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Journal of Function Spaces and Applications
VolumeΒ 2012, Article IDΒ 819321, 15 pages
http://dx.doi.org/10.1155/2012/819321
Research Article

Embedding Operators in Vector-Valued Weighted Besov Spaces and Applications

Department of Electronics Engineering and Communication, Okan University, Akfirat Beldesi, Tuzla, 34959 Istanbul, Turkey

Received 12 July 2011; Accepted 9 January 2012

Academic Editor: Lars-ErikΒ Persson

Copyright Β© 2012 Veli Shakhmurov. 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

The embedding theorems in weighted Besov-Lions type spaces 𝐡𝑙,𝑠𝑝,π‘ž,𝛾 (Ξ©;𝐸0,𝐸) in which 𝐸0,𝐸 are two Banach spaces and 𝐸0βŠ‚πΈ are studied. The most regular class of interpolation space 𝐸𝛼 between 𝐸0 and E is found such that the mixed differential operator 𝐷𝛼 is bounded from 𝐡𝑙,𝑠𝑝,π‘ž,𝛾 (Ξ©;𝐸0,𝐸) to 𝐡𝑠𝑝,π‘ž,𝛾 (Ξ©;𝐸𝛼) and Ehrling-Nirenberg-Gagliardo type sharp estimates are established. By using these results, the uniform separability of degenerate abstract differential equations with parameters and the maximal B-regularity of Cauchy problem for abstract parabolic equations are obtained. The infinite systems of the degenerate partial differential equations and Cauchy problem for system of parabolic equations are further studied in applications.

1. Introduction

Embedding theorems in function spaces have been elaborated in [1–3]. A comprehensive introduction to the theory of embedding of function spaces and historical references may also be found in [4, 5]. Embedding theorems in abstract function spaces have been studied in [2, 6–18]. The anisotropic Sobolev spaces π‘Šπ‘™π‘(Ξ©;𝐻0,𝐻), Ξ©βŠ‚π‘…π‘›, and corresponding weighted spaces have been investigated in [11, 13–16, 18], respectively. Embedding theorems in Banach-valued Besov spaces have been studied in [6–8, 17, 19]. Moreover, boundary value problems (BVPs) for differential-operator equations (DOEs) have been studied in [4, 5, 20, 21]. The solvability and the spectrum of BVPs for elliptic DOEs have also been refined in [7, 13–18, 22–26]. A comprehensive introduction to the differential-operator equations and historical references may be found in [4, 5]. In these works, Hilbert-valued function spaces essentially have been considered.

Let 𝑙=(𝑙1,𝑙2,…,𝑙𝑛) and 𝑠=(𝑠1,𝑠2,…,𝑠𝑛). Let 𝐸0 and 𝐸 be Banach spaces such that 𝐸0 is continuously and densely embedded in 𝐸. In the present paper, the weighted Banach-valued Besov space 𝐡𝑙,𝑠𝑝,πœƒ,𝛾(Ξ©;𝐸0,𝐸) is to be introduced. The smoothest interpolation class 𝐸𝛼 between 𝐸0, 𝐸 (i.e., to find the possible small πœŽπ›Ό for 𝐸𝛼=(𝐸0,𝐸)πœŽπ›Ό,𝑝) is found such that the appropriate mixed differential operators 𝐷𝛼 are bounded from 𝐡𝑙,𝑠𝑝,πœƒ,𝛾(Ξ©;𝐸0,𝐸) to 𝐡𝑠𝑝,πœƒ,𝛾(Ξ©;𝐸𝛼). By applying these results, the maximal 𝐡-regularity of certain classes of anisotropic partial DOE with parameters is derived. The paper is organized as follows. Section 2 collects notations and definitions. Section 3 presents embedding theorems in Besov-Lions space 𝐡𝑙,𝑠𝑝,πœƒ,𝛾(Ξ©;𝐸0,𝐸). Section 4 contains applications of corresponding embedding theorems to vector-valued function spaces, and Section 5 is devoted to applications of these embedding theorems to anisotropic DOE with parameters for which the uniformly maximal 𝐡-regularity is obtained. Then, in Section 6, by using these results, the maximal 𝐡-regularity of parabolic Cauchy problem is shown. In Section 7, this DOE is applied to BVP and Cauchy problem for infinite systems of quasielliptic and parabolic PDE, respectively.

2. Notations and Definitions

Let 𝐸 be a Banach space and 𝛾=𝛾(π‘₯) a nonnegative measurable weighted function defined on a domain Ξ©βŠ‚π‘…π‘›. Let 𝐿𝑝,𝛾(Ξ©;𝐸) denote the space of strongly measurable 𝐸-valued functions that are defined on Ξ© with the norm ‖𝑓‖𝐿𝑝,𝛾(Ξ©;𝐸)=ξ‚΅ξ€œβ€–π‘“(π‘₯)‖𝑝𝐸𝛾(π‘₯)𝑑π‘₯1/𝑝,1≀𝑝<∞,β€–π‘“β€–πΏβˆž,𝛾(Ξ©;𝐸)=esssupπ‘₯βˆˆΞ©ξ€Ίβ€–π‘“(π‘₯)‖𝐸.𝛾(π‘₯)(2.1)

Let β„Žβˆˆπ‘…,π‘šβˆˆπ‘, and 𝑒𝑖,𝑖=1,2,…,𝑛 be the standard unit vectors in 𝑅𝑛. Let (see [1, Section  16])Δ𝑖(β„Ž)𝑓(π‘₯)=𝑓π‘₯+β„Žπ‘’π‘–ξ€Έβˆ’π‘“(π‘₯),…,Ξ”π‘šπ‘–(β„Ž)𝑓(π‘₯)=Δ𝑖Δ(β„Ž)π‘–π‘šβˆ’1ξ€»=(β„Ž)𝑓(π‘₯)π‘šξ“π‘˜=0(βˆ’1)π‘š+π‘˜πΆπ‘˜π‘šπ‘“ξ€·π‘₯+π‘˜β„Žπ‘’π‘–ξ€Έ.(2.2)

Let Ξ”π‘šπ‘–βŽ§βŽͺ⎨βŽͺβŽ©Ξ”(Ξ©,β„Ž)=π‘šπ‘–ξ€Ί(β„Ž),forπ‘₯,π‘₯+π‘šπ‘¦π‘’π‘–ξ€»ξ€ΊβŠ‚Ξ©,0,forπ‘₯,π‘₯+π‘šπ‘¦π‘’π‘–ξ€»βŠ‚π‘…π‘›Ξ©βŽ«βŽͺ⎬βŽͺ⎭.(2.3)

Let πΏβˆ—πœƒ(𝐸) be a 𝐸-valued function space such thatβ€–π‘’β€–πΏβˆ—πœƒ(𝐸)=ξ‚΅ξ€œβˆž0‖𝑒(𝑑)β€–πœƒπΈπ‘‘π‘‘π‘‘ξ‚Ά1/πœƒ<∞.(2.4)

Let π‘šπ‘– be positive integers, π‘˜π‘– nonnegative integers, 𝑠𝑖 positive numbers, and π‘šπ‘–>π‘ π‘–βˆ’π‘˜π‘–>0,𝑖=1,2,…,𝑛,𝑠=(𝑠1,𝑠2,…,𝑠𝑛),  1β‰€π‘β‰€βˆž,1β‰€πœƒβ‰€βˆž,0<𝑦0<∞. Let 𝐹 denote the Fourier transform. The Banach-valued Besov space 𝐡𝑠𝑝,πœƒ,𝛾(Ξ©;𝐸) is defined as 𝐡𝑠𝑝,πœƒ,π›ΎβŽ§βŽͺ⎨βŽͺ⎩(Ξ©;𝐸)=π‘“βˆΆπ‘“βˆˆπΏπ‘(Ξ©;𝐸),‖𝑓‖𝐡𝑠𝑝,πœƒ(Ξ©;𝐸)=‖𝑓‖𝐿𝑝,𝛾(Ξ©;𝐸)+𝑛𝑖=1ξ‚΅ξ€œβ„Ž00β„Žβˆ’[(π‘ π‘–βˆ’π‘˜π‘–)π‘ž+1]β€–β€–Ξ”π‘šπ‘–π‘–(β„Ž,Ξ©)π·π‘˜π‘–π‘–π‘“β€–β€–πœƒπΏπ‘,𝛾(Ξ©;𝐸)𝑑𝑦1/πœƒβŽ«βŽͺ⎬βŽͺ⎭<∞,1β‰€πœƒ<∞,‖𝑓‖𝐡𝑠𝑝,πœƒ,𝛾(Ξ©;𝐸)=𝑛𝑖=1sup0<β„Ž<β„Ž0β€–β€–Ξ”π‘ši𝑖(β„Ž,Ξ©)π·π‘˜π‘–π‘–π‘“β€–β€–πΏπ‘,𝛾(Ξ©;𝐸)β„Žπ‘ π‘–βˆ’π‘˜π‘–forπœƒ=∞.(2.5)

For 𝐸=𝑅 and 𝛾(π‘₯)≑1, we obtain a scalar-valued anisotropic Besov space 𝐡𝑠𝑝,πœƒ,𝛾(Ξ©) [1, Section  18].

Let 𝐂 be the set of complex numbers andπ‘†πœ‘=ξ€½||||ξ€Ύβˆͺπœ†;πœ†βˆˆπ‚,argπœ†β‰€πœ‘{0},0β‰€πœ‘<πœ‹.(2.6)

A linear operator 𝐴 is said to be a πœ‘-positive in a Banach space 𝐸 with bound 𝑀>0 if 𝐷(𝐴) is dense on 𝐸 and β€–(𝐴+πœ†πΌ)βˆ’1‖𝐿(𝐸)≀𝑀(1+|πœ†|)βˆ’1with πœ†βˆˆπ‘†πœ‘,πœ‘βˆˆ[0πœ‹), where 𝐼 is the identity operator in 𝐸 and 𝐿(𝐸) is the space of bounded linear operators in 𝐸.

It is known [3, Section  1.15.1] that there exist the fractional powers π΄πœƒof the positive operator 𝐴. Let 𝐸(π΄πœƒ) denote the space 𝐷(π΄πœƒ) with a graph norm defined as‖𝑒‖𝐸(π΄πœƒ)=‖𝑒‖𝑝+β€–β€–π΄πœƒπ‘’β€–β€–π‘ξ€Έ1/𝑝,1≀𝑝<∞,βˆ’βˆž<πœƒ<∞.(2.7)

The operator 𝐴(𝑑) is said to be πœ‘-positive in 𝐸 uniformly with respect to 𝑑 with bound 𝑀>0 if 𝐷(𝐴(𝑑)) is independent of 𝑑, 𝐷(𝐴(𝑑)) is dense in 𝐸, and β€–(𝐴(𝑑)+πœ†πΌ)βˆ’1‖≀𝑀(1+|πœ†|)βˆ’1 for all πœ†βˆˆπ‘†πœ‘, 0β‰€πœ‘<πœ‹,where 𝑀 does not depend on 𝑑 and πœ†.

Let 𝑙=(𝑙1,𝑙2,…,𝑙𝑛),𝑠=(𝑠1,𝑠2,…,𝑠𝑛), where π‘™π‘˜ are positive integers. Let 𝐡𝑙,𝑠𝑝,πœƒ,𝛾(Ξ©;𝐸) denote a 𝐸-valued weighted Sobolev-Besov space of functions π‘’βˆˆπ΅π‘ π‘,πœƒ,𝛾(Ξ©;𝐸) that have generalized derivatives π·π‘™π‘˜π‘˜π‘’=(πœ•π‘™π‘˜/πœ•π‘₯π‘™π‘˜π‘˜)π‘’βˆˆπ΅π‘ π‘,πœƒ,𝛾(Ξ©;𝐸),π‘˜=1,2,…,𝑛, with the norm‖𝑒‖𝐡𝑙,𝑠𝑝,πœƒ,𝛾(Ξ©;𝐸)=‖𝑒‖𝐡𝑠𝑝,πœƒ,𝛾(Ξ©;𝐸)+π‘›ξ“π‘˜=1β€–β€–π·π‘™π‘˜π‘˜π‘’β€–β€–π΅π‘ π‘,πœƒ,𝛾(Ξ©;𝐸)<∞.(2.8)

Suppose 𝐸0 is continuously and densely embedded into 𝐸. Let 𝐡𝑙,𝑠𝑝,πœƒ,𝛾(Ξ©;𝐸0,𝐸) denote the space with the norm‖𝑒‖𝐡𝑙,𝑠𝑝,πœƒ,𝛾=‖𝑒‖𝐡𝑙,𝑠𝑝,πœƒ,𝛾(Ξ©;𝐸0,𝐸)=‖𝑒‖𝐡𝑠𝑝,πœƒ,𝛾(Ξ©;𝐸0)+π‘›ξ“π‘˜=1β€–β€–π·π‘™π‘˜π‘˜π‘’β€–β€–π΅π‘ π‘,πœƒ,𝛾(Ξ©;𝐸)<∞.(2.9)

Let 𝑑=(𝑑1,𝑑2,…,𝑑𝑛), where π‘‘π‘˜ are parameters. We define the following parameterized norm in 𝐡𝑙,𝑠𝑝,πœƒ(Ξ©;𝐸0,𝐸):‖𝑒‖𝐡𝑙,𝑠𝑝,πœƒ,𝛾,𝑑(Ξ©;𝐸0,𝐸)=‖𝑒‖𝐡𝑠𝑝,πœƒ,𝛾(Ξ©;𝐸0)+π‘›ξ“π‘˜=1β€–β€–π‘‘π‘˜π·π‘™π‘˜π‘˜π‘’β€–β€–π΅π‘ π‘,πœƒ,𝛾(Ξ©;𝐸)<∞.(2.10)

Let π‘š be a positive integer. 𝐢(π‘š)(Ξ©;𝐸) denotes the spaces of 𝐸-valued bounded and π‘š-times continuously differentiable functions on Ξ©. For two sequences {π‘Žπ‘—}∞1 and {𝑏𝑗}∞1 of positive numbers, the expression π‘Žπ‘—βˆΌπ‘π‘— means that there exist positive numbers 𝐢1 and 𝐢2 such that𝐢1π‘Žπ‘—β‰€π‘π‘—β‰€πΆ2π‘Žπ‘—.(2.11)

Let 𝐸1,  and 𝐸2 be two Banach spaces. Let 𝐹 denote the Fourier transformation and let β„Ž be some parameter. We say that the function Ξ¨β„Ž dependent of β„Ž is a uniform collection of multipliers if there exists a positive constant 𝑀 independent of β„Ž such that β€–πΉβˆ’1Ξ¨β„ŽπΉπ‘’β€–π΅π‘ π‘ž,πœƒ,𝛾(𝑅𝑛;𝐸2)≀𝑀‖𝑒‖𝐡𝑠𝑝,πœƒ,𝛾(𝑅𝑛;𝐸1)for all π‘’βˆˆπ΅π‘ π‘,πœƒ,𝛾(𝑅𝑛;𝐸1). The set of all multipliers from 𝐡𝑠𝑝,πœƒ,𝛾(𝑅𝑛;𝐸1) to π΅π‘ π‘ž,πœƒ,𝛾(𝑅𝑛;𝐸2) will be denoted by π‘€π‘ž,πœƒ,𝛾𝑝,πœƒ,𝛾(𝐸1,𝐸2). For 𝐸1=𝐸2=𝐸, it will be denoted by π‘€π‘ž,πœƒ,𝛾𝑝,πœƒ,𝛾(𝐸). The exposition of the theory of Fourier multipliers and some related references can be found in [3, Sections  2.2.1–2.2.4]. In weighted 𝐿𝑝 spaces, Fourier multipliers have been investigated in several studies like [27, 28]. Operator-valued Fourier multipliers in Banach-valued 𝐿𝑝 spaces studied, for example, in [4, 6, 25, 27–33].

Let 𝛽=(𝛽1,𝛽2,…,𝛽𝑛) be multi-indexes andπœ‰π›½=πœ‰π›½11πœ‰π›½22,…,πœ‰π›½π‘›π‘›,π‘ˆπ‘›=ξ€½||𝛽||ξ€Ύ1π›½βˆΆβ‰€π‘›,πœ‚=π‘βˆ’1π‘ž.(2.12)

Definition 2.1. A Banach space 𝐸 satisfies a 𝐡-multiplier condition with respect to 𝑝,π‘ž,πœƒ,𝑠 (or with respect to 𝑝,πœƒ,𝑠 for 𝑝=π‘ž), and the weight 𝛾, when Ξ¨βˆˆπΆπ‘›(𝑅𝑛;𝐡(𝐸)), 1β‰€π‘β‰€π‘žβ‰€βˆž, π›½βˆˆπ‘ˆπ‘› and πœ‰βˆˆπ‘‰π‘›, if the estimate (1+|πœ‰|)|𝛽|+πœ‚β€–π·π›½Ξ¨(πœ‰)‖𝐿(𝐸)≀𝐢,π‘˜=0,1,…,|𝛽| implies Ξ¨βˆˆπ‘€π‘ž,πœƒ,𝛾𝑝,πœƒ,𝛾(𝐸).
It is well known (e.g., see [32]) that any Hilbert space satisfies the 𝐡-multiplier condition. There are, however, Banach spaces which are not Hilbert spaces but satisfy the 𝐡-multiplier condition (see [7, 30]). However, additional conditions are needed for operator-valued multipliers in 𝐿𝑝 spaces, for example, UMD spaces (e.g., see [25, 33]). Let 𝛼1,𝛼2,…,𝛼𝑛 be nonnegative and 𝑙1,𝑙2,…,𝑙𝑛 positive integers: ||||=π›ΌβˆΆπ‘™π‘›ξ“π‘˜=1π›Όπ‘˜π‘™π‘˜ξ€·π›Ό,𝛼=1,𝛼2,…,𝛼𝑛𝑙,𝑙=1,𝑙2,…,𝑙𝑛,𝐷𝛼=𝐷𝛼11𝐷𝛼22β‹―D𝛼𝑛𝑛=πœ•|𝛼|πœ•π‘₯𝛼11πœ•π‘₯𝛼22β‹―πœ•π‘₯𝛼𝑛𝑛,|𝛼|=π‘›ξ“π‘˜=!π›Όπ‘˜.(2.13)
Consider the following differential-operator equation: 𝐿𝑒=π‘›ξ“π‘˜=1(βˆ’1)π‘™π‘˜π‘‘π‘˜π·2π‘™π‘˜π‘˜π‘’+π΄πœ†ξ“π‘’+||||π›ΌβˆΆ2𝑙<1𝛼(𝑑)𝐴𝛼(π‘₯)𝐷𝛼𝑒=𝑓,(2.14) where 𝐴(π‘₯), 𝐴𝛼(π‘₯) are linear operators in a Banach space 𝐸,β€‰β€‰π‘Žπ‘˜ are complex-valued functions and π‘‘π‘˜ are some parameters βˆπ›Ό(𝑑)=π‘›π‘˜=1π‘‘π›Όπ‘˜/2π‘™π‘˜π‘˜. For 𝑙1=𝑙2=β‹―=𝑙𝑛=π‘š, we obtain the elliptic class of DOE.
The function belonging to 𝐡𝑙,𝑠𝑝,πœƒ,𝛾(𝑅𝑛;𝐸(𝐴),𝐸) and satisfying (2.14) a.e. on 𝑅𝑛 is said to be a solution of (2.14) on 𝑅𝑛.

Definition 2.2. The problem (2.14) is said to be uniform weighted 𝐡-separable (or weighted 𝐡𝑠𝑝,πœƒ,𝛾(Ξ©;𝐸)-separable) if, for all π‘“βˆˆπ΅π‘ π‘,πœƒ,𝛾(Ξ©;𝐸), the problem (2.14) has a unique solution π‘’βˆˆπ΅π‘™,𝑠𝑝,πœƒ,𝛾(Ξ©;𝐸(𝐴),𝐸) and the following estimate holds: ‖𝐴𝑒‖𝐡𝑠𝑝,πœƒ,𝛾(Ξ©;𝐸)+π‘›ξ“π‘˜=1π‘‘π‘˜β€–β€–π·2π‘™π‘˜π‘˜π‘’β€–β€–π΅π‘ π‘,πœƒ,𝛾(Ξ©;𝐸)≀𝐢‖𝑓‖𝐡𝑠𝑝,πœƒ,𝛾(Ξ©;𝐸).(2.15)
Consider the following degenerate DOE: 𝐿𝑒=π‘›ξ“π‘˜=1(βˆ’1)π‘™π‘˜π‘‘π‘˜π·[2π‘™π‘˜]π‘˜π‘’+π΄πœ†ξ“π‘’+||||π›ΌβˆΆ2𝑙<1𝛼(𝑑)𝐴𝛼(π‘₯)𝐷[𝛼]𝑒=𝑓,(2.16) where 𝐴(π‘₯), 𝐴𝛼(π‘₯) are linear operators in a Banach space 𝐸,π‘Žπ‘˜ are complex-valued functions, π‘‘π‘˜ are some parameters and 𝐷π‘₯[𝑖]π‘˜=𝛾π‘₯π‘˜ξ€Έπœ•πœ•π‘₯π‘˜ξ‚Άπ‘–,π‘˜=1,2,…,𝑛.(2.17)

Remark 2.3. Under the substitution πœπ‘˜=ξ€œπ‘₯π‘˜0π›Ύβˆ’1(𝑦)𝑑𝑦,(2.18)𝐡𝑠𝑝,πœƒ,𝛾(𝑅𝑛;𝐸),  𝐡[𝑙],𝑠𝑝,πœƒ,𝛾(𝑅𝑛;𝐸(𝐴),𝐸) are mapped isomorphically onto the spaces 𝐡𝑠𝑝,πœƒ,̃𝛾(𝑅𝑛;𝐸),  𝐡𝑙,𝑠𝑝,πœƒ,̃𝛾(𝑅𝑛;𝐸(𝐴),𝐸), respectively, where 𝛾=π‘›ξ‘π‘˜=1𝛾π‘₯π‘˜ξ€Έ,̃𝛾=̃𝛾(𝜏)=π‘›ξ‘π‘˜=1𝛾π‘₯π‘˜ξ€·πœπ‘˜.ξ€Έξ€Έ(2.19) Moreover, under the substitution (2.18), the degenerate problem (2.16) is mapped to the undegenerate problem (2.14).

3. Embedding Theorems

Let𝜘=π‘›ξ“π‘˜=1π›Όπ‘˜+1/π‘βˆ’1/π‘žπ‘™π‘˜,𝜈(𝑙)=maxπ‘˜,π‘—βˆˆ{1,2,…,𝑛}ξ‚Έ1π‘™π‘˜βˆ’1𝑙𝑗,𝜎(𝑑)=π‘›ξ‘π‘˜=1𝑑(π›Όπ‘˜+1/π‘βˆ’1/π‘ž)/π‘™π‘˜π‘˜.(3.1)

Theorem 3.1. Suppose the following conditions hold:(1)𝐸 is a Banach space satisfying the 𝐡-multiplier condition with respect to 𝑝, π‘ž, 𝑠;(2)𝑑=(𝑑1,𝑑2,…,𝑑𝑛),  0<π‘‘π‘˜β‰€π‘‡<∞, π‘˜=1,2,…,𝑛, 1<π‘β‰€π‘ž<∞, πœƒβˆˆ[1,∞];(3)π‘™π‘˜ are positive, π›Όπ‘˜ nonnegative integers such that 0<𝜘+𝜈(𝑙)≀1 and 0β‰€πœ‡β‰€1βˆ’πœ˜βˆ’πœˆ(𝑙);(4)𝐴 is a πœ‘-positive operator in 𝐸.
Then, the embedding 𝐷𝛼𝐡𝑙,𝑠𝑝,πœƒ,𝛾(𝑅𝑛;𝐸(𝐴),𝐸)βŠ‚π΅π‘ π‘ž,πœƒ,𝛾(𝑅𝑛;𝐸(𝐴1βˆ’πœ˜βˆ’πœ‡)) is continuous, and there exists a constant πΆπœ‡>0, depending only on πœ‡ such that 𝜎(𝑑)β€–π·π›Όπ‘’β€–π΅π‘ π‘ž,πœƒ,𝛾(𝑅𝑛;𝐸(𝐴1βˆ’πœ˜βˆ’πœ‡))β‰€πΆπœ‡ξ‚ƒβ„Žπœ‡β€–π‘’β€–π΅π‘™,𝑠𝑝,πœƒ,𝛾,𝑑+β„Žβˆ’(1βˆ’πœ‡)‖𝑒‖𝐡𝑠𝑝,πœƒ,𝛾(𝑅𝑛;𝐸)ξ‚„(3.2) for all π‘’βˆˆπ΅π‘™,𝑠𝑝,πœƒ,𝛾(𝑅𝑛;𝐸(𝐴),𝐸) and 0<β„Žβ‰€β„Ž0<∞.

Proof. Denoting 𝐹𝑒 by ̂𝑒, it is clear that β€–π·π›Όπ‘’β€–π΅π‘ π‘ž,πœƒ,𝛾(𝑅𝑛;𝐸(𝐴1βˆ’πœ˜βˆ’πœ‡))βˆΌβ€–β€–πΉβˆ’1(π‘–πœ‰)𝛼𝐴1βˆ’πœ˜βˆ’πœ‡β€–β€–Μ‚π‘’π΅π‘ π‘ž,πœƒ,𝛾(𝑅𝑛;𝐸).(3.3) Similarly, from the definition of 𝐡𝑙,𝑠𝑝,πœƒ,𝛾(𝑅𝑛;𝐸(𝐴),𝐸), we have ‖𝑒‖𝐡𝑙,𝑠𝑝,πœƒ,𝛾,𝑑(𝑅𝑛;𝐸(𝐴),𝐸)=‖𝑒‖𝐡𝑠𝑝,πœƒ,𝛾(𝑅𝑛;𝐸(𝐴))+π‘›ξ“π‘˜=1β€–β€–π‘‘π‘˜π·π‘™π‘˜π‘˜π‘’β€–β€–π΅π‘ π‘,πœƒ,𝛾(𝑅𝑛;𝐸)βˆΌβ€–β€–πΉβˆ’1‖‖𝐴̂𝑒𝐡𝑠𝑝,πœƒ,𝛾(𝑅𝑛;𝐸)+π‘›ξ“π‘˜=1β€–β€–π‘‘π‘˜πΉβˆ’1ξ‚ƒξ€·π‘–πœ‰π‘˜ξ€Έπ‘™π‘˜ξ‚„β€–β€–Μ‚π‘’π΅π‘ π‘,πœƒ,𝛾(𝑅𝑛;𝐸).(3.4) Thus, proving the inequality (3.2) is equivalent to proving β€–β€–πΉπœŽ(𝑑)βˆ’1ξ€Ί(π‘–πœ‰)𝛼𝐴1βˆ’πœ˜βˆ’πœ‡ξ€»β€–β€–Μ‚π‘’π‘‹π‘ β‰€β„Žπœ‡β€–β€–πΉβˆ’1‖‖𝐴̂𝑒𝑋𝑠+β„Žπœ‡π‘›ξ“π‘˜=1β€–β€–π‘‘π‘˜πΉβˆ’1ξ‚ƒξ€·π‘–πœ‰π‘˜ξ€Έπ‘™π‘˜ξ‚„β€–β€–Μ‚π‘’π‘‹π‘ +β„Žβˆ’(1βˆ’πœ‡)‖𝑒‖𝑋𝑠.(3.5) So, the inequality (3.2) will be followed if we prove the following inequality: β€–β€–πΉπœŽ(𝑑)βˆ’1ξ€Ί(π‘–πœ‰)𝛼𝐴1βˆ’πœ˜βˆ’πœ‡ξ€»β€–β€–Μ‚π‘’π‘‹π‘ β‰€πΆπœ‡β€–β€–πΉβˆ’1ξ€Ίβ„Žπœ‡ξ€»β€–β€–(𝐴+πœ“(𝑑,πœ‰))̂𝑒𝑋𝑠(3.6) for a suitable πΆπœ‡>0 and for all π‘’βˆˆπ΅π‘™,𝑠𝑝,πœƒ,𝛾(𝑅𝑛;𝐸(𝐴),𝐸), where πœ“=πœ“(𝑑,πœ‰)=π‘›ξ“π‘˜=1π‘‘π‘˜||πœ‰π‘˜||π‘™π‘˜+β„Žβˆ’1,𝑋𝑠=𝐡𝑠𝑝,πœƒ,𝛾(𝑅𝑛;𝐸).(3.7)
Let us express the left-hand side of (3.6) as β€–β€–πΉπœŽ(𝑑)βˆ’1ξ€Ί(π‘–πœ‰)𝛼𝐴1βˆ’πœ˜βˆ’πœ‡ξ€»β€–β€–Μ‚π‘’π΅π‘ π‘ž,πœƒ,𝛾(𝑅𝑛;𝐸)=β€–β€–πΉπœŽ(𝑑)βˆ’1(π‘–πœ‰)𝛼𝐴1βˆ’πœ˜βˆ’πœ‡ξ€Ίβ„Žπœ‡ξ€»(𝐴+πœ“)βˆ’1ξ€Ίβ„Žπœ‡ξ€»β€–β€–(𝐴+πœ“)π΅π‘ π‘ž,πœƒ,𝛾(𝑅𝑛;𝐸).(3.8) (Since 𝐴 is a positive operator in 𝐸 and βˆ’πœ“(𝑑,πœ‰)βˆˆπ‘†(πœ‘), it is possible.) By virtue of Definition 2.1, it is clear that the inequality (3.6) will be followed immediately from (3.8) if we can prove that the operator-function Ψ𝑑=Ψ𝑑,β„Ž,πœ‡=𝜎(𝑑)(π‘–πœ‰)𝛼𝐴1βˆ’πœ˜βˆ’πœ‡[β„Žπœ‡(𝐴+πœ“)]βˆ’1 is a multiplier in π‘€π‘ž,πœƒ,𝛾𝑝,πœƒ,𝛾(𝐸), which is uniform with respect to β„Ž and 𝑑. Since 𝐸 satisfies the multiplier condition with respect to 𝑝 and π‘ž, it suffices to show the following estimate: ||πœ‰||π‘˜+πœ‚β€–β€–π·π›½Ξ¨π‘‘β€–β€–(πœ‰)𝐿(𝐸)||𝛽||≀𝐢,π‘˜=0,1,…,(3.9) for all π›½βˆˆπ‘ˆπ‘›, πœ‰βˆˆπ‘…π‘›/{πœ‰π‘˜=0} and πœ‚=1/π‘βˆ’1/π‘ž. In a way similar to [18,Lemma  3.1], we obtain that |πœ‰|βˆ’πœ‚β€–Ξ¨π‘‘(πœ‰)‖𝐿(𝐸)β‰€π‘€πœ‡for all πœ‰βˆˆπ‘…π‘›. This shows that the inequality (3.9) is satisfied for 𝛽=(0,…,0). We next consider (3.9) for 𝛽=(𝛽1,…,𝛽𝑛), where π›½π‘˜=1 and 𝛽=0 for π‘—β‰ π‘˜. By using the condition 𝜘+𝜈(𝑙)≀1 and well-known inequality 𝑦𝛼11𝑦𝛼22β‹―π‘¦π›Όπ‘›π‘›βˆ‘β‰€πΆ(1+π‘›π‘˜=1π‘¦π‘™π‘˜π‘˜),π‘¦π‘˜β‰₯0 and by reasoning according to [18, Theorem  3.1], we have ||πœ‰||1+πœ‚||πœ‰π‘˜||β€–β€–π·π‘˜Ξ¨π‘‘β€–β€–(πœ‰)𝐿(𝐸)β‰€π‘€πœ‡,π‘˜=1,2,…,𝑛.(3.10)
Repeating the above process, we obtain the estimate (3.9). Thus, the operator-function Ψ𝑑,β„Ž,πœ‡(πœ‰) is a uniform collection of multiplier, that is, Ψ𝑑,β„Ž,πœ‡βˆˆΞ¦β„ŽβŠ‚π‘€π‘ž,πœƒ,𝛾𝑝,πœƒ,𝛾(𝐸). This completes the proof of the Theorem 3.1.

It is possible to state Theorem 3.1 in a more general setting. For this, we use the extension operator in 𝐡𝑙,𝑠𝑝,πœƒ,𝛾(Ξ©;𝐸(𝐴),𝐸).

Condition 1. Let 𝐴 be a πœ‘-positive operator in Banach spaces 𝐸 satisfying the 𝐡-multiplier condition. Let a region Ξ©βŠ‚π‘…π‘› be such that there exists a bounded linear extension operatorfrom 𝐡𝑙,𝑠𝑝,πœƒ,𝛾(Ξ©;𝐸(𝐴),𝐸) to 𝐡𝑙,𝑠𝑝,πœƒ,𝛾(𝑅𝑛;𝐸(𝐴),𝐸), for 1≀𝑝,β€‰β€‰πœƒβ‰€βˆž.

Remark 3.2. If Ξ©βŠ‚π‘…π‘› is a region satisfying a strong 𝑙-horn condition (see [23, Section  18]) 𝐸=𝑅,𝐴=𝐼, then there exists a bounded linear extension operator from 𝐡𝑠𝑝,πœƒ,𝛾(Ξ©)=𝐡𝑠𝑝,πœƒ,𝛾(Ξ©;𝑅,𝑅) to 𝐡𝑠𝑝,πœƒ,𝛾(𝑅𝑛)=𝐡𝑠𝑝,πœƒ,𝛾(𝑅𝑛;𝑅,𝑅).

Theorem 3.3. Suppose all conditions of Theorem 3.1 and Condition 1 are satisfied. Then, the embedding 𝐷𝛼𝐡𝑙,𝑠𝑝,πœƒ,𝛾(Ξ©;𝐸(𝐴),𝐸)βŠ‚π΅π‘ π‘ž,πœƒ,𝛾(Ξ©;𝐸(𝐴1βˆ’πœ˜βˆ’πœ‡)) is continuous and there exists a constant πΆπœ‡ depending only on πœ‡ such that 𝜎(𝑑)β€–π·π›Όπ‘’β€–π΅π‘ π‘ž,πœƒ,𝛾(Ξ©;𝐸(𝐴1βˆ’πœ˜βˆ’πœ‡))β‰€πΆπœ‡ξ‚ƒβ„Žπœ‡β€–π‘’β€–π΅π‘™,𝑠𝑝,πœƒ,𝛾,𝑑+β„Žβˆ’(1βˆ’πœ‡)‖𝑒‖𝐡𝑠𝑝,πœƒ,𝛾(Ξ©;𝐸)ξ‚„(3.11) for all π‘’βˆˆπ΅π‘™,𝑠𝑝,πœƒ,𝛾(Ξ©;𝐸(𝐴),𝐸) and 0<β„Žβ‰€β„Ž0<∞.

Proof. It suffices to prove the estimate (3.11). Let 𝑃 be a bounded linear extension operator from π΅π‘ π‘ž,πœƒ,𝛾(Ξ©;𝐸) to π΅π‘ π‘ž,πœƒ,𝛾(𝑅𝑛;𝐸) and also from 𝐡𝑙,𝑠𝑝,πœƒ,𝛾(Ξ©;𝐸(𝐴),𝐸) to 𝐡𝑙,𝑠𝑝,πœƒ,𝛾(𝑅𝑛;𝐸(𝐴),𝐸). Let 𝑃Ω be a restriction operator from 𝑅𝑛 to Ξ©. Then, for any π‘’βˆˆπ΅π‘™,𝑠𝑝,πœƒ(Ξ©;𝐸(𝐴),𝐸), we have β€–π·π›Όπ‘’β€–π΅π‘ π‘ž,πœƒ,𝛾(Ξ©;𝐸(𝐴1βˆ’πœ˜βˆ’πœ‡))=β€–β€–π·π›Όπ‘ƒΞ©β€–β€–π‘ƒπ‘’π΅π‘ π‘ž,πœƒ,𝛾(Ξ©;𝐸(𝐴1βˆ’πœ˜βˆ’πœ‡))ξ‚ƒβ„Žβ‰€πΆπœ‡πœ‡β€–π‘’β€–π΅π‘™,𝑠𝑝,πœƒ,𝛾(Ξ©;𝐸(𝐴),𝐸)+β„Žβˆ’(1βˆ’πœ‡)‖𝑒‖𝐡𝑠𝑝,πœƒ,𝛾(Ξ©;𝐸)ξ‚„.(3.12)

Result 1. Let all conditions of Theorem 3.3 hold. Then, for all π‘’βˆˆπ΅π‘™,𝑠𝑝,πœƒ,𝛾(Ξ©;𝐸(𝐴),𝐸) we get β€–π·π›Όπ‘’β€–π΅π‘ π‘ž,πœƒ,𝛾(Ξ©;𝐸(𝐴1βˆ’πœ˜βˆ’πœ‡))β‰€πΆπœ‡β€–π‘’β€–π΅1βˆ’πœ‡π‘™,𝑠𝑝,πœƒ,𝛾(Ξ©;𝐸(𝐴),𝐸)β€–π‘’β€–πœ‡π΅π‘ π‘,πœƒ,𝛾(Ξ©;𝐸).(3.13) Indeed, setting β„Ž=‖𝑒‖𝐡𝑠𝑝,πœƒ,𝛾(Ξ©;𝐸)β‹…β€–π‘’β€–π΅βˆ’1𝑙,𝑠𝑝,πœƒ,𝛾(Ξ©;𝐸(𝐴),𝐸) in (3.13), we obtain (3.11).

Result 2. If 𝑙1=𝑙2=β‹―=𝑙𝑛=π‘š and 𝑠1=𝑠2=β‹―=𝑠𝑛=𝜎, then we obtain that embedding π·π›Όπ΅π‘š,πœŽπ‘,πœƒ,𝛾(Ξ©;𝐸(𝐴),𝐸)βŠ‚π΅πœŽπ‘ž,πœƒ,𝛾(Ξ©;𝐸(𝐴1βˆ’πœ˜)) for 𝜘=|𝛼|/π‘š and the corresponding estimate (3.11). For 𝐸=𝐂,  𝐴=𝐼, we obtain the embedding of weighted Besov spaces 𝐷𝛼𝐡𝑙,𝑠𝑝,πœƒ,𝛾(Ξ©)βŠ‚π΅π‘ π‘ž,πœƒ,𝛾(Ξ©).

4. Application to Vector-Valued Functions

Let 𝑠>0, and consider the space [3,Section  1.18.2]π‘™πœŽπ‘ž=𝑒𝑒;𝑒=π‘–ξ€Ύβˆž1,π‘’π‘–ξ€ΎβˆˆπΆ,β€–π‘’β€–π‘™πœŽπ‘ž=ξƒ©βˆžξ“π‘–=12π‘–π‘žπœŽ||𝑒𝑖||π‘žξƒͺ1/π‘ž<∞.(4.1)

Note that 𝑙0π‘ž=π‘™π‘ž. Let 𝐴 be an infinite matrix defined in π‘™π‘ž such that 𝐷(𝐴)=π‘™πœŽπ‘ž,𝐴=[𝛿𝑖𝑗2πœŽπ‘–],where 𝛿𝑖𝑗=0, when 𝑖≠𝑗,𝛿𝑖𝑗=1, when 𝑖=𝑗,  𝑖, 𝑗=1,2,…,∞. It is clear to see that 𝐴 is positive in π‘™π‘ž. Then, by Theorem 3.3, we obtain the embedding𝐷𝛼𝐡𝑝𝑙,𝑠1,πœƒ,𝛾Ω;π‘™πœŽπ‘ž,π‘™π‘žξ€ΈβŠ‚π΅π‘ π‘2,πœƒ,𝛾Ω;π‘™π‘žπœŽ(1βˆ’πœ˜βˆ’πœ‡),𝜘=π‘›ξ“π‘˜=1π›Όπ‘˜+1/𝑝1βˆ’1/𝑝2π‘™π‘˜,(4.2) and the corresponding estimate (3.11), where 0β‰€πœ‡+𝜈(𝑙)≀1βˆ’πœ˜.

It should be noted that the above embedding has not been obtained with classical methods up to this time.

5. 𝐡-Separable DOE in 𝑅𝑛 with Parameters

Let us consider the differential-operator equation (2.14). Let𝑋𝑠=𝐡𝑠𝑝,πœƒ,𝛾(𝑅𝑛;𝐸),π‘Œπ‘ =𝐡𝑠𝑝,πœƒ(𝑅𝑛;𝐸).(5.1)

Theorem 5.1. Suppose the following conditions hold:
(1)π‘ π‘˜>0, 1<𝑝<∞,1β‰€πœƒ<∞, π‘‘π‘˜>0, π‘˜=1,2,…,𝑛; (2)𝐸 is a Banach space satisfying the 𝐡-multiplier condition;(3)𝐴 is a πœ‘-positive operator in 𝐸 and𝐴𝛼(π‘₯)π΄βˆ’(1βˆ’|π›ΌβˆΆ2𝑙|βˆ’πœ‡)∈𝐿∞(𝑅𝑛||||.;𝐿(𝐸)),0<πœ‡<1βˆ’π›ΌβˆΆ2𝑙(5.2)

Then, for all π‘“βˆˆπ΅π‘ π‘,πœƒ,𝛾(𝑅𝑛;𝐸) and for sufficiently large |πœ†|>0, πœ†βˆˆπ‘†(πœ‘), the equation (2.18) has a unique solution 𝑒(π‘₯) that belongs to space 𝐡2𝑙,𝑠𝑝,πœƒ,𝛾(𝑅𝑛;𝐸(𝐴),𝐸) and the following uniform coercive estimate holds:π‘›ξ“π‘˜=1π‘‘π‘˜β€–β€–π·2π‘™π‘˜π‘˜π‘’β€–β€–π΅π‘ π‘,πœƒ,𝛾(𝑅𝑛;𝐸)+‖𝐴𝑒‖𝐡𝑠𝑝,πœƒ,𝛾(𝑅𝑛;𝐸)≀𝐢‖𝑓‖𝐡𝑠𝑝,πœƒ,𝛾(𝑅𝑛;𝐸).(5.3)

Proof. At first, we will consider the principal part of (2.14), that is, the differential-operator equation 𝐿0ξ€Έ+πœ†π‘’=π‘›ξ“π‘˜=1(βˆ’1)π‘™π‘˜π‘‘π‘˜π·2π‘™π‘˜π‘˜π‘’+(𝐴+πœ†)𝑒=𝑓.(5.4) Then, by applying the Fourier transform to (5.4), we obtain π‘›ξ“π‘˜=1π‘‘π‘˜πœ‰2π‘™π‘˜π‘˜Μ‚π‘’(πœ‰)+(𝐴+πœ†)̂𝑒(πœ‰)=𝑓̂(πœ‰).(5.5)
Since βˆ‘π‘›π‘˜=1π‘‘π‘˜πœ‰2π‘™π‘˜π‘˜β‰₯0 for all πœ‰=(πœ‰1,…,πœ‰π‘›)βˆˆπ‘…π‘›, we can say that βˆ‘πœ”=πœ”(𝑑,πœ†,πœ‰)=πœ†+π‘›π‘˜=1π‘‘π‘˜πœ‰2π‘™π‘˜π‘˜βˆˆπ‘†(πœ‘) for all πœ‰βˆˆπ‘…π‘›, that is, operator 𝐴+πœ” is invertible in 𝐸. Hence, (5.5) implies that the solution of (5.4) can be represented in the form 𝑒(π‘₯)=πΉβˆ’1(𝐴+πœ”)βˆ’1𝑓̂. It is clear to see that the operator-functionπœ‘πœ†,𝑑(πœ‰)=[𝐴+πœ”]βˆ’1 is a multiplier in 𝐡𝑠𝑝,πœƒ,𝛾(𝑅𝑛;𝐸) uniformly with respect to πœ†βˆˆπ‘†(πœ‘). Actually, by definition of the positive operator, for all πœ‰βˆˆπ‘…π‘› andπœ†β‰₯0, we get β€–β€–πœ‘πœ†β€–β€–(πœ‰)𝐿(𝐸)=β€–β€–(𝐴+πœ”)βˆ’1‖‖≀𝑀(1+|πœ”|)βˆ’1≀𝑀0.(5.6) Moreover, since π·π‘˜πœ‘πœ†,𝑑(πœ‰)=2π‘™π‘˜π‘‘π‘˜(𝐴+πœ”)βˆ’2πœ‰2π‘™π‘˜π‘˜βˆ’1, then β€–πœ‰π‘˜π·π‘˜πœ‘πœ†,𝑑‖𝐿(𝐸)≀𝑀. By using this estimate for π›½βˆˆπ‘ˆπ‘›, we get ||πœ‰||π›½β€–β€–π·π›½πœ‰πœ‘πœ†,𝑑‖‖(πœ‰)𝐿(𝐸)≀𝐢.(5.7) In a similar way to Theorem 3.1, we prove that πœ‘π‘˜,πœ†,𝑑(πœ‰)=πœ‰2π‘™π‘˜π‘˜πœ‘πœ†,𝑑,β€‰β€‰π‘˜=1,2,…,𝑛, and πœ‘0,πœ†,𝑑=π΄πœ‘πœ†,𝑑 satisfy the estimates ξ€·||πœ‰||ξ€Έ1+|𝛽|β€–β€–π·π›½πœ‰πœ‘π‘˜,πœ†,𝑑‖‖(πœ‰)𝐿(𝐸)ξ€·||πœ‰||≀𝐢,1+|𝛽|β€–β€–π·π›½πœ‰πœ‘0,πœ†,𝑑‖‖(πœ‰)𝐿(𝐸)≀𝐢.(5.8) Since the space 𝐸 satisfies the multiplier condition with respect to 𝑝, then, in view of estimates (5.7) and (5.8), we obtain that the operator-functions πœ‘πœ†,𝑑,πœ‘π‘˜,πœ†,𝑑,πœ‘0,πœ†,𝑑 are multipliers in 𝐡𝑠𝑝,πœƒ,𝛾(𝑅𝑛;𝐸). Then, we obtain that there exists a unique solution of (5.4) for π‘“βˆˆπ΅π‘ π‘,πœƒ,𝛾(𝑅𝑛;𝐸) and the following estimate holds: π‘›ξ“π‘˜=1π‘‘π‘˜β€–β€–π·2π‘™π‘˜π‘˜π‘’β€–β€–π΅π‘ π‘,πœƒ,𝛾+‖𝐴𝑒‖𝐡𝑠𝑝,πœƒ,𝛾≀𝐢‖𝑓‖𝐡𝑠𝑝,πœƒ,𝛾.(5.9) Consider now the differential operator 𝐺0𝑑 generated by problem (5.4), that is, 𝐷𝐺0𝑑=𝐡2𝑙,𝑠𝑝,πœƒ,𝛾(𝑅𝑛;𝐸(𝐴),𝐸),𝐺0𝑑𝑒=π‘›ξ“π‘˜=1(βˆ’1)π‘™π‘˜π‘‘π‘˜π·2π‘™π‘˜π‘˜π‘’+𝐴𝑒.(5.10) The estimate (5.9) implies that the operator 𝐺0𝑑+𝜘 has a bounded inverse from 𝐡𝑠𝑝,πœƒ,𝛾(𝑅𝑛;𝐸) into 𝐡2𝑙,𝑠𝑝,πœƒ,𝛾(𝑅𝑛;𝐸(𝐴),𝐸) for all 𝜘β‰₯0. Let 𝐺𝑑 denote the differential operator in 𝐡𝑠𝑝,πœƒ,𝛾(𝑅𝑛;𝐸) generated by problem (2.14). In view of (2.18) condition, by virtue of Theorem 3.1, for all π‘’βˆˆπ΅2𝑙,𝑠𝑝,πœƒ,𝛾(𝑅𝑛;𝐸(𝐴),𝐸), we have ‖‖𝐿1𝑒‖‖𝐡𝑠𝑝,πœƒ,𝛾≀||||π›ΌβˆΆ2𝑙<1‖‖𝐴𝛼(𝑑)1βˆ’|π›ΌβˆΆ2𝑙|βˆ’πœ‡π·π›Όπ‘’β€–β€–π΅π‘ π‘,πœƒ,π›Ύξƒ¬β„Žβ‰€πΆπœ‡ξƒ©π‘›ξ“π‘˜=1π‘‘π‘˜β€–β€–π·2π‘™π‘˜π‘˜π‘’β€–β€–π΅π‘ π‘,πœƒ,𝛾+‖𝐴𝑒‖𝐡𝑠𝑝,πœƒ,𝛾ξƒͺ+β„Žβˆ’(1βˆ’πœ‡)‖𝑒‖𝐡𝑠𝑝,πœƒ,𝛾.(5.11) Then, from (5.11), we have ‖‖𝐿1𝑒‖‖𝐡𝑠𝑝,πœƒ,π›Ύξ‚ƒβ„Žβ‰€πΆπœ‡β€–β€–(𝐺0𝑑‖‖+πœ†)𝑒𝐡𝑠𝑝,πœƒ,𝛾+β„Žβˆ’(1βˆ’πœ‡)‖𝑒‖𝐡𝑠𝑝,πœƒ,𝛾.(5.12) Since ‖𝑒‖𝐡𝑠𝑝,πœƒ,𝛾=(1/πœ†)β€–(𝐺0𝑑+πœ†)π‘’βˆ’πΊ0𝑑𝑒‖𝐡𝑠𝑝,πœƒ,𝛾 for all π‘’βˆˆπ΅2𝑙,𝑠𝑝,πœƒ,𝛾(𝑅𝑛;𝐸(𝐴),𝐸), we get ‖𝑒‖𝐡𝑠𝑝,πœƒ,𝛾≀1||πœ†||‖‖𝐺0𝑑𝑒‖‖+πœ†π΅π‘ π‘,πœƒ,𝛾+‖‖𝐺0𝑑𝑒‖‖𝐡𝑠𝑝,πœƒ,𝛾.(5.13) From estimates (5.11)–(5.13), we obtain ‖‖𝐿1π‘’β€–β€–π‘‹π‘ β‰€πΆβ„Žπœ‡β€–β€–ξ€·πΊ0𝑑𝑒‖‖+πœ†π‘‹π‘ +𝐢1||πœ†||βˆ’1β„Žβˆ’(1βˆ’πœ‡)‖‖𝐺0𝑑𝑒‖‖+πœ†π‘‹π‘ .(5.14) Then, by choosing β„Ž and πœ†, such that πΆβ„Žπœ‡<1,𝐢1|πœ†|βˆ’1β„Žβˆ’(1βˆ’πœ‡)<1 from (5.14), we get the following uniform estimate: ‖‖𝐿1𝐺0𝑑+πœ†βˆ’1‖‖𝐿(𝐸)<1.(5.15) Then, using the estimates of (5.9), (5.15) and the perturbation theory of linear operators, we obtain that the operator 𝐺𝑑+πœ† is invertible from 𝐡𝑠𝑝,πœƒ,𝛾(𝑅𝑛,𝐸) into 𝐡2𝑙,𝑠𝑝,πœƒ,𝛾(𝑅𝑛;𝐸(𝐴),𝐸). This implies the estimate (5.3).

Result 1. Let all conditions of Theorem 5.1 hold. Then,(1)forπ‘“βˆˆπ΅π‘ π‘,πœƒ(𝑅𝑛,𝐸),β€‰β€‰πœ†βˆˆπ‘†(πœ‘), (2.16) has a unique solution π‘’βˆˆπ΅[2𝑙],𝑠𝑝,πœƒ,𝛾(𝑅𝑛;𝐸(𝐴),𝐸) and π‘›ξ“π‘˜=1π‘‘π‘˜β€–β€–π·[2π‘™π‘˜]π‘˜π‘’β€–β€–π΅π‘ π‘,πœƒ(𝑅𝑛;𝐸)+‖𝐴𝑒‖𝐡𝑠𝑝,πœƒ(𝑅𝑛;𝐸)≀𝐢‖𝑓‖𝐡𝑠𝑝,πœƒ(𝑅𝑛;𝐸),(5.16)(2)the operator 𝑄𝑑 has a resolvent (𝑄𝑑+πœ†)βˆ’1 for |argπœ†|β‰€πœ‘, and the following uniform estimate holds: ||||π›ΌβˆΆ2𝑙≀1||πœ†||𝛼(𝑑)1βˆ’|π›ΌβˆΆ2𝑙|‖‖𝐷[𝛼]𝑄𝑑+πœ†βˆ’1‖‖𝐿(π‘Œπ‘ )+‖‖𝐴𝑄𝑑+πœ†βˆ’1‖‖𝐿(π‘Œπ‘ )≀𝐢.(5.17)

Remark 5.2. Result 1 implies that operator 𝑄𝑑 is uniformly positive in 𝐡𝑠𝑝,πœƒ(𝑅𝑛;𝐸). Then, by virtue of [3, Section  1.14.5], the operator 𝑄𝑑 is a generator of an analytic semigroup in 𝐡𝑠𝑝,πœƒ(𝑅𝑛;𝐸).

6. Cauchy Problem for Degenerate Parabolic DOE with Parameters

Consider the Cauchy problem for the degenerate parabolic DOEπœ•π‘’+πœ•π‘¦π‘›ξ“π‘˜=1(βˆ’1)π‘™π‘˜π‘‘π‘˜π·[2π‘™π‘˜]π‘˜ξ“π‘’+𝐴𝑒+||||π›ΌβˆΆ2𝑙<1𝛼(𝑑)𝐴𝛼(π‘₯)𝐷[𝛼]𝑒=𝑓(𝑦,π‘₯),𝑒(0,π‘₯)=0,(6.1) where 𝐴 and 𝐴𝛼(π‘₯) are linear operators in a Banach space in 𝐸. Let 𝐹=𝐡𝑠𝑝,πœƒ(𝑅𝑛;𝐸).

Theorem 6.1. Assume all conditions of Theorem 5.1 hold for πœ‘βˆˆ(πœ‹/2,πœ‹) and 𝑠>0. Then, for π‘“βˆˆπ΅π‘ π‘,πœƒ(𝑅+;𝐹), (6.1) has a unique solution π‘’βˆˆπ΅1𝑝,π‘ž(𝑅+;𝐷(𝑄𝑑),𝐹) satisfying β€–β€–β€–πœ•π‘’β€–β€–β€–πœ•π‘¦π΅π‘ π‘,πœƒ(𝑅+;𝐹)+π‘›ξ“π‘˜=1β€–β€–π‘‘π‘˜π·[2π‘™π‘˜]π‘˜π‘’β€–β€–π΅π‘ π‘,πœƒ(𝑅+;𝐹)+‖𝐴𝑒‖𝐡𝑠𝑝,πœƒ(𝑅+;𝐹)≀𝐢‖𝑓‖𝐡𝑠𝑝,πœƒ(𝑅+;𝐹).(6.2)

Proof. The problem (6.1) can be expressed as 𝑑𝑒𝑑𝑦+𝑄𝑑𝑒(𝑦)=𝑓(𝑦),𝑒(0)=0,π‘¦βˆˆ(0,∞).(6.3)
Result 1 implies the uniform positivity of 𝐺𝑑. So, by [6, Application D], we obtain that, for π‘“βˆˆπ΅π‘ π‘,πœƒ(𝑅+;𝐹), the Cauchy problem (6.3) has a unique solution π‘’βˆˆπ΅1+𝑠𝑝,πœƒ(𝑅+;𝐷(𝑄𝑑),𝐹) satisfying ‖‖𝐷𝑑𝑒‖‖𝐡𝑠𝑝,πœƒ(𝑅+;𝐹)+‖‖𝑄𝑑𝑒‖‖𝐡𝑠𝑝,πœƒ(𝑅+;𝐹)≀𝐢‖𝑓‖𝐡𝑠𝑝,π‘ž(𝑅+;𝐹).(6.4)
In view of Result 1, the operator 𝑄𝑑 is uniform separable in 𝐹; therefore, the estimate (6.4) implies (6.2).

7. Infinite Systems of the Quasielliptic Equation

Consider the following infinity systems:(𝐿+πœ†)π‘’π‘š=π‘›ξ“π‘˜=1(βˆ’1)π‘™π‘˜π‘‘π‘˜π·2π‘™π‘˜π‘˜π‘’π‘šξ€·π‘‘(π‘₯)+π‘šξ€Έπ‘’+πœ†π‘šξ“(π‘₯)+||||π›ΌβˆΆπ‘™<1𝛼(𝑑)βˆžξ“π‘˜=1π‘‘π›Όπ‘˜π‘š(π‘₯)π·π›Όπ‘’π‘š=π‘“π‘š(π‘₯),π‘₯βˆˆπ‘…π‘›,π‘š=1,2,…,∞.(7.1)

Let 𝑑𝑑(π‘₯)=π‘šξ€Ύ(π‘₯),π‘‘π‘šξ€½π‘’>0,𝑒=π‘šξ€Ύξ€½π‘‘,𝑑𝑒=π‘šπ‘’π‘šξ€Ύ,𝑑𝑄(π‘₯)=π‘šξ€Ύ(π‘₯),π‘‘π‘šξ€½π‘’>0,𝑒=π‘šξ€Ύξ€½π‘‘,𝑄𝑒=π‘šπ‘’π‘šξ€Ύ,π‘™π‘žξƒ―(𝑄)==π‘’βˆΆπ‘’βˆˆπ‘™π‘ž,β€–π‘’β€–π‘™π‘ž(𝑄)=β€–π‘„π‘’β€–π‘™π‘ž=ξ‚΅βˆžβˆ‘π‘š=1||π‘‘π‘šπ‘’π‘š||π‘žξ‚Ά1/π‘žξƒ°,<∞(7.2) and let π‘‘π‘˜ be positive parameters. Let 𝑂𝑑 denote the differential operator in 𝐡=𝐿(𝐡𝑠𝑝,πœƒ,𝛾(𝑅𝑛;π‘™π‘ž)) generated by problem (7.1).

Theorem 7.1. Let π‘Žπ›ΌβˆˆπΆπ‘(𝑅𝑛), π‘‘π‘šβˆˆπΆπ‘(𝑅𝑛),β€‰β€‰π‘‘π›Όπ‘˜π‘šβˆˆπΏβˆž(𝑅𝑛) such that max𝛼supπ‘šβˆžβˆ‘π‘˜=1π‘‘π›Όπ‘˜π‘š(π‘₯)π‘‘π‘˜βˆ’(1βˆ’|π›ΌβˆΆπ‘™|βˆ’πœ‡)<𝑀 for all π‘₯βˆˆπ‘…π‘›,  𝑝,β€‰π‘žβˆˆ(1,∞),β€‰β€‰πœƒβˆˆ[1,∞] and 0<πœ‡<1βˆ’|π›ΌβˆΆπ‘™|.
Then,(a)for all 𝑓(π‘₯)={π‘“π‘š(π‘₯)}∞1βˆˆπ΅π‘ π‘,πœƒ,𝛾(𝑅𝑛;π‘™π‘ž), for |argπœ†|β‰€πœ‘ and for sufficiently large |πœ†|, the problem (7.1) has a unique solution 𝑒={π‘’π‘š(π‘₯)}∞1 that belongs to space 𝐡𝑠+2𝑙𝑝,πœƒ,𝛾(𝑅𝑛,π‘™π‘ž(𝑑),π‘™π‘ž) and the uniform coercive estimate holds ||||π›ΌβˆΆ2𝑙≀1‖𝐷𝛼𝑒‖𝐡𝑠𝑝,πœƒ,𝛾(𝑅𝑛;π‘™π‘ž)+‖𝑄𝑒‖𝐡𝑠𝑝,πœƒ,𝛾(𝑅𝑛;π‘™π‘ž)≀𝐢‖𝑓‖𝐡𝑠𝑝,πœƒ,𝛾(𝑅𝑛;π‘™π‘ž),(7.3)(b)for |argπœ†|β‰€πœ‘ and for sufficiently large |πœ†|, there exists a resolvent (𝑂𝑑+πœ†)βˆ’1 of operator 𝑂𝑑 and ||||π›ΌβˆΆ2𝑙≀1||πœ†||𝛼(𝑑)1βˆ’|π›ΌβˆΆπ‘™|‖‖𝐷𝛼𝑂𝑑+πœ†βˆ’1‖‖𝐡+‖‖𝑄𝑂𝑑+πœ†βˆ’1‖‖𝐡≀𝑀.(7.4)

Proof. Really, let 𝐸=π‘™π‘ž,  𝐴(π‘₯), and let 𝐴𝛼(π‘₯) be infinite matrices such that 𝑑𝐴=π‘š(π‘₯)π›Ώπ‘˜π‘šξ€»,𝐴𝛼𝑑(π‘₯)=π›Όπ‘˜π‘šξ€»(π‘₯),π‘˜,π‘š=1,2,…,∞.(7.5) It is clear that the operator 𝐴 is positive in π‘™π‘ž. Therefore, from Theorem 6.1, we obtain that the problem (7.1) has a unique solution π‘’βˆˆπ΅π‘ +2𝑙𝑝,πœƒ,𝛾(𝑅𝑛;π‘™π‘ž(𝑄),π‘™π‘ž) for all π‘“βˆˆπ΅π‘ π‘,πœƒ,𝛾(𝑅𝑛;π‘™π‘ž), |argπœ†|β‰€πœ‘, sufficiently large |πœ†| and estimate (7.3) holds. From estimate (7.3), we obtain (7.4).

8. Cauchy Problem for Infinite Systems of Parabolic Equations

Consider the following infinity systems of parabolic Cauchy problemπœ•π‘’π‘š+πœ•π‘¦π‘›ξ“π‘˜=1(βˆ’1)π‘™π‘˜π‘‘π‘˜π·2π‘™π‘˜π‘˜π‘’π‘š+π‘‘π‘š(π‘₯)π‘’π‘š+||||π›ΌβˆΆπ‘™<1𝛼(𝑑)π‘‘π›Όπ‘˜π‘š(π‘₯)π·π›Όπ‘’π‘š=π‘“π‘š(𝑦,π‘₯),π‘¦βˆˆπ‘…+,π‘₯βˆˆπ‘…π‘›,π‘’π‘š(0,π‘₯)=0,π‘š=1,2,…,∞.(8.1)

Theorem 8.1. Let all conditions of Theorem 7.1 hold. Then, the parabolic systems (8.1) for sufficiently large 𝜘>0 have a unique solution π‘’βˆˆπ΅1,𝑠+2𝑙𝑝,πœƒ,𝛾(𝑅𝑛;π‘™π‘ž(𝑄),π‘™π‘ž), and the following estimate holds: β€–β€–β€–πœ•π‘’β€–β€–β€–πœ•π‘¦π΅π‘ π‘,πœƒ(𝑅+𝑛+1;π‘™π‘ž)+π‘›ξ“π‘˜=1β€–β€–π‘‘π‘˜π·[2π‘™π‘˜]π‘˜π‘’β€–β€–π΅π‘ π‘,πœƒ(𝑅+𝑛+1;π‘™π‘ž)+‖𝑄𝑒‖𝐡𝑠𝑝,πœƒ(𝑅+𝑛+1;π‘™π‘ž)≀𝐢‖𝑓‖𝐡𝑠𝑝,πœƒ(𝑅+𝑛+1;π‘™π‘ž).(8.2)

Proof. Really, let 𝐸=π‘™π‘ž,  and let 𝐴 and 𝐴𝛼(π‘₯) be infinite matrices, such that 𝑑𝐴=π‘š(π‘₯)π›Ώπ‘˜π‘šξ€»,𝐴𝛼𝑑(π‘₯)=π›Όπ‘˜π‘šξ€»(π‘₯),π‘˜,π‘š=1,2,β€¦βˆž.(8.3)
Then, the problem (8.1) can be expressed in the form (6.3), where 𝑑𝐴=π‘š(π‘₯)π›Ώπ‘˜π‘šξ€»,𝐴𝛼𝑑(π‘₯)=π›Όπ‘˜π‘šξ€»(π‘₯),π‘˜,π‘š=1,2,β€¦βˆž.(8.4) Then, by virtue of Theorems 6.1 and 7.1, we obtain the assertion.

Acknowledgment

The author would like to express gratitude to proofreader Amy Spangler for her useful advice while preparing this paper.

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