confined to elliptic differential operators with constant coefficients. One of the limitations is that the resolvent sets of generators must contain a right half-plane; however, it is known that there are many nonelliptic operators whose resolvent sets are empty (see, e.g., [4]). On the other hand, the resolvent sets of the generators of regularized semigroups need not be nonempty; this makes it possible to apply the theory of regularized semigroups to nonelliptic operators, such as coercive operators and hypoelliptic operators (see [5–8]). Moreover, for second-order equations, Zheng [9] considered coercive differential operators with constant coefficients generating integrated cosine functions. The aim of this paper is to consider fractional evolution equations associated with coercive differential operators.

Let X be a Banach space, and let A be a closed linear unbounded operator with densely defined domain D(A). A family of strongly continuous bounded linear operators on X, {R(t)}t≥0, is called a resolvent family for A with kernel a(t)∈Lloc1(ℝ+) if R(t)A⊂AR(t) and the resolvent equationR(t)x=x+∫0ta(t-s)AR(s)xds,t≥0,x∈D(A)
holds. It is obvious that a C0-semigroup is a resolvent family for its generator with kernel a1(t)≡1; a cosine function is a resolvent family for its generator with kernel a2(t)=t. If we define the α-times resolvent family for A as being a resolvent family with kernel gα(t):=tα-1/Γ(α), then such resolvent families interpolate C0-semigroups and cosine functions.

Recently Bazhlekova studied classes of such resolvent families (see [10]). Let 0<α≤2, and let m be the smallest integer greater than or equal to α. It was shown in [10] that the fractional evolution equation of order α,
Dαu(t)=Au(t),t>0;u(k)(0)=xk,k=0,1,…,m-1,
is well posed if and only if there exists an α-times resolvent family for A. Here Dα is the Caputo fractional derivative of order α>0 defined by
Dαf(t):=∫0tgm-α(t-s)dmdsmf(s)ds,
where f∈Wm,1(I) for every interval I. The hypothesis on f can be relaxed; see [10] for details. Fujita in [11] studied (1.3) for the case that A=Δ, the Laplacian (∂/∂x)2 on ℝ, which interpolates the heat equation and the wave equation. Since α-times resolvent families interpolate C0-semigroups and cosine functions, this motivates us to consider the existence of fractional resolvent families for PDOs.

There are several examples of the existence of α-times resolvent families for concrete PDOs in [10], but Bazhlekova did not develop the theory of α-times resolvent families for general PDOs. The authors showed in [12] that there exist fractional resolvent families for elliptic operators. In this paper we will consider coercive operators. Since α-times resolvent families are not sufficient for applications we have in mind, we first extend, in Section 2, such a notion to the setting of C-regularized resolvent families which was introduced in [13]. To do this, we use methods of the Fourier multiplier theory.

This paper is organized as follows. Section 2 contains the definition and some basic properties of α-times regularized resolvent families. Section 3 prepares for the proof of the main result of this paper. Our main result, Theorem 4.1, shows that there are α-times regularized resolvent families for PDOs corresponding to coercive polynomials taking values in a sector of angle less than π. Some examples are also given in Section 4.

2. <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M56"><mml:mrow><mml:mi mathvariant="bold-italic">α</mml:mi></mml:mrow></mml:math></inline-formula>-Times Regularized Resolvent Family

Throughout this paper, X is a complex Banach space, and we denote by B(X) the algebra of all bounded linear operators on X. Let A be a closed densely defined operator on X, let D(A) and R(A) be its domain and range, respectively, and let α∈(0,2], C∈B(X) be injective. Define ρC(A):={λ∈ℂ:λ-Ais injective andR(C)⊂R(λ-A)}. Let Σθ:={λ∈ℂ:|argλ|<θ} be the open sector of angle 2θ in the complex plane, where arg is the branch of the argument between -π and π.

Definition 2.1.

A strongly continuous family {Sα(t)}t≥0⊂B(X) is called an α-times C-regularized resolvent family for A if

Sα(0)=C;

Sα(t)A⊂ASα(t) for t≥0;

C-1AC=A;

for x∈D(A), Sα(t)x=Cx+∫0t((t-s)α-1/Γ(α))Sα(s)Axds.

{Sα(t)}t≥0 is called analytic if it can be extended analytically to some sector Σθ.

If ∥Sα(t)∥≤Meωt (t≥0) for some constants M≥1 and ω∈ℝ+, we will write A∈𝒞Cα(M,ω), and 𝒞Cα(ω):=∪{𝒞Cα(M,ω);M≥1}, 𝒞Cα:=∪{𝒞Cα(ω);ω≥0}.

Define the operator Ã by
Ãx=C-1(limt↓0Γ(α+1)tα(Sα(t)x-Cx)),x∈D(Ã),
with
D(Ã)={x∈X:limt↓0Sα(t)x-CxtαexistsandisinR(C)}.

Proposition 2.2.

Suppose that there exists an α-times C-regularized resolvent family, {Sα(t)}t≥0, for the operator A, and let Ã be defined as above. Then A=Ã.

Proof.

By the strong continuity of Sα(t), we have for every x∈X,
∥gα+1(t)-1∫0tgα(t-s)Sα(s)xds-Cx∥≤gα+1(t)-1∫0tgα(t-s)∥Sα(s)x-Cx∥ds≤sup0≤s≤t∥Sα(s)x-Cx∥→0ast→0.
Thus for x∈D(A), by Definition 2.1,
limt↓0gα+1(t)-1(Sα(t)x-Cx)=limt↓0gα+1(t)-1∫0tgα(t-s)Sα(s)Axds=CAx,
which means that x∈D(Ã) and Ãx=Ax. On the other hand, for x∈D(Ã), by the definition of Ã and Definition 2.1,
CÃx=limt↓0gα+1(t)-1(Sα(t)x-Cx)=limt↓0gα+1(t)-1A∫0tgα(t-s)Sα(s)xds,
but limt→0gα+1(t)-1∫0tgα(t-s)Sα(s)xds=Cx, by (d) of Definition 2.1. Thus it follows from the closedness of A that Cx∈D(A) with ACx=CÃx. This implies that x∈D(C-1AC)=D(A), so we have Ã=A.

The following generation theorem and subordination principle for α-times C-regularized resolvent families can be proved similarly as those for α-times resolvent families (see [10]).

Theorem 2.3.

Let α∈(0,2]. Then the following statements are equivalent:

A∈𝒞Cα(M,ω);

A=C-1AC, (ωα,∞)⊆ρC(A) and
∥dndλn(λα-1(λα-A)-1C)∥≤Mn!(λ-ω)n+1,λ>ω,n∈ℕ0:=ℕ∪{0};

A=C-1AC, (ωα,∞)⊆ρC(A) and there exists a strongly continuous family {Sα(t)}t≥0⊂B(X) satisfying ∥Sα(t)∥≤Meωt such that
λα-1(λα-A)-1Cx=∫0∞e-λtSα(t)xdt,λ>ω,x∈X.

Theorem 2.4.

Suppose that 0<α<β≤2, γ=α/β. If A∈𝒞Cβ(ω) then A∈𝒞Cα(ω1/γ) and the α-times C-regularized resolvent family for A, {Sα(t)}t≥0, can be extended analytically to Σmin{θ(γ),π}, where θ(γ):=(1/γ-1)π/2.

3. Coercive Operators and Mittag-Leffler Functions

We now introduce a functional calculus for generators of bounded C0-groups (cf. [14]), which will play a key role in our proof.

Let iAj(1≤j≤n) be commuting generators of bounded C0-groups on a Banach space X. Write A=(A1,…,An) and Aμ=A1μ1⋯Anμn for μ=(μ1,…,μn)∈ℕ0n. Similarly, write Dμ=D1μ1⋯Dnμn, where Dj=-i∂/∂xj for j=1,…,n. For a polynomial P(ξ):=∑|μ|≤maμξμ(ξ∈ℝn)(|μ|:=∑j=1nμj) with constant coefficients, we define P(A)=∑|μ|≤maμAμ(ξ∈ℝn) with maximal domain. Then P(A) is closable. Let ℱ be the Fourier transform, that is, (ℱu)(η)=∫ℝnu(ξ)e-i(ξ,η)dξ for u∈L1(ℝn), where (ξ,η)=∑j=1nξjηj. If u∈ℱL1(ℝn):={ℱv:v∈L1(ℝn)}, then there exists a unique function in L1(ℝn), written ℱ-1u, such that u=ℱ(ℱ-1u). In particular, ℱ-1u is the inverse Fourier transform of u if u∈𝒮(ℝn) (the space of rapidly decreasing functions on ℝn). We define u(A)∈B(X) by
u(A)x=∫ℝn(ℱ-1u)(ξ)e-i(ξ,A)xdξ,x∈X,
where (ξ,A)=∑j=1nξjAj.

We will need the following lemma, in which the statements (a) and (b) are well-known, (c) and (d) can be found in [14] and [6], respectively.

Lemma 3.1.

(a) ℱL1(ℝn) is a Banach algebra under pointwise multiplication and addition with norm ∥u∥ℱL1:=∥ℱ-1u∥L1.

(b) u↦u(A) is an algebra homomorphism from ℱL1(ℝn) into B(X), and there exists a constant M>0 such that ∥u(A)∥≤M∥u∥ℱL1.

(c) E:={ϕ(A)x:ϕ∈𝒮(ℝn),x∈X}⊂∩μ∈ℕ0nD(Aμ), E¯=X, P(A)|E¯=P(A)¯ and ϕ(A)P(A)⊂P(A)ϕ(A)=(Pϕ)(A) for ϕ∈𝒮(ℝn).

(d) Let u∈Cj(ℝn)(j>n/2). Suppose that there exist constants L, M0, a>0, and b∈[-1,2a/n-1) such that
|Dku(ξ)|≤{M0|k||ξ|b|k|-a,for|ξ|≥L,|k|≤j,M0|k|,for|ξ|<L,|k|≤j,
where k∈ℕ0n, then u∈ℱL1(ℝn) and ∥u∥ℱL1≤MM0n/2 for some constant M>0.

Recall that the Mittag-Leffler function (see [15, 16]) is defined by
Eα,β(z):=∑n=0∞znΓ(αn+β)=12πi∫𝒯μα-βeμμα-zdμ,α,β>0,z∈ℂ,
where the path 𝒯 is a loop which starts and ends at -∞ and encircles the disc |t|≤|z|1/α in the positive sense. The most interesting properties of the Mittag-Leffler functions are associated with their Laplace integral
∫0∞e-λttβ-1Eα,β(ωtα)dt=λα-βλα-ω,Reλ>ω1/α,ω>0
and with their asymptotic expansion as z→∞. If 0<α<2, β>0, then
Eα,β(z)=1αz(1-β)/αexp(z1/α)+εα,β(z),|arg z|≤12απ,Eα,β(z)=εα,β(z),|arg(-z)|<(1-12α)π,
where
εα,β(z)=-∑n=1N-1z-nΓ(β-αn)+O(|z|-N)
as z→∞, and the O-term is uniform in argz if |arg(-z)|≤(1-α/2-ϵ)π. Note that for β>0,
|Eα,β(z)|≤Eα,β(|z|),z∈ℂ.

The following two lemmas are about derivatives of the Mittag-Leffler functions.

Lemma 3.2.

Eα,β′(z)=1αEα,α+β-1(z)-β-1αEα,α+β(z).

Proof.

By the definition of Eα,β(z),
Eα,β′(z)=(∑n=0∞znΓ(αn+β))′=∑n=1∞nzn-1Γ(αn+β)=∑n=1∞nzn-1Γ(αn+β-1+1)=∑n=1∞zn-1·(αn+β-1)α(αn+β-1)Γ(αn+β-1)-∑n=1∞β-1α·zn-1Γ(αn+β)=1αEα,α+β-1(z)-β-1αEα,α+β(z),
as we wanted to show.

For short, Eα(z):=Eα,1(z).

Lemma 3.3.

Suppose that 1<α<2. For every n∈ℕ and ϵ>0 there exist constants M>0 and L>0 such that for k=0,…,n,
|Eα(k)(z)|≤M|z|,if|z|≥L,|arg(-z)|≤(1-α2-ϵ)π.

Proof.

First note that Eα′(z)=(1/α)Eα,α(z), and by induction on k one can prove that
Eα(k)(z)=∑j=1kajEα,αk-(k-j)(z),
where aj only depend on α and k. Since α>1 we have that αk-(k-j)>0 whence, by the asymptotic formula for Mittag-Leffler functions (3.6), we obtain (3.11).

Now let us recall the definition of coercive polynomials. For fixed r>0, a polynomial P(ξ) is called r-coercive if |P(ξ)|-1=O(|ξ|-r) as |ξ|→∞. In the sequel, M is a generic constant independent of t which may vary from line to line.

Lemma 3.4.

Suppose that P(ξ) is an r-coercive polynomial of order m and {P(ξ):ξ∈ℝn}⊂ℂ∖Σα′π/2, where 1<α′<2. Let k0=[n/2]+1. Then for 1<α<α′, γ>0, a∈Σα′π/2, there exist constants M,L≥0 such that
|Dμ[Eα(tαP)(a-P)-γ]|≤M(1+tα|μ|)|ξ|(m-1)|μ|-rγ,|ξ|≥L,|μ|≤k0,t≥0.

Proof.

Suppose that for |ξ|≥L, (3.11) holds up to order k0 and
|P(ξ)|≥M|ξ|r,|a-P(ξ)|≥M|ξ|r.
By induction, one can show that
DμEα(tαP)=∑j=1|μ|tαjEα(j)(tαP)Qj,
where degQj≤mj-|μ|. Thus if |ξ|≥L and |tαP|≥L,
|DμEα(tαP)|≤M(1+tα(|μ|-1))|ξ|m|μ|-|μ|-r≤M(1+tα|μ|)|ξ|(m-1)|μ|-r,
and if |ξ|≥L with |tαP|≤L, by (3.8) and (3.12) we know that
|DμEα(tαP)|≤M(tα+tα|μ|)|ξ|(m-1)|μ|.
Altogether, we have
|DμEα(tαP)|≤M(1+tα|μ|)|ξ|(m-1)|μ|,|ξ|≥L.
And by
|Dμ(a-P)-γ|≤M|ξ|(m-r-1)|μ|-rγ,|ξ|≥L
and Leibniz's formula we have
|Dμ(Eα(tαP)(a-P)-γ)|≤M(1+tα|μ|)|ξ|(m-1)|μ|-rγ,|ξ|≥L.

Lemma 3.5.

This proves (3.13). Suppose that the assumptions of Lemma 3.4 are satisfied. Let γ>nm/2r. Then Eα(tαP)(a-P)-γ∈ℱL1(ℝn) and
∥Eα(tαP)(a-P)-γ∥ℱL1≤M(1+tαn/2),t≥0.
The same result holds with Eα(tαP) replaced by Eα,α(tαP).

Proof.

By Lemma 3.1(d), it remains to prove that for |ξ|≤L,
|Dμ(Eα(tαP)(a-P)-γ)|≤M(1+tα|μ|),|μ|≤k0,t≥0.
To show this we can use (3.15) and then give the estimates according to the values tαP. For |ξ|≤L with |tαP|≥L the estimate (3.8) can be applied, and for |ξ|≤L with |tαP|≤L note that all the functions Eα(j)(tαP) are uniformly bounded.

For the second part of the lemma, note that Eα,α(z)=αEα′(z).

4. Existence of <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M275"><mml:mrow><mml:mi mathvariant="bold-italic">α</mml:mi></mml:mrow></mml:math></inline-formula>-Times Regularized Resolvents for Operator Polynomials

In this section, we will construct the fractional regularized resolvent families for coercive differential operators on Banach spaces.

Theorem 4.1.

Suppose that P is an r-coercive polynomial of order m, and {P(ξ):ξ∈ℝn}⊂ℂ∖Σα′π/2, where 1<α′<2. Then for 1<α<α′, a∈Σα′π/2, γ>nm/2r, C=(a-P)-γ(A), there exists an analytic α-times C-regularized resolvent family Sα(t) for P(A)¯, and Sα(t)=(Eα(tαP)(a-P)-γ)(A) with
∥Sα(t)∥≤M(1+tαn/2),t≥0.

Proof.

Let ut=Eα(tαP)(a-P)-γ, t≥0. By Lemma 3.5, ut∈ℱL1 and ∥ut∥ℱL1≤M(1+tαn/2). Define Sα(t)=ut(A). Then by Lemma 3.1(b), Sα(t)∈B(X), ∥Sα(t)∥≤M(1+tαn/2), and in particular C=Sα(0)∈B(X). To check the strong continuity of Sα(t), take ϕ∈𝒮(ℝn). Then for t,t+h≥0, by Lemma 3.5∥Sα(t+h)ϕ(A)-Sα(t)ϕ(A)∥≤M∥(Eα((t+h)αP)-Eα(tαP))(a-P)-γϕ∥ℱL1≤M∥∫tt+hsα-1Eα,α(sαP)(a-P)-γPϕds∥ℱL1≤M∫tt+hsα-1(1+sαn/2)ds·∥Pϕ∥ℱL1→0,ash→0.
Since the set E of Lemma 3.1 is dense in X, we have done. Next we will show that
λα-1(λα-P(A)¯)-1C=∫0∞e-λtSα(t)dt,λ>0.
In fact, for ϕ∈𝒮(ℝn), by Lemma 3.1(b) and (c) we have
P(A)Sα(t)ϕ(A)=(Putϕ)(A)=Sα(t)P(A)ϕ(A).
Since ℱL1(ℝn) is a Banach algebra, it follows that ut,ut(λ-P)ϕ∈ℱL1. Thus by Lemmas 3.1, 3.5, (3.4), and Fubini's theorem one obtains that for x∈X, λ>0,
∫0∞e-λtSα(t)(λα-P(A))ϕ(A)xdt=∫0∞e-λt(ut(λα-P)ϕ)(A)xdt=(∫0∞e-λtutdt(λα-P)ϕ)(A)x=λα-1Cϕ(A)x.
This implies that
∫0∞e-λtSα(t)(λα-P(A)¯)xdt=λα-1Cx,x∈D(P(A)¯),
once again by the density of the set E of Lemma 3.1. A similar argument works to get
(λα-P(A)¯)∫0∞e-λtSα(t)xdt=λα-1Cx,x∈X.
Therefore, we have proved (4.3). And it is routine to show that C-1P(A)¯C=P(A)¯, thus by Theorem 2.3 we know that Sα(t) is the α-times C-regularized resolvent family for P(A)¯. Moreover, since α<α′ is arbitrary, by the subordination principle (Theorem 2.4) we know that Sα(t) is analytic.

We can extend this result to a more general case.

Theorem 4.2.

Let P(ξ) be an r-coercive polynomial of order m such that {P(ξ):ξ∈ℝn}⊂ℂ∖(ω+Σα′π/2) for some ω≥0, and let α′>1. Then for 1<α<α′, a∈ω+Σα′π/2, γ>nm/2r, and C=(a-P)-γ(A), there exists an analytic α-times C-regularized resolvent family Sα(t) for P(A)¯ with
∥Sα(t)∥≤M(1+tαn/2)exp(ω1/αt),t≥0.

Proof.

We only consider the area above the x-axis, the lower area can be treated similarly.

Let Ray 1 :={ω+ρeiα′π/2:0≤ρ<∞}, and let Ray 2 :={ρeiαπ/2:0≤ρ<∞}, where 1<α<α′<2. Let G be the point (ω,0), and set B to denote the intersection point of the two above rays. Let Ω denote the region to the left side of Ray 1 and 2 (see Figure 1).

If P(ξ) falls into Ω, the asymptotic formula (3.6) can be applied to get estimates similarly as in the proof of Theorem 4.1. It remains to consider the values P(ξ) within the triangle ΔGOB. To estimate DμEα(tαP(ξ)) for such values P(ξ), we use (3.12), (3.15), and (3.5) to obtain
|DμEα(tαP(ξ))|≤M(1+tα|μ|)exp(ω1/αt)
if Re((ρeiθ)1/α)<ω1/α, where ρeiθ denotes an arbitrary point on the line segment from G to B.

Since
ρsin(α′π/2)=ωsin(α′π/2-θ),
we have
Re((ρeiθ)1/α)=ω1/α(sin(α′π/2)sin(α′π/2-θ))1/αcos(θ/α).
Thus, to show that Re((ρeiθ)1/α)<ω1/α (0<θ≤απ/2) one needs to check that
cosα(θ/α)<cosθ+sinθ·tan(α′-12π),0<θ≤απ2;
and this is true if
cos(θ/α)≤cosθ+sinθ·tan(α-12π),0<θ≤απ2,
since 1<α<α′<2.

We first consider the case when π/2≤θ≤απ/2. Let g(θ)=cosθ+sinθ·tan(((α-1)/2)π)-cos(θ/α), then g′(θ)=-sinθ+(1/α)sin(θ/α)+cosθ·tan(((α-1)/2)π)≤0 since sinθ>(1/α)sin(θ/α) and cosθ≤0 for π/2≤θ≤απ/2. So g(θ) decreases with respect to θ, which means that g(θ)≥0 since g(απ/2)=0.

For 0<θ<π/2, we will show that
cos(θ/α)≤cosθ+sinθ·α-12π,
which implies (4.13). Now for fixed θ∈(0,π/2), denote by h(α)=cosθ+sinθ·((α-1)/2)π-cos(θ/α). Since α>1, we have h′(α)=(π/2)sinθ-(θ/α2)sin(θ/α)>0; it thus follows that h(α)≥h(1)=0. Therefore we have proved (4.14).

Now by (3.19) and (4.9) one obtains, for |ξ|≥L,
|Dμ(Eα(tαP)(a-P)-γ)|≤M(1+tα|μ|)|ξ|(m-1)|μ|-rγexp(ω1/αt),
and for |ξ|≤L,
|Dμ(Eα(tαP)(a-P)-γ)|≤M(1+tα|μ|)exp(ω1/αt).
An argument similar to that one of the proof of Theorem 4.1 gives our claim.

In the following theorem, we do not assume that P is coercive, but the choice of C is different.

Theorem 4.3.

Suppose that P(ξ) is a polynomial of order m, and {P(ξ):ξ∈ℝn}⊂ℂ∖(ω+Σα′π/2), where 1<α′<2. Then for 1<α<α′<2, β>n/2, C=(1+|A|2)-mβ/2 (which is defined by (3.1) with u(x)=(1+|x|2)-mβ/2), there exists an analytic α-times C-regularized resolvent family Sα(t) for P(A)¯ such that
∥Sα(t)∥≤M(1+tαn/2)exp(ω1/αt),t≥0.

Proof.

From (4.9) and
|Dμ(1+|ξ|2)-β/2|≤M|ξ|-|μ|-β,|ξ|≥L,μ∈ℕ0n,
we have for |ξ|≥L,
|Dμ(Eα(tαP)(1+ξ|2)-β/2)|≤M(1+tα|μ|)|ξ|(m-1)|μ|-βexp(ω1/αt),
and for |ξ|≤L,
|Dμ(Eα(tαP)(1+|ξ|2)-β/2)|≤M(1+tα|μ|)exp(ω1/αt).
It thus follows from Lemma 3.1 that when β>nm/2, Eα(tαP)(1+|ξ|2)-β/2∈ℱL1(ℝn). Similarly as in the proof of Theorem 4.1 we can show that there is an analytic α-times C-regularized resolvent family for P(A)¯.

From now on X will be Lp(ℝn)(1≤p<∞) or C0(ℝn):={f∈C(ℝn):lim|x|→∞f(x)=0}. The partial differential operator P(D) defined by
P(D)f=ℱ-1(Pℱf)
with
D(P(D))={f∈X:ℱ-1(Pℱf)∈X}
is closed and densely defined on X. Since iDj=∂/∂xj(1≤j≤n) is the generator of the bounded C0-group {Tj(t)}t∈ℝ given by
Tj(t)f(x1,…,xn)=f(x1,…,xj-1,xj+t,xj+1,…,xn)t∈ℝ
on X, we can apply the above results to P(D) on X. It is remarkable that when X=Lp(ℝn) (1<p<∞) these results can be improved. In fact, if A=D=(D1,…,Dn), then the functions ut's in the proofs of the above theorems give rise to Fourier multipliers on Lp(ℝn) having norm of polynomial growth tnp at infinity, where np=n|1/2-1/p|. For details we refer to [3, 8]). We summarize these conclusions in the following two theorems.

Theorem 4.4.

Suppose that the assumptions of Theorem 4.2 are satisfied.

For X=L1(ℝn) or C0(ℝn), C=(a-P)-γ(D), where γ>nm/2r, there exists an analytic α-times C-regularized resolvent family Sα(t) for P(D) and
∥Sα(t)∥≤M(1+tαn/2)exp(ω1/αt),t≥0.

For X=Lp(ℝn), C=(a-P)-γ(D), where γ>npm/r, np=n|1/2-1/p|, there exists an analytic α-times C-regularized resolvent family Sα(t) for P(D) and

∥Sα(t)∥≤M(1+tαnp)exp(ω1/αt),t≥0.Theorem 4.5.

Suppose that the assumptions of Theorem 4.3 are satisfied.

For X=L1(ℝn) or C0(ℝn), C=(1-Δ)-mβ/2, where β>n/2, there exists an analytic α-times C-regularized resolvent family Sα(t) for P(D) and
∥Sα(t)∥≤M(1+tαn/2)exp(ω1/αt),t≥0.

For X=Lp(ℝn), C=(1-Δ)-mβ/2, where β>np, there exists an analytic α-times C-regularized resolvent family Sα(t) for P(D) and
∥Sα(t)∥≤M(1+tαnp)exp(ω1/αt),t≥0.

We end this paper with some examples to demonstrate the applications of our results.

Example 4.6.

(a) The polynomial corresponding to the Laplacian Δ on Lp(ℝn) (n>1,p≠2) is P(ξ)=-|ξ|2. By Theorem 4.4, for every 1<α<2 there exists an analytic α-times (1-Δ)-γ-regularized resolvent family for the operator Δ, where γ>np.

Consider P(D) on Lp(ℝ2) (1<p<∞) with
P(ξ)=-(1+ξ12)(1+(ξ2-ξ1l)2)(l∈ℕ).

Then P(ξ)≤-1(ξ∈ℝ2). We claim that P is (2/l)-coercive. Indeed, if |ξ2|≥2|ξ1l|, then
|P(ξ)|≥(1+ξ12)(1+14ξ22)≥14|ξ|2.
If |ξ2|<2|ξ1l|, then
|P(ξ)|≥1+|ξ1|2≥c|ξ|2/lfor|ξ|≥1,
for some proper constant c, as desired. By Theorems 4.4 and 4.5, for every 1<α<2 there exists an analytic α-times C-regularized resolvent family for P(D), where C=(1-P)-γ(D) with γ>2(l2+l)|1/2-1/p| or C=(1-Δ)-(l+1)β with β>2|1/2-1/p|. We remark that if l≥5 and |1/2-1/p|≥1/4+1/l, then 0∈σ(P(D)) (see [17]). Since 0∈P(ℝ2), it follows from [18, Theorem 1] that ρ(P(D))=∅. Consequently, in this case there is no α-times resolvent family for P(D) for any α.Acknowledgments

The authors are very grateful to the referees for many helpful suggestions to improve this paper. The first and second authors were supported by the NSF of China (Grant no. 10501032) and NSFC-RFBR Programm (Grant no. 108011120015), and the third by TRAPOYT and the NSF of China (Grant no. 10671079).

KellermanH.HieberM.Integrated semigroupsArendtW.KellermanH.Da PratoG.IannelliM.Integrated solutions of Volterra integrodifferential equations and applicationsHieberM.Integrated semigroups and differential operators on Lp spacesSchechterM.HieberM.HolderriethA.NeubranderF.Regularized semigroups and systems of linear partial differential equationsZhengQ.Cauchy problems for polynomials of generators of bounded C0-semigroups and for differential operatorsZhengQ.Applications of semigroups of operators to non-elliptic differential operatorsZhengQ.LiY.Abstract parabolic systems and regularized semigroupsZhengQ.Coercive differential operators and fractionally integrated cosine functionsBazhlekovaE. G.FujitaY.Integrodifferential equation which interpolates the heat equation and the wave equationLiM.LiF.-B.ZhengQ.Elliptic operators with variable coefficients generating fractional resolvent familiesLiM.ZhengQ.ZhangJ.Regularized resolvent familiesLeiY. S.YiW. H.ZhengQ.Semigroups of operators and polynomials of generators of bounded strongly continuous groupsErdélyiA.MagnusW.OberhettingerF.TricomiF. G.ErdélyiA.MagnusW.OberhettingerF.TricomiF. G.RuizA.Lp-boundedness of a certain class of multipliers associated with curves on the plane—IIIhaF. T.SchubertC. F.The spectrum of partial differential operators on Lp(Rn)