Abstract

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(1.1) 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(𝑛) (𝑝2) 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 [13] 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 [58]). 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)}𝑡0, is called a resolvent family for 𝐴 with kernel 𝑎(𝑡)𝐿1loc(+) if 𝑅(𝑡)𝐴𝐴𝑅(𝑡) and the resolvent equation󵐐𝑅(𝑡)𝑥=𝑥+𝑡0𝑎(𝑡𝑠)𝐴𝑅(𝑠)𝑥𝑑𝑠,𝑡0,𝑥𝐷(𝐴)(1.2) 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)=𝑡𝛼1/Γ(𝛼), then such resolvent families interpolate 𝐶0-semigroups and cosine functions.

Recently Bazhlekova studied classes of such resolvent families (see [10]). Let 0<𝛼2, 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,,𝑚1,(1.3) 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 𝐃𝛼󵐐𝑓(𝑡)=𝑡0𝑔𝑚𝛼𝑑(𝑡𝑠)𝑚𝑑𝑠𝑚𝑓(𝑠)𝑑𝑠,(1.4) where 𝑓𝑊𝑚,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 , 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. 𝜶-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], 𝐶𝐁(𝑋) be injective. Define 𝜌𝐶(𝐴)={𝜆𝜆𝐴isinjectiveand𝑅(𝐶)𝑅(𝜆𝐴)}. Let Σ𝜃={𝜆|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)}𝑡0𝐁(𝑋) is called an 𝛼-times 𝐶-regularized resolvent family for 𝐴 if(a)𝑆𝛼(0)=𝐶;(b)𝑆𝛼(𝑡)𝐴𝐴𝑆𝛼(𝑡) for 𝑡0;(c)𝐶1𝐴𝐶=𝐴;(d)for 𝑥𝐷(𝐴), 𝑆𝛼(𝑡)𝑥=𝐶𝑥+𝑡0((𝑡𝑠)𝛼1/Γ(𝛼))𝑆𝛼(𝑠)𝐴𝑥𝑑𝑠.{𝑆𝛼(𝑡)}𝑡0 is called analytic if it can be extended analytically to some sector Σ𝜃.

If 𝑆𝛼(𝑡)𝑀𝑒𝜔𝑡 (𝑡0) for some constants 𝑀1 and 𝜔+, we will write 𝐴𝒞𝛼𝐶(𝑀,𝜔), and 𝒞𝛼𝐶(𝜔)={𝒞𝛼𝐶(𝑀,𝜔);𝑀1}, 𝒞𝛼𝐶={𝒞𝛼𝐶(𝜔);𝜔0}.

Define the operator 󵰒𝐴 by 󵰒𝐴𝑥=𝐶1(lim𝑡0Γ(𝛼+1)𝑡𝛼(𝑆𝛼󵰒(𝑡)𝑥𝐶𝑥)),𝑥𝐷(𝐴),(2.1) with 󵰒𝐷(𝐴)={𝑥𝑋lim𝑡0𝑆𝛼(𝑡)𝑥𝐶𝑥𝑡𝛼existsandisin𝑅(𝐶)}.(2.2)

Proposition 2.2. Suppose that there exists an 𝛼-times 𝐶-regularized resolvent family, {𝑆𝛼(𝑡)}𝑡0, for the operator 𝐴, and let 󵰒𝐴 be defined as above. Then 󵰒𝐴𝐴=.

Proof. By the strong continuity of 𝑆𝛼(𝑡), we have for every 𝑥𝑋, 𝑔𝛼+1(𝑡)1󵐐𝑡0𝑔𝛼(𝑡𝑠)𝑆𝛼(𝑠)𝑥𝑑𝑠𝐶𝑥𝑔𝛼+1(𝑡)1󵐐𝑡0𝑔𝛼(𝑡𝑠)𝑆𝛼(𝑠)𝑥𝐶𝑥𝑑𝑠sup0𝑠𝑡𝑆𝛼(𝑠)𝑥𝐶𝑥0as𝑡0.(2.3) Thus for 𝑥𝐷(𝐴), by Definition 2.1, lim𝑡0𝑔𝛼+1(𝑡)1(𝑆𝛼(𝑡)𝑥𝐶𝑥)=lim𝑡0𝑔𝛼+1(𝑡)1󵐐𝑡0𝑔𝛼(𝑡𝑠)𝑆𝛼(𝑠)𝐴𝑥𝑑𝑠=𝐶𝐴𝑥,(2.4) which means that 󵰒𝑥𝐷(𝐴) and 󵰒𝐴𝑥=𝐴𝑥. On the other hand, for 󵰒𝑥𝐷(𝐴), by the definition of 󵰒𝐴 and Definition 2.1, 𝐶󵰒𝐴𝑥=lim𝑡0𝑔𝛼+1(𝑡)1(𝑆𝛼(𝑡)𝑥𝐶𝑥)=lim𝑡0𝑔𝛼+1(𝑡)1𝐴󵐐𝑡0𝑔𝛼(𝑡𝑠)𝑆𝛼(𝑠)𝑥𝑑𝑠,(2.5) but lim𝑡0𝑔𝛼+1(𝑡)1𝑡0𝑔𝛼(𝑡𝑠)𝑆𝛼(𝑠)𝑥𝑑𝑠=𝐶𝑥, by (d) of Definition 2.1. Thus it follows from the closedness of 𝐴 that 𝐶𝑥𝐷(𝐴) with 󵰒𝐴𝐶𝑥=𝐶𝐴𝑥. This implies that 𝑥𝐷(𝐶1𝐴𝐶)=𝐷(𝐴), so we have 󵰒𝐴=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)𝐴𝒞𝛼𝐶(𝑀,𝜔);(b)𝐴=𝐶1𝐴𝐶, (𝜔𝛼,)𝜌𝐶(𝐴) and 𝑑𝑛𝑑𝜆𝑛(𝜆𝛼1(𝜆𝛼𝐴)1𝐶)𝑀𝑛!(𝜆𝜔)𝑛+1,𝜆>𝜔,𝑛0={0};(2.6)(c)𝐴=𝐶1𝐴𝐶, (𝜔𝛼,)𝜌𝐶(𝐴) and there exists a strongly continuous family {𝑆𝛼(𝑡)}𝑡0𝐁(𝑋) satisfying 𝑆𝛼(𝑡)𝑀𝑒𝜔𝑡 such that 𝜆𝛼1(𝜆𝛼𝐴)1𝐶𝑥=󵐐0𝑒𝜆𝑡𝑆𝛼(𝑡)𝑥𝑑𝑡,𝜆>𝜔,𝑥𝑋.(2.7)

Theorem 2.4. Suppose that 0<𝛼<𝛽2, 𝛾=𝛼/𝛽. If 𝐴𝒞𝛽𝐶(𝜔) then 𝐴𝒞𝛼𝐶(𝜔1/𝛾) and the 𝛼-times 𝐶-regularized resolvent family for 𝐴, {𝑆𝛼(𝑡)}𝑡0, can be extended analytically to Σmin{𝜃(𝛾),𝜋}, where 𝜃(𝛾)=(1/𝛾1)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(1𝑗𝑛) be commuting generators of bounded 𝐶0-groups on a Banach space 𝑋. Write 𝐴=(𝐴1,,𝐴𝑛) and 𝐴𝜇=𝐴𝜇11𝐴𝜇𝑛𝑛 for 𝜇=(𝜇1,,𝜇𝑛)𝑛0. Similarly, write 𝐷𝜇=𝐷𝜇11𝐷𝜇𝑛𝑛, where 𝐷𝑗=𝑖𝜕/𝜕𝑥𝑗 for 𝑗=1,,𝑛. For a polynomial 𝑃(𝜉)=|𝜇|𝑚𝑎𝜇𝜉𝜇(𝜉𝑛)(|𝜇|=𝑛𝑗=1𝜇𝑗) with constant coefficients, we define 𝑃(𝐴)=|𝜇|𝑚𝑎𝜇𝐴𝜇(𝜉𝑛) with maximal domain. Then 𝑃(𝐴) is closable. Let be the Fourier transform, that is, (𝑢)(𝜂)=𝑛𝑢(𝜉)𝑒𝑖(𝜉,𝜂)𝑑𝜉 for 𝑢𝐿1(𝑛), where (𝜉,𝜂)=𝑛𝑗=1𝜉𝑗𝜂𝑗. If 𝑢𝐿1(𝑛)={𝑣𝑣𝐿1(𝑛)}, then there exists a unique function in 𝐿1(𝑛), written 1𝑢, such that 𝑢=(1𝑢). In particular, 1𝑢 is the inverse Fourier transform of 𝑢 if 𝑢𝒮(𝑛) (the space of rapidly decreasing functions on 𝑛). We define 𝑢(𝐴)𝐵(𝑋) by 󵐐𝑢(𝐴)𝑥=𝑛(1𝑢)(𝜉)𝑒𝑖(𝜉,𝐴)𝑥𝑑𝜉,𝑥𝑋,(3.1) where (𝜉,𝐴)=𝑛𝑗=1𝜉𝑗𝐴𝑗.

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) 𝐿1(𝑛) is a Banach algebra under pointwise multiplication and addition with norm 𝑢𝐿1=1𝑢𝐿1.
(b) 𝑢𝑢(𝐴) is an algebra homomorphism from 𝐿1(𝑛) into 𝐁(𝑋), and there exists a constant 𝑀>0 such that 𝑢(𝐴)𝑀𝑢𝐿1.
(c) 𝐸={𝜙(𝐴)𝑥𝜙𝒮(𝑛),𝑥𝑋}𝜇𝑛0𝐷(𝐴𝜇), 𝐸=𝑋, 𝑃(𝐴)|𝐸=𝑃(𝐴) and 𝜙(𝐴)𝑃(𝐴)𝑃(𝐴)𝜙(𝐴)=(𝑃𝜙)(𝐴) for 𝜙𝒮(𝑛).
(d) Let 𝑢𝐶𝑗(𝑛)(𝑗>𝑛/2). Suppose that there exist constants 𝐿, 𝑀0, 𝑎>0, and 𝑏[1,2𝑎/𝑛1) such that |𝐷𝑘󶁆𝑀𝑢(𝜉)|0|𝑘||𝜉|𝑏|𝑘|𝑎𝑀,for|𝜉|𝐿,|𝑘|𝑗,0|𝑘|,for|𝜉|<𝐿,|𝑘|𝑗,(3.2) where 𝑘𝑛0, then 𝑢𝐿1(𝑛) and 𝑢𝐿1𝑀𝑀0𝑛/2 for some constant 𝑀>0.

Recall that the Mittag-Leffler function (see [15, 16]) is defined by 𝐸𝛼,𝛽(𝑧)=󵠈𝑛=0𝑧𝑛=1Γ(𝛼𝑛+𝛽)󵐐2𝜋𝑖𝒯𝜇𝛼𝛽𝑒𝜇𝜇𝛼𝑧𝑑𝜇,𝛼,𝛽>0,𝑧,(3.3) 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 󵐐0𝑒𝜆𝑡𝑡𝛽1𝐸𝛼,𝛽(𝜔𝑡𝛼𝜆)𝑑𝑡=𝛼𝛽𝜆𝛼𝜔,Re𝜆>𝜔1/𝛼,𝜔>0(3.4) and with their asymptotic expansion as 𝑧. If 0<𝛼<2, 𝛽>0, then 𝐸𝛼,𝛽1(𝑧)=𝛼𝑧(1𝛽)/𝛼exp(𝑧1/𝛼)+𝜀𝛼,𝛽1(𝑧),|arg𝑧|2𝐸𝛼𝜋,(3.5)𝛼,𝛽(𝑧)=𝜀𝛼,𝛽1(𝑧),|arg(𝑧)|<(12𝛼)𝜋,(3.6) where 𝜀𝛼,𝛽(𝑧)=𝑁1󵠈𝑛=1𝑧𝑛Γ(𝛽𝛼𝑛)+𝑂(|𝑧|𝑁)(3.7) as 𝑧, and the 𝑂-term is uniform in arg𝑧 if |arg(𝑧)|(1𝛼/2𝜖)𝜋. Note that for 𝛽>0, |𝐸𝛼,𝛽(𝑧)|𝐸𝛼,𝛽(|𝑧|),𝑧.(3.8)

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

Lemma 3.2. 𝐸𝛼,𝛽1(𝑧)=𝛼𝐸𝛼,𝛼+𝛽1(𝑧)𝛽1𝛼𝐸𝛼,𝛼+𝛽(𝑧).(3.9)

Proof. By the definition of 𝐸𝛼,𝛽(𝑧), 𝐸𝛼,𝛽(𝑧)=(󵠈𝑛=0𝑧𝑛)Γ(𝛼𝑛+𝛽)=󵠈𝑛=1𝑛𝑧𝑛1=Γ(𝛼𝑛+𝛽)󵠈𝑛=1𝑛𝑧𝑛1=Γ(𝛼𝑛+𝛽1+1)󵠈𝑛=1𝑧𝑛1(𝛼𝑛+𝛽1)𝛼(𝛼𝑛+𝛽1)Γ(𝛼𝑛+𝛽1)󵠈𝑛=1𝛽1𝛼𝑧𝑛1=1Γ(𝛼𝑛+𝛽)𝛼𝐸𝛼,𝛼+𝛽1(𝑧)𝛽1𝛼𝐸𝛼,𝛼+𝛽(𝑧),(3.10) as we wanted to show.

For short, 𝐸𝛼(𝑧)=𝐸𝛼,1(𝑧).

Lemma 3.3. Suppose that 1<𝛼<2. For every 𝑛 and 𝜖>0 there exist constants 𝑀>0 and 𝐿>0 such that for 𝑘=0,,𝑛, |𝐸𝛼(𝑘)𝑀(𝑧)|𝛼|𝑧|,if|𝑧|𝐿,|arg(𝑧)|(12𝜖)𝜋.(3.11)

Proof. First note that 𝐸𝛼(𝑧)=(1/𝛼)𝐸𝛼,𝛼(𝑧), and by induction on 𝑘 one can prove that 𝐸𝛼(𝑘)(𝑧)=𝑘󵠈𝑗=1𝑎𝑗𝐸𝛼,𝛼𝑘(𝑘𝑗)(𝑧),(3.12) where 𝑎𝑗 only depend on 𝛼 and 𝑘. Since 𝛼>1 we have that 𝛼𝑘(𝑘𝑗)>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)𝜉𝑛}Σ𝛼𝜋/2, where 1<𝛼<2. Let 𝑘0=[𝑛/2]+1. Then for 1<𝛼<𝛼, 𝛾>0, 𝑎Σ𝛼𝜋/2, there exist constants 𝑀,𝐿0 such that |𝐷𝜇[𝐸𝛼(𝑡𝛼𝑃)(𝑎𝑃)𝛾]|𝑀(1+𝑡𝛼|𝜇|)|𝜉|(𝑚1)|𝜇|𝑟𝛾,|𝜉|𝐿,|𝜇|𝑘0,𝑡0.(3.13)

Proof. Suppose that for |𝜉|𝐿, (3.11) holds up to order 𝑘0 and |𝑃(𝜉)|𝑀|𝜉|𝑟,|𝑎𝑃(𝜉)|𝑀|𝜉|𝑟.(3.14) By induction, one can show that 𝐷𝜇𝐸𝛼(𝑡𝛼𝑃)=|𝜇|󵠈𝑗=1𝑡𝛼𝑗𝐸𝛼(𝑗)(𝑡𝛼𝑃)𝑄𝑗,(3.15) where deg𝑄𝑗𝑚𝑗|𝜇|. Thus if |𝜉|𝐿 and |𝑡𝛼𝑃|𝐿, |𝐷𝜇𝐸𝛼(𝑡𝛼𝑃)|𝑀(1+𝑡𝛼(|𝜇|1))|𝜉|𝑚|𝜇||𝜇|𝑟𝑀(1+𝑡𝛼|𝜇|)|𝜉|(𝑚1)|𝜇|𝑟,(3.16) and if |𝜉|𝐿 with |𝑡𝛼𝑃|𝐿, by (3.8) and (3.12) we know that |𝐷𝜇𝐸𝛼(𝑡𝛼𝑃)|𝑀(𝑡𝛼+𝑡𝛼|𝜇|)|𝜉|(𝑚1)|𝜇|.(3.17) Altogether, we have |𝐷𝜇𝐸𝛼(𝑡𝛼𝑃)|𝑀(1+𝑡𝛼|𝜇|)|𝜉|(𝑚1)|𝜇|,|𝜉|𝐿.(3.18) And by |𝐷𝜇(𝑎𝑃)𝛾|𝑀|𝜉|(𝑚𝑟1)|𝜇|𝑟𝛾,|𝜉|𝐿(3.19) and Leibniz's formula we have |𝐷𝜇(𝐸𝛼(𝑡𝛼𝑃)(𝑎𝑃)𝛾)|𝑀(1+𝑡𝛼|𝜇|)|𝜉|(𝑚1)|𝜇|𝑟𝛾,|𝜉|𝐿.(3.20)

Lemma 3.5. This proves (3.13). Suppose that the assumptions of Lemma 3.4 are satisfied. Let 𝛾>𝑛𝑚/2𝑟. Then 𝐸𝛼(𝑡𝛼𝑃)(𝑎𝑃)𝛾𝐿1(𝑛) and 𝐸𝛼(𝑡𝛼𝑃)(𝑎𝑃)𝛾𝐿1𝑀(1+𝑡𝛼𝑛/2),𝑡0.(3.21) 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)(𝑎𝑃)𝛾)|𝑀(1+𝑡𝛼|𝜇|),|𝜇|𝑘0,𝑡0.(3.22) 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 𝜶-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)𝜉𝑛}Σ𝛼𝜋/2, where 1<𝛼<2. Then for 1<𝛼<𝛼, 𝑎Σ𝛼𝜋/2, 𝛾>𝑛𝑚/2𝑟, 𝐶=(𝑎𝑃)𝛾(𝐴), there exists an analytic 𝛼-times 𝐶-regularized resolvent family 𝑆𝛼(𝑡) for 𝑃(𝐴), and 𝑆𝛼(𝑡)=(𝐸𝛼(𝑡𝛼𝑃)(𝑎𝑃)𝛾)(𝐴) with 𝑆𝛼(𝑡)𝑀(1+𝑡𝛼𝑛/2),𝑡0.(4.1)

Proof. Let 𝑢𝑡=𝐸𝛼(𝑡𝛼𝑃)(𝑎𝑃)𝛾, 𝑡0. By Lemma 3.5, 𝑢𝑡𝐿1 and 𝑢𝑡𝐿1𝑀(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 𝜙𝒮(𝑛). Then for 𝑡,𝑡+0, by Lemma 3.5𝑆𝛼(𝑡+)𝜙(𝐴)𝑆𝛼(𝑡)𝜙(𝐴)𝑀(𝐸𝛼((𝑡+)𝛼𝑃)𝐸𝛼(𝑡𝛼𝑃))(𝑎𝑃)𝛾𝜙𝐿1𝑀󵐐𝑡𝑡+𝑠𝛼1𝐸𝛼,𝛼(𝑠𝛼𝑃)(𝑎𝑃)𝛾𝑃𝜙𝑑𝑠𝐿1󵐐𝑀𝑡𝑡+𝑠𝛼1(1+𝑠𝛼𝑛/2)𝑑𝑠𝑃𝜙𝐿10,as0.(4.2) Since the set 𝐸 of Lemma 3.1 is dense in 𝑋, we have done. Next we will show that 𝜆𝛼1(𝜆𝛼𝑃(𝐴))1𝐶=󵐐0𝑒𝜆𝑡𝑆𝛼(𝑡)𝑑𝑡,𝜆>0.(4.3) In fact, for 𝜙𝒮(𝑛), by Lemma 3.1(b) and (c) we have 𝑃(𝐴)𝑆𝛼(𝑡)𝜙(𝐴)=(𝑃𝑢𝑡𝜙)(𝐴)=𝑆𝛼(𝑡)𝑃(𝐴)𝜙(𝐴).(4.4) Since 𝐿1(𝑛) is a Banach algebra, it follows that 𝑢𝑡,𝑢𝑡(𝜆𝑃)𝜙𝐿1. Thus by Lemmas 3.1, 3.5, (3.4), and Fubini's theorem one obtains that for 𝑥𝑋, 𝜆>0, 󵐐0𝑒𝜆𝑡𝑆𝛼(𝑡)(𝜆𝛼𝑃(𝐴))𝜙(𝐴)𝑥𝑑𝑡=󵐐0𝑒𝜆𝑡(𝑢𝑡(𝜆𝛼󵐐𝑃)𝜙)(𝐴)𝑥𝑑𝑡=(0𝑒𝜆𝑡𝑢𝑡𝑑𝑡(𝜆𝛼𝑃)𝜙)(𝐴)𝑥=𝜆𝛼1𝐶𝜙(𝐴)𝑥.(4.5) This implies that 󵐐0𝑒𝜆𝑡𝑆𝛼(𝑡)(𝜆𝛼𝑃(𝐴))𝑥𝑑𝑡=𝜆𝛼1𝐶𝑥,𝑥𝐷(𝑃(𝐴)),(4.6) once again by the density of the set 𝐸 of Lemma 3.1. A similar argument works to get (𝜆𝛼𝑃(𝐴))󵐐0𝑒𝜆𝑡𝑆𝛼(𝑡)𝑥𝑑𝑡=𝜆𝛼1𝐶𝑥,𝑥𝑋.(4.7) Therefore, we have proved (4.3). And it is routine to show that 𝐶1𝑃(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)𝜉𝑛}(𝜔+Σ𝛼𝜋/2) for some 𝜔0, and let 𝛼>1. Then for 1<𝛼<𝛼, 𝑎𝜔+Σ𝛼𝜋/2, 𝛾>𝑛𝑚/2𝑟, and 𝐶=(𝑎𝑃)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),𝑡0.(4.8)

Proof. We only consider the area above the 𝑥-axis, the lower area can be treated similarly.
Let Ray 1 ={𝜔+𝜌𝑒𝑖𝛼𝜋/20𝜌<}, and let Ray 2 ={𝜌𝑒𝑖𝛼𝜋/20𝜌<}, 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)(4.9) if Re((𝜌𝑒𝑖𝜃)1/𝛼)<𝜔1/𝛼, where 𝜌𝑒𝑖𝜃 denotes an arbitrary point on the line segment from 𝐺 to 𝐵.
Since 𝜌sin(𝛼=𝜔𝜋/2)sin(𝛼𝜋/2𝜃),(4.10) we have Re((𝜌𝑒𝑖𝜃)1/𝛼)=𝜔1/𝛼(sin(𝛼𝜋/2)sin(𝛼)𝜋/2𝜃)1/𝛼cos(𝜃/𝛼).(4.11) Thus, to show that Re((𝜌𝑒𝑖𝜃)1/𝛼)<𝜔1/𝛼 (0<𝜃𝛼𝜋/2) one needs to check that cos𝛼𝛼(𝜃/𝛼)<cos𝜃+sin𝜃tan(12𝜋),0<𝜃𝛼𝜋2;(4.12) and this is true if cos(𝜃/𝛼)cos𝜃+sin𝜃tan(𝛼12𝜋),0<𝜃𝛼𝜋2,(4.13) since 1<𝛼<𝛼<2.
We first consider the case when 𝜋/2𝜃𝛼𝜋/2. Let 𝑔(𝜃)=cos𝜃+sin𝜃tan(((𝛼1)/2)𝜋)cos(𝜃/𝛼), then 𝑔(𝜃)=sin𝜃+(1/𝛼)sin(𝜃/𝛼)+cos𝜃tan(((𝛼1)/2)𝜋)0 since sin𝜃>(1/𝛼)sin(𝜃/𝛼) and cos𝜃0 for 𝜋/2𝜃𝛼𝜋/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+sin𝜃𝛼12𝜋,(4.14) which implies (4.13). Now for fixed 𝜃(0,𝜋/2), denote by (𝛼)=cos𝜃+sin𝜃((𝛼1)/2)𝜋cos(𝜃/𝛼). Since 𝛼>1, we have (𝛼)=(𝜋/2)sin𝜃(𝜃/𝛼2)sin(𝜃/𝛼)>0; it thus follows that (𝛼)(1)=0. Therefore we have proved (4.14).
Now by (3.19) and (4.9) one obtains, for |𝜉|𝐿, |𝐷𝜇(𝐸𝛼(𝑡𝛼𝑃)(𝑎𝑃)𝛾)|𝑀(1+𝑡𝛼|𝜇|)|𝜉|(𝑚1)|𝜇|𝑟𝛾exp(𝜔1/𝛼𝑡),(4.15) and for |𝜉|𝐿, |𝐷𝜇(𝐸𝛼(𝑡𝛼𝑃)(𝑎𝑃)𝛾)|𝑀(1+𝑡𝛼|𝜇|)exp(𝜔1/𝛼𝑡).(4.16) 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)𝜉𝑛}(𝜔+Σ𝛼𝜋/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),𝑡0.(4.17)

Proof. From (4.9) and |𝐷𝜇(1+|𝜉|2)𝛽/2|𝑀|𝜉||𝜇|𝛽,|𝜉|𝐿,𝜇𝑛0,(4.18) we have for |𝜉|𝐿, |𝐷𝜇(𝐸𝛼(𝑡𝛼𝑃)(1+𝜉|2)𝛽/2)|𝑀(1+𝑡𝛼|𝜇|)|𝜉|(𝑚1)|𝜇|𝛽exp(𝜔1/𝛼𝑡),(4.19) and for |𝜉|𝐿, |𝐷𝜇(𝐸𝛼(𝑡𝛼𝑃)(1+|𝜉|2)𝛽/2)|𝑀(1+𝑡𝛼|𝜇|)exp(𝜔1/𝛼𝑡).(4.20) It thus follows from Lemma 3.1 that when 𝛽>𝑛𝑚/2, 𝐸𝛼(𝑡𝛼𝑃)(1+|𝜉|2)𝛽/2𝐿1(𝑛). 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(𝑛)(1𝑝<) or 𝐶0(𝑛)={𝑓𝐶(𝑛)lim|𝑥|𝑓(𝑥)=0}. The partial differential operator 𝑃(𝐷) defined by 𝑃(𝐷)𝑓=1(𝑃𝑓)(4.21) with 𝐷(𝑃(𝐷))={𝑓𝑋1(𝑃𝑓)𝑋}(4.22) is closed and densely defined on 𝑋. Since 𝑖𝐷𝑗=𝜕/𝜕𝑥𝑗(1𝑗𝑛) is the generator of the bounded 𝐶0-group {𝑇𝑗(𝑡)}𝑡 given by 𝑇𝑗(𝑡)𝑓(𝑥1,,𝑥𝑛)=𝑓(𝑥1,,𝑥𝑗1,𝑥𝑗+𝑡,𝑥𝑗+1,,𝑥𝑛)𝑡(4.23) on 𝑋, we can apply the above results to 𝑃(𝐷) on 𝑋. It is remarkable that when 𝑋=𝐿𝑝(𝑛) (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(𝑛) having norm of polynomial growth 𝑡𝑛𝑝 at infinity, where 𝑛𝑝=𝑛|1/21/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. (a)For 𝑋=𝐿1(𝑛) or 𝐶0(𝑛), 𝐶=(𝑎𝑃)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),𝑡0.(4.24)(b)For 𝑋=𝐿𝑝(𝑛), 𝐶=(𝑎𝑃)𝛾(𝐷), where 𝛾>𝑛𝑝𝑚/𝑟, 𝑛𝑝=𝑛|1/21/𝑝|, there exists an analytic 𝛼-times 𝐶-regularized resolvent family 𝑆𝛼(𝑡) for 𝑃(𝐷) and𝑆𝛼(𝑡)𝑀(1+𝑡𝛼𝑛𝑝)exp(𝜔1/𝛼𝑡),𝑡0.(4.25)

Theorem 4.5. Suppose that the assumptions of Theorem 4.3 are satisfied.(a)For 𝑋=𝐿1(𝑛) or 𝐶0(𝑛), 𝐶=(1Δ)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),𝑡0.(4.26)(b)For 𝑋=𝐿𝑝(𝑛), 𝐶=(1Δ)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),𝑡0.(4.27)

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(𝑛) (𝑛>1,𝑝2) is 𝑃(𝜉)=|𝜉|2. By Theorem 4.4, for every 1<𝛼<2 there exists an analytic 𝛼-times (1Δ)𝛾-regularized resolvent family for the operator Δ, where 𝛾>𝑛𝑝.(b) Consider 𝑃(𝐷) on 𝐿𝑝(2) (1<𝑝<) with 𝑃(𝜉)=(1+𝜉21)(1+(𝜉2𝜉𝑙1)2)(𝑙).(4.28)Then 𝑃(𝜉)1(𝜉2). We claim that 𝑃 is (2/𝑙)-coercive. Indeed, if |𝜉2|2|𝜉𝑙1|, then |𝑃(𝜉)|(1+𝜉211)(1+4𝜉221)4|𝜉|2.(4.29) If |𝜉2|<2|𝜉𝑙1|, then |𝑃(𝜉)|1+|𝜉1|2𝑐|𝜉|2/𝑙for|𝜉|1,(4.30) 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=(1𝑃)𝛾(𝐷) with 𝛾>2(𝑙2+𝑙)|1/21/𝑝| or 𝐶=(1Δ)(𝑙+1)𝛽 with 𝛽>2|1/21/𝑝|. We remark that if 𝑙5 and |1/21/𝑝|1/4+1/𝑙, then 0𝜎(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).