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 A comprehensive introduction to the theory of embedding of function spaces and historical references may also be found in [4, 5].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 , , 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 A comprehensive introduction to the differential-operator equations and historical references may be found in [4, 5].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 and . Let and be Banach spaces such that is continuously and densely embedded in . In the present paper, the weighted Banach-valued Besov space is to be introduced. The smoothest interpolation class between , (i.e., to find the possible small for ) is found such that the appropriate mixed differential operators are bounded from to . By applying these results, the maximal -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 . 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 -regularity is obtained. Then, in Section 6, by using these results, the maximal -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 be a Banach space and a nonnegative measurable weighted function defined on a domain . Let denote the space of strongly measurable -valued functions that are defined on with the norm

Let , and be the standard unit vectors in . Let (see [1, Section  16])

Let

Let be a -valued function space such that

Let be positive integers, nonnegative integers, positive numbers, and ,  . Let denote the Fourier transform. The Banach-valued Besov space is defined as

For and , we obtain a scalar-valued anisotropic Besov space [1, Section  18].

Let be the set of complex numbers and

A linear operator is said to be a -positive in a Banach space with bound if is dense on and with ,, where is the identity operator in and is the space of bounded linear operators in .

It is known [3, Section  1.15.1] that there exist the fractional powers of the positive operator . Let denote the space with a graph norm defined as

The operator is said to be -positive in uniformly with respect to with bound if is independent of , is dense in , and for all , where does not depend on and .

Let , where are positive integers. Let denote a -valued weighted Sobolev-Besov space of functions that have generalized derivatives , with the norm

Suppose is continuously and densely embedded into . Let denote the space with the norm

Let , where are parameters. We define the following parameterized norm in :

Let be a positive integer. denotes the spaces of -valued bounded and -times continuously differentiable functions on . For two sequences and of positive numbers, the expression means that there exist positive numbers and such that

Let ,  and be two Banach spaces. Let denote the Fourier transformation and let be some parameter. We say that the function dependent of is a uniform collection of multipliers if there exists a positive constant independent of such that for all . The set of all multipliers from to will be denoted by . For , it will be denoted by . 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 spaces, Fourier multipliers have been investigated in several studies like [27, 28]. Operator-valued Fourier multipliers in Banach-valued spaces studied, for example, in [4, 6, 25, 27–33].

Let be multi-indexes and

Definition 2.1. A Banach space satisfies a -multiplier condition with respect to (or with respect to for ), and the weight , when , , and , if the estimate , implies .
It is well known (e.g., see [32]) that any Hilbert space satisfies the -multiplier condition. There are, however, Banach spaces which are not Hilbert spaces but satisfy the -multiplier condition (see [7, 30]). However, additional conditions are needed for operator-valued multipliers in spaces, for example, UMD spaces (e.g., see [25, 33]). Let be nonnegative and positive integers:
Consider the following differential-operator equation: where , are linear operators in a Banach space ,   are complex-valued functions and are some parameters For , we obtain the elliptic class of DOE.
The function belonging to and satisfying (2.14) a.e. on is said to be a solution of (2.14) on .

Definition 2.2. The problem (2.14) is said to be uniform weighted -separable (or weighted -separable) if, for all , the problem (2.14) has a unique solution and the following estimate holds:
Consider the following degenerate DOE: where , are linear operators in a Banach space , are complex-valued functions, are some parameters and

Remark 2.3. Under the substitution ,   are mapped isomorphically onto the spaces ,  , respectively, where Moreover, under the substitution (2.18), the degenerate problem (2.16) is mapped to the undegenerate problem (2.14).

3. Embedding Theorems

Let

Theorem 3.1. Suppose the following conditions hold:(1) is a Banach space satisfying the -multiplier condition with respect to , , ;(2),   , , , ;(3) are positive, nonnegative integers such that and ;(4) is a -positive operator in .
Then, the embedding is continuous, and there exists a constant , depending only on such that for all and .

Proof. Denoting by , it is clear that Similarly, from the definition of , we have Thus, proving the inequality (3.2) is equivalent to proving So, the inequality (3.2) will be followed if we prove the following inequality: for a suitable and for all , where
Let us express the left-hand side of (3.6) as (Since is a positive operator in and , 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 is a multiplier in , which is uniform with respect to and . Since satisfies the multiplier condition with respect to and , it suffices to show the following estimate: for all , and . In a way similar to [18,Lemma  3.1], we obtain that for all . This shows that the inequality (3.9) is satisfied for . We next consider (3.9) for , where and for . By using the condition and well-known inequality , and by reasoning according to [18, Theorem  3.1], we have
Repeating the above process, we obtain the estimate (3.9). Thus, the operator-function is a uniform collection of multiplier, that is, . 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 .

Condition 1. Let be a -positive operator in Banach spaces satisfying the -multiplier condition. Let a region be such that there exists a bounded linear extension operatorfrom to , for ,  .

Remark 3.2. If is a region satisfying a strong -horn condition (see [23, Section  18]) ,, then there exists a bounded linear extension operator from to .

Theorem 3.3. Suppose all conditions of Theorem 3.1 and Condition 1 are satisfied. Then, the embedding is continuous and there exists a constant depending only on such that for all and .

Proof. It suffices to prove the estimate (3.11). Let be a bounded linear extension operator from to and also from to . Let be a restriction operator from to . Then, for any , we have

Result 1. Let all conditions of Theorem 3.3 hold. Then, for all we get Indeed, setting in (3.13), we obtain (3.11).

Result 2. If and , then we obtain that embedding for and the corresponding estimate (3.11). For ,  , we obtain the embedding of weighted Besov spaces .

4. Application to Vector-Valued Functions

Let , and consider the space [3,Section  1.18.2]

Note that . Let be an infinite matrix defined in such that where , when , when ,  , . It is clear to see that is positive in . Then, by Theorem 3.3, we obtain the embedding and the corresponding estimate (3.11), where .

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

5. -Separable DOE in with Parameters

Let us consider the differential-operator equation (2.14). Let

Theorem 5.1. Suppose the following conditions hold:
(1), , , ; (2) is a Banach space satisfying the -multiplier condition;(3) is a -positive operator in and

Then, for all and for sufficiently large , , the equation (2.18) has a unique solution that belongs to space and the following uniform coercive estimate holds:

Proof. At first, we will consider the principal part of (2.14), that is, the differential-operator equation Then, by applying the Fourier transform to (5.4), we obtain
Since for all , we can say that for all , that is, operator is invertible in . Hence, (5.5) implies that the solution of (5.4) can be represented in the form . It is clear to see that the operator-function is a multiplier in uniformly with respect to . Actually, by definition of the positive operator, for all and, we get Moreover, since , then . By using this estimate for , we get In a similar way to Theorem 3.1, we prove that ,  , and satisfy the estimates Since the space satisfies the multiplier condition with respect to , then, in view of estimates (5.7) and (5.8), we obtain that the operator-functions are multipliers in . Then, we obtain that there exists a unique solution of (5.4) for and the following estimate holds: Consider now the differential operator generated by problem (5.4), that is, The estimate (5.9) implies that the operator has a bounded inverse from into for all . Let denote the differential operator in generated by problem (2.14). In view of (2.18) condition, by virtue of Theorem 3.1, for all , we have Then, from (5.11), we have Since for all , we get From estimates (5.11)–(5.13), we obtain Then, by choosing and , such that , from (5.14), we get the following uniform estimate: Then, using the estimates of (5.9), (5.15) and the perturbation theory of linear operators, we obtain that the operator is invertible from into . This implies the estimate (5.3).