This paper is concerned with evolution equations of fractional order D๐›ผ๐‘ข(๐‘ก)=๐ด๐‘ข(๐‘ก);๐‘ข(0)=๐‘ข0,๐‘ขโ€ฒ(0)=0, where ๐ด is a differential operator corresponding to a coercive polynomial taking values in a sector of angle less than ๐œ‹ and 1<๐›ผ<2. We show that such equations are well posed in the sense that there always exists an ๐›ผ-times resolvent family for the operator ๐ด.

1. Introduction

It is well known that the abstract Cauchy problem of first order ๐‘ขโ€ฒ(๐‘ก)=๐ด๐‘ข(๐‘ก),๐‘ก>0;๐‘ข(0)=๐‘ฅ(1.1) is well posed if and only if ๐ด is the generator of a ๐ถ0-semigroup. However, many partial differential operators (PDOs) such as the Schrรถdinger operator ๐‘–ฮ” on ๐ฟ๐‘(โ„๐‘›) (๐‘โ‰ 2) cannot generate ๐ถ0-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 [1โ€“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 [5โ€“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 ๐‘‹ be a Banach space, and let ๐ด be a closed linear unbounded operator with densely defined domain ๐ท(๐ด). A family of strongly continuous bounded linear operators on ๐‘‹, {๐‘…(๐‘ก)}๐‘กโ‰ฅ0, is called a resolvent family for ๐ด with kernel ๐‘Ž(๐‘ก)โˆˆ๐ฟ1loc(โ„+) if ๐‘…(๐‘ก)๐ดโŠ‚๐ด๐‘…(๐‘ก) and the resolvent equation๓ต๐‘…(๐‘ก)๐‘ฅ=๐‘ฅ+๐‘ก0๐‘Ž(๐‘กโˆ’๐‘ )๐ด๐‘…(๐‘ )๐‘ฅ๐‘‘๐‘ ,๐‘กโ‰ฅ0,๐‘ฅโˆˆ๐ท(๐ด)(1.2) holds. It is obvious that a ๐ถ0-semigroup is a resolvent family for its generator with kernel ๐‘Ž1(๐‘ก)โ‰ก1; a cosine function is a resolvent family for its generator with kernel ๐‘Ž2(๐‘ก)=๐‘ก. If we define the ๐›ผ-times resolvent family for ๐ด as being a resolvent family with kernel ๐‘”๐›ผ(๐‘ก)โˆถ=๐‘ก๐›ผโˆ’1/ฮ“(๐›ผ), then such resolvent families interpolate ๐ถ0-semigroups and cosine functions.

Recently Bazhlekova studied classes of such resolvent families (see [10]). Let 0<๐›ผโ‰ค2, and let ๐‘š be the smallest integer greater than or equal to ๐›ผ. It was shown in [10] that the fractional evolution equation of order ๐›ผ, ๐ƒ๐›ผ๐‘ข(๐‘ก)=๐ด๐‘ข(๐‘ก),๐‘ก>0;๐‘ข(๐‘˜)(0)=๐‘ฅ๐‘˜,๐‘˜=0,1,โ€ฆ,๐‘šโˆ’1,(1.3) is well posed if and only if there exists an ๐›ผ-times resolvent family for ๐ด. Here ๐ƒ๐›ผ is the Caputo fractional derivative of order ๐›ผ>0 defined by ๐ƒ๐›ผ๓ต๐‘“(๐‘ก)โˆถ=๐‘ก0๐‘”๐‘šโˆ’๐›ผ๐‘‘(๐‘กโˆ’๐‘ )๐‘š๐‘‘๐‘ ๐‘š๐‘“(๐‘ )๐‘‘๐‘ ,(1.4) where ๐‘“โˆˆ๐‘Š๐‘š,1(๐ผ) for every interval ๐ผ. The hypothesis on ๐‘“ can be relaxed; see [10] for details. Fujita in [11] studied (1.3) for the case that ๐ด=ฮ”, the Laplacian (๐œ•/๐œ•๐‘ฅ)2 on โ„, which interpolates the heat equation and the wave equation. Since ๐›ผ-times resolvent families interpolate ๐ถ0-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 ๐›ผ-times resolvent families for concrete PDOs in [10], but Bazhlekova did not develop the theory of ๐›ผ-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 ๐›ผ-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 ๐ถ-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 ๐›ผ-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 ๐›ผ-times regularized resolvent families for PDOs corresponding to coercive polynomials taking values in a sector of angle less than ๐œ‹. Some examples are also given in Section 4.

2. ๐œถ-Times Regularized Resolvent Family

Throughout this paper, ๐‘‹ is a complex Banach space, and we denote by ๐(๐‘‹) the algebra of all bounded linear operators on ๐‘‹. Let ๐ด be a closed densely defined operator on ๐‘‹, let ๐ท(๐ด) and ๐‘…(๐ด) be its domain and range, respectively, and let ๐›ผโˆˆ(0,2], ๐ถโˆˆ๐(๐‘‹) be injective. Define ๐œŒ๐ถ(๐ด)โˆถ={๐œ†โˆˆโ„‚โˆถ๐œ†โˆ’๐ดisinjectiveand๐‘…(๐ถ)โŠ‚๐‘…(๐œ†โˆ’๐ด)}. Let ฮฃ๐œƒโˆถ={๐œ†โˆˆโ„‚โˆถ|arg๐œ†|<๐œƒ} be the open sector of angle 2๐œƒ in the complex plane, where arg is the branch of the argument between โˆ’๐œ‹ and ๐œ‹.

Definition 2.1. A strongly continuous family {๐‘†๐›ผ(๐‘ก)}๐‘กโ‰ฅ0โŠ‚๐(๐‘‹) is called an ๐›ผ-times ๐ถ-regularized resolvent family for ๐ด if(a)๐‘†๐›ผ(0)=๐ถ;(b)๐‘†๐›ผ(๐‘ก)๐ดโŠ‚๐ด๐‘†๐›ผ(๐‘ก) for ๐‘กโ‰ฅ0;(c)๐ถโˆ’1๐ด๐ถ=๐ด;(d)for ๐‘ฅโˆˆ๐ท(๐ด), ๐‘†๐›ผ(๐‘ก)๐‘ฅ=๐ถ๐‘ฅ+โˆซ๐‘ก0((๐‘กโˆ’๐‘ )๐›ผโˆ’1/ฮ“(๐›ผ))๐‘†๐›ผ(๐‘ )๐ด๐‘ฅ๐‘‘๐‘ .{๐‘†๐›ผ(๐‘ก)}๐‘กโ‰ฅ0 is called analytic if it can be extended analytically to some sector ฮฃ๐œƒ.

If โ€–๐‘†๐›ผ(๐‘ก)โ€–โ‰ค๐‘€๐‘’๐œ”๐‘ก (๐‘กโ‰ฅ0) for some constants ๐‘€โ‰ฅ1 and ๐œ”โˆˆโ„+, we will write ๐ดโˆˆ๐’ž๐›ผ๐ถ(๐‘€,๐œ”), and ๐’ž๐›ผ๐ถ(๐œ”)โˆถ=โˆช{๐’ž๐›ผ๐ถ(๐‘€,๐œ”);๐‘€โ‰ฅ1}, ๐’ž๐›ผ๐ถโˆถ=โˆช{๐’ž๐›ผ๐ถ(๐œ”);๐œ”โ‰ฅ0}.

Define the operator ๓ตฐ’๐ด by ๓ตฐ’๐ด๐‘ฅ=๐ถโˆ’1(lim๐‘กโ†“0ฮ“(๐›ผ+1)๐‘ก๐›ผ(๐‘†๐›ผ๓ตฐ’(๐‘ก)๐‘ฅโˆ’๐ถ๐‘ฅ)),๐‘ฅโˆˆ๐ท(๐ด),(2.1) with ๓ตฐ’๐ท(๐ด)={๐‘ฅโˆˆ๐‘‹โˆถlim๐‘กโ†“0๐‘†๐›ผ(๐‘ก)๐‘ฅโˆ’๐ถ๐‘ฅ๐‘ก๐›ผexistsandisin๐‘…(๐ถ)}.(2.2)

Proposition 2.2. Suppose that there exists an ๐›ผ-times ๐ถ-regularized resolvent family, {๐‘†๐›ผ(๐‘ก)}๐‘กโ‰ฅ0, for the operator ๐ด, and let ๓ตฐ’๐ด be defined as above. Then ๓ตฐ’๐ด๐ด=.

Proof. By the strong continuity of ๐‘†๐›ผ(๐‘ก), we have for every ๐‘ฅโˆˆ๐‘‹, โ€–๐‘”๐›ผ+1(๐‘ก)โˆ’1๓ต๐‘ก0๐‘”๐›ผ(๐‘กโˆ’๐‘ )๐‘†๐›ผ(๐‘ )๐‘ฅ๐‘‘๐‘ โˆ’๐ถ๐‘ฅโ€–โ‰ค๐‘”๐›ผ+1(๐‘ก)โˆ’1๓ต๐‘ก0๐‘”๐›ผ(๐‘กโˆ’๐‘ )โ€–๐‘†๐›ผ(๐‘ )๐‘ฅโˆ’๐ถ๐‘ฅโ€–๐‘‘๐‘ โ‰คsup0โ‰ค๐‘ โ‰ค๐‘กโ€–๐‘†๐›ผ(๐‘ )๐‘ฅโˆ’๐ถ๐‘ฅโ€–โ†’0as๐‘กโ†’0.(2.3) Thus for ๐‘ฅโˆˆ๐ท(๐ด), by Definition 2.1, lim๐‘กโ†“0๐‘”๐›ผ+1(๐‘ก)โˆ’1(๐‘†๐›ผ(๐‘ก)๐‘ฅโˆ’๐ถ๐‘ฅ)=lim๐‘กโ†“0๐‘”๐›ผ+1(๐‘ก)โˆ’1๓ต๐‘ก0๐‘”๐›ผ(๐‘กโˆ’๐‘ )๐‘†๐›ผ(๐‘ )๐ด๐‘ฅ๐‘‘๐‘ =๐ถ๐ด๐‘ฅ,(2.4) which means that ๓ตฐ’๐‘ฅโˆˆ๐ท(๐ด) and ๓ตฐ’๐ด๐‘ฅ=๐ด๐‘ฅ. On the other hand, for ๓ตฐ’๐‘ฅโˆˆ๐ท(๐ด), by the definition of ๓ตฐ’๐ด and Definition 2.1, ๐ถ๓ตฐ’๐ด๐‘ฅ=lim๐‘กโ†“0๐‘”๐›ผ+1(๐‘ก)โˆ’1(๐‘†๐›ผ(๐‘ก)๐‘ฅโˆ’๐ถ๐‘ฅ)=lim๐‘กโ†“0๐‘”๐›ผ+1(๐‘ก)โˆ’1๐ด๓ต๐‘ก0๐‘”๐›ผ(๐‘กโˆ’๐‘ )๐‘†๐›ผ(๐‘ )๐‘ฅ๐‘‘๐‘ ,(2.5) but lim๐‘กโ†’0๐‘”๐›ผ+1(๐‘ก)โˆ’1โˆซ๐‘ก0๐‘”๐›ผ(๐‘กโˆ’๐‘ )๐‘†๐›ผ(๐‘ )๐‘ฅ๐‘‘๐‘ =๐ถ๐‘ฅ, by (d) of Definition 2.1. Thus it follows from the closedness of ๐ด that ๐ถ๐‘ฅโˆˆ๐ท(๐ด) with ๓ตฐ’๐ด๐ถ๐‘ฅ=๐ถ๐ด๐‘ฅ. This implies that ๐‘ฅโˆˆ๐ท(๐ถโˆ’1๐ด๐ถ)=๐ท(๐ด), so we have ๓ตฐ’๐ด=๐ด.

The following generation theorem and subordination principle for ๐›ผ-times ๐ถ-regularized resolvent families can be proved similarly as those for ๐›ผ-times resolvent families (see [10]).

Theorem 2.3. Let ๐›ผโˆˆ(0,2]. Then the following statements are equivalent:(a)๐ดโˆˆ๐’ž๐›ผ๐ถ(๐‘€,๐œ”);(b)๐ด=๐ถโˆ’1๐ด๐ถ, (๐œ”๐›ผ,โˆž)โŠ†๐œŒ๐ถ(๐ด) and โ€–๐‘‘๐‘›๐‘‘๐œ†๐‘›(๐œ†๐›ผโˆ’1(๐œ†๐›ผโˆ’๐ด)โˆ’1๐ถ)โ€–โ‰ค๐‘€๐‘›!(๐œ†โˆ’๐œ”)๐‘›+1,๐œ†>๐œ”,๐‘›โˆˆโ„•0โˆถ=โ„•โˆช{0};(2.6)(c)๐ด=๐ถโˆ’1๐ด๐ถ, (๐œ”๐›ผ,โˆž)โŠ†๐œŒ๐ถ(๐ด) and there exists a strongly continuous family {๐‘†๐›ผ(๐‘ก)}๐‘กโ‰ฅ0โŠ‚๐(๐‘‹) satisfying โ€–๐‘†๐›ผ(๐‘ก)โ€–โ‰ค๐‘€๐‘’๐œ”๐‘ก such that ๐œ†๐›ผโˆ’1(๐œ†๐›ผโˆ’๐ด)โˆ’1๐ถ๐‘ฅ=๓ตโˆž0๐‘’โˆ’๐œ†๐‘ก๐‘†๐›ผ(๐‘ก)๐‘ฅ๐‘‘๐‘ก,๐œ†>๐œ”,๐‘ฅโˆˆ๐‘‹.(2.7)

Theorem 2.4. Suppose that 0<๐›ผ<๐›ฝโ‰ค2, ๐›พ=๐›ผ/๐›ฝ. If ๐ดโˆˆ๐’ž๐›ฝ๐ถ(๐œ”) then ๐ดโˆˆ๐’ž๐›ผ๐ถ(๐œ”1/๐›พ) and the ๐›ผ-times ๐ถ-regularized resolvent family for ๐ด, {๐‘†๐›ผ(๐‘ก)}๐‘กโ‰ฅ0, can be extended analytically to ฮฃmin{๐œƒ(๐›พ),๐œ‹}, where ๐œƒ(๐›พ)โˆถ=(1/๐›พโˆ’1)๐œ‹/2.

3. Coercive Operators and Mittag-Leffler Functions

We now introduce a functional calculus for generators of bounded ๐ถ0-groups (cf. [14]), which will play a key role in our proof.

Let ๐‘–๐ด๐‘—(1โ‰ค๐‘—โ‰ค๐‘›) be commuting generators of bounded ๐ถ0-groups on a Banach space ๐‘‹. Write ๐ด=(๐ด1,โ€ฆ,๐ด๐‘›) and ๐ด๐œ‡=๐ด๐œ‡11โ‹ฏ๐ด๐œ‡๐‘›๐‘› for ๐œ‡=(๐œ‡1,โ€ฆ,๐œ‡๐‘›)โˆˆโ„•๐‘›0. Similarly, write ๐ท๐œ‡=๐ท๐œ‡11โ‹ฏ๐ท๐œ‡๐‘›๐‘›, where ๐ท๐‘—=โˆ’๐‘–๐œ•/๐œ•๐‘ฅ๐‘— for ๐‘—=1,โ€ฆ,๐‘›. For a polynomial ๐‘ƒ(๐œ‰)โˆถ=โˆ‘|๐œ‡|โ‰ค๐‘š๐‘Ž๐œ‡๐œ‰๐œ‡(๐œ‰โˆˆโ„๐‘›)(|๐œ‡|โˆถ=โˆ‘๐‘›๐‘—=1๐œ‡๐‘—) with constant coefficients, we define ๐‘ƒ(๐ด)=โˆ‘|๐œ‡|โ‰ค๐‘š๐‘Ž๐œ‡๐ด๐œ‡(๐œ‰โˆˆโ„๐‘›) with maximal domain. Then ๐‘ƒ(๐ด) is closable. Let โ„ฑ be the Fourier transform, that is, (โ„ฑ๐‘ข)(๐œ‚)=โˆซโ„๐‘›๐‘ข(๐œ‰)๐‘’โˆ’๐‘–(๐œ‰,๐œ‚)๐‘‘๐œ‰ for ๐‘ขโˆˆ๐ฟ1(โ„๐‘›), where (๐œ‰,๐œ‚)=โˆ‘๐‘›๐‘—=1๐œ‰๐‘—๐œ‚๐‘—. If ๐‘ขโˆˆโ„ฑ๐ฟ1(โ„๐‘›)โˆถ={โ„ฑ๐‘ฃโˆถ๐‘ฃโˆˆ๐ฟ1(โ„๐‘›)}, then there exists a unique function in ๐ฟ1(โ„๐‘›), written โ„ฑโˆ’1๐‘ข, such that ๐‘ข=โ„ฑ(โ„ฑโˆ’1๐‘ข). In particular, โ„ฑโˆ’1๐‘ข is the inverse Fourier transform of ๐‘ข if ๐‘ขโˆˆ๐’ฎ(โ„๐‘›) (the space of rapidly decreasing functions on โ„๐‘›). We define ๐‘ข(๐ด)โˆˆ๐ต(๐‘‹) by ๓ต๐‘ข(๐ด)๐‘ฅ=โ„๐‘›(โ„ฑโˆ’1๐‘ข)(๐œ‰)๐‘’โˆ’๐‘–(๐œ‰,๐ด)๐‘ฅ๐‘‘๐œ‰,๐‘ฅโˆˆ๐‘‹,(3.1) where (๐œ‰,๐ด)=โˆ‘๐‘›๐‘—=1๐œ‰๐‘—๐ด๐‘—.

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) โ„ฑ๐ฟ1(โ„๐‘›) is a Banach algebra under pointwise multiplication and addition with norm โ€–๐‘ขโ€–โ„ฑ๐ฟ1โˆถ=โ€–โ„ฑโˆ’1๐‘ขโ€–๐ฟ1.
(b) ๐‘ขโ†ฆ๐‘ข(๐ด) is an algebra homomorphism from โ„ฑ๐ฟ1(โ„๐‘›) into ๐(๐‘‹), and there exists a constant ๐‘€>0 such that โ€–๐‘ข(๐ด)โ€–โ‰ค๐‘€โ€–๐‘ขโ€–โ„ฑ๐ฟ1.
(c) ๐ธโˆถ={๐œ™(๐ด)๐‘ฅโˆถ๐œ™โˆˆ๐’ฎ(โ„๐‘›),๐‘ฅโˆˆ๐‘‹}โŠ‚โˆฉ๐œ‡โˆˆโ„•๐‘›0๐ท(๐ด๐œ‡), ๐ธ=๐‘‹, ๐‘ƒ(๐ด)|๐ธ=๐‘ƒ(๐ด) and ๐œ™(๐ด)๐‘ƒ(๐ด)โŠ‚๐‘ƒ(๐ด)๐œ™(๐ด)=(๐‘ƒ๐œ™)(๐ด) for ๐œ™โˆˆ๐’ฎ(โ„๐‘›).
(d) Let ๐‘ขโˆˆ๐ถ๐‘—(โ„๐‘›)(๐‘—>๐‘›/2). Suppose that there exist constants ๐ฟ, ๐‘€0, ๐‘Ž>0, and ๐‘โˆˆ[โˆ’1,2๐‘Ž/๐‘›โˆ’1) such that |๐ท๐‘˜๓ถ†๐‘€๐‘ข(๐œ‰)|โ‰ค0|๐‘˜||๐œ‰|๐‘|๐‘˜|โˆ’๐‘Ž๐‘€,for|๐œ‰|โ‰ฅ๐ฟ,|๐‘˜|โ‰ค๐‘—,0|๐‘˜|,for|๐œ‰|<๐ฟ,|๐‘˜|โ‰ค๐‘—,(3.2) where ๐‘˜โˆˆโ„•๐‘›0, then ๐‘ขโˆˆโ„ฑ๐ฟ1(โ„๐‘›) and โ€–๐‘ขโ€–โ„ฑ๐ฟ1โ‰ค๐‘€๐‘€0๐‘›/2 for some constant ๐‘€>0.

Recall that the Mittag-Leffler function (see [15, 16]) is defined by ๐ธ๐›ผ,๐›ฝ(๐‘ง)โˆถ=โˆž๓ต ˆ๐‘›=0๐‘ง๐‘›=1ฮ“(๐›ผ๐‘›+๐›ฝ)๓ต2๐œ‹๐‘–๐’ฏ๐œ‡๐›ผโˆ’๐›ฝ๐‘’๐œ‡๐œ‡๐›ผโˆ’๐‘ง๐‘‘๐œ‡,๐›ผ,๐›ฝ>0,๐‘งโˆˆโ„‚,(3.3) where the path ๐’ฏ is a loop which starts and ends at โˆ’โˆž and encircles the disc |๐‘ก|โ‰ค|๐‘ง|1/๐›ผ in the positive sense. The most interesting properties of the Mittag-Leffler functions are associated with their Laplace integral ๓ตโˆž0๐‘’โˆ’๐œ†๐‘ก๐‘ก๐›ฝโˆ’1๐ธ๐›ผ,๐›ฝ(๐œ”๐‘ก๐›ผ๐œ†)๐‘‘๐‘ก=๐›ผโˆ’๐›ฝ๐œ†๐›ผโˆ’๐œ”,Re๐œ†>๐œ”1/๐›ผ,๐œ”>0(3.4) and with their asymptotic expansion as ๐‘งโ†’โˆž. If 0<๐›ผ<2, ๐›ฝ>0, then ๐ธ๐›ผ,๐›ฝ1(๐‘ง)=๐›ผ๐‘ง(1โˆ’๐›ฝ)/๐›ผexp(๐‘ง1/๐›ผ)+๐œ€๐›ผ,๐›ฝ1(๐‘ง),|arg๐‘ง|โ‰ค2๐ธ๐›ผ๐œ‹,(3.5)๐›ผ,๐›ฝ(๐‘ง)=๐œ€๐›ผ,๐›ฝ1(๐‘ง),|arg(โˆ’๐‘ง)|<(1โˆ’2๐›ผ)๐œ‹,(3.6) where ๐œ€๐›ผ,๐›ฝ(๐‘ง)=โˆ’๐‘โˆ’1๓ต ˆ๐‘›=1๐‘งโˆ’๐‘›ฮ“(๐›ฝโˆ’๐›ผ๐‘›)+๐‘‚(|๐‘ง|โˆ’๐‘)(3.7) as ๐‘งโ†’โˆž, and the ๐‘‚-term is uniform in arg๐‘ง if |arg(โˆ’๐‘ง)|โ‰ค(1โˆ’๐›ผ/2โˆ’๐œ–)๐œ‹. Note that for ๐›ฝ>0, |๐ธ๐›ผ,๐›ฝ(๐‘ง)|โ‰ค๐ธ๐›ผ,๐›ฝ(|๐‘ง|),๐‘งโˆˆโ„‚.(3.8)

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

Lemma 3.2. ๐ธโ€ฒ๐›ผ,๐›ฝ1(๐‘ง)=๐›ผ๐ธ๐›ผ,๐›ผ+๐›ฝโˆ’1(๐‘ง)โˆ’๐›ฝโˆ’1๐›ผ๐ธ๐›ผ,๐›ผ+๐›ฝ(๐‘ง).(3.9)

Proof. By the definition of ๐ธ๐›ผ,๐›ฝ(๐‘ง), ๐ธโ€ฒ๐›ผ,๐›ฝ(๐‘ง)=(โˆž๓ต ˆ๐‘›=0๐‘ง๐‘›)ฮ“(๐›ผ๐‘›+๐›ฝ)โ€ฒ=โˆž๓ต ˆ๐‘›=1๐‘›๐‘ง๐‘›โˆ’1=ฮ“(๐›ผ๐‘›+๐›ฝ)โˆž๓ต ˆ๐‘›=1๐‘›๐‘ง๐‘›โˆ’1=ฮ“(๐›ผ๐‘›+๐›ฝโˆ’1+1)โˆž๓ต ˆ๐‘›=1๐‘ง๐‘›โˆ’1โ‹…(๐›ผ๐‘›+๐›ฝโˆ’1)โˆ’๐›ผ(๐›ผ๐‘›+๐›ฝโˆ’1)ฮ“(๐›ผ๐‘›+๐›ฝโˆ’1)โˆž๓ต ˆ๐‘›=1๐›ฝโˆ’1๐›ผโ‹…๐‘ง๐‘›โˆ’1=1ฮ“(๐›ผ๐‘›+๐›ฝ)๐›ผ๐ธ๐›ผ,๐›ผ+๐›ฝโˆ’1(๐‘ง)โˆ’๐›ฝโˆ’1๐›ผ๐ธ๐›ผ,๐›ผ+๐›ฝ(๐‘ง),(3.10) as we wanted to show.

For short, ๐ธ๐›ผ(๐‘ง)โˆถ=๐ธ๐›ผ,1(๐‘ง).

Lemma 3.3. Suppose that 1<๐›ผ<2. For every ๐‘›โˆˆโ„• and ๐œ–>0 there exist constants ๐‘€>0 and ๐ฟ>0 such that for ๐‘˜=0,โ€ฆ,๐‘›, |๐ธ๐›ผ(๐‘˜)๐‘€(๐‘ง)|โ‰ค๐›ผ|๐‘ง|,if|๐‘ง|โ‰ฅ๐ฟ,|arg(โˆ’๐‘ง)|โ‰ค(1โˆ’2โˆ’๐œ–)๐œ‹.(3.11)

Proof. First note that ๐ธโ€ฒ๐›ผ(๐‘ง)=(1/๐›ผ)๐ธ๐›ผ,๐›ผ(๐‘ง), and by induction on ๐‘˜ one can prove that ๐ธ๐›ผ(๐‘˜)(๐‘ง)=๐‘˜๓ต ˆ๐‘—=1๐‘Ž๐‘—๐ธ๐›ผ,๐›ผ๐‘˜โˆ’(๐‘˜โˆ’๐‘—)(๐‘ง),(3.12) where ๐‘Ž๐‘— only depend on ๐›ผ and ๐‘˜. Since ๐›ผ>1 we have that ๐›ผ๐‘˜โˆ’(๐‘˜โˆ’๐‘—)>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 ๐‘Ÿ>0, a polynomial ๐‘ƒ(๐œ‰) is called ๐‘Ÿ-coercive if |๐‘ƒ(๐œ‰)|โˆ’1=๐‘‚(|๐œ‰|โˆ’๐‘Ÿ) as |๐œ‰|โ†’โˆž. In the sequel, ๐‘€ is a generic constant independent of ๐‘ก which may vary from line to line.

Lemma 3.4. Suppose that ๐‘ƒ(๐œ‰) is an ๐‘Ÿ-coercive polynomial of order ๐‘š and {๐‘ƒ(๐œ‰)โˆถ๐œ‰โˆˆโ„๐‘›}โŠ‚โ„‚โงตฮฃ๐›ผโ€ฒ๐œ‹/2, where 1<๐›ผโ€ฒ<2. Let ๐‘˜0=[๐‘›/2]+1. Then for 1<๐›ผ<๐›ผโ€ฒ, ๐›พ>0, ๐‘Žโˆˆฮฃ๐›ผโ€ฒ๐œ‹/2, there exist constants ๐‘€,๐ฟโ‰ฅ0 such that |๐ท๐œ‡[๐ธ๐›ผ(๐‘ก๐›ผ๐‘ƒ)(๐‘Žโˆ’๐‘ƒ)โˆ’๐›พ]|โ‰ค๐‘€(1+๐‘ก๐›ผ|๐œ‡|)|๐œ‰|(๐‘šโˆ’1)|๐œ‡|โˆ’๐‘Ÿ๐›พ,|๐œ‰|โ‰ฅ๐ฟ,|๐œ‡|โ‰ค๐‘˜0,๐‘กโ‰ฅ0.(3.13)

Proof. Suppose that for |๐œ‰|โ‰ฅ๐ฟ, (3.11) holds up to order ๐‘˜0 and |๐‘ƒ(๐œ‰)|โ‰ฅ๐‘€|๐œ‰|๐‘Ÿ,|๐‘Žโˆ’๐‘ƒ(๐œ‰)|โ‰ฅ๐‘€|๐œ‰|๐‘Ÿ.(3.14) By induction, one can show that ๐ท๐œ‡๐ธ๐›ผ(๐‘ก๐›ผ๐‘ƒ)=|๐œ‡|๓ต ˆ๐‘—=1๐‘ก๐›ผ๐‘—๐ธ๐›ผ(๐‘—)(๐‘ก๐›ผ๐‘ƒ)๐‘„๐‘—,(3.15) where deg๐‘„๐‘—โ‰ค๐‘š๐‘—โˆ’|๐œ‡|. Thus if |๐œ‰|โ‰ฅ๐ฟ and |๐‘ก๐›ผ๐‘ƒ|โ‰ฅ๐ฟ, |๐ท๐œ‡๐ธ๐›ผ(๐‘ก๐›ผ๐‘ƒ)|โ‰ค๐‘€(1+๐‘ก๐›ผ(|๐œ‡|โˆ’1))|๐œ‰|๐‘š|๐œ‡|โˆ’|๐œ‡|โˆ’๐‘Ÿโ‰ค๐‘€(1+๐‘ก๐›ผ|๐œ‡|)|๐œ‰|(๐‘šโˆ’1)|๐œ‡|โˆ’๐‘Ÿ,(3.16) and if |๐œ‰|โ‰ฅ๐ฟ with |๐‘ก๐›ผ๐‘ƒ|โ‰ค๐ฟ, by (3.8) and (3.12) we know that |๐ท๐œ‡๐ธ๐›ผ(๐‘ก๐›ผ๐‘ƒ)|โ‰ค๐‘€(๐‘ก๐›ผ+๐‘ก๐›ผ|๐œ‡|)|๐œ‰|(๐‘šโˆ’1)|๐œ‡|.(3.17) Altogether, we have |๐ท๐œ‡๐ธ๐›ผ(๐‘ก๐›ผ๐‘ƒ)|โ‰ค๐‘€(1+๐‘ก๐›ผ|๐œ‡|)|๐œ‰|(๐‘šโˆ’1)|๐œ‡|,|๐œ‰|โ‰ฅ๐ฟ.(3.18) And by |๐ท๐œ‡(๐‘Žโˆ’๐‘ƒ)โˆ’๐›พ|โ‰ค๐‘€|๐œ‰|(๐‘šโˆ’๐‘Ÿโˆ’1)|๐œ‡|โˆ’๐‘Ÿ๐›พ,|๐œ‰|โ‰ฅ๐ฟ(3.19) and Leibniz's formula we have |๐ท๐œ‡(๐ธ๐›ผ(๐‘ก๐›ผ๐‘ƒ)(๐‘Žโˆ’๐‘ƒ)โˆ’๐›พ)|โ‰ค๐‘€(1+๐‘ก๐›ผ|๐œ‡|)|๐œ‰|(๐‘šโˆ’1)|๐œ‡|โˆ’๐‘Ÿ๐›พ,|๐œ‰|โ‰ฅ๐ฟ.(3.20)

Lemma 3.5. This proves (3.13). Suppose that the assumptions of Lemma 3.4 are satisfied. Let ๐›พ>๐‘›๐‘š/2๐‘Ÿ. Then ๐ธ๐›ผ(๐‘ก๐›ผ๐‘ƒ)(๐‘Žโˆ’๐‘ƒ)โˆ’๐›พโˆˆโ„ฑ๐ฟ1(โ„๐‘›) and โ€–๐ธ๐›ผ(๐‘ก๐›ผ๐‘ƒ)(๐‘Žโˆ’๐‘ƒ)โˆ’๐›พโ€–โ„ฑ๐ฟ1โ‰ค๐‘€(1+๐‘ก๐›ผ๐‘›/2),๐‘กโ‰ฅ0.(3.21) The same result holds with ๐ธ๐›ผ(๐‘ก๐›ผ๐‘ƒ) replaced by ๐ธ๐›ผ,๐›ผ(๐‘ก๐›ผ๐‘ƒ).

Proof. By Lemma 3.1(d), it remains to prove that for |๐œ‰|โ‰ค๐ฟ, |๐ท๐œ‡(๐ธ๐›ผ(๐‘ก๐›ผ๐‘ƒ)(๐‘Žโˆ’๐‘ƒ)โˆ’๐›พ)|โ‰ค๐‘€(1+๐‘ก๐›ผ|๐œ‡|),|๐œ‡|โ‰ค๐‘˜0,๐‘กโ‰ฅ0.(3.22) To show this we can use (3.15) and then give the estimates according to the values ๐‘ก๐›ผ๐‘ƒ. For |๐œ‰|โ‰ค๐ฟ with |๐‘ก๐›ผ๐‘ƒ|โ‰ฅ๐ฟ the estimate (3.8) can be applied, and for |๐œ‰|โ‰ค๐ฟ with |๐‘ก๐›ผ๐‘ƒ|โ‰ค๐ฟ note that all the functions ๐ธ๐›ผ(๐‘—)(๐‘ก๐›ผ๐‘ƒ) are uniformly bounded.
For the second part of the lemma, note that ๐ธ๐›ผ,๐›ผ(๐‘ง)=๐›ผ๐ธโ€ฒ๐›ผ(๐‘ง).

4. Existence of ๐œถ-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 ๐‘ƒ is an ๐‘Ÿ-coercive polynomial of order ๐‘š, and {๐‘ƒ(๐œ‰)โˆถ๐œ‰โˆˆโ„๐‘›}โŠ‚โ„‚โงตฮฃ๐›ผโ€ฒ๐œ‹/2, where 1<๐›ผโ€ฒ<2. Then for 1<๐›ผ<๐›ผโ€ฒ, ๐‘Žโˆˆฮฃ๐›ผโ€ฒ๐œ‹/2, ๐›พ>๐‘›๐‘š/2๐‘Ÿ, ๐ถ=(๐‘Žโˆ’๐‘ƒ)โˆ’๐›พ(๐ด), there exists an analytic ๐›ผ-times ๐ถ-regularized resolvent family ๐‘†๐›ผ(๐‘ก) for ๐‘ƒ(๐ด), and ๐‘†๐›ผ(๐‘ก)=(๐ธ๐›ผ(๐‘ก๐›ผ๐‘ƒ)(๐‘Žโˆ’๐‘ƒ)โˆ’๐›พ)(๐ด) with โ€–๐‘†๐›ผ(๐‘ก)โ€–โ‰ค๐‘€(1+๐‘ก๐›ผ๐‘›/2),๐‘กโ‰ฅ0.(4.1)

Proof. Let ๐‘ข๐‘ก=๐ธ๐›ผ(๐‘ก๐›ผ๐‘ƒ)(๐‘Žโˆ’๐‘ƒ)โˆ’๐›พ, ๐‘กโ‰ฅ0. By Lemma 3.5, ๐‘ข๐‘กโˆˆโ„ฑ๐ฟ1 and โ€–๐‘ข๐‘กโ€–โ„ฑ๐ฟ1โ‰ค๐‘€(1+๐‘ก๐›ผ๐‘›/2). Define ๐‘†๐›ผ(๐‘ก)=๐‘ข๐‘ก(๐ด). Then by Lemma 3.1(b), ๐‘†๐›ผ(๐‘ก)โˆˆ๐(๐‘‹), โ€–๐‘†๐›ผ(๐‘ก)โ€–โ‰ค๐‘€(1+๐‘ก๐›ผ๐‘›/2), and in particular ๐ถ=๐‘†๐›ผ(0)โˆˆ๐(๐‘‹). To check the strong continuity of ๐‘†๐›ผ(๐‘ก), take ๐œ™โˆˆ๐’ฎ(โ„๐‘›). Then for ๐‘ก,๐‘ก+โ„Žโ‰ฅ0, by Lemma 3.5โ€–๐‘†๐›ผ(๐‘ก+โ„Ž)๐œ™(๐ด)โˆ’๐‘†๐›ผ(๐‘ก)๐œ™(๐ด)โ€–โ‰ค๐‘€โ€–(๐ธ๐›ผ((๐‘ก+โ„Ž)๐›ผ๐‘ƒ)โˆ’๐ธ๐›ผ(๐‘ก๐›ผ๐‘ƒ))(๐‘Žโˆ’๐‘ƒ)โˆ’๐›พ๐œ™โ€–โ„ฑ๐ฟ1โ‰ค๐‘€โ€–๓ต๐‘ก๐‘ก+โ„Ž๐‘ ๐›ผโˆ’1๐ธ๐›ผ,๐›ผ(๐‘ ๐›ผ๐‘ƒ)(๐‘Žโˆ’๐‘ƒ)โˆ’๐›พ๐‘ƒ๐œ™๐‘‘๐‘ โ€–โ„ฑ๐ฟ1๓ตโ‰ค๐‘€๐‘ก๐‘ก+โ„Ž๐‘ ๐›ผโˆ’1(1+๐‘ ๐›ผ๐‘›/2)๐‘‘๐‘ โ‹…โ€–๐‘ƒ๐œ™โ€–โ„ฑ๐ฟ1โ†’0,asโ„Žโ†’0.(4.2) Since the set ๐ธ of Lemma 3.1 is dense in ๐‘‹, we have done. Next we will show that ๐œ†๐›ผโˆ’1(๐œ†๐›ผโˆ’๐‘ƒ(๐ด))โˆ’1๐ถ=๓ตโˆž0๐‘’โˆ’๐œ†๐‘ก๐‘†๐›ผ(๐‘ก)๐‘‘๐‘ก,๐œ†>0.(4.3) In fact, for ๐œ™โˆˆ๐’ฎ(โ„๐‘›), by Lemma 3.1(b) and (c) we have ๐‘ƒ(๐ด)๐‘†๐›ผ(๐‘ก)๐œ™(๐ด)=(๐‘ƒ๐‘ข๐‘ก๐œ™)(๐ด)=๐‘†๐›ผ(๐‘ก)๐‘ƒ(๐ด)๐œ™(๐ด).(4.4) Since โ„ฑ๐ฟ1(โ„๐‘›) is a Banach algebra, it follows that ๐‘ข๐‘ก,๐‘ข๐‘ก(๐œ†โˆ’๐‘ƒ)๐œ™โˆˆโ„ฑ๐ฟ1. Thus by Lemmas 3.1, 3.5, (3.4), and Fubini's theorem one obtains that for ๐‘ฅโˆˆ๐‘‹, ๐œ†>0, ๓ตโˆž0๐‘’โˆ’๐œ†๐‘ก๐‘†๐›ผ(๐‘ก)(๐œ†๐›ผโˆ’๐‘ƒ(๐ด))๐œ™(๐ด)๐‘ฅ๐‘‘๐‘ก=๓ตโˆž0๐‘’โˆ’๐œ†๐‘ก(๐‘ข๐‘ก(๐œ†๐›ผ๓ตโˆ’๐‘ƒ)๐œ™)(๐ด)๐‘ฅ๐‘‘๐‘ก=(โˆž0๐‘’โˆ’๐œ†๐‘ก๐‘ข๐‘ก๐‘‘๐‘ก(๐œ†๐›ผโˆ’๐‘ƒ)๐œ™)(๐ด)๐‘ฅ=๐œ†๐›ผโˆ’1๐ถ๐œ™(๐ด)๐‘ฅ.(4.5) This implies that ๓ตโˆž0๐‘’โˆ’๐œ†๐‘ก๐‘†๐›ผ(๐‘ก)(๐œ†๐›ผโˆ’๐‘ƒ(๐ด))๐‘ฅ๐‘‘๐‘ก=๐œ†๐›ผโˆ’1๐ถ๐‘ฅ,๐‘ฅโˆˆ๐ท(๐‘ƒ(๐ด)),(4.6) once again by the density of the set ๐ธ of Lemma 3.1. A similar argument works to get (๐œ†๐›ผโˆ’๐‘ƒ(๐ด))๓ตโˆž0๐‘’โˆ’๐œ†๐‘ก๐‘†๐›ผ(๐‘ก)๐‘ฅ๐‘‘๐‘ก=๐œ†๐›ผโˆ’1๐ถ๐‘ฅ,๐‘ฅโˆˆ๐‘‹.(4.7) Therefore, we have proved (4.3). And it is routine to show that ๐ถโˆ’1๐‘ƒ(๐ด)๐ถ=๐‘ƒ(๐ด), thus by Theorem 2.3 we know that ๐‘†๐›ผ(๐‘ก) is the ๐›ผ-times ๐ถ-regularized resolvent family for ๐‘ƒ(๐ด). Moreover, since ๐›ผ<๐›ผโ€ฒ is arbitrary, by the subordination principle (Theorem 2.4) we know that ๐‘†๐›ผ(๐‘ก) is analytic.

We can extend this result to a more general case.

Theorem 4.2. Let ๐‘ƒ(๐œ‰) be an ๐‘Ÿ-coercive polynomial of order ๐‘š such that {๐‘ƒ(๐œ‰)โˆถ๐œ‰โˆˆโ„๐‘›}โŠ‚โ„‚โงต(๐œ”+ฮฃ๐›ผโ€ฒ๐œ‹/2) for some ๐œ”โ‰ฅ0, and let ๐›ผโ€ฒ>1. Then for 1<๐›ผ<๐›ผโ€ฒ, ๐‘Žโˆˆ๐œ”+ฮฃ๐›ผโ€ฒ๐œ‹/2, ๐›พ>๐‘›๐‘š/2๐‘Ÿ, and ๐ถ=(๐‘Žโˆ’๐‘ƒ)โˆ’๐›พ(๐ด), there exists an analytic ๐›ผ-times ๐ถ-regularized resolvent family ๐‘†๐›ผ(๐‘ก) for ๐‘ƒ(๐ด) with โ€–๐‘†๐›ผ(๐‘ก)โ€–โ‰ค๐‘€(1+๐‘ก๐›ผ๐‘›/2)exp(๐œ”1/๐›ผ๐‘ก),๐‘กโ‰ฅ0.(4.8)

Proof. We only consider the area above the ๐‘ฅ-axis, the lower area can be treated similarly.
Let Ray 1 โˆถ={๐œ”+๐œŒ๐‘’๐‘–๐›ผโ€ฒ๐œ‹/2โˆถ0โ‰ค๐œŒ<โˆž}, and let Ray 2 โˆถ={๐œŒ๐‘’๐‘–๐›ผ๐œ‹/2โˆถ0โ‰ค๐œŒ<โˆž}, where 1<๐›ผ<๐›ผโ€ฒ<2. Let ๐บ be the point (๐œ”,0), and set ๐ต to denote the intersection point of the two above rays. Let ฮฉ denote the region to the left side of Ray 1 and 2 (see Figure 1).
If ๐‘ƒ(๐œ‰) falls into ฮฉ, 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 ๐‘ƒ(๐œ‰) within the triangle ฮ”๐บ๐‘‚๐ต. To estimate ๐ท๐œ‡๐ธ๐›ผ(๐‘ก๐›ผ๐‘ƒ(๐œ‰)) for such values ๐‘ƒ(๐œ‰), we use (3.12), (3.15), and (3.5) to obtain |๐ท๐œ‡๐ธ๐›ผ(๐‘ก๐›ผ๐‘ƒ(๐œ‰))|โ‰ค๐‘€(1+๐‘ก๐›ผ|๐œ‡|)exp(๐œ”1/๐›ผ๐‘ก)(4.9) if Re((๐œŒ๐‘’๐‘–๐œƒ)1/๐›ผ)<๐œ”1/๐›ผ, where ๐œŒ๐‘’๐‘–๐œƒ denotes an arbitrary point on the line segment from ๐บ to ๐ต.
Since ๐œŒsin(๐›ผโ€ฒ=๐œ”๐œ‹/2)sin(๐›ผโ€ฒ๐œ‹/2โˆ’๐œƒ),(4.10) we have Re((๐œŒ๐‘’๐‘–๐œƒ)1/๐›ผ)=๐œ”1/๐›ผ(sin(๐›ผโ€ฒ๐œ‹/2)sin(๐›ผโ€ฒ)๐œ‹/2โˆ’๐œƒ)1/๐›ผcos(๐œƒ/๐›ผ).(4.11) Thus, to show that Re((๐œŒ๐‘’๐‘–๐œƒ)1/๐›ผ)<๐œ”1/๐›ผ (0<๐œƒโ‰ค๐›ผ๐œ‹/2) one needs to check that cos๐›ผ๐›ผ(๐œƒ/๐›ผ)<cos๐œƒ+sin๐œƒโ‹…tan(โ€ฒโˆ’12๐œ‹),0<๐œƒโ‰ค๐›ผ๐œ‹2;(4.12) and this is true if cos(๐œƒ/๐›ผ)โ‰คcos๐œƒ+sin๐œƒโ‹…tan(๐›ผโˆ’12๐œ‹),0<๐œƒโ‰ค๐›ผ๐œ‹2,(4.13) since 1<๐›ผ<๐›ผโ€ฒ<2.
We first consider the case when ๐œ‹/2โ‰ค๐œƒโ‰ค๐›ผ๐œ‹/2. Let ๐‘”(๐œƒ)=cos๐œƒ+sin๐œƒโ‹…tan(((๐›ผโˆ’1)/2)๐œ‹)โˆ’cos(๐œƒ/๐›ผ), then ๐‘”โ€ฒ(๐œƒ)=โˆ’sin๐œƒ+(1/๐›ผ)sin(๐œƒ/๐›ผ)+cos๐œƒโ‹…tan(((๐›ผโˆ’1)/2)๐œ‹)โ‰ค0 since sin๐œƒ>(1/๐›ผ)sin(๐œƒ/๐›ผ) and cos๐œƒโ‰ค0 for ๐œ‹/2โ‰ค๐œƒโ‰ค๐›ผ๐œ‹/2. So ๐‘”(๐œƒ) decreases with respect to ๐œƒ, which means that ๐‘”(๐œƒ)โ‰ฅ0 since ๐‘”(๐›ผ๐œ‹/2)=0.
For 0<๐œƒ<๐œ‹/2, we will show that cos(๐œƒ/๐›ผ)โ‰คcos๐œƒ+sin๐œƒโ‹…๐›ผโˆ’12๐œ‹,(4.14) which implies (4.13). Now for fixed ๐œƒโˆˆ(0,๐œ‹/2), denote by โ„Ž(๐›ผ)=cos๐œƒ+sin๐œƒโ‹…((๐›ผโˆ’1)/2)๐œ‹โˆ’cos(๐œƒ/๐›ผ). Since ๐›ผ>1, we have โ„Žโ€ฒ(๐›ผ)=(๐œ‹/2)sin๐œƒโˆ’(๐œƒ/๐›ผ2)sin(๐œƒ/๐›ผ)>0; it thus follows that โ„Ž(๐›ผ)โ‰ฅโ„Ž(1)=0. Therefore we have proved (4.14).
Now by (3.19) and (4.9) one obtains, for |๐œ‰|โ‰ฅ๐ฟ, |๐ท๐œ‡(๐ธ๐›ผ(๐‘ก๐›ผ๐‘ƒ)(๐‘Žโˆ’๐‘ƒ)โˆ’๐›พ)|โ‰ค๐‘€(1+๐‘ก๐›ผ|๐œ‡|)|๐œ‰|(๐‘šโˆ’1)|๐œ‡|โˆ’๐‘Ÿ๐›พexp(๐œ”1/๐›ผ๐‘ก),(4.15) and for |๐œ‰|โ‰ค๐ฟ, |๐ท๐œ‡(๐ธ๐›ผ(๐‘ก๐›ผ๐‘ƒ)(๐‘Žโˆ’๐‘ƒ)โˆ’๐›พ)|โ‰ค๐‘€(1+๐‘ก๐›ผ|๐œ‡|)exp(๐œ”1/๐›ผ๐‘ก).(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 ๐‘ƒ is coercive, but the choice of ๐ถ is different.

Theorem 4.3. Suppose that ๐‘ƒ(๐œ‰) is a polynomial of order ๐‘š, and {๐‘ƒ(๐œ‰)โˆถ๐œ‰โˆˆโ„๐‘›}โŠ‚โ„‚โงต(๐œ”+ฮฃ๐›ผโ€ฒ๐œ‹/2), where 1<๐›ผโ€ฒ<2. Then for 1<๐›ผ<๐›ผโ€ฒ<2, ๐›ฝ>๐‘›/2, ๐ถ=(1+|๐ด|2)โˆ’๐‘š๐›ฝ/2 (which is defined by (3.1) with ๐‘ข(๐‘ฅ)=(1+|๐‘ฅ|2)โˆ’๐‘š๐›ฝ/2), there exists an analytic ๐›ผ-times ๐ถ-regularized resolvent family ๐‘†๐›ผ(๐‘ก) for ๐‘ƒ(๐ด) such that โ€–๐‘†๐›ผ(๐‘ก)โ€–โ‰ค๐‘€(1+๐‘ก๐›ผ๐‘›/2)exp(๐œ”1/๐›ผ๐‘ก),๐‘กโ‰ฅ0.(4.17)

Proof. From (4.9) and |๐ท๐œ‡(1+|๐œ‰|2)โˆ’๐›ฝ/2|โ‰ค๐‘€|๐œ‰|โˆ’|๐œ‡|โˆ’๐›ฝ,|๐œ‰|โ‰ฅ๐ฟ,๐œ‡โˆˆโ„•๐‘›0,(4.18) we have for |๐œ‰|โ‰ฅ๐ฟ, |๐ท๐œ‡(๐ธ๐›ผ(๐‘ก๐›ผ๐‘ƒ)(1+๐œ‰|2)โˆ’๐›ฝ/2)|โ‰ค๐‘€(1+๐‘ก๐›ผ|๐œ‡|)|๐œ‰|(๐‘šโˆ’1)|๐œ‡|โˆ’๐›ฝexp(๐œ”1/๐›ผ๐‘ก),(4.19) and for |๐œ‰|โ‰ค๐ฟ, |๐ท๐œ‡(๐ธ๐›ผ(๐‘ก๐›ผ๐‘ƒ)(1+|๐œ‰|2)โˆ’๐›ฝ/2)|โ‰ค๐‘€(1+๐‘ก๐›ผ|๐œ‡|)exp(๐œ”1/๐›ผ๐‘ก).(4.20) It thus follows from Lemma 3.1 that when ๐›ฝ>๐‘›๐‘š/2, ๐ธ๐›ผ(๐‘ก๐›ผ๐‘ƒ)(1+|๐œ‰|2)โˆ’๐›ฝ/2โˆˆโ„ฑ๐ฟ1(โ„๐‘›). Similarly as in the proof of Theorem 4.1 we can show that there is an analytic ๐›ผ-times ๐ถ-regularized resolvent family for ๐‘ƒ(๐ด).

From now on ๐‘‹ will be ๐ฟ๐‘(โ„๐‘›)(1โ‰ค๐‘<โˆž) or ๐ถ0(โ„๐‘›)โˆถ={๐‘“โˆˆ๐ถ(โ„๐‘›)โˆถlim|๐‘ฅ|โ†’โˆž๐‘“(๐‘ฅ)=0}. The partial differential operator ๐‘ƒ(๐ท) defined by ๐‘ƒ(๐ท)๐‘“=โ„ฑโˆ’1(๐‘ƒโ„ฑ๐‘“)(4.21) with ๐ท(๐‘ƒ(๐ท))={๐‘“โˆˆ๐‘‹โˆถโ„ฑโˆ’1(๐‘ƒโ„ฑ๐‘“)โˆˆ๐‘‹}(4.22) is closed and densely defined on ๐‘‹. Since ๐‘–๐ท๐‘—=๐œ•/๐œ•๐‘ฅ๐‘—(1โ‰ค๐‘—โ‰ค๐‘›) is the generator of the bounded ๐ถ0-group {๐‘‡๐‘—(๐‘ก)}๐‘กโˆˆโ„ given by ๐‘‡๐‘—(๐‘ก)๐‘“(๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘›)=๐‘“(๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘—โˆ’1,๐‘ฅ๐‘—+๐‘ก,๐‘ฅ๐‘—+1,โ€ฆ,๐‘ฅ๐‘›)๐‘กโˆˆโ„(4.23) on ๐‘‹, we can apply the above results to ๐‘ƒ(๐ท) on ๐‘‹. It is remarkable that when ๐‘‹=๐ฟ๐‘(โ„๐‘›) (1<๐‘<โˆž) these results can be improved. In fact, if ๐ด=๐ท=(๐ท1,โ€ฆ,๐ท๐‘›), then the functions ๐‘ข๐‘ก's in the proofs of the above theorems give rise to Fourier multipliers on ๐ฟ๐‘(โ„๐‘›) having norm of polynomial growth ๐‘ก๐‘›๐‘ at infinity, where ๐‘›๐‘=๐‘›|1/2โˆ’1/๐‘|. 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 ๐‘‹=๐ฟ1(โ„๐‘›) or ๐ถ0(โ„๐‘›), ๐ถ=(๐‘Žโˆ’๐‘ƒ)โˆ’๐›พ(๐ท), where ๐›พ>๐‘›๐‘š/2๐‘Ÿ, there exists an analytic ๐›ผ-times ๐ถ-regularized resolvent family ๐‘†๐›ผ(๐‘ก) for ๐‘ƒ(๐ท) and โ€–๐‘†๐›ผ(๐‘ก)โ€–โ‰ค๐‘€(1+๐‘ก๐›ผ๐‘›/2)exp(๐œ”1/๐›ผ๐‘ก),๐‘กโ‰ฅ0.(4.24)(b)For ๐‘‹=๐ฟ๐‘(โ„๐‘›), ๐ถ=(๐‘Žโˆ’๐‘ƒ)โˆ’๐›พ(๐ท), where ๐›พ>๐‘›๐‘๐‘š/๐‘Ÿ, ๐‘›๐‘=๐‘›|1/2โˆ’1/๐‘|, there exists an analytic ๐›ผ-times ๐ถ-regularized resolvent family ๐‘†๐›ผ(๐‘ก) for ๐‘ƒ(๐ท) andโ€–๐‘†๐›ผ(๐‘ก)โ€–โ‰ค๐‘€(1+๐‘ก๐›ผ๐‘›๐‘)exp(๐œ”1/๐›ผ๐‘ก),๐‘กโ‰ฅ0.(4.25)

Theorem 4.5. Suppose that the assumptions of Theorem 4.3 are satisfied.(a)For ๐‘‹=๐ฟ1(โ„๐‘›) or ๐ถ0(โ„๐‘›), ๐ถ=(1โˆ’ฮ”)โˆ’๐‘š๐›ฝ/2, where ๐›ฝ>๐‘›/2, there exists an analytic ๐›ผ-times ๐ถ-regularized resolvent family ๐‘†๐›ผ(๐‘ก) for ๐‘ƒ(๐ท) and โ€–๐‘†๐›ผ(๐‘ก)โ€–โ‰ค๐‘€(1+๐‘ก๐›ผ๐‘›/2)exp(๐œ”1/๐›ผ๐‘ก),๐‘กโ‰ฅ0.(4.26)(b)For ๐‘‹=๐ฟ๐‘(โ„๐‘›), ๐ถ=(1โˆ’ฮ”)โˆ’๐‘š๐›ฝ/2, where ๐›ฝ>๐‘›๐‘, there exists an analytic ๐›ผ-times ๐ถ-regularized resolvent family ๐‘†๐›ผ(๐‘ก) for ๐‘ƒ(๐ท) and โ€–๐‘†๐›ผ(๐‘ก)โ€–โ‰ค๐‘€(1+๐‘ก๐›ผ๐‘›๐‘)exp(๐œ”1/๐›ผ๐‘ก),๐‘กโ‰ฅ0.(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 ฮ” on ๐ฟ๐‘(โ„๐‘›) (๐‘›>1,๐‘โ‰ 2) is ๐‘ƒ(๐œ‰)=โˆ’|๐œ‰|2. By Theorem 4.4, for every 1<๐›ผ<2 there exists an analytic ๐›ผ-times (1โˆ’ฮ”)โˆ’๐›พ-regularized resolvent family for the operator ฮ”, where ๐›พ>๐‘›๐‘.(b) Consider ๐‘ƒ(๐ท) on ๐ฟ๐‘(โ„2) (1<๐‘<โˆž) with ๐‘ƒ(๐œ‰)=โˆ’(1+๐œ‰21)(1+(๐œ‰2โˆ’๐œ‰๐‘™1)2)(๐‘™โˆˆโ„•).(4.28)Then ๐‘ƒ(๐œ‰)โ‰คโˆ’1(๐œ‰โˆˆโ„2). We claim that ๐‘ƒ is (2/๐‘™)-coercive. Indeed, if |๐œ‰2|โ‰ฅ2|๐œ‰๐‘™1|, then |๐‘ƒ(๐œ‰)|โ‰ฅ(1+๐œ‰211)(1+4๐œ‰221)โ‰ฅ4|๐œ‰|2.(4.29) If |๐œ‰2|<2|๐œ‰๐‘™1|, then |๐‘ƒ(๐œ‰)|โ‰ฅ1+|๐œ‰1|2โ‰ฅ๐‘|๐œ‰|2/๐‘™for|๐œ‰|โ‰ฅ1,(4.30) for some proper constant ๐‘, as desired. By Theorems 4.4 and 4.5, for every 1<๐›ผ<2 there exists an analytic ๐›ผ-times ๐ถ-regularized resolvent family for ๐‘ƒ(๐ท), where ๐ถ=(1โˆ’๐‘ƒ)โˆ’๐›พ(๐ท) with ๐›พ>2(๐‘™2+๐‘™)|1/2โˆ’1/๐‘| or ๐ถ=(1โˆ’ฮ”)โˆ’(๐‘™+1)๐›ฝ with ๐›ฝ>2|1/2โˆ’1/๐‘|. We remark that if ๐‘™โ‰ฅ5 and |1/2โˆ’1/๐‘|โ‰ฅ1/4+1/๐‘™, then 0โˆˆ๐œŽ(๐‘ƒ(๐ท)) (see [17]). Since 0โˆˆ๐‘ƒ(โ„2), it follows from [18, Theorem 1] that ๐œŒ(๐‘ƒ(๐ท))=โˆ…. Consequently, in this case there is no ๐›ผ-times resolvent family for ๐‘ƒ(๐ท) for any ๐›ผ.


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).