EMBEDDING THEOREMS IN BANACH-VALUED B-SPACES AND MAXIMAL B-REGULAR DIFFERENTIAL-OPERATOR EQUATIONS

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 spacesB^l_p,θ(Rⁿ;E0,E) are stud- ied; hereE0 and Eare two Banach spaces. The most regular spacesEαare found such that the mixed differential operatorsD^αare bounded fromB^l_p,θ(Rⁿ;E0,E) toB^s_q,θ(Rⁿ;Eα), whereEαare interpolation spaces betweenE0andEdepending onα=(α1,α2,. . .,αn) and l=(l1,l2,. . .,ln). 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 A com- prehensive introduction to the theory of embedding of function spaces and historical references may be also found in [37].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∈L2

0,T;H0

, u^(m)∈L2(0,T;H), (1.1)

then

u⁽ⁱ⁾∈L2

0,T;H,H0

i/m

, i=1, 2,. . .,m−1, (1.2) whereH0,H are Hilbert spaces,H0is continuously and densely embedded inH, where [H0,H]θare interpolation spaces betweenH0andH for 0≤θ≤1. The similar questions for anisotropic Sobolev spacesW_p^l(Ω;H0,H),Ω⊂Rⁿand for corresponding weighted

Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2006, Article ID 16192, Pages1–22 DOI10.1155/JIA/2006/16192

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 A comprehensive introduction to DOE’s and historical references may be found in [15,18,40].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 maximalL_pregularity 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].

LetE0,Ebe Banach spaces such thatE0is continuously and densely embedded inE.

In the present paper,E-valued Besov spacesB^l+s_p,θ(Rⁿ;E0,E)=B^s_p,θ(Rⁿ;E0)∩B^l+s_p,θ(Rⁿ;E) are introduced and called Besov-Lions type spaces. The most regular interpolation classEα

betweenE0 and Eis found such that the appropriate mixed differential operatorsD^α are bounded fromB^l+s_p,q(Rⁿ;E0,E) toB^s_p,q(Rⁿ;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 2collects notations and definitions.Section 3presents the embedding theorems in Besov-Lions type spaces

B^s+l_p,qRⁿ;E0,E. (1.3)

Section 4contains applications of the underlying embedding theorem to vector-valued function spaces. Section 5is devoted to the maximal regularity (in B^s_p,q(Rⁿ;E)) of the certain class of anisotropic DOE with variable coefficients in principal part. Then by us- ing these results the maximalB-regularity of the parabolic Cauchy problem is shown. In Section 6these 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

LetEbe a Banach space. LetLp(Ω;E) denote the space of all strongly measurableE-valued functions that are defined onΩ⊂Rⁿwith the norm

fLp(Ω;E)= f(x)_E^pdx

1/ p

, 1≤p <∞, fL∞(Ω;E)=ess sup

x∈Ω

f(x)_E, x=

x1,x2,. . .,xn

.

(2.1)

The Banach spaceEis said to be aζ-convex space (see [9,11,12,19]) if there exists onE×Ea symmetric real-valued functionζ(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, foru ≤1≤ v. (2.2)

Aζ-convex spaceEis often called a UMD-space and written asE∈UMD. It is shown in [9] that the Hilbert operator

(H f)(x)=lim

ε→0

|x−y|>ε

f(y)

x−yd y (2.3)

is bounded inLp(R;E), p∈(1,∞) for those and only those spacesE, which possess the property of UMD spaces. The UMD spaces include, for example,L_p,l_p spaces and the Lorentz spacesLpq,p,q∈(1,∞).

Let C be the set of complex numbers and let Sϕ=

λ;λ∈C,|argλ−π| ≤π−ϕ∪ {0}, 0< ϕ≤π. (2.4) A linear operatorAis said to be aϕ-positive in a Banach spaceE, with boundM >0 if D(A) is dense onEand

(A−λI)⁻¹_L(E)≤M1 +|λ|−1

(2.5) withλ∈Sϕ,ϕ∈(0,π],Iis identity operator inE, andL(E) is the space of all bounded linear operators inE. SometimesA+λIwill be written asA+λand denoted byAλ. It is known [37, Section 1.15.1] that there exist fractional powersA^θof the positive operator A. LetE(A^θ) denote the spaceD(A^θ) with the graphical norm

uE(A^θ)=

u^p+A^θu^{p1/ p}, 1≤p <∞,−∞< θ <∞. (2.6) LetE0andEbe two Banach spaces. By (E0,E)σ,p, 0< σ <1, 1≤p≤ ∞we will denote the interpolation spaces obtained from{E0,E}by theK-method (see, e.g., [37, Section 1.3.1] or [10]).

LetS(Rⁿ;E) denote a Schwartz class, that is, the space of allE-valued rapidly decreasing smooth functionsϕonRⁿ.E=C will be denoted byS(Rⁿ). LetS(Rⁿ;E) denote the space ofE-valued tempered distributions, that is, the space of continuous linear operators from S(Rⁿ) toE.

Letα=(α1,α2,. . .,αn),αiare integers. AnE-values generalized functionD^αf is called a generalized derivative in the sense of Schwartz distributions of the generalized function

f ∈S(Rⁿ,E) if the equality

D^αf,ϕ=(−1)^|α|f,D^αϕ (2.7) holds for allϕ∈S(Rⁿ).

By using (2.7) the following relations FD^α_xf=

iξ1

α1

,. . .,iξn

αnf, D^α_ξF(f)=F−ixn

α1

,. . .,−ixn

αn

f (2.8) are obtained for all f ∈S(Rⁿ;E).

LetL^∗_θ(E) denote the space of allE-valued function spaces such that uL^∗_θ(E)=

_∞

0

u(t)^θ_Edt t

1/θ

<∞, 1≤θ <∞, uL^∗∞(E)= sup

0<t<∞

u(t)_E. (2.9)

Lets=(s1,s2,. . .,sn) andsk>0. LetFdenote the Fourier transform. Fourier-analytic rep- resentation ofE-valued Besov space onRⁿis defined as

B^s_p,θRⁿ;E=

u∈SRⁿ;E,uB^s_p,θ(Rⁿ;E)

= F⁻¹

n k=1

t^κk−sk1 +ξk^κk

e^−t|ξ|2Fu

L^∗_θ(Lp(Rⁿ;E))

,

p∈(1,∞), θ∈[1,∞],κk> s_k

.

(2.10)

It should be noted that the norm of Besov space do not depend onκk. Sometimes we will writeuB^s_p,θin place ofuB^s_p,θ(Rⁿ;E).

Letl=(l1,l2,. . .,ln),s=(s1,s2,. . .,sn), wherelkare integers andskare positive numbers.

LetW^lB^s_p,θ(Rⁿ;E) denote anE-valued Sobolev-Besov space of all functionsu∈B^s_p,θ(Rⁿ;E) such that they have the generalized derivativesD^l_k^ku=∂^lku/∂x_k^lk∈B^s_p,θ(Rⁿ;E),k=1, 2,. . .,n with the norm

uW^lB^s_p,θ(Rⁿ;E)= uB^s_p,θ(Rⁿ;E)+ n k=1

D^l_k^ku_Bs

p,θ(Rⁿ;E)<∞. (2.11) LetE0is continuously and densely embedded intoE.W^lB^s_p,θ(Rⁿ;E0,E) denotes a space of all functionsu∈B^s_p,θ(Rⁿ;E0)∩W^lB^s_p,θ(Rⁿ;E) with the norm

uW^lB^s_p,θ= uW^lB^s_p,θ(Rⁿ;E0,E)= uB^s_p,θ(Rⁿ;E0)+ n k=1

D_k^lku

B^s_p,θ(Rⁿ;E)<∞. (2.12) Letl=(l1,l2,. . .,l_n),s=(s1,s2,. . .,s_n), wheres_kare real numbers andl_kare positive num- bers.B^l+s_p,θ(Rⁿ;E0,E) denotes a space of all functionsu∈B^s_p,θ(Rⁿ;E0)∩B^l+s_p,θ(Rⁿ;E) with the norm

u_B^s+l_p,θ(Rⁿ_;E0,E)= uB^s_p,θ(Rⁿ;E0)+u_B^l+s_p,θ(Rⁿ_;E). (2.13) ForE0=Ethe spaceB^l+s_p,θ(Rⁿ;E0,E) will be denoted byB^l+s_p,θ(Rⁿ;E).

Letmbe a positive integer.C(Ω;E) andC^m(Ω;E) will denote the spaces of allE-valued bounded continuous andm-times continuously differentiable functions onΩ, respec- tively. We set

C_b(Ω;E)₌

u_∈C(Ω;E), lim

|x|→∞u(x) exists

. (2.14)

LetE1andE2be two Banach spaces. A functionΨ∈C^m(Rⁿ;L(E1,E2)) is called a multi- plier fromB^s_p,θ(Rⁿ;E1) toB^s_q,θ(Rⁿ;E2) forp∈(1,∞) andq∈[1,∞] if the mapu→Ku= F⁻¹Ψ(ξ)Fu,u∈S(Rⁿ;E1), is well defined and extends to a bounded linear operator

K:B^s_p,θRⁿ;E1

−→B^s_q,θRⁿ;E2

. (2.15)

The set of all multipliers fromB^s_p,θ(Rⁿ;E1) toB^s_q,θ(Rⁿ;E2) will be denoted byM^q,θ_p,θ(s,E1, E2).E1=E2=Ewill be denoted byM^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=

Ψh∈M^q,θ_p,θs,E1,E2

,h=

h1h2,. . .,h_n∈K (2.16) be a collection of multipliers inM^q,θ_p,θ(s,E1,E2). We say thatHkis a uniform collection of multipliers if there exists a constantM0>0, independent onh∈K, such that

F⁻¹ΨhFu_Bs

p,θ(Rⁿ;E2)≤M0uB_q,θ^s (Rⁿ;E1) (2.17) for allh∈Kandu∈S(Rⁿ;E1).

Letβ=(β1,β2,. . .,β_n) be multiindexes. We also define V_n=

ξ=

ξ1,ξ2,. . .,ξ_n∈Rⁿ,ξ_i=0,i=1, 2,. . .,n, Un=

β:|β| ≤n, ξ^β=ξ1^β1ξ2^β2,. . .,ξn^βn, ν= 1 p⁻

1

q. (2.18)

Definition 2.1. A Banach spaceEsatisfies aB-multiplier condition with respect to p,q, θ, ands(or with respect to p,θ, ands for the case of p=q) whenΨ∈Cⁿ(Rⁿ;L(E)), 1≤p≤q≤ ∞,β∈U_n, andξ∈V_nif the estimate

ξ1^β1+νξ2^β2+ν,. . .,ξn^βn+νD^βΨ(ξ)_L(E)≤C (2.19)

impliesΨ∈M^q,θ_p,θ(s,E).

Remark 2.2. Definition 2.1is a combined restriction toE,p,q,θ, ands. This condition is sufficient for our main aim. Nevertheless, it is well known that there are Banach spaces satisfying theB-multiplier condition for isotropic case andp=q, for example, the UMD spaces (see [4,14]).

A Banach spaceEis said to have a local unconditional structure (l.u.st.) if there exists a constantC <∞such that for any finite-dimensional subspaceE0ofEthere exists a finite- dimensional spaceFwith an unconditional basis such that the natural embeddingE0⊂E factors asABwithB:E0→F,A:F→E, andAB ≤C. All Banach lattices (e.g.,L_p, Lp,q, Orlicz spaces,C[0, 1]) have l.u.st.

The expressionuE1∼uE2means that there exist the positive constantsC1andC2

such that

C1u_E1≤ u_E2≤C2u_E1 (2.20) for allu∈E1∩E2.

Letα1,α2,. . .,αnbe nonnegative and letl1,l2,. . .,lnbe positive integers and let 1≤p≤q≤ ∞, 1≤θ≤ ∞, |α:.l| =

n k=1

αk

lk, κ= n k=1

αk+ 1/ p−1/q lk , D^α=D^α1¹D^α2²,. . .,D_n^αn= ∂^|α|

∂x1^α1∂x2^α2,. . .,∂x^αnⁿ, |α| = n k=!

αk.

(2.21)

Consider in general, the anisotropic differential-operator equation (L+λ)u=

|α:.l|=1

aα(x)D^αu+Aλ(x)u+

|α:.l|<1

Aα(x)D^αu= f (2.22) inB^s_p,θ(Rⁿ;E), whereaαare complex-valued functions andA(x),Aα(x) are possibly un- bounded operators in a Banach spaceE, here the domain definitionD(A)=D(A(x)) of operatorA(x) does not depend onx. Forl1=l2=,. . .,=l_nwe obtain isotropic equations containing the elliptic class of DOE.

The function belonging to spaceB^s+l_p,θ(Rⁿ;E(A),E) and satisfying (2.22) a.e. onRⁿ is said to be a solution of (2.22) onRⁿ.

Definition 2.3. The problem (2.22) is said to be aB-separable (orB^s_p,θ(Rⁿ;E)-separable) if the problem (2.22) for all f ∈B^s_p,θ(Rⁿ;E) has a unique solutionu∈B^s+l_p,θ(Rⁿ;E(A),E) and

AuB^s_p,θ(Rⁿ;E)+

|α:l|=1

D^αu_Bs p,θ

Rⁿ;E≤CfB^s_p,θ(Rⁿ;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∈Rⁿ, (2.24) whereLis a realization differential operator inB^s_p,θ(Rⁿ;E) generated by problem (2.22), that is,

D(L)=B^s+l_p,θRⁿ;E(A),E, Lu=

|α:.l|=1

aα(x)D^αu+A(x)u+

|α:.l|<1

Aα(x)D^αu. (2.25) We say that the parabolic Cauchy problem (2.24) is said to be a maximalB-regular, if for all f ∈B^s_p,θ(Rⁿ⁺¹₊ ;E) there exists a unique solutionusatisfying (2.24) almost every- where onRⁿ⁺¹₊ and there exists a positive constantCindependent on f, such that it has the estimate

∂u(y,x)

∂y

B^s_p,θ(Rⁿ⁺¹+ ;E)+LuB^s_p,θ(Rⁿ⁺¹+ ;E)≤CfB^s_p,θ(Rⁿ⁺¹+ ;E). (2.26) 3. Embedding theorems

In this section we prove the boundedness of the mixed differential operatorsD^α in the Besov-Lions type spaces.

Lemma 3.1. LetAbe a positive operator in a Banach spaceE, letbbe a positive number, r=(r1,r2,. . .,rn),α=(α1,α2,. . .,αn), andl=(l1,l2,. . .,ln), whereϕ∈(0,π],rk∈[0,b],lk

are positive andα_k,k=1, 2,. . .,n, are nonnegative integers such thatκ= |(α+r) :l| ≤1.

For 0< h≤h0<∞and 0≤μ≤1−κthe operator-function

Ψ(ξ)=Ψh,μ(ξ)=ξ1^r1ξ2^r2,. . .,ξ_n^rn(iξ)^αA^1−κ−μh^−μA+η(ξ)⁻¹ (3.1) is a bounded operator inEuniformly with respect toξandh, that is, there is a constantC_μ such that

Ψh,μ(ξ)_L(E)≤C_μ (3.2)

for allξ∈Rⁿ, where

η=η(ξ)= n k=1

ξk^lk+h⁻¹. (3.3)

Proof. Since−η(ξ)∈S(ϕ), for allϕ∈(0,π] andAis aϕ-positive inE, then the operator A+η(ξ) is invertiable inE. Let

u=h^−μA+η(ξ)⁻¹f . (3.4)

Then

Ψ(ξ)f_E=(hA)^1−κ−μu_Eh^−(1−μ)h^1/l1ξ1^α1+r1,. . .,h^1/lnξn^αn+rn. (3.5) Using the moment inequality for powers of positive operators, we get a constantCμde- pending only onμsuch that

Ψ(ξ)f_E≤Cμh^−(1−μ)hAu^1−κ−μu^κ+μh^1/l1ξ1^α1+r1,. . .,h^1/lnξn^αn+rn. (3.6) Now, we apply the Young inequality, which states thatab≤a^k1/k1+b^k2/k2for any positive real numbersa,bandk1,k2with 1/k1+ 1/k2=1 to the product

hAu^1−κ−μ

u^κ+μh^1/l1ξ1^α1+r1,. . .,h^1/lnξn^αn+rn

(3.7) withk1=1/(1−κ−μ),k2=1/(κ+μ) to get

Ψ(ξ)f_E≤C_μh^−(1−μ)(1−κ−μ)hAu

+(κ+μ)h^1/l1ξ1^{(α1+r1)/(κ+μ)},. . .,h^1/lnξn^{(αn+rn)/(κ+μ)}u . (3.8) Since

n i=1

αi+ri

(κ+μ)⁼ 1 κ+μ

n i=1

αi+ri

l_i ⁼ κ

κ+μ^≤1, (3.9)

there exists a constantM0independent onξ, such that ξ1^{(α1+r1)/(κ+μ)},. . .,ξ_n^{(αn+rn)/(κ+μ)}≤M0

1 +

n k=1

ξ_k^lk

(3.10) for allξ∈Rⁿ. Substituting this on the inequality (3.8) and absorbing the constant coeffi- cients inC_μ, we obtain

ψ(ξ)f≤Cμ

h^μ

Au+

n k=1

ξk^lku

+h^−(1−μ)u

. (3.11)

Substituting the value ofuwe get ψ(ξ)f≤C_μh^μ

AA+η(ξ)⁻¹f+ n k=1

ξ_k^lkA+η(ξ)⁻¹f

+h^−(1−μ)

A+η(ξ)⁻¹f.

(3.12)

By using the properties of the positive operatorAfor allf ∈Ewe obtain from (3.12)

Ψ(ξ)f_E≤CμfE. (3.13)

Lemma 3.2. LetEbe a UMD space with l.u.st.,p∈(1,∞),θ∈[1,∞] and let for allk,j∈ (1,n)

s_k

lk+sk+ sj

lj+sj ≤1. (3.14)

Then the spacesB^l+s_p,θ(Rⁿ;E) andW^lB^s_p,θ(Rⁿ;E) are coincided.

Proof. In the first step we show that the continuous embeddingW^lB^s_p,θ(Rⁿ;E)⊂B^l+s_p,θ(Rⁿ; E) holds, that is, there is a positive constantCsuch that

u_B^l+s_p,θ(Rⁿ_;E)≤CuW^lB^s_p,θ(Rⁿ;E) (3.15) for allu∈W^lB^s_p,θ(Rⁿ;E). For this aim by using the Fourier-analytic definition of anE- valued Besov space and the space W^lB^s_p,θ(Rⁿ;E) it is sufficient to prove the following estimate:

F⁻¹

n k=1

t^{κk−lk−sk}1 +ξk^κk

e^−t|ξ|2Fu

Lθp

≤CF⁻¹ n k=1

t^κk−sk1 +ξk^κk

e^−t|ξ|2Fυ

Lθp

, (3.16) where

Lθ p=L^∗_θLp

Rⁿ;E, υ=F⁻¹

1 + n k=1

ξ_k^lk

Fu. (3.17)

To see this, it is sufficient to show that the function φ(ξ)=

n k=1

1 +ξk^lk+sk+δn

k=1

1 +ξk^sk+δ−1 1 +

n k=1

ξk^lk−1

, δ >0 (3.18) is Fourier multiplier inLp(Rⁿ;E). It is clear to see that forβ∈Unandξ∈Vn

ξ1^β1ξ2^β2,. . .,ξn^βnD^βφ(ξ)_L(E)≤C. (3.19) Then in view of [41, Proposition 3] we obtain that the functionφis Fourier multiplier in L_p(Rⁿ;E).

In the second step we prove that the embeddingB^l+s_p,θ(Rⁿ;E)⊂W^lB^s_p,θ(Rⁿ;E) is contin- uous. In a similar way as in the first step we show that fors_k/(l_k+s_k) +s_j/(l_j+s_j)≤1 the function

ψ(ξ)= _n

k=1

1 +ξ_k^sk+δ1 + n k=1

ξ_k^lk _n

k=1

1 +ξ_k^lk+sk+δ

−1

(3.20) is Fourier multiplier inLp(Rⁿ;E). So, we obtain for allu∈B^l+s_p,θ(Rⁿ;E) the estimate

F⁻¹

n k=1

t^κk−sk1 +ξk^κk 1 +

n k=1

ξ_k^lk

e^−t|ξ|2Fu Lθp

≤CF⁻¹ n k=1

t^{κk−lk−sk}1 +ξk^κk

e^−t|ξ|2Fu

Lθp

.

(3.21)

It implies the second embedding. This completes the prove ofLemma 3.2.

Theorem 3.3. Suppose the following conditions hold:

(1)Eis a UMD space with l.u.st. satisfying theB-multiplier condition with respect to p,q∈(1,∞),θ∈[1,∞], ands=(s1,s2,. . .,sn), whereskare positive numbers;

(2)α=(α1,α2,. . .,α_n),l=(l1,l2,. . .,l_n), whereα_kare nonnegative,l_kare positive integers, andsk such thatsk/(lk+sk) +sj/(lj+sj)≤1 fork,j=1, 2,. . .,nand 0≤μ≤1−κ,κ=

|(α+ 1/ p−1/q) :l|;

(3)Ais aϕ-positive operator inE, whereϕ∈(0,π] and 0< h≤h0<∞. Then the following embedding

D^αB^l+s_p,θRⁿ;E(A),E⊂B_q,θ^s Rⁿ;EA^1−κ−μ (3.22) is continuous and there exists a positive constantC_μdepending only onμ, such that

D^αu_Bs

q,θ(Rⁿ;E(A^1−κ−μ))≤C_μh^μu_B^l+s

p,θ(Rⁿ;E(A),E)+h^−(1−μ)uB^s_p,θ(Rⁿ;E)

(3.23)

for allu∈B^l+s_p,θ(Rⁿ;E(A),E).