Fractional Evolution Equations Governed by Coercive Differential Operators

Abstract

This paper is concerned with evolution equations of fractional order D^𝛼𝑢(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 𝐿^𝑝(ℝ^𝑛) (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)∈𝐿¹_loc(ℝ₊) if 𝑅(𝑡)𝐴⊂𝐴𝑅(𝑡) and the resolvent equation𝑅(𝑡)𝑥=𝑥+󵐐^𝑡0𝑎(𝑡−𝑠)𝐴𝑅(𝑠)𝑥𝑑𝑠,𝑡≥0,𝑥∈𝐷(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 𝑔_𝛼(𝑡)∶=𝑡^𝛼−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, 𝐃^𝛼𝑢(𝑡)=𝐴𝑢(𝑡),𝑡>0;𝑢^(𝑘)(0)=𝑥_𝑘,𝑘=0,1,…,𝑚−1,(1.3) is well posed if and only if there exists an 𝛼-times resolvent family for 𝐴. Here 𝐃^𝛼 is the Caputo fractional derivative of order 𝛼>0 defined by 𝐃^𝛼𝑓(𝑡)∶=󵐐^𝑡0𝑔_𝑚−𝛼(𝑡−𝑠)𝑑^𝑚𝑑𝑠^𝑚𝑓(𝑠)𝑑𝑠,(1.4) where 𝑓∈𝑊^𝑚,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. 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], 𝐶∈𝐁(𝑋) be injective. Define 𝜌_𝐶(𝐴)∶={𝜆∈ℂ∶𝜆−𝐴isinjectiveand𝑅(𝐶)⊂𝑅(𝜆−𝐴)}. Let Σ_𝜃∶={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 {𝑆_𝛼(𝑡)}_𝑡≥0⊂𝐁(𝑋) is called an 𝛼-times 𝐶-regularized resolvent family for 𝐴 if(a)𝑆_𝛼(0)=𝐶;(b)𝑆_𝛼(𝑡)𝐴⊂𝐴𝑆_𝛼(𝑡) for 𝑡≥0;(c)𝐶⁻¹𝐴𝐶=𝐴;(d)for 𝑥∈𝐷(𝐴), 𝑆_𝛼(𝑡)𝑥=𝐶𝑥+∫^𝑡0((𝑡−𝑠)^𝛼−1/Γ(𝛼))𝑆_𝛼(𝑠)𝐴𝑥𝑑𝑠.{𝑆_𝛼(𝑡)}_𝑡≥0 is called analytic if it can be extended analytically to some sector Σ_𝜃.

If ‖𝑆_𝛼(𝑡)‖≤𝑀𝑒^𝜔𝑡 (𝑡≥0) for some constants 𝑀≥1 and 𝜔∈ℝ₊, we will write 𝐴∈𝒞^𝛼𝐶(𝑀,𝜔), and 𝒞^𝛼𝐶(𝜔)∶=∪{𝒞^𝛼𝐶(𝑀,𝜔);𝑀≥1}, 𝒞^𝛼𝐶∶=∪{𝒞^𝛼𝐶(𝜔);𝜔≥0}.

Define the operator 𝐴󵰒 by 𝐴󵰒𝑥=𝐶⁻¹(lim_𝑡↓0Γ(𝛼+1)𝑡^𝛼(𝑆_𝛼(𝑡)𝑥−𝐶𝑥)),𝑥∈𝐷(𝐴󵰒),(2.1) with 𝐷(𝐴󵰒)={𝑥∈𝑋∶lim_𝑡↓0𝑆_𝛼(𝑡)𝑥−𝐶𝑥𝑡^𝛼existsandisin𝑅(𝐶)}.(2.2)

Proposition 2.2. Suppose that there exists an 𝛼-times 𝐶-regularized resolvent family, {𝑆_𝛼(𝑡)}_𝑡≥0, for the operator 𝐴, and let 𝐴󵰒 be defined as above. Then 𝐴=𝐴󵰒.

Proof. By the strong continuity of 𝑆_𝛼(𝑡), we have for every 𝑥∈𝑋, ‖𝑔_𝛼+1(𝑡)⁻¹󵐐^𝑡0𝑔_𝛼(𝑡−𝑠)𝑆_𝛼(𝑠)𝑥𝑑𝑠−𝐶𝑥‖≤≤𝑔sup_{𝛼+10≤𝑠≤𝑡}(𝑡)⁻¹󵐐^𝑡0𝑔_𝛼(𝑡−𝑠)‖𝑆_𝛼(𝑠)𝑥−𝐶𝑥‖𝑑𝑠‖𝑆_𝛼(𝑠)𝑥−𝐶𝑥‖→0as𝑡→0.(2.3) Thus for 𝑥∈𝐷(𝐴), by Definition 2.1, lim_𝑡↓0𝑔_𝛼+1(𝑡)⁻¹(𝑆_𝛼(𝑡)𝑥−𝐶𝑥)==𝐶lim𝐴_𝑡↓0𝑥𝑔,_𝛼+1(𝑡)⁻¹󵐐^𝑡0𝑔_𝛼(𝑡−𝑠)𝑆_𝛼(𝑠)𝐴𝑥𝑑𝑠(2.4) which means that 𝑥∈𝐷(𝐴󵰒) and 𝐴󵰒𝑥=𝐴𝑥. On the other hand, for 𝑥∈𝐷(𝐴󵰒), by the definition of 𝐴󵰒 and Definition 2.1, 𝐶𝐴󵰒𝑥==lliimm_{𝑡↓0𝑡↓0}𝑔_𝛼+1(𝑡)⁻¹(𝑆_𝛼(𝑡)𝑥−𝐶𝑥)𝑔_𝛼+1(𝑡)⁻¹𝐴󵐐^𝑡0𝑔_𝛼(𝑡−𝑠)𝑆_𝛼(𝑠)𝑥𝑑𝑠,(2.5) but lim_𝑡→0𝑔_𝛼+1(𝑡)⁻¹∫^𝑡0𝑔_𝛼(𝑡−𝑠)𝑆_𝛼(𝑠)𝑥𝑑𝑠=𝐶𝑥, by (d) of Definition 2.1. Thus it follows from the closedness of 𝐴 that 𝐶𝑥∈𝐷(𝐴) with 𝐴𝐶𝑥=𝐶𝐴󵰒𝑥. This implies that 𝑥∈𝐷(𝐶⁻¹𝐴𝐶)=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)𝐴∈𝒞^𝛼𝐶(𝑀,𝜔);(b)𝐴=𝐶⁻¹𝐴𝐶, (𝜔^𝛼,∞)⊆𝜌_𝐶(𝐴) and ‖𝑑^𝑛𝑑𝜆^𝑛(𝜆^𝛼−1(𝜆^𝛼−𝐴)⁻¹𝐶)‖≤𝑀𝑛!(𝜆−𝜔)^𝑛+1,𝜆>𝜔,𝑛∈ℕ₀∶=ℕ∪{0};(2.6)(c)𝐴=𝐶⁻¹𝐴𝐶, (𝜔^𝛼,∞)⊆𝜌_𝐶(𝐴) and there exists a strongly continuous family {𝑆_𝛼(𝑡)}_𝑡≥0⊂𝐁(𝑋) satisfying ‖𝑆_𝛼(𝑡)‖≤𝑀𝑒^𝜔𝑡 such that 𝜆^𝛼−1(𝜆^𝛼−𝐴)⁻¹𝐶𝑥=󵐐^∞0𝑒^−𝜆𝑡𝑆_𝛼(𝑡)𝑥𝑑𝑡,𝜆>𝜔,𝑥∈𝑋.(2.7)

Theorem 2.4. Suppose that 0<𝛼<𝛽≤2, 𝛾=𝛼/𝛽. If 𝐴∈𝒞^𝛽𝐶(𝜔) then 𝐴∈𝒞^𝛼𝐶(𝜔^1/𝛾) and the 𝛼-times 𝐶-regularized resolvent family for 𝐴, {𝑆_𝛼(𝑡)}_𝑡≥0, can be extended analytically to Σ_{min{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 𝑖𝐴_𝑗(1≤𝑗≤𝑛) be commuting generators of bounded 𝐶₀-groups on a Banach space 𝑋. Write 𝐴=(𝐴₁,…,𝐴_𝑛) and 𝐴^𝜇=𝐴^𝜇1₁⋯𝐴^𝜇𝑛_𝑛 for 𝜇=(𝜇₁,…,𝜇_𝑛)∈ℕ^𝑛0. Similarly, write 𝐷^𝜇=𝐷^𝜇1₁⋯𝐷^𝜇𝑛_𝑛, where 𝐷_𝑗=−𝑖𝜕/𝜕𝑥_𝑗 for 𝑗=1,…,𝑛. For a polynomial 𝑃(𝜉)∶=∑_{|𝜇|≤𝑚}𝑎_𝜇𝜉^𝜇(𝜉∈ℝ^𝑛)(|𝜇|∶=∑^𝑛𝑗=1𝜇_𝑗) with constant coefficients, we define 𝑃(𝐴)=∑_{|𝜇|≤𝑚}𝑎_𝜇𝐴^𝜇(𝜉∈ℝ^𝑛) 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(ℝ^𝑛), where (𝜉,𝜂)=∑^𝑛𝑗=1𝜉_𝑗𝜂_𝑗. If 𝑢∈ℱ𝐿¹(ℝ^𝑛)∶={ℱ𝑣∶𝑣∈𝐿¹(ℝ^𝑛)}, then there exists a unique function in 𝐿¹(ℝ^𝑛), written ℱ⁻¹𝑢, such that 𝑢=ℱ(ℱ⁻¹𝑢). In particular, ℱ⁻¹𝑢 is the inverse Fourier transform of 𝑢 if 𝑢∈𝒮(ℝ^𝑛) (the space of rapidly decreasing functions on ℝ^𝑛). We define 𝑢(𝐴)∈𝐵(𝑋) by 𝑢(𝐴)𝑥=󵐐_ℝ𝑛(ℱ⁻¹𝑢)(𝜉)𝑒^{−𝑖(𝜉,𝐴)}𝑥𝑑𝜉,𝑥∈𝑋,(3.1) where (𝜉,𝐴)=∑^𝑛𝑗=1𝜉_𝑗𝐴_𝑗.

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(ℝ^𝑛) 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(ℝ^𝑛) into 𝐁(𝑋), and there exists a constant 𝑀>0 such that ‖𝑢(𝐴)‖≤𝑀‖𝑢‖_ℱ𝐿¹.
(c) 𝐸∶={𝜙(𝐴)𝑥∶𝜙∈𝒮(ℝ^𝑛),𝑥∈𝑋}⊂∩_𝜇∈ℕ^𝑛0𝐷(𝐴^𝜇), 𝐸=𝑋, 𝑃(𝐴)|_𝐸=𝑃(𝐴) and 𝜙(𝐴)𝑃(𝐴)⊂𝑃(𝐴)𝜙(𝐴)=(𝑃𝜙)(𝐴) for 𝜙∈𝒮(ℝ^𝑛).
(d) Let 𝑢∈𝐶^𝑗(ℝ^𝑛)(𝑗>𝑛/2). Suppose that there exist constants 𝐿, 𝑀₀, 𝑎>0, and 𝑏∈[−1,2𝑎/𝑛−1) such that |𝐷^𝑘𝑢(𝜉)|≤󶁆𝑀^0|𝑘|𝑀|𝜉|^{𝑏|𝑘|−𝑎},for|𝜉|≥𝐿,|𝑘|≤𝑗,_0|𝑘|,for|𝜉|<𝐿,|𝑘|≤𝑗,(3.2) where 𝑘∈ℕ^𝑛0, then 𝑢∈ℱ𝐿¹(ℝ^𝑛) and ‖𝑢‖_ℱ𝐿¹≤𝑀𝑀^0𝑛/2 for some constant 𝑀>0.

Recall that the Mittag-Leffler function (see [15, 16]) is defined by 𝐸_𝛼,𝛽(𝑧)∶=󵠈^∞_𝑛=0𝑧^𝑛Γ(𝛼𝑛+𝛽)=21𝜋𝑖󵐐_𝒯𝜇^𝛼−𝛽𝑒^𝜇𝜇^𝛼−𝑧𝑑𝜇,𝛼,𝛽>0,𝑧∈ℂ,(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 󵐐^∞0𝑒^−𝜆𝑡𝑡^𝛽−1𝐸_𝛼,𝛽(𝜔𝑡^𝛼)𝑑𝑡=𝜆^𝛼−𝛽𝜆^𝛼−𝜔,Re𝜆>𝜔^1/𝛼,𝜔>0(3.4) and with their asymptotic expansion as 𝑧→∞. If 0<𝛼<2, 𝛽>0, then 𝐸_𝛼,𝛽(𝑧)=1𝛼𝑧^{(1−𝛽)/𝛼}exp(𝑧^1/𝛼)+𝜀_𝛼,𝛽(𝑧),|arg𝑧|≤1𝐸2𝛼𝜋,(3.5)_𝛼,𝛽(𝑧)=𝜀_𝛼,𝛽(𝑧),|arg(−𝑧)|<(1−12𝛼)𝜋,(3.6) where 𝜀_𝛼,𝛽(𝑧)=−󵠈^𝑁−1_𝑛=1𝑧^−𝑛Γ(𝛽−𝛼𝑛)+𝑂(|𝑧|^−𝑁)(3.7) as 𝑧→∞, and the 𝑂-term is uniform in arg𝑧 if |arg(−𝑧)|≤(1−𝛼/2−𝜖)𝜋. Note that for 𝛽>0, |𝐸_𝛼,𝛽(𝑧)|≤𝐸_𝛼,𝛽(|𝑧|),𝑧∈ℂ.(3.8)

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

Lemma 3.2. 𝐸^′𝛼,𝛽(𝑧)=1𝛼𝐸^{𝛼,𝛼+𝛽−1}(𝑧)−𝛽−1𝛼𝐸^{𝛼,𝛼+𝛽}(𝑧).(3.9)

Proof. By the definition of 𝐸_𝛼,𝛽(𝑧), 𝐸^′𝛼,𝛽(𝑧)=(󵠈^∞_𝑛=0𝑧^𝑛Γ(𝛼𝑛+𝛽))^′=󵠈^∞_𝑛=1𝑛𝑧^𝑛−1=Γ(𝛼𝑛+𝛽)_∞𝑛󵠈₌₁𝑛𝑧^𝑛−1=Γ(𝛼𝑛+𝛽−1+1)_∞𝑛󵠈₌₁𝑧^𝑛−1⋅(𝛼𝑛+𝛽−1)𝛼(𝛼𝑛+𝛽−1)Γ(𝛼𝑛+𝛽−1)−^∞_𝑛󵠈₌₁𝛽−1=1𝛼⋅Γ(𝛼𝑧𝑛+^𝑛−1𝛽)𝛼𝐸^{𝛼,𝛼+𝛽−1}(𝑧)−𝛽−1𝛼𝐸^{𝛼,𝛼+𝛽}(𝑧),(3.10) as we wanted to show.

For short, 𝐸_𝛼(𝑧)∶=𝐸_𝛼,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, |𝐸^𝛼(𝑘)(𝑧)|≤𝑀|𝑧|,if|𝑧|≥𝐿,|arg(−𝑧)|≤(1−𝛼2−𝜖)𝜋.(3.11)

Proof. First note that 𝐸^′𝛼(𝑧)=(1/𝛼)𝐸_𝛼,𝛼(𝑧), and by induction on 𝑘 one can prove that 𝐸^𝛼(𝑘)(𝑧)=󵠈^𝑘_𝑗=1𝑎_𝑗𝐸_{𝛼,𝛼𝑘−(𝑘−𝑗)}(𝑧),(3.12) where 𝑎_𝑗 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∈ℝ^𝑛}⊂ℂ⧵Σ_{𝛼′𝜋/2}, where 1<𝛼^′<2. Let 𝑘₀=[𝑛/2]+1. Then for 1<𝛼<𝛼^′, 𝛾>0, 𝑎∈Σ_{𝛼′𝜋/2}, there exist constants 𝑀,𝐿≥0 such that |𝐷^𝜇[𝐸_𝛼(𝑡^𝛼𝑃)(𝑎−𝑃)^−𝛾]|≤𝑀(1+𝑡^𝛼|𝜇|)|𝜉|^{(𝑚−1)|𝜇|−𝑟𝛾},|𝜉|≥𝐿,|𝜇|≤𝑘₀,𝑡≥0.(3.13)

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.(3.14) By induction, one can show that 𝐷^𝜇𝐸_𝛼(𝑡^𝛼𝑃)=󵠈^|𝜇|_𝑗=1𝑡^𝛼𝑗𝐸^𝛼(𝑗)(𝑡^𝛼𝑃)𝑄_𝑗,(3.15) where deg𝑄_𝑗≤𝑚𝑗−|𝜇|. Thus if |𝜉|≥𝐿 and |𝑡^𝛼𝑃|≥𝐿, |𝐷^𝜇𝐸_𝛼(𝑡^𝛼𝑃)|≤𝑀(1+𝑡^{𝛼(|𝜇|−1)})|𝜉|^{𝑚|𝜇|−|𝜇|−𝑟}≤𝑀(1+𝑡^𝛼|𝜇|)|𝜉|^{(𝑚−1)|𝜇|−𝑟},(3.16) and if |𝜉|≥𝐿 with |𝑡^𝛼𝑃|≤𝐿, by (3.8) and (3.12) we know that |𝐷^𝜇𝐸_𝛼(𝑡^𝛼𝑃)|≤𝑀(𝑡^𝛼+𝑡^𝛼|𝜇|)|𝜉|^{(𝑚−1)|𝜇|}.(3.17) Altogether, we have |𝐷^𝜇𝐸_𝛼(𝑡^𝛼𝑃)|≤𝑀(1+𝑡^𝛼|𝜇|)|𝜉|^{(𝑚−1)|𝜇|},|𝜉|≥𝐿.(3.18) And by |𝐷^𝜇(𝑎−𝑃)^−𝛾|≤𝑀|𝜉|^{(𝑚−𝑟−1)|𝜇|−𝑟𝛾},|𝜉|≥𝐿(3.19) and Leibniz's formula we have |𝐷^𝜇(𝐸_𝛼(𝑡^𝛼𝑃)(𝑎−𝑃)^−𝛾)|≤𝑀(1+𝑡^𝛼|𝜇|)|𝜉|^{(𝑚−1)|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 𝐸_𝛼(𝑡^𝛼𝑃)(𝑎−𝑃)^−𝛾∈ℱ𝐿¹(ℝ^𝑛) and ‖𝐸_𝛼(𝑡^𝛼𝑃)(𝑎−𝑃)^−𝛾‖_ℱ𝐿1≤𝑀(1+𝑡^𝛼𝑛/2),𝑡≥0.(3.21) The same result holds with 𝐸_𝛼(𝑡^𝛼𝑃) replaced by 𝐸_𝛼,𝛼(𝑡^𝛼𝑃).

Proof. By Lemma 3.1(d), it remains to prove that for |𝜉|≤𝐿, |𝐷^𝜇(𝐸_𝛼(𝑡^𝛼𝑃)(𝑎−𝑃)^−𝛾)|≤𝑀(1+𝑡^𝛼|𝜇|),|𝜇|≤𝑘₀,𝑡≥0.(3.22) To show this we can use (3.15) and then give the estimates according to the values 𝑡^𝛼𝑃. For |𝜉|≤𝐿 with |𝑡^𝛼𝑃|≥𝐿 the estimate (3.8) can be applied, and for |𝜉|≤𝐿 with |𝑡^𝛼𝑃|≤𝐿 note that all the functions 𝐸^𝛼(𝑗)(𝑡^𝛼𝑃) are uniformly bounded.
For the second part of the lemma, note that 𝐸_𝛼,𝛼(𝑧)=𝛼𝐸^′𝛼(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)∶𝜉∈ℝ^𝑛}⊂ℂ⧵Σ_{𝛼′𝜋/2}, where 1<𝛼^′<2. Then for 1<𝛼<𝛼^′, 𝑎∈Σ_{𝛼′𝜋/2}, 𝛾>𝑛𝑚/2𝑟, 𝐶=(𝑎−𝑃)^−𝛾(𝐴), there exists an analytic 𝛼-times 𝐶-regularized resolvent family 𝑆_𝛼(𝑡) for 𝑃(𝐴), and 𝑆_𝛼(𝑡)=(𝐸_𝛼(𝑡^𝛼𝑃)(𝑎−𝑃)^−𝛾)(𝐴) with ‖𝑆_𝛼(𝑡)‖≤𝑀(1+𝑡^𝛼𝑛/2),𝑡≥0.(4.1)

Proof. Let 𝑢_𝑡=𝐸_𝛼(𝑡^𝛼𝑃)(𝑎−𝑃)^−𝛾, 𝑡≥0. By Lemma 3.5, 𝑢_𝑡∈ℱ𝐿¹ and ‖𝑢_𝑡‖_ℱ𝐿1≤𝑀(1+𝑡^𝛼𝑛/2). Define 𝑆_𝛼(𝑡)=𝑢_𝑡(𝐴). Then by Lemma 3.1(b), 𝑆_𝛼(𝑡)∈𝐁(𝑋), ‖𝑆_𝛼(𝑡)‖≤𝑀(1+𝑡^𝛼𝑛/2), and in particular 𝐶=𝑆_𝛼(0)∈𝐁(𝑋). To check the strong continuity of 𝑆_𝛼(𝑡), take 𝜙∈𝒮(ℝ^𝑛). Then for 𝑡,𝑡+ℎ≥0, by Lemma 3.5‖𝑆_𝛼(𝑡≤+𝑀ℎ)‖𝜙(𝐸(𝐴_𝛼)−𝑆_𝛼(𝑡)𝜙(𝐴)‖((𝑡+ℎ)^𝛼𝑃)−𝐸_𝛼(𝑡^𝛼𝑃))(𝑎−𝑃)^−𝛾𝜙‖_ℱ𝐿1≤𝑀‖󵐐^𝑡𝑡+ℎ𝑠^𝛼−1𝐸_𝛼,𝛼(𝑠^𝛼𝑃)(𝑎−𝑃)^−𝛾𝑃𝜙𝑑𝑠‖_ℱ𝐿1≤𝑀󵐐^𝑡𝑡+ℎ𝑠^𝛼−1(1+𝑠^𝛼𝑛/2)𝑑𝑠⋅‖𝑃𝜙‖_ℱ𝐿1→0,asℎ→0.(4.2) Since the set 𝐸 of Lemma 3.1 is dense in 𝑋, we have done. Next we will show that 𝜆^𝛼−1(𝜆^𝛼−𝑃(𝐴))⁻¹𝐶=󵐐^∞0𝑒^−𝜆𝑡𝑆_𝛼(𝑡)𝑑𝑡,𝜆>0.(4.3) In fact, for 𝜙∈𝒮(ℝ^𝑛), by Lemma 3.1(b) and (c) we have 𝑃(𝐴)𝑆_𝛼(𝑡)𝜙(𝐴)=(𝑃𝑢_𝑡𝜙)(𝐴)=𝑆_𝛼(𝑡)𝑃(𝐴)𝜙(𝐴).(4.4) Since ℱ𝐿¹(ℝ^𝑛) is a Banach algebra, it follows that 𝑢_𝑡,𝑢_𝑡(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, 󵐐^∞0𝑒^−𝜆𝑡𝑆_𝛼(𝑡)(𝜆^𝛼−𝑃(𝐴))𝜙(𝐴)𝑥𝑑𝑡==󵐐(󵐐^∞0_∞0𝑒^−𝜆𝑡(𝑢_𝑡(𝜆^𝛼−𝑃)𝜙)(𝐴)𝑥𝑑𝑡=𝜆_𝛼−1𝑒^−𝜆𝑡𝑢_𝑡𝑑𝑡(𝜆^𝛼−𝑃)𝜙)(𝐴)𝑥𝐶𝜙(𝐴)𝑥.(4.5) This implies that 󵐐^∞0𝑒^−𝜆𝑡𝑆_𝛼(𝑡)(𝜆^𝛼−𝑃(𝐴))𝑥𝑑𝑡=𝜆^𝛼−1𝐶𝑥,𝑥∈𝐷(𝑃(𝐴)),(4.6) once again by the density of the set 𝐸 of Lemma 3.1. A similar argument works to get (𝜆^𝛼−𝑃(𝐴))󵐐^∞0𝑒^−𝜆𝑡𝑆_𝛼(𝑡)𝑥𝑑𝑡=𝜆^𝛼−1𝐶𝑥,𝑥∈𝑋.(4.7) 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 𝑆_𝛼(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 𝑆_𝛼(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∈ℝ^𝑛}⊂ℂ⧵(𝜔+Σ_{𝛼′𝜋/2}) for some 𝜔≥0, and let 𝛼^′>1. Then for 1<𝛼<𝛼^′, 𝑎∈𝜔+Σ_{𝛼′𝜋/2}, 𝛾>𝑛𝑚/2𝑟, and 𝐶=(𝑎−𝑃)^−𝛾(𝐴), there exists an analytic 𝛼-times 𝐶-regularized resolvent family 𝑆_𝛼(𝑡) for 𝑃(𝐴) with ‖𝑆_𝛼(𝑡)‖≤𝑀(1+𝑡^𝛼𝑛/2)exp(𝜔^1/𝛼𝑡),𝑡≥0.(4.8)

Proof. We only consider the area above the 𝑥-axis, the lower area can be treated similarly.
Let Ray 1 ∶={𝜔+𝜌𝑒^{𝑖𝛼′𝜋/2}∶0≤𝜌<∞}, and let Ray 2 ∶={𝜌𝑒^{𝑖𝛼𝜋/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 𝐷^𝜇𝐸_𝛼(𝑡^𝛼𝑃(𝜉)) for such values 𝑃(𝜉), we use (3.12), (3.15), and (3.5) to obtain |𝐷^𝜇𝐸_𝛼(𝑡^𝛼𝑃(𝜉))|≤𝑀(1+𝑡^𝛼|𝜇|)exp(𝜔^1/𝛼𝑡)(4.9) if Re((𝜌𝑒^𝑖𝜃)^1/𝛼)<𝜔^1/𝛼, where 𝜌𝑒^𝑖𝜃 denotes an arbitrary point on the line segment from 𝐺 to 𝐵.
Since 𝜌sin(𝛼^′𝜋/2)=sin(𝛼^′𝜋𝜔/2−𝜃),(4.10) we have Re((𝜌𝑒^𝑖𝜃)^1/𝛼)=𝜔^1/𝛼(sin(𝛼^′𝜋/2)sin(𝛼^′𝜋/2−𝜃))^1/𝛼cos(𝜃/𝛼).(4.11) Thus, to show that Re((𝜌𝑒^𝑖𝜃)^1/𝛼)<𝜔^1/𝛼 (0<𝜃≤𝛼𝜋/2) one needs to check that cos^𝛼(𝜃/𝛼)<cos𝜃+sin𝜃⋅tan(𝛼^′−12𝜋),0<𝜃≤𝛼2𝜋;(4.12) and this is true if cos(𝜃/𝛼)≤cos𝜃+sin𝜃⋅tan(𝛼−12𝜋),0<𝜃≤𝛼2𝜋,(4.13) 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,(4.14) 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, |𝐷^𝜇(𝐸_𝛼(𝑡^𝛼𝑃)(𝑎−𝑃)^−𝛾)|≤𝑀(1+𝑡^𝛼|𝜇|)|𝜉|^{(𝑚−1)|𝜇|−𝑟𝛾}exp(𝜔^1/𝛼𝑡),(4.15) and for |𝜉|≤𝐿, |𝐷^𝜇(𝐸_𝛼(𝑡^𝛼𝑃)(𝑎−𝑃)^−𝛾)|≤𝑀(1+𝑡^𝛼|𝜇|)exp(𝜔^1/𝛼𝑡).(4.16) An argument similar to that one of the proof of Theorem 4.1 gives our claim.

Figure 1 

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∈ℝ^𝑛}⊂ℂ⧵(𝜔+Σ_{𝛼′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 𝑆_𝛼(𝑡) for 𝑃(𝐴) such that ‖𝑆_𝛼(𝑡)‖≤𝑀(1+𝑡^𝛼𝑛/2)exp(𝜔^1/𝛼𝑡),𝑡≥0.(4.17)

Proof. From (4.9) and |𝐷^𝜇(1+|𝜉|²)^−𝛽/2|≤𝑀|𝜉|^{−|𝜇|−𝛽},|𝜉|≥𝐿,𝜇∈ℕ^𝑛0,(4.18) we have for |𝜉|≥𝐿, |𝐷^𝜇(𝐸_𝛼(𝑡^𝛼𝑃)(1+𝜉|²)^−𝛽/2)|≤𝑀(1+𝑡^𝛼|𝜇|)|𝜉|^{(𝑚−1)|𝜇|−𝛽}exp(𝜔^1/𝛼𝑡),(4.19) and for |𝜉|≤𝐿, |𝐷^𝜇(𝐸_𝛼(𝑡^𝛼𝑃)(1+|𝜉|²)^−𝛽/2)|≤𝑀(1+𝑡^𝛼|𝜇|)exp(𝜔^1/𝛼𝑡).(4.20) It thus follows from Lemma 3.1 that when 𝛽>𝑛𝑚/2, 𝐸_𝛼(𝑡^𝛼𝑃)(1+|𝜉|²)^−𝛽/2∈ℱ𝐿¹(ℝ^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 𝐿^𝑝(ℝ^𝑛)(1≤𝑝<∞) or 𝐶₀(ℝ^𝑛)∶={𝑓∈𝐶(ℝ^𝑛)∶lim_|𝑥|→∞𝑓(𝑥)=0}. The partial differential operator 𝑃(𝐷) defined by 𝑃(𝐷)𝑓=ℱ⁻¹(𝑃ℱ𝑓)(4.21) with 𝐷(𝑃(𝐷))={𝑓∈𝑋∶ℱ⁻¹(𝑃ℱ𝑓)∈𝑋}(4.22) is closed and densely defined on 𝑋. Since 𝑖𝐷_𝑗=𝜕/𝜕𝑥_𝑗(1≤𝑗≤𝑛) is the generator of the bounded 𝐶₀-group {𝑇_𝑗(𝑡)}_𝑡∈ℝ given by 𝑇_𝑗(𝑡)𝑓(𝑥₁,…,𝑥_𝑛)=𝑓(𝑥₁,…,𝑥_𝑗−1,𝑥_𝑗+𝑡,𝑥_𝑗+1,…,𝑥_𝑛)𝑡∈ℝ(4.23) on 𝑋, we can apply the above results to 𝑃(𝐷) on 𝑋. It is remarkable that when 𝑋=𝐿^𝑝(ℝ^𝑛) (1<𝑝<∞) these results can be improved. In fact, if 𝐴=𝐷=(𝐷₁,…,𝐷_𝑛), then the functions 𝑢_𝑡's in the proofs of the above theorems give rise to Fourier multipliers on 𝐿^𝑝(ℝ^𝑛) having norm of polynomial growth 𝑡^𝑛𝑝 at infinity, where 𝑛_𝑝=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. (a)For 𝑋=𝐿¹(ℝ^𝑛) or 𝐶₀(ℝ^𝑛), 𝐶=(𝑎−𝑃)^−𝛾(𝐷), where 𝛾>𝑛𝑚/2𝑟, there exists an analytic 𝛼-times 𝐶-regularized resolvent family 𝑆_𝛼(𝑡) for 𝑃(𝐷) and ‖𝑆_𝛼(𝑡)‖≤𝑀(1+𝑡^𝛼𝑛/2)exp(𝜔^1/𝛼𝑡),𝑡≥0.(4.24)(b)For 𝑋=𝐿^𝑝(ℝ^𝑛), 𝐶=(𝑎−𝑃)^−𝛾(𝐷), where 𝛾>𝑛_𝑝𝑚/𝑟, 𝑛_𝑝=𝑛|1/2−1/𝑝|, there exists an analytic 𝛼-times 𝐶-regularized resolvent family 𝑆_𝛼(𝑡) for 𝑃(𝐷) and‖𝑆_𝛼(𝑡)‖≤𝑀(1+𝑡^𝛼𝑛𝑝)exp(𝜔^1/𝛼𝑡),𝑡≥0.(4.25)

Theorem 4.5. Suppose that the assumptions of Theorem 4.3 are satisfied.(a)For 𝑋=𝐿¹(ℝ^𝑛) or 𝐶₀(ℝ^𝑛), 𝐶=(1−Δ)^{−𝑚𝛽/2}, where 𝛽>𝑛/2, there exists an analytic 𝛼-times 𝐶-regularized resolvent family 𝑆_𝛼(𝑡) for 𝑃(𝐷) and ‖𝑆_𝛼(𝑡)‖≤𝑀(1+𝑡^𝛼𝑛/2)exp(𝜔^1/𝛼𝑡),𝑡≥0.(4.26)(b)For 𝑋=𝐿^𝑝(ℝ^𝑛), 𝐶=(1−Δ)^{−𝑚𝛽/2}, where 𝛽>𝑛_𝑝, there exists an analytic 𝛼-times 𝐶-regularized resolvent family 𝑆_𝛼(𝑡) for 𝑃(𝐷) and ‖𝑆_𝛼(𝑡)‖≤𝑀(1+𝑡^𝛼𝑛𝑝)exp(𝜔^1/𝛼𝑡),𝑡≥0.(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 𝐿^𝑝(ℝ^𝑛) (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>𝑛_𝑝.(b) Consider 𝑃(𝐷) on 𝐿^𝑝(ℝ²) (1<𝑝<∞) with 𝑃(𝜉)=−(1+𝜉²¹)(1+(𝜉₂−𝜉^𝑙1)²)(𝑙∈ℕ).(4.28)Then 𝑃(𝜉)≤−1(𝜉∈ℝ²). We claim that 𝑃 is (2/𝑙)-coercive. Indeed, if |𝜉₂|≥2|𝜉^𝑙1|, then |𝑃(𝜉)|≥(1+𝜉²¹)(1+14𝜉²²)≥14|𝜉|².(4.29) If |𝜉₂|<2|𝜉^𝑙1|, then |𝑃(𝜉)|≥1+|𝜉₁|²≥𝑐|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=(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).