# Embedding Operators in Vector-Valued Weighted Besov Spaces and Applications

Authors: Veli Shakhmurov
Journal: Journal of Function Spaces and Applications
 (2012)

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## Abstract

The embedding theorems in weighted Besov-Lions type spacesBp,q,γl,s (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 Dα is bounded from Bp,q,γl,s (Ω;E0,E) to Bp,q,γs (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.


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## Body

## 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 A comprehensive introduction to the theory of embedding of function spaces and historical references may also be found in [4, 5].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 Wpl(Ω;H0,H), Ω⊂Rn, 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 A comprehensive introduction to the differential-operator equations and historical references may be found in [4, 5].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.Letl=(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 Bp,θ,γl,s(Ω;E0,E) is to be introduced. The smoothest interpolation class Eα between E0, E (i.e., to find the possible small σα for Eα=(E0,E)σα,p) is found such that the appropriate mixed differential operators Dα are bounded from Bp,θ,γl,s(Ω;E0,E) to Bp,θ,γs(Ω;Eα). By applying these results, the maximal B-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 Bp,θ,γl,s(Ω;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 B-regularity is obtained. Then, in Section 6, by using these results, the maximal B-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

LetE be a Banach space and γ=γ(x) a nonnegative measurable weighted function defined on a domain Ω⊂Rn. Let Lp,γ(Ω;E) denote the space of strongly measurable E-valued functions that are defined on Ω with the norm
(2.1)‖f‖Lp,γ(Ω;E)=(∫‖f(x)‖Epγ(x)dx)1/p,1≤p<∞,‖f‖L∞,γ(Ω;E)=esssup⁡x∈Ω[‖f(x)‖Eγ(x)].Leth∈R,m∈N, and ei,i=1,2,…,n be the standard unit vectors in Rn. Let (see [1, Section  16])(2.2)Δi(h)f(x)=f(x+hei)-f(x),…,Δim(h)f(x)=Δi(h)[Δim-1(h)f(x)]=∑k=0m(-1)m+kCmkf(x+khei).Let(2.3)Δim(Ω,h)={Δim(h),for[x,x+myei]⊂Ω,0,for[x,x+myei]⊂RnΩ}.LetLθ*(E) be a E-valued function space such that(2.4)‖u‖Lθ*(E)=(∫0∞‖u(t)‖Eθdtt)1/θ<∞.Letmi be positive integers, ki nonnegative integers, si positive numbers, and mi>si-ki>0,i=1,2,…,n,s=(s1,s2,…,sn),  1≤p≤∞,1≤θ≤∞,0<y0<∞. Let F denote the Fourier transform. The Banach-valued Besov space Bp,θ,γs(Ω;E) is defined as
(2.5)Bp,θ,γs(Ω;E)={∫0h0h-[(si-ki)q+1]‖Δimi(h,Ω)Dikif‖Lp,γ(Ω;E)θdy(∫0h0h-[(si-ki)q+1]‖Δimi(h,Ω)Dikif‖Lp,γ(Ω;E)θdy)1θf:f∈Lp(Ω;E),‖f‖Bp,θs(Ω;E)=‖f‖Lp,γ(Ω;E)+∑i=1n(∫0h0h-[(si-ki)q+1]‖Δimi(h,Ω)Dikif‖Lp,γ(Ω;E)θdy)1/θ<∞(∫0h0h-[(si-ki)q+1]‖Δimi(h,Ω)Dikif‖Lp,γ(Ω;E)θdy)1θ},1≤θ<∞,‖f‖Bp,θ,γs(Ω;E)=∑i=1nsup⁡0<h<h0‖Δimi(h,Ω)Dikif‖Lp,γ(Ω;E)hsi-kiforθ=∞.ForE=R and γ(x)≡1, we obtain a scalar-valued anisotropic Besov space Bp,θ,γs(Ω) [1, Section  18].LetC be the set of complex numbers and(2.6)Sφ={λ;λ∈C,|argλ|≤φ}∪{0},0≤φ<π.A linear operatorA 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|)-1with λ∈Sφ,φ∈[0π), 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 Aθof the positive operator A. Let E(Aθ) denote the space D(Aθ) with a graph norm defined as(2.7)‖u‖E(Aθ)=(‖u‖p+‖Aθu‖p)1/p,1≤p<∞,-∞<θ<∞.The operatorA(t) is said to be φ-positive in E uniformly with respect to t with bound M>0 if D(A(t)) is independent of t, D(A(t)) is dense in E, and ∥(A(t)+λI)-1∥≤M(1+|λ|)-1 for all λ∈Sφ, 0≤φ<π,where M does not depend on t and λ.Letl=(l1,l2,…,ln),s=(s1,s2,…,sn), where lk are positive integers. Let Bp,θ,γl,s(Ω;E) denote a E-valued weighted Sobolev-Besov space of functions u∈Bp,θ,γs(Ω;E) that have generalized derivatives Dklku=(∂lk/∂xklk)u∈Bp,θ,γs(Ω;E),k=1,2,…,n, with the norm(2.8)‖u‖Bp,θ,γl,s(Ω;E)=‖u‖Bp,θ,γs(Ω;E)+∑k=1n‖Dklku‖Bp,θ,γs(Ω;E)<∞.SupposeE0 is continuously and densely embedded into E. Let Bp,θ,γl,s(Ω;E0,E) denote the space with the norm(2.9)‖u‖Bp,θ,γl,s=‖u‖Bp,θ,γl,s(Ω;E0,E)=‖u‖Bp,θ,γs(Ω;E0)+∑k=1n‖Dklku‖Bp,θ,γs(Ω;E)<∞.Lett=(t1,t2,…,tn), where tk are parameters. We define the following parameterized norm in Bp,θl,s(Ω;E0,E):(2.10)‖u‖Bp,θ,γ,tl,s(Ω;E0,E)=‖u‖Bp,θ,γs(Ω;E0)+∑k=1n‖tkDklku‖Bp,θ,γs(Ω;E)<∞.Letm 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 {aj}1∞ and {bj}1∞ of positive numbers, the expression aj~bj means that there exist positive numbers C1 and C2 such that(2.11)C1aj≤bj≤C2aj.LetE1,  and E2 be two Banach spaces. Let F denote the Fourier transformation and let h be some parameter. We say that the function Ψh dependent of h is a uniform collection of multipliers if there exists a positive constant M independent of h such that ∥F-1ΨhFu∥Bq,θ,γs(Rn;E2)≤M∥u∥Bp,θ,γs(Rn;E1)for all u∈Bp,θ,γs(Rn;E1). The set of all multipliers from Bp,θ,γs(Rn;E1) to Bq,θ,γs(Rn;E2) will be denoted by Mp,θ,γq,θ,γ(E1,E2). For E1=E2=E, it will be denoted by Mp,θ,γq,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 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].Letβ=(β1,β2,…,βn) be multi-indexes and(2.12)ξβ=ξ1β1ξ2β2,…,ξnβn,Un={β:|β|≤n},η=1p-1q.Definition 2.1.
A Banach spaceE 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 Ψ∈Cn(Rn;B(E)), 1≤p≤q≤∞, β∈Un and ξ∈Vn, if the estimate (1+|ξ|)|β|+η∥DβΨ(ξ)∥L(E)≤C,k=0,1,…,|β| implies Ψ∈Mp,θ,γq,θ,γ(E).
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 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:
(2.13)|α:l|=∑k=1nαklk,α=(α1,α2,…,αn),l=(l1,l2,…,ln),Dα=D1α1D2α2⋯Dnαn=∂|α|∂x1α1∂x2α2⋯∂xnαn,|α|=∑k=!nαk.
Consider the following differential-operator equation:(2.14)Lu=∑k=1n(-1)lktkDk2lku+Aλu+∑|α:2l|<1α(t)Aα(x)Dαu=f,
where A(x), Aα(x) are linear operators in a Banach space E,  ak are complex-valued functions and tk are some parameters α(t)=∏k=1ntkαk/2lk. For l1=l2=⋯=ln=m, we obtain the elliptic class of DOE.
The function belonging toBp,θ,γl,s(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 B-separable (or weighted Bp,θ,γs(Ω;E)-separable) if, for all f∈Bp,θ,γs(Ω;E), the problem (2.14) has a unique solution u∈Bp,θ,γl,s(Ω;E(A),E) and the following estimate holds:
(2.15)‖Au‖Bp,θ,γs(Ω;E)+∑k=1ntk‖Dk2lku‖Bp,θ,γs(Ω;E)≤C‖f‖Bp,θ,γs(Ω;E).
Consider the following degenerate DOE:(2.16)Lu=∑k=1n(-1)lktkDk[2lk]u+Aλu+∑|α:2l|<1α(t)Aα(x)D[α]u=f,
where A(x), Aα(x) are linear operators in a Banach space E,ak are complex-valued functions, tk are some parameters and
(2.17)Dxk[i]=(γ(xk)∂∂xk)i,k=1,2,…,n.Remark 2.3.
Under the substitution(2.18)τk=∫0xkγ-1(y)dy,Bp,θ,γs(Rn;E),  Bp,θ,γ[l],s(Rn;E(A),E) are mapped isomorphically onto the spaces Bp,θ,γ̃s(Rn;E),  Bp,θ,γ̃l,s(Rn;E(A),E), respectively, where
(2.19)γ=∏k=1nγ(xk),γ̃=γ̃(τ)=∏k=1nγ(xk(τk)).
Moreover, under the substitution (2.18), the degenerate problem (2.16) is mapped to the undegenerate problem (2.14).


## 3. Embedding Theorems

Let(3.1)ϰ=∑k=1nαk+1/p-1/qlk,ν(l)=max⁡k,j∈{1,2,…,n}[1lk-1lj],σ(t)=∏k=1ntk(αk+1/p-1/q)/lk.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;(2)
t=(t1,t2,…,tn),  0<tk≤T<∞,  k=1,2,…,n,  1<p≤q<∞, θ∈[1,∞];(3)
lk are positive, αk nonnegative integers such that 0<ϰ+ν(l)≤1 and 0≤μ≤1-ϰ-ν(l);(4)
A is a φ-positive operator in E.
Then, the embeddingDαBp,θ,γl,s(Rn;E(A),E)⊂Bq,θ,γs(Rn;E(A1-ϰ-μ)) is continuous, and there exists a constant Cμ>0, depending only on μ such that
(3.2)σ(t)‖Dαu‖Bq,θ,γs(Rn;E(A1-ϰ-μ))≤Cμ[hμ‖u‖Bp,θ,γ,tl,s+h-(1-μ)‖u‖Bp,θ,γs(Rn;E)]
for all u∈Bp,θ,γl,s(Rn;E(A),E) and 0<h≤h0<∞.Proof.
DenotingFu by û, it is clear that
(3.3)‖Dαu‖Bq,θ,γs(Rn;E(A1-ϰ-μ))~‖F-1(iξ)αA1-ϰ-μû‖Bq,θ,γs(Rn;E).
Similarly,  from the definition of Bp,θ,γl,s(Rn;E(A),E), we have
(3.4)‖u‖Bp,θ,γ,tl,s(Rn;E(A),E)=‖u‖Bp,θ,γs(Rn;E(A))+∑k=1n‖tkDklku‖Bp,θ,γs(Rn;E)~‖F-1Aû‖Bp,θ,γs(Rn;E)+∑k=1n‖tkF-1[(iξk)lkû]‖Bp,θ,γs(Rn;E).
Thus, proving the inequality (3.2) is equivalent to proving
(3.5)σ(t)‖F-1[(iξ)αA1-ϰ-μû]‖Xs≤hμ‖F-1Aû‖Xs+hμ∑k=1n‖tkF-1[(iξk)lkû]‖Xs+h-(1-μ)‖u‖Xs.
So, the inequality (3.2) will be followed if we prove the following inequality:
(3.6)σ(t)‖F-1[(iξ)αA1-ϰ-μû]‖Xs≤Cμ‖F-1[hμ(A+ψ(t,ξ))]û‖Xs
for a suitable Cμ>0 and for all u∈Bp,θ,γl,s(Rn;E(A),E), where
(3.7)ψ=ψ(t,ξ)=∑k=1ntk|ξk|lk+h-1,Xs=Bp,θ,γs(Rn;E).
Let us express the left-hand side of (3.6) as
(3.8)σ(t)‖F-1[(iξ)αA1-ϰ-μû]‖Bq,θ,γs(Rn;E)=σ(t)‖F-1(iξ)αA1-ϰ-μ[hμ(A+ψ)]-1[hμ(A+ψ)]‖Bq,θ,γs(Rn;E).
(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 Ψt=Ψt,h,μ=σ(t)(iξ)αA1-ϰ-μ[hμ(A+ψ)]-1 is a multiplier in Mp,θ,γq,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:
(3.9)|ξ|k+η‖DβΨt(ξ)‖L(E)≤C,k=0,1,…,|β|
for all β∈Un, ξ∈Rn/{ξk=0} and η=1/p-1/q. In a way similar to [18,Lemma  3.1], we obtain that |ξ|-η∥Ψt(ξ)∥L(E)≤Mμfor all ξ∈Rn. This shows that the inequality (3.9) is satisfied for β=(0,…,0). We next consider (3.9) for β=(β1,…,βn), where βk=1 and β=0 for j≠k. By using the condition ϰ+ν(l)≤1 and well-known inequality y1α1y2α2⋯ynαn≤C(1+∑k=1nyklk),yk≥0 and by reasoning according to [18, Theorem  3.1], we have
(3.10)|ξ|1+η|ξk|‖DkΨt(ξ)‖L(E)≤Mμ,k=1,2,…,n.
Repeating the above process, we obtain the estimate (3.9). Thus, the operator-function Ψt,h,μ(ξ) is a uniform collection of multiplier, that is, Ψt,h,μ∈Φh⊂Mp,θ,γq,θ,γ(E). This completes the proof of the Theorem 3.1.It is possible to state Theorem3.1 in a more general setting. For this, we use the extension operator in Bp,θ,γl,s(Ω;E(A),E).Condition 1.
LetA be a φ-positive operator in Banach spaces E satisfying the B-multiplier condition. Let a region Ω⊂Rn be such that there exists a bounded linear extension operatorfrom Bp,θ,γl,s(Ω;E(A),E) to Bp,θ,γl,s(Rn;E(A),E), for 1≤p,  θ≤∞.Remark 3.2.
IfΩ⊂Rn 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 Bp,θ,γs(Ω)=Bp,θ,γs(Ω;R,R) to Bp,θ,γs(Rn)=Bp,θ,γs(Rn;R,R).Theorem 3.3.
Suppose all conditions of Theorem3.1 and Condition 1 are satisfied. Then, the embedding DαBp,θ,γl,s(Ω;E(A),E)⊂Bq,θ,γs(Ω;E(A1-ϰ-μ)) is continuous and there exists a constant Cμ depending only on μ such that
(3.11)σ(t)‖Dαu‖Bq,θ,γs(Ω;E(A1-ϰ-μ))≤Cμ[hμ‖u‖Bp,θ,γ,tl,s+h-(1-μ)‖u‖Bp,θ,γs(Ω;E)]
for all u∈Bp,θ,γl,s(Ω;E(A),E) and 0<h≤h0<∞.Proof.
It suffices to prove the estimate (3.11). Let P be a bounded linear extension operator from Bq,θ,γs(Ω;E) to Bq,θ,γs(Rn;E) and also from Bp,θ,γl,s(Ω;E(A),E) to Bp,θ,γl,s(Rn;E(A),E). Let PΩ be a restriction operator from Rn to Ω. Then, for any u∈Bp,θl,s(Ω;E(A),E), we have
(3.12)‖Dαu‖Bq,θ,γs(Ω;E(A1-ϰ-μ))=‖DαPΩPu‖Bq,θ,γs(Ω;E(A1-ϰ-μ))≤Cμ[hμ‖u‖Bp,θ,γl,s(Ω;E(A),E)+h-(1-μ)‖u‖Bp,θ,γs(Ω;E)].Result 1.
Let all conditions of Theorem3.3 hold. Then, for all u∈Bp,θ,γl,s(Ω;E(A),E) we get
(3.13)‖Dαu‖Bq,θ,γs(Ω;E(A1-ϰ-μ))≤Cμ‖u‖Bp,θ,γl,s(Ω;E(A),E)1-μ‖u‖Bp,θ,γs(Ω;E)μ.
Indeed, setting h=∥u∥Bp,θ,γs(Ω;E)·∥u∥Bp,θ,γl,s(Ω;E(A),E)-1 in (3.13), we obtain (3.11).Result 2.
Ifl1=l2=⋯=ln=m and s1=s2=⋯=sn=σ, then we obtain that embedding DαBp,θ,γm,σ(Ω;E(A),E)⊂Bq,θ,γσ(Ω;E(A1-ϰ)) for ϰ=|α|/m and the corresponding estimate (3.11). For E=C,  A=I, we obtain the embedding of weighted Besov spaces DαBp,θ,γl,s(Ω)⊂Bq,θ,γs(Ω).


## 4.  Application to Vector-Valued Functions

Lets>0, and consider the space [3,Section  1.18.2](4.1)lqσ={u;u={ui}1∞,ui∈C},‖u‖lqσ=(∑i=1∞2iqσ|ui|q)1/q<∞.Note that lq0=lq. Let A be an infinite matrix defined in lq such that D(A)=lqσ,A=[δij2σi],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 lq. Then, by Theorem 3.3, we obtain the embedding(4.2)DαBp1,θ,γl,s(Ω;lqσ,lq)⊂Bp2,θ,γs(Ω;lqσ(1-ϰ-μ)),ϰ=∑k=1nαk+1/p1-1/p2lk,
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.


## 5.B-Separable DOE in Rn with Parameters

Let us consider the differential-operator equation (2.14). Let(5.1)Xs=Bp,θ,γs(Rn;E),Ys=Bp,θs(Rn;E).Theorem 5.1.
Suppose the following conditions hold:
(1)
sk>0,
1<p<∞,1≤θ<∞,
tk>0,
k=1,2,…,n;(2)
E is a Banach space satisfying the B-multiplier condition;(3)
A is a φ-positive operator in E and(5.2)Aα(x)A-(1-|α:2l|-μ)∈L∞(Rn;L(E)),0<μ<1-|α:2l|.Then, for allf∈Bp,θ,γs(Rn;E) and for sufficiently large |λ|>0, λ∈S(φ), the equation (2.18) has a unique solution u(x) that belongs to space Bp,θ,γ2l,s(Rn;E(A),E) and the following uniform coercive estimate holds:(5.3)∑k=1ntk‖Dk2lku‖Bp,θ,γs(Rn;E)+‖Au‖Bp,θ,γs(Rn;E)≤C‖f‖Bp,θ,γs(Rn;E).Proof.
At first, we will consider the principal part of (2.14), that is, the differential-operator equation
(5.4)(L0+λ)u=∑k=1n(-1)lktkDk2lku+(A+λ)u=f.
Then, by applying the Fourier transform to (5.4), we obtain
(5.5)∑k=1ntkξk2lkû(ξ)+(A+λ)û(ξ)=f̂(ξ).
Since∑k=1ntkξk2lk≥0 for all ξ=(ξ1,…,ξn)∈Rn, we can say that ω=ω(t,λ,ξ)=λ+∑k=1ntkξk2lk∈S(φ) for all ξ∈Rn, 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)-1f̂. It is clear to see that the operator-functionφλ,t(ξ)=[A+ω]-1 is a multiplier in Bp,θ,γs(Rn;E) uniformly with respect to λ∈S(φ). Actually, by definition of the positive operator, for all ξ∈Rn andλ≥0, we get
(5.6)‖φλ(ξ)‖L(E)=‖(A+ω)-1‖≤M(1+|ω|)-1≤M0.
Moreover, since Dkφλ,t(ξ)=2lktk(A+ω)-2ξk2lk-1, then ∥ξkDkφλ,t∥L(E)≤M. By using this estimate for β∈Un, we get
(5.7)|ξ|β‖Dξβφλ,t(ξ)‖L(E)≤C.
In a similar way to Theorem 3.1, we prove that φk,λ,t(ξ)=ξk2lkφλ,t,  k=1,2,…,n, and φ0,λ,t=Aφλ,t satisfy the estimates
(5.8)(1+|ξ|)|β|‖Dξβφk,λ,t(ξ)‖L(E)≤C,(1+|ξ|)|β|‖Dξβφ0,λ,t(ξ)‖L(E)≤C.
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 φλ,t,φk,λ,t,φ0,λ,t are multipliers in Bp,θ,γs(Rn;E). Then, we obtain that there exists a unique solution of (5.4) for f∈Bp,θ,γs(Rn;E) and the following estimate holds:
(5.9)∑k=1ntk‖Dk2lku‖Bp,θ,γs+‖Au‖Bp,θ,γs≤C‖f‖Bp,θ,γs.
Consider now the differential operator G0t generated by problem (5.4), that is,
(5.10)D(G0t)=Bp,θ,γ2l,s(Rn;E(A),E),G0tu=∑k=1n(-1)lktkDk2lku+Au.
The estimate (5.9) implies that the operator G0t+ϰ has a bounded inverse from Bp,θ,γs(Rn;E) into Bp,θ,γ2l,s(Rn;E(A),E) for all ϰ≥0. Let Gt denote the differential operator in Bp,θ,γs(Rn;E) generated by problem (2.14). In view of (2.18) condition, by virtue of Theorem 3.1, for all u∈Bp,θ,γ2l,s(Rn;E(A),E), we have
(5.11)‖L1u‖Bp,θ,γs≤∑|α:2l|<1α(t)‖A1-|α:2l|-μDαu‖Bp,θ,γs≤C[hμ(∑k=1ntk‖Dk2lku‖Bp,θ,γs+‖Au‖Bp,θ,γs)+h-(1-μ)‖u‖Bp,θ,γs].
Then, from (5.11), we have
(5.12)‖L1u‖Bp,θ,γs≤C[hμ‖(G0t+λ)u‖Bp,θ,γs+h-(1-μ)‖u‖Bp,θ,γs].
Since ∥u∥Bp,θ,γs=(1/λ)∥(G0t+λ)u-G0tu∥Bp,θ,γs for all u∈Bp,θ,γ2l,s(Rn;E(A),E), we get
(5.13)‖u‖Bp,θ,γs≤1|λ|[‖(G0t+λ)u‖Bp,θ,γs+‖G0tu‖Bp,θ,γs].
From estimates (5.11)–(5.13), we obtain
(5.14)‖L1u‖Xs≤Chμ‖(G0t+λ)u‖Xs+C1|λ|-1h-(1-μ)‖(G0t+λ)u‖Xs.
Then, by choosing h and λ, such that Chμ<1,C1|λ|-1h-(1-μ)<1 from (5.14), we get the following uniform estimate:
(5.15)‖L1(G0t+λ)-1‖L(E)<1.
Then, using the estimates of (5.9), (5.15) and the perturbation theory of linear operators, we obtain that the operator Gt+λ is invertible from Bp,θ,γs(Rn,E) into Bp,θ,γ2l,s(Rn;E(A),E). This implies the estimate (5.3).Result 1.
Let all conditions of Theorem5.1 hold. Then,(1)
forf∈Bp,θs(Rn,E),  λ∈S(φ), (2.16) has a unique solution u∈Bp,θ,γ[2l],s(Rn;E(A),E) and
(5.16)∑k=1ntk‖Dk[2lk]u‖Bp,θs(Rn;E)+‖Au‖Bp,θs(Rn;E)≤C‖f‖Bp,θs(Rn;E),(2)
the operatorQt has a resolvent (Qt+λ)-1 for |argλ|≤φ, and the following uniform estimate holds:
(5.17)∑|α:2l|≤1α(t)|λ|1-|α:2l|‖D[α](Qt+λ)-1‖L(Ys)+‖A(Qt+λ)-1‖L(Ys)≤C.Remark 5.2.
Result1 implies that operator Qt is uniformly positive in Bp,θs(Rn;E). Then, by virtue of [3, Section  1.14.5], the operator Qt is a generator of an analytic semigroup in Bp,θs(Rn;E).


## 6. Cauchy Problem for Degenerate Parabolic DOE with Parameters

Consider the Cauchy problem for the degenerate parabolic DOE(6.1)∂u∂y+∑k=1n(-1)lktkDk[2lk]u+Au+∑|α:2l|<1α(t)Aα(x)D[α]u=f(y,x),u(0,x)=0,
where A and Aα(x) are linear operators in a Banach space in E. Let F=Bp,θs(Rn;E).Theorem 6.1.
Assume all conditions of Theorem5.1 hold for φ∈(π/2,π) and s>0. Then, for f∈Bp,θs(R+;F), (6.1) has a unique solution u∈Bp,q1(R+;D(Qt),F) satisfying
(6.2)‖∂u∂y‖Bp,θs(R+;F)+∑k=1n‖tkDk[2lk]u‖Bp,θs(R+;F)+‖Au‖Bp,θs(R+;F)≤C‖f‖Bp,θs(R+;F).Proof.
The problem (6.1) can be expressed as
(6.3)dudy+Qtu(y)=f(y),u(0)=0,y∈(0,∞).
Result1 implies the uniform positivity of Gt. So, by [6, Application D], we obtain that, for f∈Bp,θs(R+;F), the Cauchy problem (6.3) has a unique solution u∈Bp,θ1+s(R+;D(Qt),F) satisfying
(6.4)‖Dtu‖Bp,θs(R+;F)+‖Qtu‖Bp,θs(R+;F)≤C‖f‖Bp,qs(R+;F).
In view of Result1, the operator Qt 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:(7.1)(L+λ)um=∑k=1n(-1)lktkDk2lkum(x)+(dm+λ)um(x)+∑|α:l|<1α(t)∑k=1∞dαkm(x)Dαum=fm(x),x∈Rn,m=1,2,…,∞.Let(7.2)d(x)={dm(x)},dm>0,u={um},du={dmum},Q(x)={dm(x)},dm>0,u={um},Qu={dmum},lq(Q)=={u:u∈lq,‖u‖lq(Q)=‖Qu‖lq=(∑m=1∞|dmum|q)1/q<∞},
and let tk be positive parameters. Let Ot denote the differential operator in B=L(Bp,θ,γs(Rn;lq)) generated by problem (7.1).Theorem 7.1.
Letaα∈Cb(Rn), dm∈Cb(Rn),  dαkm∈L∞(Rn) such that maxαsupm∑k=1∞dαkm(x)dk-(1-|α:l|-μ)<M for all x∈Rn,  p, q∈(1,∞),  θ∈[1,∞] and 0<μ<1-|α:l|.
Then,(a)
for allf(x)={fm(x)}1∞∈Bp,θ,γs(Rn;lq), for |argλ|≤φ and for sufficiently large |λ|, the problem (7.1) has a unique solution u={um(x)}1∞ that belongs to space Bp,θ,γs+2l(Rn,lq(d),lq) and the uniform coercive estimate holds
(7.3)∑|α:2l|≤1‖Dαu‖Bp,θ,γs(Rn;lq)+‖Qu‖Bp,θ,γs(Rn;lq)≤C‖f‖Bp,θ,γs(Rn;lq),(b)
for|argλ|≤φ and for sufficiently large |λ|, there exists a resolvent (Ot+λ)-1 of operator Ot and
(7.4)∑|α:2l|≤1α(t)|λ|1-|α:l|‖Dα(Ot+λ)-1‖B+‖Q(Ot+λ)-1‖B≤M.Proof.
Really, letE=lq,  A(x), and let Aα(x) be infinite matrices such that
(7.5)A=[dm(x)δkm],Aα(x)=[dαkm(x)],k,m=1,2,…,∞.
It is clear that the operator A is positive in lq. Therefore, from Theorem 6.1, we obtain that the problem (7.1) has a unique solution u∈Bp,θ,γs+2l(Rn;lq(Q),lq) for all f∈Bp,θ,γs(Rn;lq), |argλ|≤φ, sufficiently large |λ| 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 problem(8.1)∂um∂y+∑k=1n(-1)lktkDk2lkum+dm(x)um+∑|α:l|<1α(t)dαkm(x)Dαum=fm(y,x),y∈R+,x∈Rn,um(0,x)=0,m=1,2,…,∞.Theorem 8.1.
Let all conditions of Theorem7.1 hold. Then, the parabolic systems (8.1) for sufficiently large ϰ>0 have a unique solution u∈Bp,θ,γ1,s+2l(Rn;lq(Q),lq), and the following estimate holds:
(8.2)‖∂u∂y‖Bp,θs(R+n+1;lq)+∑k=1n‖tkDk[2lk]u‖Bp,θs(R+n+1;lq)+‖Qu‖Bp,θs(R+n+1;lq)≤C‖f‖Bp,θs(R+n+1;lq).Proof.
Really, letE=lq,  and let A and Aα(x) be infinite matrices, such that
(8.3)A=[dm(x)δkm],Aα(x)=[dαkm(x)],k,m=1,2,…∞.
Then, the problem (8.1) can be expressed in the form (6.3), where
(8.4)A=[dm(x)δkm],Aα(x)=[dαkm(x)],k,m=1,2,…∞.
Then, by virtue of Theorems 6.1 and 7.1, we obtain the assertion.



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*Source: 819321-2012-03-08.xml*