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EMBEDDING THEOREMS IN BANACH-VALUED B-SPACES AND MAXIMAL B-REGULAR DIFFERENTIAL-OPERATOR EQUATIONS VELI B. SHAKHMUROV Received 28 September 2004; Revised 8 November 2005; Accepted 4 May 2006 The embedding theorems in anisotropic Besov-Lions type spaces B l p,θ (R n ;E 0 ,E)arestud- ied; here E 0 and E are two Banach spaces. The most regular spaces E α are found such that the mixed differential operators D α are bounded from B l p,θ (R n ;E 0 ,E)toB s q,θ (R n ;E α ), where E α are interpolation spaces between E 0 and E depending on α = (α 1 ,α 2 , ,α n )and l = (l 1 ,l 2 , ,l n ). By using these results the separability of anisotropic differential-operator equations with dependent coefficients in principal part and the maximal B-regularity of parabolic Cauchy problem are obtained. In applications, the infinite systems of the quasielliptic partial differential equations and the parabolic Cauchy problems are stud- ied. Copyright © 2006 Veli B. 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. 1. Introduction Embedding theorems in function spaces have been studied in [8, 35, 37, 38]. A com- prehensive introduction to the theory of embedding of function spaces and historical references may be also found in [37]. In abstract function spaces embedding theorems have been investigated in [4, 5, 10, 17, 21, 27, 34, 40]. Lions and Peetre [21] showed that if u ∈ L 2  0,T;H 0  , u (m) ∈ L 2 (0,T;H), (1.1) then u (i) ∈ L 2  0,T;  H,H 0  i/m  , i = 1,2, ,m − 1, (1.2) where H 0 , H are Hilbert spaces, H 0 is continuously and densely embedded in H,where [H 0 ,H] θ are interpolation spaces between H 0 and H for 0 ≤ θ ≤ 1. The similar questions for anisotropic Sobolev spaces W l p (Ω;H 0 ,H), Ω ⊂ R n and for corresponding weighted Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2006, Article ID 16192, Pages 1–22 DOI 10.1155/JIA/2006/16192 2 Embedding and B-regular operators spaces have been investigated in [28–31]and[23, 24], respectively. Embedding theorems in Banach-valued Besov spaces have been studied in [4, 5, 27, 32]. The solvability and spectrum of boundary value problems for elliptic differential-operator equations (DOE’s) have been refined in [3–7, 13, 28–33, 39, 40]. A comprehensive introduction to DOE’s and historical references may be found in [15, 18, 40]. In these works, Hilbert-valued function spaces essentially have been considered. The maximal L p regularity and Fredholmness of partial elliptic equations in smooth regions have been studied, for example, in [1, 2, 20] and for nonsmooth domains studied, for example, in [16, 26]. For DOE’s the similar problems have been investigated in [13, 28–32, 36, 39, 40]. Let E 0 , E be Banach spaces such that E 0 is continuously and densely embedded in E. In the present paper, E-valued Besov spaces B l+s p,θ (R n ;E 0 ,E) = B s p,θ (R n ;E 0 ) ∩ B l+s p,θ (R n ;E)are introduced and cal led Besov-Lions type spaces. The most regular interpolation class E α between E 0 and E is found such that the appropriate mixed differential operators D α are bounded from B l+s p,q (R n ;E 0 ,E)toB s p,q (R n ;E α ). By applying these results the maximal regularity of certain class of anisotropic partial DOE with varying coefficients in Banach- valued Besov spaces is derived. The paper is organized as follows. Section 2 collects notations and definitions. Section 3 presents the embedding theorems in Besov-Lions type spaces B s+l p,q  R n ;E 0 ,E  . (1.3) Section 4 contains applications of the underlying embedding theorem to vector-valued function spaces. Section 5 is devoted to the maximal regularity (in B s p,q (R n ;E)) of the certain class of anisotropic DOE with var iable coefficients in principal part. Then by us- ing these results the maximal B-regularity of the parabolic Cauchy problem is shown. In Section 6 these DOE are applied to BVP’s and Cauchy problem for the finite and infinite systems of quasielliptic and parabolic PDEs, respectively. 2. Notations and definitions Let E be a Banach space. Let L p (Ω;E) denote the space of all strongly measurable E-valued functions that are defined on Ω ⊂ R n with the norm  f  L p (Ω;E) =     f (x)   p E dx  1/p ,1≤ p<∞,  f  L ∞ (Ω;E) = esssup x∈Ω    f (x)   E  , x =  x 1 ,x 2 , ,x n  . (2.1) The Banach space E is said to be a ζ-convex space (see [9, 11, 12, 19]) if there exists on E × E a symmetric real-valued function ζ(u,v) which is convex with respect to each of the variables, and satisfies the conditions ζ(0,0) > 0, ζ(u,v) ≤u + v,foru≤1 ≤v. (2.2) Veli B. Shakhmurov 3 A ζ-convex space E is often called a UMD-space and written as E ∈ UMD. It is shown in [9] that the Hilbert operator (Hf)(x) = lim ε→0  |x−y|>ε f (y) x − y dy (2.3) is bounded in L p (R;E), p ∈ (1,∞) for those and only those spaces E, which possess the property of UMD spaces. The UMD spaces include, for example, L p , l p spaces and the Lorentz spaces L pq , p,q ∈ (1,∞). Let C be the set of complex numbers and let S ϕ =  λ; λ ∈ C,|argλ − π|≤π − ϕ  ∪{ 0},0<ϕ≤ π. (2.4) A linear operator A is said to be a ϕ-positive in a Banach space E, with bound M>0if D( A)isdenseonE and   (A − λI) −1   L(E) ≤ M  1+|λ|  −1 (2.5) with λ ∈ S ϕ , ϕ ∈ (0, π], I is identity operator in E,andL(E) is the space of all bounded linear operators in E.SometimesA + λI will be written as A + λ and denoted by A λ .Itis known [37, Section 1.15.1] that there exist fractional powers A θ of the positive operator A.LetE(A θ ) denote the space D(A θ ) with the graphical norm u E(A θ ) =   u p +   A θ u   p  1/p ,1≤ p<∞, −∞ <θ<∞. (2.6) Let E 0 and E be two Banach spaces. By (E 0 ,E) σ,p ,0<σ<1, 1 ≤ p ≤∞we will denote the interpolation spaces obtained from {E 0 ,E} by the K-method (see, e.g., [37,Section 1.3.1] or [10]). Let S(R n ;E) denote a Schwar tz class, that is, the space of all E-valued rapidly decreasing smooth functions ϕ on R n . E = C will be denoted by S(R n ). Let S  (R n ;E) denote the space of E-valued tempered distributions, that is, the space of continuous linear operators from S(R n )toE. Let α = (α 1 ,α 2 , ,α n ), α i are integers. An E-values generalized function D α f is called a generalized derivative in the sense of Schwartz distributions of the generalized function f ∈ S  (R n ,E) if the equality  D α f ,ϕ  = (−1) |α|  f ,D α ϕ  (2.7) holds for all ϕ ∈ S(R n ). By using (2.7) the following relations F  D α x f  =  iξ 1  α 1 , ,  iξ n  α n  f , D α ξ  F( f )  = F  − ix n  α 1 , ,  − ix n  α n f  (2.8) are obtained for all f ∈ S  (R n ;E). Let L ∗ θ (E) denote the space of all E-valued function spaces such that u L ∗ θ (E) =   ∞ 0   u(t)   θ E dt t  1/θ < ∞,1≤ θ<∞, u L ∗ ∞ (E) = sup 0<t<∞   u(t)   E . (2.9) 4 Embedding and B-regular operators Let s = (s 1 ,s 2 , ,s n )ands k > 0. Let F denote the Fourier tr a nsform. Fourier-analytic rep- resentation of E-valued Besov space on R n is defined as B s p,θ  R n ;E  =  u ∈ S   R n ;E  , u B s p,θ (R n ;E) =      F −1 n  k=1 t κ k −s k  1+   ξ k   κ k  e −t|ξ| 2 Fu      L ∗ θ (L p (R n ;E)) , p ∈ (1,∞), θ ∈ [1,∞], κ k >s k  . (2.10) It should be noted that the norm of Besov space do not depend on κ k .Sometimeswe will write u B s p,θ in place of u B s p,θ (R n ;E) . Let l = (l 1 ,l 2 , ,l n ), s = (s 1 ,s 2 , ,s n ), where l k are integers and s k are positive numbers. Let W l B s p,θ (R n ;E) denote an E-valued Sobolev-Besov space of all functions u ∈ B s p,θ (R n ;E) such that they have the generalized derivatives D l k k u=∂ l k u/∂x l k k ∈B s p,θ (R n ;E), k = 1, 2, , n with the norm u W l B s p,θ (R n ;E) =u B s p,θ (R n ;E) + n  k=1   D l k k u   B s p,θ (R n ;E) < ∞. (2.11) Let E 0 is continuously and densely embedded into E. W l B s p,θ (R n ;E 0 ,E) denotes a space of all functions u ∈ B s p,θ (R n ;E 0 ) ∩ W l B s p,θ (R n ;E) with the norm u W l B s p,θ =u W l B s p,θ (R n ;E 0 ,E) =u B s p,θ (R n ;E 0 ) + n  k=1    D l k k u    B s p,θ (R n ;E) < ∞. (2.12) Let l = (l 1 ,l 2 , ,l n ), s = (s 1 ,s 2 , ,s n ), where s k are real numbers and l k are positive num- bers. B l+s p,θ (R n ;E 0 ,E) denotes a space of all functions u ∈ B s p,θ (R n ;E 0 ) ∩ B l+s p,θ (R n ;E) with the norm u B s+l p,θ (R n ;E 0 ,E) =u B s p,θ (R n ;E 0 ) + u B l+s p,θ (R n ;E) . (2.13) For E 0 = E the space B l+s p,θ (R n ;E 0 ,E) will be denoted by B l+s p,θ (R n ;E). Let m be a positive integer. C(Ω;E)andC m (Ω;E) will denote the spaces of all E-valued bounded continuous and m-times continuously di fferentiable functions on Ω,respec- tively. We set C b (Ω;E) =  u ∈ C(Ω;E), lim |x|→∞ u(x) exists  . (2.14) Let E 1 and E 2 be two Banach spaces. A function Ψ ∈ C m (R n ;L(E 1 ,E 2 )) is called a multi- plier from B s p,θ (R n ;E 1 )toB s q,θ (R n ;E 2 )forp ∈ (1,∞)andq ∈ [1,∞]ifthemapu → Ku= F −1 Ψ(ξ)Fu, u ∈ S(R n ;E 1 ), is well defined and extends to a bounded linear operator K : B s p,θ  R n ;E 1  −→ B s q,θ  R n ;E 2  . (2.15) Veli B. Shakhmurov 5 The set of all multipliers from B s p,θ (R n ;E 1 )toB s q,θ (R n ;E 2 ) will be denoted by M q,θ p,θ (s,E 1 , E 2 ). E 1 = E 2 = E will be denoted by M q,θ p,θ (s,E). The multipliers and operator-valued mul- tipliers in Banach-valued function spaces were studied, for example, by [25], [37,Section 2.2.2.], and [4, 11, 12, 14, 22], respectively. Let H k =  Ψ h ∈ M q,θ p,θ  s,E 1 ,E 2  , h =  h 1 h 2 , ,h n  ∈ K  (2.16) be a collection of multipliers in M q,θ p,θ (s,E 1 ,E 2 ). We say that H k is a uniform collection of multipliers if there exists a constant M 0 > 0, independent on h ∈ K,suchthat   F −1 Ψ h Fu   B s p,θ (R n ;E 2 ) ≤ M 0 u B s q,θ (R n ;E 1 ) (2.17) for all h ∈ K and u ∈ S(R n ;E 1 ). Let β = (β 1 ,β 2 , ,β n ) be multiindexes. We also define V n =  ξ =  ξ 1 ,ξ 2 , ,ξ n  ∈ R n , ξ i = 0, i = 1,2, ,n  , U n =  β : |β|≤n  , ξ β = ξ β 1 1 ξ β 2 2 , ,ξ β n n , ν = 1 p − 1 q . (2.18) Definit ion 2.1. ABanachspaceE satisfies a B-multiplier condition with respect to p, q, θ,ands (or with respect to p, θ,ands for the case of p = q)whenΨ ∈ C n (R n ;L(E)), 1 ≤ p ≤ q ≤∞, β ∈ U n ,andξ ∈ V n if the estimate   ξ 1   β 1 +ν   ξ 2   β 2 +ν , ,   ξ n   β n +ν   D β Ψ(ξ)   L(E) ≤ C (2.19) implies Ψ ∈ M q,θ p,θ (s,E). Remark 2.2. Definition 2.1 is a combined restriction to E, p, q, θ,ands. This condition is sufficient for our main aim. Nevertheless, it is well known that there are Banach spaces satisfying the B-multiplier condition for isotropic case and p = q, for example, the UMD spaces (see [4, 14]). ABanachspaceE is said to have a local unconditional str ucture (l.u.st.) if there exists a constant C< ∞ such that for any finite-dimensional subspace E 0 of E there exists a finite- dimensional space F with an unconditional basis such that the natural embedding E 0 ⊂ E factors as AB with B : E 0 → F, A : F → E,andAB≤C. All Banach lattices (e.g., L p , L p,q , Orlicz spaces, C[0,1]) have l.u.st. The expression u E 1 ∼ u E 2 means that there exist the positive constants C 1 and C 2 such that C 1 u E 1 ≤u E 2 ≤ C 2 u E 1 (2.20) for all u ∈ E 1 ∩ E 2 . 6 Embedding and B-regular operators Let α 1 ,α 2 , ,α n be nonnegative and let l 1 ,l 2 , ,l n be positive integers and let 1 ≤ p ≤ q ≤∞,1≤ θ ≤∞, |α: .l|= n  k=1 α k l k , κ = n  k=1 α k +1/p− 1/q l k , D α = D α 1 1 D α 2 2 , ,D α n n = ∂ |α| ∂x α 1 1 ∂x α 2 2 , ,∂x α n n , |α|= n  k=! α k . (2.21) Consider in general, the anisotropic differential-operator equation (L + λ)u =  |α:.l|=1 a α (x) D α u + A λ (x) u +  |α:.l|<1 A α (x) D α u = f (2.22) in B s p,θ (R n ;E), where a α are complex-valued functions and A(x), A α (x) are possibly un- bounded operators in a B anach space E, here the domain definition D(A) = D(A(x)) of operator A(x) does not depend on x.Forl 1 = l 2 =, ,= l n we obtain isotropic equations containing the elliptic class of DOE. The function belonging to space B s+l p,θ (R n ;E(A),E) and satisfying (2.22)a.e.onR n is said to be a solution of (2.22)onR n . Definit ion 2.3. The problem (2.22)issaidtobeaB-separable (or B s p,θ (R n ;E)-separable) if the problem (2.22)forall f ∈ B s p,θ (R n ;E) has a unique solution u ∈ B s+l p,θ (R n ;E(A),E)and Au B s p,θ (R n ;E) +  |α:l|=1   D α u   B s p,θ  R n ;E  ≤ C f  B s p,θ (R n ;E) . (2.23) Consider the following parabolic Cauchy problem ∂u(y,x) ∂y +(L+ λ)u(y,x) = f (y,x), u(0,x) = 0, y ∈ R + , x ∈ R n , (2.24) where L is a realization differential operator in B s p,θ (R n ;E) generated by problem (2.22), that is, D( L) = B s+l p,θ  R n ;E(A),E  , Lu =  |α:.l|=1 a α (x) D α u + A(x)u +  |α:.l|<1 A α (x) D α u. (2.25) We say that the parabolic Cauchy problem (2.24)issaidtobeamaximalB-regular, if for all f ∈ B s p,θ (R n+1 + ;E) there exists a unique solution u satisfying (2.24) almost every- where on R n+1 + and there exists a positive constant C independent on f ,suchthatithas the estimate     ∂u(y,x) ∂y     B s p,θ (R n+1 + ;E) + Lu B s p,θ (R n+1 + ;E) ≤ C f  B s p,θ (R n+1 + ;E) . (2.26) 3. Embedding theorems In this section we prove the boundedness of the mixed differential operators D α in the Besov-Lions type spaces. Veli B. Shakhmurov 7 Lemma 3.1. Let A be a positive operator in a Banach space E,letb be a positive number, r = (r 1 ,r 2 , ,r n ), α = (α 1 ,α 2 , ,α n ),andl = (l 1 ,l 2 , ,l n ),whereϕ ∈ (0,π], r k ∈ [0,b], l k are positive and α k , k = 1,2, ,n, are nonnegative integers such that κ =|(α + r):l|≤1. For 0 <h ≤ h 0 < ∞ and 0 ≤ μ ≤ 1 − κ the operator-function Ψ(ξ) = Ψ h,μ (ξ) =   ξ 1   r 1   ξ 2   r 2 , ,   ξ n   r n (iξ) α A 1−κ−μ h −μ  A + η(ξ)  −1 (3.1) is a bounded operator in E uniformly with respect to ξ and h, that is, there is a constant C μ such that   Ψ h,μ (ξ)   L(E) ≤ C μ (3.2) for all ξ ∈ R n ,where η = η(ξ) = n  k=1   ξ k   l k + h −1 . (3.3) Proof. Since −η(ξ) ∈ S(ϕ), for all ϕ ∈ (0,π]andA is a ϕ-positive in E, then the operator A + η(ξ)isinvertiableinE.Let u = h −μ  A + η(ξ)  −1 f. (3.4) Then   Ψ(ξ) f   E =   (hA) 1−κ−μ u   E h −(1−μ)   h 1/l 1 ξ 1   α 1 +r 1 , ,   h 1/l n ξ n   α n +r n . (3.5) Using the moment inequality for powers of positive operators, we get a constant C μ de- pending only on μ such that   Ψ(ξ) f   E ≤ C μ h −(1−μ) hAu 1−κ−μ u κ+μ   h 1/l 1 ξ 1   α 1 +r 1 , ,   h 1/l n ξ n   α n +r n . (3.6) Now, we apply the Young inequality, which states that ab ≤ a k 1 /k 1 + b k 2 /k 2 for any positive real numbers a, b and k 1 , k 2 with 1/k 1 +1/k 2 = 1totheproduct hAu 1−κ−μ   u κ+μ   h 1/l 1 ξ 1   α 1 +r 1 , ,   h 1/l n ξ n   α n +r n  (3.7) with k 1 = 1/(1 − κ − μ), k 2 = 1/(κ + μ)toget   Ψ(ξ) f   E ≤C μ h −(1−μ)  (1 − κ − μ)   hAu   +(κ +μ)  h 1/l 1   ξ 1    (α 1 +r 1 )/(κ+μ) , ,  h 1/l n   ξ n    (α n +r n )/(κ+μ) u  . (3.8) Since n  i=1 α i + r i (κ + μ) = 1 κ + μ n  i=1 α i + r i l i = κ κ + μ ≤ 1, (3.9) 8 Embedding and B-regular operators there exists a constant M 0 independent on ξ,suchthat   ξ 1   (α 1 +r 1 )/(κ+μ) , ,   ξ n   (α n +r n )/(κ+μ) ≤ M 0  1+ n  k=1   ξ k   l k  (3.10) for all ξ ∈ R n . Substituting this on the inequality (3.8) and absorbing the constant coeffi- cients in C μ ,weobtain   ψ(ξ) f   ≤ C μ  h μ   Au + n  k=1   ξ k   l k u  + h −(1−μ) u  . (3.11) Substituting the value of u we get   ψ(ξ) f    ≤ C μ h μ    A  A + η(ξ)  −1 f   + n  k=1   ξ k   l k    A + η(ξ)  −1 f    + h −(1−μ)     A + η(ξ)  −1 f    . (3.12) By using the properties of the positive operator A for all f ∈ E we obtain from (3.12)   Ψ(ξ) f   E ≤ C μ  f  E . (3.13)  Lemma 3.2. Let E be a UMD space with l.u.st., p ∈ (1,∞), θ ∈ [1,∞] and let for all k, j ∈ (1,n) s k l k + s k + s j l j + s j ≤ 1. (3.14) Then the spaces B l+s p,θ (R n ;E) and W l B s p,θ (R n ;E) are coinc ided. Proof. In the first step we show that the continuous embedding W l B s p,θ (R n ;E) ⊂ B l+s p,θ (R n ; E) holds, that is, there is a positive constant C such that u B l+s p,θ (R n ;E) ≤ Cu W l B s p,θ (R n ;E) (3.15) for all u ∈ W l B s p,θ (R n ;E). For this aim by using the Fourier-analytic definition of an E- valued Besov space and the space W l B s p,θ (R n ;E)itissufficient to prove the following estimate:      F −1 n  k=1 t κ k −l k −s k  1+   ξ k   κ k  e −t|ξ| 2 Fu      L θ p ≤ C      F −1 n  k=1 t κ k −s k  1+   ξ k   κ k  e −t|ξ| 2 Fυ      L θ p , (3.16) where L θp = L ∗ θ  L p  R n ;E  , υ = F −1  1+ n  k=1 ξ l k k  Fu. (3.17) Veli B. Shakhmurov 9 To see this, it is sufficient to show that the function φ(ξ) = n  k=1  1+   ξ k   l k +s k +δ   n  k=1  1+   ξ k   s k +δ   −1  1+ n  k=1   ξ k   l k  −1 , δ>0 (3.18) is Fourier multiplier in L p (R n ;E). It is clear to see that for β ∈ U n and ξ ∈ V n   ξ 1   β 1   ξ 2   β 2 , ,   ξ n   β n   D β φ(ξ)   L(E) ≤ C. (3.19) Then in view of [41, Proposition 3] we obtain that the function φ is Fourier multiplier in L p (R n ;E). In the second step we prove that the embedding B l+s p,θ (R n ;E) ⊂ W l B s p,θ (R n ;E)iscontin- uous. In a similar way as in the first step we show that for s k /(l k + s k )+s j /(l j + s j ) ≤ 1the function ψ(ξ) =  n  k=1  1+   ξ k   s k +δ   1+ n  k=1   ξ k   l k  n  k=1  1+   ξ k   l k +s k +δ   −1 (3.20) is Fourier multiplier in L p (R n ;E). So, we obtain for all u ∈ B l+s p,θ (R n ;E)theestimate      F −1 n  k=1 t κ k −s k  1+   ξ k   κ k   1+ n  k=1 ξ l k k  e −t|ξ| 2 Fu      L θ p ≤ C      F −1 n  k=1 t κ k −l k −s k  1+   ξ k   κ k  e −t|ξ| 2 Fu      L θ p . (3.21) It implies the second embedding. This completes the prove of Lemma 3.2.  Theorem 3.3. Suppose the following conditions hold: (1) E is a UMD space with l.u.st. satisfying the B-multiplier condition with respect to p, q ∈ (1,∞), θ ∈ [1,∞],ands = (s 1 ,s 2 , ,s n ),wheres k are positive numbers; (2) α = (α 1 ,α 2 , ,α n ), l = (l 1 ,l 2 , ,l n ),whereα k are nonnegative, l k are positive integers, and s k such that s k /(l k + s k )+s j /(l j + s j ) ≤ 1 for k, j = 1,2, ,n and 0 ≤ μ ≤ 1 − κ , κ = | (α +1/p− 1/q):l|; (3) A is a ϕ-positive operator in E,whereϕ ∈ (0,π] and 0 <h≤ h 0 < ∞. Then the following embedding D α B l+s p,θ  R n ;E(A),E  ⊂ B s q,θ  R n ;E  A 1−κ−μ  (3.22) is continuous and there exists a positive constant C μ depending only on μ, such that   D α u   B s q,θ (R n ;E(A 1−κ−μ )) ≤ C μ  h μ u B l+s p,θ (R n ;E(A),E) + h −(1−μ) u B s p,θ (R n ;E)  (3.23) for all u ∈ B l+s p,θ (R n ;E(A),E). 10 Embedding and B-regular operators Proof. We have   D α u   B s q,θ (R n ;E(A 1−κ−μ )) =   A 1−κ−μ D α u   B s q,θ (R n ;E) (3.24) for all u such that   D α u   B s q,θ (R n ;E(A 1−κ−μ )) < ∞. (3.25) On the other hand by using the relation (2.8)wehave A 1−α−μ D α u = F − FA 1−κ−μ D α u = F − (iξ) α A 1−κ−μ Fu. (3.26) Since the operator A is closure and does not depend on ξ ∈ R n hence denoting Fu by u, from the relations (3.24), (3.26) and by definition of the space W l B s p,θ (R n ;E 0 ,E)wehave   D α u   B s q,θ (R n ;E(A 1−κ−μ ))    F − (iξ) α A 1−κ−μ u   B s q,θ (R n ;E) , u W l B s p,θ (R n ;E 0 ,E) ∼ Au B s p,θ (R n ;E) + n  k=1   F −1 ξ l k k u   B s p,θ (R n ;E) . (3.27) By virtue of Lemma 3.2 and by the above relations it is sufficienttoprovethat   F − (iξ) α A 1−κ−μ u   B s q,θ (R n ;E) ≤ C μ  h μ    F − Au   B s p,θ (R n ;E) + n  k=1   F −  ξ l k k u    B s p,θ (R n ;E)  + h −(1−μ)   F − u   B s p,θ (R n ;E)  . (3.28) The inequality (3.23) will b e followed if we prove the following inequality   F −  (iξ) α A 1−κ−μ u    B s p,θ (R n ;E) ≤ C μ   F −  h μ (A + η)   u   B s p,θ (R n ;E) (3.29) for a suitable C μ and for all u ∈ B s+l p,θ (R n ;E(A),E), where η = η(ξ) = n  k=1   ξ k   l k + h −1 . (3.30) Let us express the left-hand side of (3.29)asfollows:   F −  (iξ) α A 1−κ−μ u    B s q,θ (R n ;E) (3.31) =   F − (iξ) α A 1−κ−μ  h μ (A + η)  −1  h μ (A + η)   u   B s q,θ (R n ;E) . (3.32) (Since A is the positive operator in E and −η(ξ) ∈ S(ϕ) so it is possible). By virtue of Definition 2.1 it is clear that the inequality (3.23) will follow immediately from (3.31)if we can prove that the operator-function Ψ = (iξ) α A 1−κ−μ [h μ (A + η)] −1 is a multiplier in [...]... ), where αk are nonnegative and lk are positive integers such that κ = |(α + 1/ p − 1/q) : l| ≤ 1 and let 0 ≤ μ ≤ 1 − κ ; (3) A is a ϕ-positive operator in E, where ϕ ∈ (0,π] and 0 < h ≤ h0 < ∞ 12 Embedding and B-regular operators Then the following embedding s Dα W l B s Rn ;E(A),E ⊂ Bq,θ Rn ;E A1−κ −μ p,θ (3.38) is continuous and there exists a positive constant Cμ depending only on μ such that Dα... obtain the following embedding Dα B l+s (Rn ;Rm ) ⊂ Bq,θ (Rn ; p,θ Rm ) for 0 ≤ μ ≤ 1 − κ and a corresponding estimate (3.23) For E = R, A = I we get s the embedding Dα B l+s (Rn ) ⊂ Bq,θ (Rn ) proved in [8, Section 18] for the numerical Besov p,θ spaces Result 4.3 Let l1 = l2 = · · · = ln = m, s1 = s2 = · · · = sn = σ, and p = q Then for all E ∈ UMD and |α| ≤ m we obtain that the continuous embedding... elliptic equations, Numerical Functional Analysis and Optimization 24 (2003), no 1-2, 1–15 Veli B Shakhmurov 21 [7] J.-P Aubin, Abstract boundary-value operators and their adjoints, Rendiconti del Seminario Matematico della Universit` di Padova 43 (1970), 1–33 a [8] O V Besov, V P Il in, and S M Nikol’ski˘, Integral representations of functions, and embedding ı theorems, Izdat “Nauka”, Moscow, 1975 [9] D... Embedding theorems and their applications to degenerate equations, Differential Equa[29] tions 24 (1988), no 4, 475–482 22 Embedding and B-regular operators [30] , Coercive boundary value problems for regular degenerate differential-operator equations, Journal of Mathematical Analysis and Applications 292 (2004), no 2, 605–620 , Embedding operators and maximal regular differential-operator equations in. .. positive in lq Then from Theorem 3.3 for sk /(lk + sk ) + s j /(l j + s j ) ≤ 1, k, j = 1,2, ,n and 0 ≤ μ ≤ 1 − κ , κ = n=1 (αk + 1/ p1 − 1/ p2 )/lk we obtain the contink σ(1−κ −μ) σ ) and the corresponding estimate uous embedding Dα B l+s (Ω;lq ,lq ) ⊂ B s 2 ,θ (Ω;lq p1 ,θ p (3.23) Veli B Shakhmurov 13 It should not be that the above embedding has not been obtained with a classical method until now 5 Maximal. .. (1966), no 2, 285–346 [18] S G Kre˘n, Linear Differential Equations in Banach Space, American Mathematical Society, ı Rhode Island, 1971 [19] J Lindenstrauss and L Tzafriri, Classical Banach Spaces II Function Spaces, Results in Mathematics and Related Areas, vol 97, Springer, New York, 1979 [20] J.-L Lions and E Magenes, Probl`mes aux limites non homog´nes VI [Problems and limites non e e homogenes], Journal... 5 Maximal B-regular DOE in Rn Consider the following differential-operator equation aα (x)Dα u + Aλ (x)u + (L + λ)u = |α:.l|=1 Aα (x)Dα u = f (5.1) |α:.l| 0, (5.35) where F = L p (G;E) and L is the differential operator in Bs (Rn ;E) generated by the probp,θ lem (5.1) In view of Result 4.3 the operator L is positive in Bs (Rn ;E) for ϕ ∈ (0,π/2) p,θ Then by virtue of [4, Corollary 8.9] we obtain the assertion... obtain the assertion Remark 5.6 There are lots of positive operators in concrete Banach spaces Therefore, putting concrete Banach spaces instead of E and concrete positive differential, pseudo differential operators, or finite, in nite matrices, and so forth, instead of operator A on DOE (5.1), by virtue of Theorem 5.2 we can obtain the maximal regularity of different class of BVP’s for partial differential... References [1] S Agmon and L Nirenberg, Properties of solutions of ordinary differential equations in Banach space, Communications on Pure and Applied Mathematics 16 (1963), 121–239 [2] M S Agranoviˇ and M I Viˇik, Elliptic problems with a parameter and parabolic problems of c s general type, Uspekhi Matematicheskikh Nauk 19 (1964), no 3 (117), 53–161 [3] H Amann, Linear and Quasilinear Parabolic Problems . EMBEDDING THEOREMS IN BANACH-VALUED B-SPACES AND MAXIMAL B-REGULAR DIFFERENTIAL-OPERATOR EQUATIONS VELI B. SHAKHMUROV Received 28 September. coefficients in principal part and the maximal B-regularity of parabolic Cauchy problem are obtained. In applications, the in nite systems of the quasielliptic partial differential equations and the. distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Embedding theorems in function spaces have been studied in [8, 35, 37, 38]. A com- prehensive introduction