Embedding Operators in Vector-Valued Weighted Besov Spaces and Applications

The embedding theorems in weighted Besov-Lions type spaces B p,q,γ Ω;E0, E in which E0, E are two Banach spaces and E0 ⊂ E are studied. The most regular class of interpolation space Eα between E0 and E is found such that the mixed differential operator D is bounded from B l,s p,q,γ Ω;E0, E to B p,q,γ Ω;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.


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spaces have been elaborated in 1-3 .A comprehensive introduction to the theory of embedding of function spaces and historical references may also be found in 4, 5 .Embedding theorems in abstract function spaces have been studied in 2, 6-18 .The anisotropic Sobolev spaces W l p Ω; H 0 , H , Ω ⊂ R n , 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DOEs have also been refined in 7, 13-18, 22-26 .A comprehensive introduction to the differential-operator equations and historical references may be found in 4, 5 .In these works, Hilbert-valued function spaces essentially have been considered.Let l l 1 , l 2 , . . ., l n and s s 1 , s 2 , . . ., s n .Let E 0 and E be Banach spaces such that E 0 is continuously and densely embedded in E. In the present paper, the weighted

Notations and Definitions
Let E be a Banach space and γ γ x a nonnegative measurable weighted function defined on a domain Ω ⊂ R n .Let L p,γ Ω; E denote the space of strongly measurable E-valued functions that are defined on Ω with the norm

2.1
Let h ∈ R, m ∈ N, and e i , i 1, 2, . . ., n be the standard unit vectors in R n .Let see 1, Section 16

2.4
Let m i be positive integers, k i nonnegative integers, s i positive numbers, and Let F denote the Fourier transform.The Banach-valued Besov space B s p,θ,γ Ω; E is defined as
Let C be the set of complex numbers and A linear operator A is said to be a ϕ-positive in a Banach space E with bound M > 0 if D A is dense on E and A λI −1  L E ≤ M 1 |λ| −1 with λ ∈ S ϕ , ϕ ∈ 0 π , 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 A θ of the positive operator A. Let E A θ denote the space D A θ with a graph norm defined as

2.7
The operator A t is said to be ϕ-positive in E uniformly with respect to t with bound Let m be a positive integer.C m Ω; E denotes the spaces of E-valued bounded and m-times continuously differentiable functions on Ω.For two sequences {a j } ∞ 1 and {b j } ∞ 1 of positive numbers, the expression a j ∼ b j means that there exist positive numbers C 1 and C 2 such that

2.11
Let E 1 , and E 2 be two Banach spaces.Let F denote the Fourier transformation and let h be some parameter.We say that the function Ψ h dependent of h is a uniform collection of multipliers if there exists a positive constant M independent of h such that p,θ,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 L p spaces, Fourier multipliers have been investigated in several studies like 27, 28 .Operator-valued Fourier multipliers in Banach-valued L p spaces studied, for example, in 4, 6, 25, 27-33 .
Let β β 1 , β 2 , . . ., β n be multi-indexes and Definition 2.1.A Banach space E satisfies a B-multiplier condition with respect to p, q, θ, s or with respect to p, θ, s for p q , and the weight γ, when It is well known e.g., see 32 that any Hilbert space satisfies the B-multiplier condition.There are, however, Banach spaces which are not Hilbert spaces but satisfy the B-multiplier condition see 7, 30 .However, additional conditions are needed for operatorvalued multipliers in L p spaces, for example, UMD spaces e.g., see 25, 33 .Let α 1 , α 2 , . . ., α n be nonnegative and l 1 , l 2 , . . ., l n positive integers:

2.13
Consider the following differential-operator equation: Consider the following degenerate DOE: where A x , A α x are linear operators in a Banach space E, a k are complex-valued functions, t k are some parameters and Moreover, under the substitution 2.18 , the degenerate problem 2.16 is mapped to the undegenerate problem 2.14 .

Embedding Theorems
Let

3.1
Theorem 3.1.Suppose the following conditions hold: 1 E is a Banach space satisfying the B-multiplier condition with respect to p, q, s; Then, the embedding D α B l,s p,θ,γ R n ; E A , E ⊂ B s q,θ,γ R n ; E A 1−κ−μ is continuous, and there exists a constant C μ > 0, depending only on μ such that Proof.Denoting Fu by u, it is clear that Similarly, from the definition of B l,s p,θ,γ R n ; E A , E , we have

3.4
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 C μ > 0 and for all u ∈ B l,s p,θ,γ R n ; E A , E , where Let us express the left-hand side of 3.6 as

3.8
Since A is a positive operator in E and −ψ t, ξ ∈ S ϕ , 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 p,θ,γ E , which is uniform with respect to h and t.Since E satisfies the multiplier condition with respect to p and q, it suffices to show the following estimate: for all β ∈ U n , ξ ∈ R n /{ξ k 0} and η 1/p − 1/q.In a way similar to 18, Lemma 3.1 , we obtain that |ξ| −η Ψ t ξ L E ≤ M μ for all ξ ∈ R n .This shows that the inequality 3.9 is satisfied for β 0, . . ., 0 .We next consider 3.9 for β β 1 , . . ., β n , where β k 1 and β 0 for j / k.By using the condition κ ν l ≤ 1 and well-known inequality k , y k ≥ 0 and by reasoning according to 18, Theorem 3.1 , we have

3.10
Repeating the above process, we obtain the estimate 3.9 .Thus, the operator-function Ψ t,h,μ ξ is a uniform collection of multiplier, that is, Ψ t,h,μ ∈ Φ h ⊂ M q,θ,γ p,θ,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 B l,s p,θ,γ Ω; E A , E .

Journal of Function Spaces and
Proof.It suffices to prove the estimate 3.11 .Let P be a bounded linear extension operator from B s q,θ,γ Ω; E to B s q,θ,γ R n ; E and also from B l,s p,θ,γ Ω; E A , E to B l,s p,θ,γ R n ; E A , E .Let P Ω be a restriction operator from R n to Ω.Then, for any u ∈ B l,s p,θ Ω; E A , E , we have

3.12
Result 1.Let all conditions of Theorem 3.3 hold.Then, for all u ∈ B l,s p,θ,γ Ω; E A , E we get p,θ,γ Ω;E A ,E in 3.13 , we obtain 3.11 .

Application to Vector-Valued Functions
Let s > 0, and consider the space 3, Section 1.18.2 Note that l 0 q l q .Let A be an infinite matrix defined in l q such that D A l σ q , A δ ij 2 σi , where δ ij 0, when i / j, δ ij 1, when i j, i, j 1, 2, . . ., ∞.It is clear to see that A is positive in l q .Then, by Theorem 3.3, we obtain the embedding and the corresponding estimate 3.11 , where 0 ≤ μ ν l ≤ 1 − κ.
It should be noted that the above embedding has not been obtained with classical methods up to this time.

B-Separable DOE in R n with Parameters
Let us consider the differential-operator equation 2.14 .Let

5.1
Theorem 5.1.Suppose the following conditions hold: 2 E is a Banach space satisfying the B-multiplier condition; 3 A is a ϕ-positive operator in E and Then, for all f ∈ B s p,θ,γ R n ; E and for sufficiently large |λ| > 0, λ ∈ S ϕ , the equation 2.18 has a unique solution u x that belongs to space B 2l,s p,θ,γ R n ; E A , E 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 ≥ 0 for all ξ ξ 1 , . . ., ξ n ∈ R n , we can say that ω ω t, λ, ξ λ n k 1 t k ξ 2l k k ∈ S ϕ for all ξ ∈ R n , that is, operator A ω is invertible in E. Hence, 5.5 implies that the solution of 5.4 can be represented in the form u x F −1 A ω −1 f .It is clear to see that the operator-function ϕ λ,t ξ A ω −1 is a multiplier in B s p,θ,γ R n ; E uniformly with respect to λ ∈ S ϕ .Actually, by definition of the positive operator, for all ξ ∈ R n and λ ≥ 0, we get By using this estimate for β ∈ U n , we get In a similar way to Theorem 3.1, we prove that ϕ k,λ,t ξ ξ 2l k k ϕ λ,t , k 1, 2, . . ., n, and ϕ 0,λ,t Aϕ λ,t satisfy the estimates 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 ϕ λ,t , ϕ k,λ,t , ϕ 0,λ,t are multipliers in B s p,θ,γ R n ; E .Then, we obtain that there exists a unique solution of 5.4 for f ∈ B s p,θ,γ R n ; E and the following estimate holds:

5.9
Consider now the differential operator G 0t generated by problem 5.4 , that is,

5.11
Then, from 5.11 , we have

5.15
Then, using the estimates of 5.9 , 5. In view of Result 1, the operator Q t is uniform separable in F; therefore, the estimate 6.4 implies 6.2 .

1 d
αkm x D α u m f m x , x ∈ R n , m 1, 2, . . ., ∞.7.1 where M does not depend on t and λ.

Applications
Condition 1.Let A be a ϕ-positive operator in Banach spaces E satisfying the B-multiplier condition.Let a region Ω ⊂ R n be such that there exists a bounded linear extension operator from B l,s p,θ,γ Ω; E A , E to B l,s p,θ,γ R n ; E A , E , for 1 ≤ p, θ ≤ ∞.
generated by problem 2.14 .In view of 2.18 condition, by virtue of Theorem 3.1, for all u ∈ B 2l,s p,θ,γ R n ; E A , E , we have Au.5.10The estimate 5.9 implies that the operator G 0t κ has a bounded inverse from B s p,θ,γ R n ; E into B 2l,s p,θ,γ R n ; E A , E for all κ ≥ 0. Let G t denote the differential operator in B s p,θ,γ R n ; E

Cauchy Problem for Degenerate Parabolic DOE with Parameters
Remark 5.2.Result 1 implies that operator Q t is uniformly positive in B s p,θ R n ; E .Then, by virtue of 3, Section 1.14.5 , the operator Q t is a generator of an analytic semigroup in B s p,θ R n ; E .and A α x are linear operators in a Banach space in E. Let F B s p,θ R n ; E .Assume all conditions of Theorem 5.1 hold for ϕ ∈ π and s > 0.Then, for f ∈ B s p,θ R ; F , 6.1 has a unique solution u ∈ B 1 p,q R ; D Q t , F satisfying Result 1 implies the uniform positivity of G t .So, by 6, Application D , we obtain that, for f ∈ B s p,θ R ; F , the Cauchy problem 6.3 has a unique solution u ∈ B 1 s p,θ R ; D Q t , F satisfying n , E , λ ∈ S ϕ , 2.16 has a unique solution u ∈ B |α:2l|≤1 α t |λ| 1−|α:2l| D α Q t λ −1 6.