Abstract and Applied Analysis
Volume 2009 (2009), Article ID 438690, 14 pages
doi:10.1155/2009/438690
Research Article

Fractional Evolution Equations Governed by Coercive Differential Operators

Fu-Bo Li,1 Miao Li,1 and Quan Zheng2

1Department of Mathematics, Sichuan University, Chengdu 610064, China
2Department of Mathematics, Huazhong University of Science and Technology, Wuhan 430074, China

Received 24 November 2008; Revised 8 February 2009; Accepted 24 March 2009

Academic Editor: Paul Eloe

Copyright © 2009 Fu-Bo Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper is concerned with evolution equations of fractional order D 𝛼 𝑢 ( 𝑡 ) = 𝐴 𝑢 ( 𝑡 ) ; 𝑢 ( 0 ) = 𝑢 0 , 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 ) = 𝐴 𝑢 ( 𝑡 ) , 𝑡 > 0 ; 𝑢 ( 0 ) = 𝑥 ( 1 . 1 ) is well posed if and only if 𝐴 is the generator of a 𝐶 0 -semigroup. However, many partial differential operators (PDOs) such as the Schrödinger operator 𝑖 Δ on 𝐿 𝑝 ( ℝ 𝑛 ) ( 𝑝 ≠ 2 ) cannot generate 𝐶 0 -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 ) ∈ 𝐿 1 l o c ( ℝ + ) if 𝑅 ( 𝑡 ) 𝐴 ⊂ 𝐴 𝑅 ( 𝑡 ) and the resolvent equation 𝑅 ( 𝑡 ) 𝑥 = 𝑥 + 𝑡 0 0 𝑥 0 2 0 0 𝑑 𝑎 ( 𝑡 − 𝑠 ) 𝐴 𝑅 ( 𝑠 ) 𝑥 𝑑 𝑠 , 𝑡 ≥ 0 , 𝑥 ∈ 𝐷 ( 𝐴 ) ( 1 . 2 ) holds. It is obvious that a 𝐶 0 -semigroup is a resolvent family for its generator with kernel 𝑎 1 ( 𝑡 ) ≡ 1 ; a cosine function is a resolvent family for its generator with kernel 𝑎 2 ( 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 𝐶 0 -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 𝛼 𝑢 ( 𝑡 ) = 𝐴 𝑢 ( 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 𝛼 𝑓 ( 𝑡 ) ∶ = 𝑡 0 0 𝑥 0 2 0 0 𝑑 𝑔 𝑚 − 𝛼 ( 𝑑 𝑡 − 𝑠 ) 𝑚 𝑑 𝑠 𝑚 𝑓 ( 𝑠 ) 𝑑 𝑠 , ( 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 𝐶 0 -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 − 𝐴 i s i n j e c t i v e a n d 𝑅 ( 𝐶 ) ⊂ 𝑅 ( 𝜆 − 𝐴 ) } . Let Σ 𝜃 ∶ = { 𝜆 ∈ ℂ ∶ | a r g 𝜆 | < 𝜃 } be the open sector of angle 2 𝜃 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 ⊂ 𝐴 𝑆 𝛼 ( 𝑡 ) for 𝑡 ≥ 0 ;(c) 𝐶 − 1 𝐴 𝐶 = 𝐴 ;(d)for 𝑥 ∈ 𝐷 ( 𝐴 ) , 𝑆 𝛼 ∫ ( 𝑡 ) 𝑥 = 𝐶 𝑥 + 𝑡 0 0 𝑥 0 2 0 0 𝑑 ( ( 𝑡 − 𝑠 ) 𝛼 − 1 / Γ ( 𝛼 ) ) 𝑆 𝛼 ( 𝑠 ) 𝐴 𝑥 𝑑 𝑠 . { 𝑆 𝛼 ( 𝑡 ) } 𝑡 ≥ 0 is called analytic if it can be extended analytically to some sector Σ 𝜃 .

If ‖ 𝑆 𝛼 ( 𝑡 ) ‖ ≤ 𝑀 𝑒 𝜔 𝑡 ( 𝑡 ≥ 0 ) for some constants 𝑀 ≥ 1 and 𝜔 ∈ ℝ + , we will write 𝐴 ∈ 𝒞 𝛼 𝐶 ( 𝑀 , 𝜔 ) , and 𝒞 𝛼 𝐶 ( 𝜔 ) ∶ = ∪ { 𝒞 𝛼 𝐶 ( 𝑀 , 𝜔 ) ; 𝑀 ≥ 1 } , 𝒞 𝛼 𝐶 ∶ = ∪ { 𝒞 𝛼 𝐶 ( 𝜔 ) ; 𝜔 ≥ 0 } .

Define the operator 𝐴 by 𝐴 𝑥 = 𝐶 − 1 l i m 𝑡 ↓ 0 Γ ( 𝛼 + 1 ) 𝑡 𝛼 𝑆 𝛼 𝐴 ( 𝑡 ) 𝑥 − 𝐶 𝑥 , 𝑥 ∈ 𝐷 , ( 2 . 1 ) with 𝐷 𝐴 = 𝑥 ∈ 𝑋 ∶ l i m 𝑡 ↓ 0 𝑆 𝛼 ( 𝑡 ) 𝑥 − 𝐶 𝑥 𝑡 𝛼 e x i s t s a n d i s i n 𝑅 ( 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 𝑡 0 0 𝑥 0 2 0 0 𝑑 𝑔 𝛼 ( 𝑡 − 𝑠 ) 𝑆 𝛼 ( ‖ ‖ ‖ 𝑠 ) 𝑥 𝑑 𝑠 − 𝐶 𝑥 ≤ 𝑔 𝛼 + 1 ( 𝑡 ) − 1 𝑡 0 0 𝑥 0 2 0 0 𝑑 𝑔 𝛼 ( ‖ ‖ 𝑆 𝑡 − 𝑠 ) 𝛼 ( ‖ ‖ 𝑠 ) 𝑥 − 𝐶 𝑥 𝑑 𝑠 ≤ s u p 0 ≤ 𝑠 ≤ 𝑡 ‖ ‖ 𝑆 𝛼 ( ‖ ‖ 𝑠 ) 𝑥 − 𝐶 𝑥 ⟶ 0 a s 𝑡 ⟶ 0 . ( 2 . 3 ) Thus for 𝑥 ∈ 𝐷 ( 𝐴 ) , by Definition 2.1, l i m 𝑡 ↓ 0 𝑔 𝛼 + 1 ( 𝑡 ) − 1 𝑆 𝛼 ( 𝑡 ) 𝑥 − 𝐶 𝑥 = l i m 𝑡 ↓ 0 𝑔 𝛼 + 1 ( 𝑡 ) − 1 𝑡 0 0 𝑥 0 2 0 0 𝑑 𝑔 𝛼 ( 𝑡 − 𝑠 ) 𝑆 𝛼 ( 𝑠 ) 𝐴 𝑥 𝑑 𝑠 = 𝐶 𝐴 𝑥 , ( 2 . 4 ) which means that 𝑥 ∈ 𝐷 ( 𝐴 ) and 𝐴 𝑥 = 𝐴 𝑥 . On the other hand, for 𝑥 ∈ 𝐷 ( 𝐴 ) , by the definition of 𝐴 and Definition 2.1, 𝐶 𝐴 𝑥 = l i m 𝑡 ↓ 0 𝑔 𝛼 + 1 ( 𝑡 ) − 1 𝑆 𝛼 ( 𝑡 ) 𝑥 − 𝐶 𝑥 = l i m 𝑡 ↓ 0 𝑔 𝛼 + 1 ( 𝑡 ) − 1 𝐴 𝑡 0 0 𝑥 0 2 0 0 𝑑 𝑔 𝛼 ( 𝑡 − 𝑠 ) 𝑆 𝛼 ( 𝑠 ) 𝑥 𝑑 𝑠 , ( 2 . 5 ) but l i m 𝑡 → 0 𝑔 𝛼 + 1 ( 𝑡 ) − 1 ∫ 𝑡 0 0 𝑥 0 2 0 0 𝑑 𝑔 𝛼 ( 𝑡 − 𝑠 ) 𝑆 𝛼 ( 𝑠 ) 𝑥 𝑑 𝑠 = 𝐶 𝑥 , by (d) of Definition 2.1. Thus it follows from the closedness of 𝐴 that 𝐶 𝑥 ∈ 𝐷 ( 𝐴 ) with 𝐴 𝐶 𝑥 = 𝐶 𝐴 𝑥 . This implies that 𝑥 ∈ 𝐷 ( 𝐶 − 1 𝐴 𝐶 ) = 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 𝛼 𝐶 ( 𝑀 , 𝜔 ) ;(b) 𝐴 = 𝐶 − 1 𝐴 𝐶 , ( 𝜔 𝛼 , ∞ ) ⊆ 𝜌 𝐶 ( 𝐴 ) and ‖ ‖ ‖ 𝑑 𝑛 𝑑 𝜆 𝑛 𝜆 𝛼 − 1 ( 𝜆 𝛼 − 𝐴 ) − 1 𝐶 ‖ ‖ ‖ ≤ 𝑀 𝑛 ! ( 𝜆 − 𝜔 ) 𝑛 + 1 , 𝜆 > 𝜔 , 𝑛 ∈ ℕ 0 ∶ = ℕ ∪ { 0 } ; ( 2 . 6 ) (c) 𝐴 = 𝐶 − 1 𝐴 𝐶 , ( 𝜔 𝛼 , ∞ ) ⊆ 𝜌 𝐶 ( 𝐴 ) and there exists a strongly continuous family { 𝑆 𝛼 ( 𝑡 ) } 𝑡 ≥ 0 ⊂ 𝐁 ( 𝑋 ) satisfying ‖ 𝑆 𝛼 ( 𝑡 ) ‖ ≤ 𝑀 𝑒 𝜔 𝑡 such that 𝜆 𝛼 − 1 ( 𝜆 𝛼 − 𝐴 ) − 1 𝐶 𝑥 = ∞ 0 0 𝑥 0 2 0 0 𝑑 𝑒 − 𝜆 𝑡 𝑆 𝛼 ( 𝑡 ) 𝑥 𝑑 𝑡 , 𝜆 > 𝜔 , 𝑥 ∈ 𝑋 . ( 2 . 7 )

Theorem 2.4. Suppose that 0 < 𝛼 < 𝛽 ≤ 2 , 𝛾 = 𝛼 / 𝛽 . If 𝐴 ∈ 𝒞 𝛽 𝐶 ( 𝜔 ) then 𝐴 ∈ 𝒞 𝛼 𝐶 ( 𝜔 1 / 𝛾 ) and the 𝛼 -times 𝐶 -regularized resolvent family for 𝐴 , { 𝑆 𝛼 ( 𝑡 ) } 𝑡 ≥ 0 , can be extended analytically to Σ m i n { 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 𝐶 0 -groups (cf. [14]), which will play a key role in our proof.

Let 𝑖 𝐴 𝑗 ( 1 ≤ 𝑗 ≤ 𝑛 ) be commuting generators of bounded 𝐶 0 -groups on a Banach space 𝑋 . Write 𝐴 = ( 𝐴 1 , … , 𝐴 𝑛 ) and 𝐴 𝜇 = 𝐴 𝜇 1 1 ⋯ 𝐴 𝜇 𝑛 𝑛 for 𝜇 = ( 𝜇 1 , … , 𝜇 𝑛 ) ∈ ℕ 𝑛 0 . Similarly, write 𝐷 𝜇 = 𝐷 𝜇 1 1 ⋯ 𝐷 𝜇 𝑛 𝑛 , where 𝐷 𝑗 = − 𝑖 𝜕 / 𝜕 𝑥 𝑗 for 𝑗 = 1 , … , 𝑛 . For a polynomial ∑ 𝑃 ( 𝜉 ) ∶ = | 𝜇 | ≤ 𝑚 0 𝑥 0 2 0 0 𝑑 𝑎 𝜇 𝜉 𝜇 ( 𝜉 ∈ ℝ 𝑛 ) ∑ ( | 𝜇 | ∶ = 𝑛 𝑗 = 1 0 𝑥 0 2 0 0 𝑑 𝜇 𝑗 ) with constant coefficients, we define ∑ 𝑃 ( 𝐴 ) = | 𝜇 | ≤ 𝑚 0 𝑥 0 2 0 0 𝑑 𝑎 𝜇 𝐴 𝜇 ( 𝜉 ∈ ℝ 𝑛 ) with maximal domain. Then 𝑃 ( 𝐴 ) is closable. Let ℱ be the Fourier transform, that is, ∫ ( ℱ 𝑢 ) ( 𝜂 ) = ℝ 𝑛 0 𝑥 0 2 0 0 𝑑 𝑢 ( 𝜉 ) 𝑒 − 𝑖 ( 𝜉 , 𝜂 ) 𝑑 𝜉 for 𝑢 ∈ 𝐿 1 ( ℝ 𝑛 ) , where ∑ ( 𝜉 , 𝜂 ) = 𝑛 𝑗 = 1 0 𝑥 0 2 0 0 𝑑 𝜉 𝑗 𝜂 𝑗 . If 𝑢 ∈ ℱ 𝐿 1 ( ℝ 𝑛 ) ∶ = { ℱ 𝑣 ∶ 𝑣 ∈ 𝐿 1 ( ℝ 𝑛 ) } , then there exists a unique function in 𝐿 1 ( ℝ 𝑛 ) , written ℱ − 1 𝑢 , such that 𝑢 = ℱ ( ℱ − 1 𝑢 ) . In particular, ℱ − 1 𝑢 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 = ℝ 𝑛 ℱ 0 𝑥 0 2 0 0 𝑑 − 1 𝑢 ( 𝜉 ) 𝑒 − 𝑖 ( 𝜉 , 𝐴 ) 𝑥 𝑑 𝜉 , 𝑥 ∈ 𝑋 , ( 3 . 1 ) where ∑ ( 𝜉 , 𝐴 ) = 𝑛 𝑗 = 1 0 𝑥 0 2 0 0 𝑑 𝜉 𝑗 𝐴 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) ℱ 𝐿 1 ( ℝ 𝑛 ) is a Banach algebra under pointwise multiplication and addition with norm ‖ 𝑢 ‖ ℱ 𝐿 1 ∶ = ‖ ℱ − 1 𝑢 ‖ 𝐿 1 .
(b) 𝑢 ↦ 𝑢 ( 𝐴 ) is an algebra homomorphism from ℱ 𝐿 1 ( ℝ 𝑛 ) into 𝐁 ( 𝑋 ) , and there exists a constant 𝑀 > 0 such that ‖ 𝑢 ( 𝐴 ) ‖ ≤ 𝑀 ‖ 𝑢 ‖ ℱ 𝐿 1 .
(c) 𝐸 ∶ = { 𝜙 ( 𝐴 ) 𝑥 ∶ 𝜙 ∈ 𝒮 ( ℝ 𝑛 ) , 𝑥 ∈ 𝑋 } ⊂ ∩ 𝜇 ∈ ℕ 𝑛 0 𝐷 ( 𝐴 𝜇 ) , 𝐸 = 𝑋 , 𝑃 ( 𝐴 ) | 𝐸 = 𝑃 ( 𝐴 ) and 𝜙 ( 𝐴 ) 𝑃 ( 𝐴 ) ⊂ 𝑃 ( 𝐴 ) 𝜙 ( 𝐴 ) = ( 𝑃 𝜙 ) ( 𝐴 ) for 𝜙 ∈ 𝒮 ( ℝ 𝑛 ) .
(d) Let 𝑢 ∈ 𝐶 𝑗 ( ℝ 𝑛 ) ( 𝑗 > 𝑛 / 2 ) . Suppose that there exist constants 𝐿 , 𝑀 0 , 𝑎 > 0 , and 𝑏 ∈ [ − 1 , 2 𝑎 / 𝑛 − 1 ) such that | | 𝐷 𝑘 | | ≤ 𝑀 𝑢 ( 𝜉 ) | | 𝑘 | | 0 | | 𝜉 | | 𝑏 | 𝑘 | − 𝑎 | | 𝜉 | | | | 𝑘 | | 𝑀 , f o r ≥ 𝐿 , ≤ 𝑗 , | | 𝑘 | | 0 | | 𝜉 | | | | 𝑘 | | , f o r < 𝐿 , ≤ 𝑗 , ( 3 . 2 ) where 𝑘 ∈ ℕ 𝑛 0 , then 𝑢 ∈ ℱ 𝐿 1 ( ℝ 𝑛 ) and ‖ 𝑢 ‖ ℱ 𝐿 1 ≤ 𝑀 𝑀 0 𝑛 / 2 for some constant 𝑀 > 0 .

Recall that the Mittag-Leffler function (see [15, 16]) is defined by 𝐸 𝛼 , 𝛽 ( 𝑧 ) ∶ = ∞ 𝑛 = 0 𝑧 0 𝑥 0 2 0 0 𝑑 𝑛 = 1 Γ ( 𝛼 𝑛 + 𝛽 ) 2 𝜋 𝑖 𝒯 𝜇 0 𝑥 0 2 0 0 𝑑 𝛼 − 𝛽 𝑒 𝜇 𝜇 𝛼 − 𝑧 𝑑 𝜇 , 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 ∞ 0 0 𝑥 0 2 0 0 𝑑 𝑒 − 𝜆 𝑡 𝑡 𝛽 − 1 𝐸 𝛼 , 𝛽 ( 𝜔 𝑡 𝛼 𝜆 ) 𝑑 𝑡 = 𝛼 − 𝛽 𝜆 𝛼 − 𝜔 , R e 𝜆 > 𝜔 1 / 𝛼 , 𝜔 > 0 ( 3 . 4 ) and with their asymptotic expansion as 𝑧 → ∞ . If 0 < 𝛼 < 2 , 𝛽 > 0 , then 𝐸 𝛼 , 𝛽 1 ( 𝑧 ) = 𝛼 𝑧 ( 1 − 𝛽 ) / 𝛼 𝑧 e x p 1 / 𝛼 + 𝜀 𝛼 , 𝛽 | | | | ≤ 1 ( 𝑧 ) , a r g 𝑧 2 𝐸 𝛼 𝜋 , ( 3 . 5 ) 𝛼 , 𝛽 ( 𝑧 ) = 𝜀 𝛼 , 𝛽 | | | | < 1 ( 𝑧 ) , a r g ( − 𝑧 ) 1 − 2 𝛼 𝜋 , ( 3 . 6 ) where 𝜀 𝛼 , 𝛽 ( 𝑧 ) = − 𝑁 − 1 𝑛 = 1 𝑧 0 𝑥 0 2 0 0 𝑑 − 𝑛 Γ ( 𝛽 − 𝛼 𝑛 ) + 𝑂 | 𝑧 | − 𝑁 ( 3 . 7 ) as 𝑧 → ∞ , and the 𝑂 -term is uniform in a r g 𝑧 if | a r g ( − 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 1 ( 𝑧 ) = 𝛼 𝐸 𝛼 , 𝛼 + 𝛽 − 1 ( 𝑧 ) − 𝛽 − 1 𝛼 𝐸 𝛼 , 𝛼 + 𝛽 ( 𝑧 ) . ( 3 . 9 )

Proof. By the definition of 𝐸 𝛼 , 𝛽 ( 𝑧 ) , 𝐸 𝛼 , 𝛽 ( 𝑧 ) = ∞ 𝑛 = 0 𝑧 0 𝑥 0 2 0 0 𝑑 𝑛 Γ ( 𝛼 𝑛 + 𝛽 ) = ∞ 𝑛 = 1 0 𝑥 0 2 0 0 𝑑 𝑛 𝑧 𝑛 − 1 = Γ ( 𝛼 𝑛 + 𝛽 ) ∞ 𝑛 = 1 0 𝑥 0 2 0 0 𝑑 𝑛 𝑧 𝑛 − 1 = Γ ( 𝛼 𝑛 + 𝛽 − 1 + 1 ) ∞ 𝑛 = 1 𝑧 0 𝑥 0 2 0 0 𝑑 𝑛 − 1 ⋅ ( 𝛼 𝑛 + 𝛽 − 1 ) − 𝛼 ( 𝛼 𝑛 + 𝛽 − 1 ) Γ ( 𝛼 𝑛 + 𝛽 − 1 ) ∞ 𝑛 = 1 0 𝑥 0 2 0 0 𝑑 𝛽 − 1 𝛼 ⋅ 𝑧 𝑛 − 1 = 1 Γ ( 𝛼 𝑛 + 𝛽 ) 𝛼 𝐸 𝛼 , 𝛼 + 𝛽 − 1 ( 𝑧 ) − 𝛽 − 1 𝛼 𝐸 𝛼 , 𝛼 + 𝛽 ( 𝑧 ) , ( 3 . 1 0 ) 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 ) | | ≤ 𝑀 ( 𝑧 ) | | | | ≤ 𝛼 | 𝑧 | , i f | 𝑧 | ≥ 𝐿 , a r g ( − 𝑧 ) 1 − 2 − 𝜖 𝜋 . ( 3 . 1 1 )

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 = 1 0 𝑥 0 2 0 0 𝑑 𝑎 𝑗 𝐸 𝛼 , 𝛼 𝑘 − ( 𝑘 − 𝑗 ) ( 𝑧 ) , ( 3 . 1 2 ) where 𝑎 𝑗 only depend on 𝛼 and 𝑘 . Since 𝛼 > 1 we have that 𝛼 𝑘 − ( 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 𝑘 0 = [ 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 𝛼 𝑃 ) ( 𝑎 − 𝑃 ) − 𝛾 | | ≤ 𝑀 1 + 𝑡 𝛼 | | 𝜇 | | | | 𝜉 | | | | 𝜇 | | ( 𝑚 − 1 ) − 𝑟 𝛾 , | | 𝜉 | | | | 𝜇 | | ≥ 𝐿 , ≤ 𝑘 0 , 𝑡 ≥ 0 . ( 3 . 1 3 )

Proof. Suppose that for | 𝜉 | ≥ 𝐿 , (3.11) holds up to order 𝑘 0 and | | | | | | 𝜉 | | 𝑃 ( 𝜉 ) ≥ 𝑀 𝑟 , | | | | | | 𝜉 | | 𝑎 − 𝑃 ( 𝜉 ) ≥ 𝑀 𝑟 . ( 3 . 1 4 ) By induction, one can show that 𝐷 𝜇 𝐸 𝛼 ( 𝑡 𝛼 𝑃 ) = | 𝜇 | 𝑗 = 1 0 𝑥 0 2 0 0 𝑑 𝑡 𝛼 𝑗 𝐸 𝛼 ( 𝑗 ) ( 𝑡 𝛼 𝑃 ) 𝑄 𝑗 , ( 3 . 1 5 ) where d e g 𝑄 𝑗 ≤ 𝑚 𝑗 − | 𝜇 | . Thus if | 𝜉 | ≥ 𝐿 and | 𝑡 𝛼 𝑃 | ≥ 𝐿 , | | 𝐷 𝜇 𝐸 𝛼 ( 𝑡 𝛼 | | 𝑃 ) ≤ 𝑀 1 + 𝑡 𝛼 ( | 𝜇 | − 1 ) | | 𝜉 | | 𝑚 | 𝜇 | − | 𝜇 | − 𝑟 ≤ 𝑀 1 + 𝑡 𝛼 | | 𝜇 | | | | 𝜉 | | | | 𝜇 | | ( 𝑚 − 1 ) − 𝑟 , ( 3 . 1 6 ) and if | 𝜉 | ≥ 𝐿 with | 𝑡 𝛼 𝑃 | ≤ 𝐿 , by (3.8) and (3.12) we know that | | 𝐷 𝜇 𝐸 𝛼 ( 𝑡 𝛼 | | 𝑡 𝑃 ) ≤ 𝑀 𝛼 + 𝑡 𝛼 | 𝜇 | | | 𝜉 | | ( 𝑚 − 1 ) | 𝜇 | . ( 3 . 1 7 ) Altogether, we have | | 𝐷 𝜇 𝐸 𝛼 ( 𝑡 𝛼 | | 𝑃 ) ≤ 𝑀 1 + 𝑡 𝛼 | 𝜇 | | | 𝜉 | | ( 𝑚 − 1 ) | 𝜇 | , | | 𝜉 | | ≥ 𝐿 . ( 3 . 1 8 ) And by | | 𝐷 𝜇 ( 𝑎 − 𝑃 ) − 𝛾 | | | | 𝜉 | | ≤ 𝑀 | | 𝜇 | | ( 𝑚 − 𝑟 − 1 ) − 𝑟 𝛾 , | | 𝜉 | | ≥ 𝐿 ( 3 . 1 9 ) and Leibniz's formula we have | | 𝐷 𝜇 𝐸 𝛼 ( 𝑡 𝛼 𝑃 ) ( 𝑎 − 𝑃 ) − 𝛾 | | ≤ 𝑀 1 + 𝑡 𝛼 | 𝜇 | | | 𝜉 | | ( 𝑚 − 1 ) | 𝜇 | − 𝑟 𝛾 , | | 𝜉 | | ≥ 𝐿 . ( 3 . 2 0 )

Lemma 3.5. This proves (3.13). Suppose that the assumptions of Lemma 3.4 are satisfied. Let 𝛾 > 𝑛 𝑚 / 2 𝑟 . Then 𝐸 𝛼 ( 𝑡 𝛼 𝑃 ) ( 𝑎 − 𝑃 ) − 𝛾 ∈ ℱ 𝐿 1 ( ℝ 𝑛 ) and ‖ ‖ 𝐸 𝛼 ( 𝑡 𝛼 𝑃 ) ( 𝑎 − 𝑃 ) − 𝛾 ‖ ‖ ℱ 𝐿 1 ≤ 𝑀 1 + 𝑡 𝛼 𝑛 / 2 , 𝑡 ≥ 0 . ( 3 . 2 1 ) The same result holds with 𝐸 𝛼 ( 𝑡 𝛼 𝑃 ) replaced by 𝐸 𝛼 , 𝛼 ( 𝑡 𝛼 𝑃 ) .

Proof. By Lemma 3.1(d), it remains to prove that for | 𝜉 | ≤ 𝐿 , | | 𝐷 𝜇 𝐸 𝛼 ( 𝑡 𝛼 𝑃 ) ( 𝑎 − 𝑃 ) − 𝛾 | | ≤ 𝑀 1 + 𝑡 𝛼 | 𝜇 | , | | 𝜇 | | ≤ 𝑘 0 , 𝑡 ≥ 0 . ( 3 . 2 2 ) To show this we can use (3.15) and then give the estimates according to the values 𝑡 𝛼 𝑃 . For | 𝜉 | ≤ 𝐿 with | 𝑡 𝛼 𝑃 | ≥ 𝐿 the estimate (3.8) can be applied, and for | 𝜉 | ≤ 𝐿 with | 𝑡 𝛼 𝑃 | ≤ 𝐿 note that all the functions 𝐸 𝛼 ( 𝑗 ) ( 𝑡 𝛼 𝑃 ) are uniformly bounded.
For the second part of the lemma, note that 𝐸 𝛼 , 𝛼 ( 𝑧 ) = 𝛼 𝐸 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 > 𝑛 𝑚 / 2 𝑟 , 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 𝛼 𝑃 ) ( 𝑎 − 𝑃 ) − 𝛾 ) ( 𝐴 ) with ‖ ‖ 𝑆 𝛼 ‖ ‖ ( 𝑡 ) ≤ 𝑀 1 + 𝑡 𝛼 𝑛 / 2 , 𝑡 ≥ 0 . ( 4 . 1 )

Proof. Let 𝑢 𝑡 = 𝐸 𝛼 ( 𝑡 𝛼 𝑃 ) ( 𝑎 − 𝑃 ) − 𝛾 , 𝑡 ≥ 0 . By Lemma 3.5, 𝑢 𝑡 ∈ ℱ 𝐿 1 and ‖ 𝑢 𝑡 ‖ ℱ 𝐿 1 ≤ 𝑀 ( 1 + 𝑡 𝛼 𝑛 / 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 𝛼 𝑛 / 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 + ℎ ) 𝜙 ( 𝐴 ) − 𝑆 𝛼 ‖ ‖ ‖ ‖ ( 𝑡 ) 𝜙 ( 𝐴 ) ≤ 𝑀 ( 𝐸 𝛼 ( ( 𝑡 + ℎ ) 𝛼 𝑃 ) − 𝐸 𝛼 ( 𝑡 𝛼 𝑃 ) ) ( 𝑎 − 𝑃 ) − 𝛾 𝜙 ‖ ‖ ℱ 𝐿 1 ‖ ‖ ‖ ≤ 𝑀 𝑡 𝑡 + ℎ 0 𝑥 0 2 0 0 𝑑 𝑠 𝛼 − 1 𝐸 𝛼 , 𝛼 ( 𝑠 𝛼 𝑃 ) ( 𝑎 − 𝑃 ) − 𝛾 ‖ ‖ ‖ 𝑃 𝜙 𝑑 𝑠 ℱ 𝐿 1 ≤ 𝑀 𝑡 𝑡 + ℎ 0 𝑥 0 2 0 0 𝑑 𝑠 𝛼 − 1 1 + 𝑠 𝛼 𝑛 / 2 𝑑 𝑠 ⋅ ‖ 𝑃 𝜙 ‖ ℱ 𝐿 1 ⟶ 0 , a s ℎ ⟶ 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 ) ) − 1 𝐶 = ∞ 0 0 𝑥 0 2 0 0 𝑑 𝑒 − 𝜆 𝑡 𝑆 𝛼 ( 𝑡 ) 𝑑 𝑡 , 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 ) = 𝑃 𝑢 𝑡 𝜙 ( 𝐴 ) = 𝑆 𝛼 ( 𝑡 ) 𝑃 ( 𝐴 ) 𝜙 ( 𝐴 ) . ( 4 . 4 ) Since ℱ 𝐿 1 ( ℝ 𝑛 ) is a Banach algebra, it follows that 𝑢 𝑡 , 𝑢 𝑡 ( 𝜆 − 𝑃 ) 𝜙 ∈ ℱ 𝐿 1 . Thus by Lemmas 3.1, 3.5, (3.4), and Fubini's theorem one obtains that for 𝑥 ∈ 𝑋 , 𝜆 > 0 , ∞ 0 0 𝑥 0 2 0 0 𝑑 𝑒 − 𝜆 𝑡 𝑆 𝛼 ( 𝑡 ) ( 𝜆 𝛼 − 𝑃 ( 𝐴 ) ) 𝜙 ( 𝐴 ) 𝑥 𝑑 𝑡 = ∞ 0 0 𝑥 0 2 0 0 𝑑 𝑒 − 𝜆 𝑡 𝑢 𝑡 ( 𝜆 𝛼 = − 𝑃 ) 𝜙 ( 𝐴 ) 𝑥 𝑑 𝑡 ∞ 0 0 𝑥 0 2 0 0 𝑑 𝑒 − 𝜆 𝑡 𝑢 𝑡 𝑑 𝑡 ( 𝜆 𝛼 − 𝑃 ) 𝜙 ( 𝐴 ) 𝑥 = 𝜆 𝛼 − 1 𝐶 𝜙 ( 𝐴 ) 𝑥 . ( 4 . 5 ) This implies that ∞ 0 0 𝑥 0 2 0 0 𝑑 𝑒 − 𝜆 𝑡 𝑆 𝛼 𝜆 ( 𝑡 ) 𝛼 − 𝑃 ( 𝐴 ) 𝑥 𝑑 𝑡 = 𝜆 𝛼 − 1 𝐶 𝑥 , 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 ) ∞ 0 0 𝑥 0 2 0 0 𝑑 𝑒 − 𝜆 𝑡 𝑆 𝛼 ( 𝑡 ) 𝑥 𝑑 𝑡 = 𝜆 𝛼 − 1 𝐶 𝑥 , 𝑥 ∈ 𝑋 . ( 4 . 7 ) Therefore, we have proved (4.3). And it is routine to show that 𝐶 − 1 𝑃 ( 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 > 𝑛 𝑚 / 2 𝑟 , 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 𝛼 𝑛 / 2 𝜔 e x p 1 / 𝛼 𝑡 , 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 + 𝜌 𝑒 𝑖 𝛼 ′ 𝜋 / 2 ∶ 0 ≤ 𝜌 < ∞ } , and let Ray 2 ∶ = { 𝜌 𝑒 𝑖 𝛼 𝜋 / 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 Δ 𝐺 𝑂 𝐵 . To estimate 𝐷 𝜇 𝐸 𝛼 ( 𝑡 𝛼 𝑃 ( 𝜉 ) ) for such values 𝑃 ( 𝜉 ) , we use (3.12), (3.15), and (3.5) to obtain | | 𝐷 𝜇 𝐸 𝛼 ( 𝑡 𝛼 | | 𝑃 ( 𝜉 ) ) ≤ 𝑀 1 + 𝑡 𝛼 | 𝜇 | 𝜔 e x p 1 / 𝛼 𝑡 ( 4 . 9 ) if Re ( ( 𝜌 𝑒 𝑖 𝜃 ) 1 / 𝛼 ) < 𝜔 1 / 𝛼 , where 𝜌 𝑒 𝑖 𝜃 denotes an arbitrary point on the line segment from 𝐺 to 𝐵 .
Since 𝜌 s i n ( 𝛼 = 𝜔 𝜋 / 2 ) s i n ( 𝛼 𝜋 / 2 − 𝜃 ) , ( 4 . 1 0 ) we have R e ( 𝜌 𝑒 𝑖 𝜃 ) 1 / 𝛼 = 𝜔 1 / 𝛼 s i n ( 𝛼 𝜋 / 2 ) s i n ( 𝛼 𝜋 / 2 − 𝜃 ) 1 / 𝛼 c o s ( 𝜃 / 𝛼 ) . ( 4 . 1 1 ) Thus, to show that Re ( ( 𝜌 𝑒 𝑖 𝜃 ) 1 / 𝛼 ) < 𝜔 1 / 𝛼 ( 0 < 𝜃 ≤ 𝛼 𝜋 / 2 ) one needs to check that c o s 𝛼 𝛼 ( 𝜃 / 𝛼 ) < c o s 𝜃 + s i n 𝜃 ⋅ t a n − 1 2 𝜋 , 0 < 𝜃 ≤ 𝛼 𝜋 2 ; ( 4 . 1 2 ) and this is true if c o s ( 𝜃 / 𝛼 ) ≤ c o s 𝜃 + s i n 𝜃 ⋅ t a n 𝛼 − 1 2 𝜋 , 0 < 𝜃 ≤ 𝛼 𝜋 2 , ( 4 . 1 3 ) since 1 < 𝛼 < 𝛼 < 2 .
We first consider the case when 𝜋 / 2 ≤ 𝜃 ≤ 𝛼 𝜋 / 2 . Let 𝑔 ( 𝜃 ) = c o s 𝜃 + s i n 𝜃 ⋅ t a n ( ( ( 𝛼 − 1 ) / 2 ) 𝜋 ) − c o s ( 𝜃 / 𝛼 ) , then 𝑔 ( 𝜃 ) = − s i n 𝜃 + ( 1 / 𝛼 ) s i n ( 𝜃 / 𝛼 ) + c o s 𝜃 ⋅ t a n ( ( ( 𝛼 − 1 ) / 2 ) 𝜋 ) ≤ 0 since s i n 𝜃 > ( 1 / 𝛼 ) s i n ( 𝜃 / 𝛼 ) and c o s 𝜃 ≤ 0 for 𝜋 / 2 ≤ 𝜃 ≤ 𝛼 𝜋 / 2 . So 𝑔 ( 𝜃 ) decreases with respect to 𝜃 , which means that 𝑔 ( 𝜃 ) ≥ 0 since 𝑔 ( 𝛼 𝜋 / 2 ) = 0 .
For 0 < 𝜃 < 𝜋 / 2 , we will show that c o s ( 𝜃 / 𝛼 ) ≤ c o s 𝜃 + s i n 𝜃 ⋅ 𝛼 − 1 2 𝜋 , ( 4 . 1 4 ) which implies (4.13). Now for fixed 𝜃 ∈ ( 0 , 𝜋 / 2 ) , denote by ℎ ( 𝛼 ) = c o s 𝜃 + s i n 𝜃 ⋅ ( ( 𝛼 − 1 ) / 2 ) 𝜋 − c o s ( 𝜃 / 𝛼 ) . Since 𝛼 > 1 , we have ℎ ( 𝛼 ) = ( 𝜋 / 2 ) s i n 𝜃 − ( 𝜃 / 𝛼 2 ) s i n ( 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 𝛼 𝑃 ) ( 𝑎 − 𝑃 ) − 𝛾 | | ≤ 𝑀 1 + 𝑡 𝛼 | 𝜇 | | | 𝜉 | | ( 𝑚 − 1 ) | 𝜇 | − 𝑟 𝛾 𝜔 e x p 1 / 𝛼 𝑡 , ( 4 . 1 5 ) and for | 𝜉 | ≤ 𝐿 , | | 𝐷 𝜇 𝐸 𝛼 ( 𝑡 𝛼 𝑃 ) ( 𝑎 − 𝑃 ) − 𝛾 | | ≤ 𝑀 1 + 𝑡 𝛼 | | 𝜇 | | 𝜔 e x p 1 / 𝛼 𝑡 . ( 4 . 1 6 ) An argument similar to that one of the proof of Theorem 4.1 gives our claim.

438690.fig.001
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 ) − 𝑚 𝛽 / 2 (which is defined by (3.1) with 𝑢 ( 𝑥 ) = ( 1 + | 𝑥 | 2 ) − 𝑚 𝛽 / 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 𝛼 𝑛 / 2 𝜔 e x p 1 / 𝛼 𝑡 , 𝑡 ≥ 0 . ( 4 . 1 7 )

Proof. From (4.9) and | | | | 𝐷 𝜇 | | 𝜉 | | 1 + 2 − 𝛽 / 2 | | | | | | 𝜉 | | ≤ 𝑀 − | 𝜇 | − 𝛽 , | | 𝜉 | | ≥ 𝐿 , 𝜇 ∈ ℕ 𝑛 0 , ( 4 . 1 8 ) we have for | 𝜉 | ≥ 𝐿 , | | | 𝐷 𝜇 𝐸 𝛼 ( 𝑡 𝛼 𝑃 ) 1 + 𝜉 | 2 − 𝛽 / 2 | | | ≤ 𝑀 1 + 𝑡 𝛼 | 𝜇 | | 𝜉 | ( 𝑚 − 1 ) | 𝜇 | − 𝛽 𝜔 e x p 1 / 𝛼 𝑡 , ( 4 . 1 9 ) and for | 𝜉 | ≤ 𝐿 , | | | 𝐷 𝜇 𝐸 𝛼 ( 𝑡 𝛼 𝑃 ) ( 1 + | 𝜉 | 2 ) − 𝛽 / 2 | | | ≤ 𝑀 1 + 𝑡 𝛼 | 𝜇 | 𝜔 e x p 1 / 𝛼 𝑡 . ( 4 . 2 0 ) It thus follows from Lemma 3.1 that when 𝛽 > 𝑛 𝑚 / 2 , 𝐸 𝛼 ( 𝑡 𝛼 𝑃 ) ( 1 + | 𝜉 | 2 ) − 𝛽 / 2 ∈ ℱ 𝐿 1 ( 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 𝐶 0 ( ℝ 𝑛 ) ∶ = { 𝑓 ∈ 𝐶 ( ℝ 𝑛 ) ∶ l i m | 𝑥 | → ∞ 𝑓 ( 𝑥 ) = 0 } . The partial differential operator 𝑃 ( 𝐷 ) defined by 𝑃 ( 𝐷 ) 𝑓 = ℱ − 1 ( 𝑃 ℱ 𝑓 ) ( 4 . 2 1 ) with 𝐷 ( 𝑃 ( 𝐷 ) ) = 𝑓 ∈ 𝑋 ∶ ℱ − 1 ( 𝑃 ℱ 𝑓 ) ∈ 𝑋 ( 4 . 2 2 ) is closed and densely defined on 𝑋 . Since 𝑖 𝐷 𝑗 = 𝜕 / 𝜕 𝑥 𝑗 ( 1 ≤ 𝑗 ≤ 𝑛 ) is the generator of the bounded 𝐶 0 -group { 𝑇 𝑗 ( 𝑡 ) } 𝑡 ∈ ℝ given by 𝑇 𝑗 𝑥 ( 𝑡 ) 𝑓 1 , … , 𝑥 𝑛 𝑥 = 𝑓 1 , … , 𝑥 𝑗 − 1 , 𝑥 𝑗 + 𝑡 , 𝑥 𝑗 + 1 , … , 𝑥 𝑛 𝑡 ∈ ℝ ( 4 . 2 3 ) 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 = ( 𝐷 1 , … , 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 𝑛 𝑝 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 = 𝐿 1 ( ℝ 𝑛 ) or 𝐶 0 ( ℝ 𝑛 ) , 𝐶 = ( 𝑎 − 𝑃 ) − 𝛾 ( 𝐷 ) , where 𝛾 > 𝑛 𝑚 / 2 𝑟 , there exists an analytic 𝛼 -times 𝐶 -regularized resolvent family 𝑆 𝛼 ( 𝑡 ) for 𝑃 ( 𝐷 ) and ‖ ‖ 𝑆 𝛼 ‖ ‖ ( 𝑡 ) ≤ 𝑀 1 + 𝑡 𝛼 𝑛 / 2 𝜔 e x p 1 / 𝛼 𝑡 , 𝑡 ≥ 0 . ( 4 . 2 4 ) (b)For 𝑋 = 𝐿 𝑝 ( ℝ 𝑛 ) , 𝐶 = ( 𝑎 − 𝑃 ) − 𝛾 ( 𝐷 ) , where 𝛾 > 𝑛 𝑝 𝑚 / 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 𝛼 𝑛 𝑝 𝜔 ) e x p 1 / 𝛼 𝑡 , 𝑡 ≥ 0 . ( 4 . 2 5 )

Theorem 4.5. Suppose that the assumptions of Theorem 4.3 are satisfied. (a)For 𝑋 = 𝐿 1 ( ℝ 𝑛 ) or 𝐶 0 ( ℝ 𝑛 ) , 𝐶 = ( 1 − Δ ) − 𝑚 𝛽 / 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 𝛼 𝑛 / 2 𝜔 e x p 1 / 𝛼 𝑡 , 𝑡 ≥ 0 . ( 4 . 2 6 ) (b)For 𝑋 = 𝐿 𝑝 ( ℝ 𝑛 ) , 𝐶 = ( 1 − Δ ) − 𝑚 𝛽 / 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 𝛼 𝑛 𝑝 𝜔 ) e x p 1 / 𝛼 𝑡 , 𝑡 ≥ 0 . ( 4 . 2 7 )

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 ( ℝ 2 ) ( 1 < 𝑝 < ∞ ) with 𝑃 ( 𝜉 ) = − 1 + 𝜉 2 1 1 + ( 𝜉 2 − 𝜉 𝑙 1 ) 2 ( 𝑙 ∈ ℕ ) . ( 4 . 2 8 ) Then 𝑃 ( 𝜉 ) ≤ − 1 ( 𝜉 ∈ ℝ 2 ) . We claim that 𝑃 is ( 2 / 𝑙 ) -coercive. Indeed, if | 𝜉 2 | ≥ 2 | 𝜉 𝑙 1 | , then | | | | ≥ 𝑃 ( 𝜉 ) 1 + 𝜉 2 1 1 1 + 4 𝜉 2 2 ≥ 1 4 | | 𝜉 | | 2 . ( 4 . 2 9 ) If | 𝜉 2 | < 2 | 𝜉 𝑙 1 | , then | | | | | | 𝜉 𝑃 ( 𝜉 ) ≥ 1 + 1 | | 2 | | 𝜉 | | ≥ 𝑐 2 / 𝑙 | | 𝜉 | | f o r ≥ 1 , ( 4 . 3 0 ) 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 ( 𝑙 2 + 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 ( ℝ 2 ) , 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).

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