EMBEDDING THEOREMS IN BANACH-VALUED
B
-SPACES
AND MAXIMAL
B
-REGULAR DIFFERENTIAL-OPERATOR
EQUATIONS
VELI B. SHAKHMUROVReceived 28 September 2004; Revised 8 November 2005; Accepted 4 May 2006
The embedding theorems in anisotropic Besov-Lions type spacesBlp,θ(Rn;E
0,E) are
stud-ied; hereE0 and Eare two Banach spaces. The most regular spacesEαare found such
that the mixed differential operatorsDαare bounded fromBl
p,θ(Rn;E0,E) toBsq,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 A com-prehensive introduction to the theory of embedding of function spaces and historical references may be also found in [37].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)∈L
2(0,T;H), (1.1)
then
u(i)∈L20,T;H,H0i/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 spacesWl
p(Ω;H0,H),Ω⊂Rnand for corresponding weighted
Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2006, Article ID 16192, Pages1–22
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 A comprehensive introduction to DOE’s and historical references may be found in [15,18,40].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 maximalLpregularity 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 spacesBl+sp,θ(Rn;E
0,E)=Bsp,θ(Rn;E0)∩Bl+sp,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 operatorsDα
are bounded fromBl+s
p,q(Rn;E0,E) toBsp,q(Rn;Eα). 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
Bs+lp,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 Bs
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)
p Edx
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
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,Lp,lp spaces and the
Lorentz spacesLpq,p,q∈(1,∞).
LetCbe 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)≤M
1 +|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 powersAθof the positive operator
A. LetE(Aθ) denote the spaceD(Aθ) with the graphical norm
uE(Aθ)=up+Aθup1/ 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=Cwill 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 functionDαf is called
a generalized derivative in the sense of Schwartz distributions of the generalized function f ∈S(Rn,E) if the equality
Dαf,ϕ=(−1)|α|f,Dαϕ (2.7)
holds for allϕ∈S(Rn).
By using (2.7) the following relations
FDαxf
=iξ1
α1
,. . .,iξn
αn
f, DαξF(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 E
dt t
1/θ
<∞, 1≤θ <∞, uL∗∞(E)= sup
0<t<∞ u(t)
Lets=(s1,s2,. . .,sn) andsk>0. LetFdenote the Fourier transform. Fourier-analytic
rep-resentation ofE-valued Besov space onRnis defined as
Bsp,θRn;E=
u∈SRn;E,uBs p,θ(Rn;E)
=F−1
n
k=1
tκk−sk1 +ξ
kκk
e−t|ξ|2Fu
L∗θ(Lp(Rn;E)) ,
p∈(1,∞), θ∈[1,∞],κk> sk
.
(2.10)
It should be noted that the norm of Besov space do not depend onκk. Sometimes we
will writeuBs
p,θin place ofuBsp,θ(Rn;E).
Letl=(l1,l2,. . .,ln),s=(s1,s2,. . .,sn), wherelkare integers andskare positive numbers.
LetWlBs
p,θ(Rn;E) denote anE-valued Sobolev-Besov space of all functionsu∈Bsp,θ(Rn;E)
such that they have the generalized derivativesDlk
ku=∂lku/∂xklk∈Bsp,θ(Rn;E),k=1, 2,. . .,n
with the norm
uWlBs
p,θ(Rn;E)= uBsp,θ(Rn;E)+
n
k=1
Dlk
kuBs
p,θ(Rn;E)<∞. (2.11)
LetE0is continuously and densely embedded intoE.WlBsp,θ(Rn;E0,E) denotes a space of
all functionsu∈Bsp,θ(Rn;E
0)∩WlBsp,θ(Rn;E) with the norm
uWlBs
p,θ= uWlBsp,θ(Rn;E0,E)= uBsp,θ(Rn;E0)+ n
k=1
Dlk
kuBs p,θ(Rn;E)
<∞. (2.12)
Letl=(l1,l2,. . .,ln),s=(s1,s2,. . .,sn), whereskare real numbers andlkare positive
num-bers.Bl+sp,θ(Rn;E
0,E) denotes a space of all functionsu∈Bsp,θ(Rn;E0)∩Bl+sp,θ(Rn;E) with the
norm
uBs+l
p,θ(Rn;E0,E)= uBsp,θ(Rn;E0)+uBl+s
p,θ(Rn;E). (2.13) ForE0=Ethe spaceBl+sp,θ(Rn;E0,E) will be denoted byBl+sp,θ(Rn;E).
Letmbe a positive integer.C(Ω;E) andCm(Ω;E) will denote the spaces of allE-valued
bounded continuous andm-times continuously differentiable functions onΩ, respec-tively. We set
Cb(Ω;E)=
u∈C(Ω;E), lim
|x|→∞u(x) exists
. (2.14)
LetE1andE2be two Banach spaces. A functionΨ∈Cm(Rn;L(E1,E2)) is called a
multi-plier fromBsp,θ(Rn;E
1) toBsq,θ(Rn;E2) forp∈(1,∞) andq∈[1,∞] if the mapu→Ku=
F−1Ψ(ξ)Fu,u∈S(Rn;E
1), is well defined and extends to a bounded linear operator
The set of all multipliers fromBs
p,θ(Rn;E1) toBsq,θ(Rn;E2) will be denoted byMq,θp,θ(s,E1,
E2).E1=E2=Ewill be denoted byMq,θp,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
Hk=
Ψh∈Mq,θp,θ
s,E1,E2
,h=h1h2,. . .,hn
∈K (2.16)
be a collection of multipliers inMq,θp,θ(s,E1,E2). We say thatHkis a uniform collection of
multipliers if there exists a constantM0>0, independent onh∈K, such that
F−1Ψ hFuBs
p,θ(Rn;E2)≤M0uBqs,θ(Rn;E1) (2.17)
for allh∈Kandu∈S(Rn;E 1).
Letβ=(β1,β2,. . .,βn) be multiindexes. We also define
Vn=
ξ=ξ1,ξ2,. . .,ξn
∈Rn,ξ
i=0,i=1, 2,. . .,n
,
Un=
β:|β| ≤n, ξβ=ξβ1 1 ξ
β2 2 ,. . .,ξ
βn
n , 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∈Un, andξ∈Vnif the estimate
ξ1β1+ν
ξ2β2 +ν
,. . .,ξnβn +ν
DβΨ(ξ)L(E)≤C (2.19)
impliesΨ∈Mq,θp,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.,Lp,
Lp,q, Orlicz spaces,C[0, 1]) have l.u.st.
The expressionuE1∼uE2means that there exist the positive constantsC1andC2
such that
C1uE1≤ uE2≤C2uE1 (2.20)
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 ,
Dα=Dα1
1 Dα22,. . .,Dnαn=
∂|α| ∂xα1
1 ∂x2α2,. . .,∂xαnn, |α| = n
k=!
αk.
(2.21)
Consider in general, the anisotropic differential-operator equation
(L+λ)u= |α:.l|=1
aα(x)Dαu+Aλ(x)u+
|α:.l|<1
Aα(x)Dαu= f (2.22)
inBsp,θ(Rn;E), wherea
αare 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=,. . .,=lnwe obtain isotropic equations
containing the elliptic class of DOE.
The function belonging to spaceBs+lp,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 (orBsp,θ(Rn;E)-separable) if
the problem (2.22) for all f ∈Bsp,θ(Rn;E) has a unique solutionu∈Bs+l
p,θ(Rn;E(A),E) and
AuBs p,θ(Rn;E)+
|α:l|=1
Dαu Bs
p,θ
Rn;E≤CfBsp,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∈R
n, (2.24)
whereLis a realization differential operator inBs
p,θ(Rn;E) generated by problem (2.22),
that is,
D(L)=Bs+lp,θRn;E(A),E, Lu= |α:.l|=1
aα(x)Dαu+A(x)u+
|α:.l|<1
Aα(x)Dαu. (2.25)
We say that the parabolic Cauchy problem (2.24) is said to be a maximalB-regular, if for all f ∈Bsp,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
Bs p,θ(Rn++1;E)
+LuBs
p,θ(Rn++1;E)≤CfBsp,θ(Rn++1;E). (2.26)
3. Embedding theorems
In this section we prove the boundedness of the mixed differential operatorsDα in the
Lemma3.1. 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 andαk,k=1, 2,. . .,n, are nonnegative integers such thatκ= |(α+r) :l| ≤1.
For0< h≤h0<∞and0≤μ≤1−κthe operator-function
Ψ(ξ)=Ψh,μ(ξ)=ξ1r1ξ2r2,. . .,ξnrn(iξ)αA1−κ−μh−μ
A+η(ξ)−1 (3.1)
is a bounded operator inEuniformly with respect toξandh, that is, there is a constantCμ
such that
Ψh,μ(ξ)
L(E)≤Cμ (3.2)
for allξ∈Rn, where
η=η(ξ)=
n
k=1
ξklk
+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−μuEh−(1−μ)h1/l1ξ1α1+r1,. . .,h1/lnξnαn+rn. (3.5)
Using the moment inequality for powers of positive operators, we get a constantCμ
de-pending only onμsuch that Ψ(ξ)f
E≤Cμh−(1−μ)hAu1−κ−μuκ+μh1/l1ξ1 α1+r1
,. . .,h1/lnξ
nαn+rn. (3.6)
Now, we apply the Young inequality, which states thatab≤ak1/k
1+bk2/k2for any positive
real numbersa,bandk1,k2with 1/k1+ 1/k2=1 to the product
hAu1−κ−μuκ+μh1/l1ξ
1α1+r1,. . .,h1/lnξnαn+rn
(3.7)
withk1=1/(1−κ−μ),k2=1/(κ+μ) to get
Ψ(ξ)f
E≤Cμh−(1−μ)
(1−κ−μ)hAu
+(κ+μ)h1/l1ξ
1(α1+r1)/(κ+μ),. . .,
h1/lnξ
n(αn+rn)/(κ+μ)u
. (3.8)
Since
n
i=1
αi+ri
(κ+μ)= 1 κ+μ
n
i=1
αi+ri
li =
κ
there exists a constantM0independent onξ, such that
ξ1(α1+r1)/(κ+μ)
,. . .,ξn(αn+rn)/(κ+μ)≤M0
1 +
n
k=1
ξklk
(3.10)
for allξ∈Rn. Substituting this on the inequality (3.8) and absorbing the constant coeffi
-cients inCμ, we obtain
ψ(ξ)f≤Cμhμ
Au+
n
k=1
ξklk u
+h−(1−μ)u
. (3.11)
Substituting the value ofuwe get ψ(ξ)f≤Cμhμ
A
A+η(ξ)−1f+
n
k=1
ξklkA
+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)
Lemma3.2. LetEbe a UMD space with l.u.st.,p∈(1,∞),θ∈[1,∞]and let for allk,j∈ (1,n)
sk
lk+sk+
sj
lj+sj ≤1. (3.14)
Then the spacesBl+sp,θ(Rn;E)andWlBs
p,θ(Rn;E)are coincided.
Proof. In the first step we show that the continuous embeddingWlBs
p,θ(Rn;E)⊂Bl+sp,θ(Rn;
E) holds, that is, there is a positive constantCsuch that uBl+s
p,θ(Rn;E)≤CuWlB s
p,θ(Rn;E) (3.15)
for allu∈WlBs
p,θ(Rn;E). For this aim by using the Fourier-analytic definition of an
E-valued Besov space and the space WlBs
p,θ(Rn;E) it is sufficient to prove the following
estimate: F−1
n
k=1
tκk−lk−sk1 +ξ
kκke−t|ξ| 2
Fu
Lθp
≤CF−1 n
k=1
tκk−sk1 +ξ
kκke−t|ξ| 2
Fυ
Lθp ,
(3.16)
where
Lθ p=L∗θ
Lp
Rn;E, υ=F−1
1 +
n
k=1
ξlk
k
To see this, it is sufficient to show that the function
φ(ξ)=
n
k=1
1 +ξklk+sk+δ
n
k=1
1 +ξksk+δ
−1
1 +
n
k=1
ξklk −1
, δ >0 (3.18)
is Fourier multiplier inLp(Rn;E). It is clear to see that forβ∈Unandξ∈Vn
ξ1β1
ξ2β2,. . .,ξnβnDβφ(ξ)L(E)≤C. (3.19)
Then in view of [41, Proposition 3] we obtain that the functionφis Fourier multiplier in Lp(Rn;E).
In the second step we prove that the embeddingBl+s
p,θ(Rn;E)⊂WlBsp,θ(Rn;E) is
contin-uous. In a similar way as in the first step we show that forsk/(lk+sk) +sj/(lj+sj)≤1 the
function
ψ(ξ)= n
k=1
1 +ξksk+δ
1 +
n
k=1
ξklk
n
k=1
1 +ξklk+sk+δ
−1
(3.20)
is Fourier multiplier inLp(Rn;E). So, we obtain for allu∈Bl+sp,θ(Rn;E) the estimate
F−1
n
k=1
tκk−sk1 +ξ
kκk
1 +
n
k=1
ξlk
k
e−t|ξ|2Fu
Lθp
≤CF−1 n
k=1
tκk−lk−sk1 +ξ
kκke−t|ξ| 2
Fu
Lθp .
(3.21)
It implies the second embedding. This completes the prove ofLemma 3.2.
Theorem3.3. 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,. . .,αn),l=(l1,l2,. . .,ln), whereαkare nonnegative,lkare positive integers,
andsk such thatsk/(lk+sk) +sj/(lj+sj)≤1 fork,j=1, 2,. . .,nand0≤μ≤1−κ,κ=
|(α+ 1/ p−1/q) :l|;
(3)Ais aϕ-positive operator inE, whereϕ∈(0,π]and0< h≤h0<∞.
Then the following embedding
DαBl+s p,θ
Rn;E(A),E⊂Bs q,θ
Rn;EA1−κ−μ (3.22)
is continuous and there exists a positive constantCμdepending only onμ, such that
Dαu Bs
q,θ(Rn;E(A1−κ−μ))≤Cμ
hμu Bl+s
p,θ(Rn;E(A),E)+h
−(1−μ)u Bs
p,θ(Rn;E)
(3.23)
Proof. We have
Dαu Bs
q,θ(Rn;E(A1−κ−μ))=
A1−κ−μDαu Bs
q,θ(Rn;E) (3.24)
for allusuch that
Dαu Bs
q,θ(Rn;E(A1−κ−μ))<∞. (3.25) On the other hand by using the relation (2.8) we have
A1−α−μDαu=F−FA1−κ−μDαu=F−(iξ)αA1−κ−μFu. (3.26)
Since the operatorAis closure and does not depend onξ∈Rnhence denotingFubyu,
from the relations (3.24), (3.26) and by definition of the spaceWlBs
p,θ(Rn;E0,E) we have
Dαu Bs
q,θ(Rn;E(A1−κ−μ))
F−(iξ)αA1−κ−μu Bs
q,θ(Rn;E),
uWlBs
p,θ(Rn;E0,E)∼AuBsp,θ(Rn;E)+
n
k=1
F−1ξlk
kuBs p,θ(Rn;E).
(3.27)
By virtue ofLemma 3.2and by the above relations it is sufficient to prove that F−(iξ)αA1−κ−μu
Bs q,θ(Rn;E)
≤Cμ
hμ
F−Au Bs
p,θ(Rn;E)+
n
k=1
F−ξlk
kuBs p,θ(Rn;E)
+h−(1−μ)F−u Bs
p,θ(Rn;E)
. (3.28)
The inequality (3.23) will be followed if we prove the following inequality F−(iξ)αA1−κ−μu
Bs
p,θ(Rn;E)≤Cμ
F−hμ(A+η)u Bs
p,θ(Rn;E) (3.29)
for a suitableCμand for allu∈Bs+lp,θ(Rn;E(A),E), where
η=η(ξ)=
n
k=1
ξklk+h−1. (3.30)
Let us express the left-hand side of (3.29) as follows: F−(iξ)αA1−κ−μu
Bs
q,θ(Rn;E) (3.31)
=F−(iξ)αA1−κ−μhμ(A+η)−1hμ(A+η)uBs
q,θ(Rn;E). (3.32) (SinceAis the positive operator inEand−η(ξ)∈S(ϕ) so it is possible). By virtue of
Mq,θp,θ(s,E), which is uniform with respect toh. SinceEsatisfies the multiplier condition with respect top,q,θ, ands, then byDefinition 2.1in order to show thatΨ∈Mq,θp,θ(s,E), it suffices to show that there exists a constantMμ>0 with
ξ1β1+ν
ξ2β2+ν,. . .,ξnβn+νDξβΨ(ξ)L(E)≤Mμ (3.33)
for allβ∈Un,ξ∈Vn, and 0< h≤h0<∞. To see this, we applyLemma 3.1and get a
constantMμ>0 depending only onμsuch that
ξ1νξ
2ν,. . .,ξnνΨ(ξ)L(E)≤Mμ (3.34)
for allξ ∈Rn and ν=1/ p−1/q. This shows that the inequality (3.33) is satisfied for
β=(0,. . ., 0). We next consider (3.33) forβ=(β1,. . .,βn) whereβk=1 andβj=0 for
j=k. By differentiation of the operator-functionΨ(ξ), by virtue of the positivity ofA, and by using (3.34) we have
∂ξ∂
kΨ
(ξ)
L(E)≤
Mμξk− (1+ν)
, k=1, 2. . .,n. (3.35)
Repeating the above process we obtain the estimate (3.33). Thus the operator-function
Ψh,μ(ξ) is a uniform multiplier with respect toh, that is,
Ψh,μ∈HK⊂Mq,θp,θ(s,E), K=R+. (3.36)
This completes the proof ofTheorem 3.3.
Result 3.4. Let all conditions ofTheorem 3.3hold. Then for allu∈Bl+s
p,θ(Rn;E(A),E) we
have a multiplicative estimate Dαu
Bs
q,θ(Rn;E(A1−κ−μ))≤Cμu
1−μ Bl+s
p,θ(Rn;E(A),E)u
μ Bs
p,θ(Rn;E). (3.37) Indeed settingh= uBs
p,θ(Rn;E)· u −1 Bl+s
p,θ(Rn;E(A),E)in the estimate (3.23) we obtain the above estimate.
Remark 3.5. It seems from the proof ofTheorem 3.3that the extra condition to space E(Eis UMD space with l.u.st.) and the conditionsk/(lk+sk) +sj/(lj+sj)≤1 fork,j=
1, 2,. . .,nare due toLemma 3.2(here the l.u.st. condition for the spaceEis required due to using of Marcinkiewicz-Lizorkin type multiplier theorem [41] inLp(Rn;E) space).
There-fore, the proof ofTheorem 3.3implies the following.
Result 3.6. Suppose the following conditions hold:
(1)Eis a Banach space satisfying theB-multiplier condition with respect to p,q∈ (1,∞),θ∈[1,∞] ands=(s1,s2,. . .,sn), whereskare positive numbers;
(2)α=t(α1,α2,. . .,αn),l=(l1,l2,. . .,ln), whereαkare nonnegative andlk are positive
Then the following embedding
DαWlBs p,θ
Rn;E(A),E⊂Bs q,θ
Rn;EA1−κ−μ (3.38)
is continuous and there exists a positive constantCμdepending only onμsuch that
Dαu Bs
q,θ(Rn;E(A1−κ−μ))≤Cμ
hμuWlBs
p,θ(Rn;E(A),E)+h
−(1−μ)u Bs
p,θ(Rn;E)
(3.39)
for allu∈WlBs
p,θ(Rn;E(A),E).
Remark 3.7. The conditionsk/(lk+sk) +sj/(lj+sj)≤1 fork,j=1, 2,. . .,ninTheorem 3.3
arise due to anisotropic nature of spaceBsp,θ. For an isotropic case the above conditions hold without any assumptions.
4. Application to vector-valued function spaces
By virtue ofTheorem 3.3we obtain the following.
Result 4.1. ForA=I we obtain the continuous embeddingDαBl+s
p,θ(Rn;E)⊂Bsp,θ(Rn;E)
and corresponding estimate (3.23) for 0≤μ≤1−κin spaceBs+lp,θ(Rn;E).
Result 4.2. ForE=Rm,A=Iwe obtain the following embeddingDαBl+s
p,θ(Rn;Rm)⊂Bsq,θ(Rn;
Rm) for 0≤μ≤1−κ and a corresponding estimate (3.23). For E=R, A=I we get
the embeddingDαBl+s
p,θ(Rn)⊂Bsq,θ(Rn) proved in [8, Section 18] for the numerical Besov
spaces.
Result 4.3. Letl1=l2= ··· =ln=m,s1=s2= ··· =sn=σ, and p=q. Then for allE
∈UMD and|α| ≤mwe obtain that the continuous embeddingDαBσ+m
p,θ (Rn;E(A),E)⊂
Bσ
p,θ(Rn;E(A1−|α|/m)) and a corresponding estimate (3.23) for the isotropic Besov-Lions
spacesBσ+m
p,θ (Rn;E(A),E).
Result 4.4. Letσbe a positive number. Consider the following space [37, Section 1.18.2]:
lqσ=u;u=ui
,i=1, 2,. . .,∞,ui∈C
(4.1)
with the norm
ulσ q=
∞
i=1
2iqσuiq
1/q
<∞. (4.2)
Note thatl0
q=lq. LetAbe an infinite matrix defined inlqsuch that
D(A)=lqσ, A=δi j2si, (4.3)
where δi j=0, when i= j,δi j =1, when i= j, i,j=1, 2,. . .,∞. It is clear to see that
this operatorAis positive inlq. Then fromTheorem 3.3forsk/(lk+sk) +sj/(lj+sj)≤1,
k,j=1, 2,. . .,nand 0≤μ≤1−κ,κ=n
k=1(αk+ 1/ p1−1/ p2)/lk we obtain the
contin-uous embeddingDαBl+s p1,θ(Ω;l
σ
q,lq)⊂Bsp2,θ(Ω;l σ(1−κ−μ)
q ) and the corresponding estimate
It should not be that the above embedding has not been obtained with a classical method until now.
5. MaximalB-regular DOE inRn
Consider the following differential-operator equation
(L+λ)u= |α:.l|=1
aα(x)Dαu+Aλ(x)u+
|α:.l|<1
Aα(x)Dαu= f (5.1)
inBs
p,q(Rn;E), whereA(x),Aα(x) are possible unbounded operators in a Banach spaceE,
akare complex-valued functions,l=(l1,l2,. . .,ln) andliare positive integers. The
max-imal regularity for DOE was investigated, for example, in [12,14,30]. Let us consider DOE with constant coefficients
L0+λ
u=
|α:.l|=1
bαDαu+Aλu=f, (5.2)
whereAis a possible unbounded operator inE,Aλ=A+λandbαare complex numbers. Theorem5.1. Suppose the following conditions hold:
(1)E is UMD space with l.u.st. satisfyingB-multiplier condition with respect to p∈ (1,∞),q∈[1,∞], ands=(s1,s2,. . .,sn), whereskare positive numbers;
(2)Ais aϕ-positive operator inEwithϕ∈(0,π]and
K(ξ)= − |α:.l|=1
bα(iξ1)α1·
iξ2
α2
,. . .,iξnαn∈S(ϕ), K(ξ)≥C n
k=1
ξklk
, ξ∈Rn;
(5.3)
(3)sk/(lk+sk) +sj/(lj+sj)≤1fork,j=1, 2,. . .,n.
Then for all f ∈Bs
p,q(Rn;E), for|argλ| ≤π−ϕand sufficiently large|λ|>0(5.2) has
a unique solution u(x)that belongs to space Bl+s
p,q(Rn;E(A),E), and the coercive uniform
estimate
|α:.l|≤1
|λ|1−|α:.l|Dαu Bs
p,q+AuBsp,q≤CfBsp,q (5.4)
holds with respect to the parameterλ.
Proof. By applying the Fourier transform to (5.2) we obtain
K(ξ) +Aλ
ˆ
u(ξ)= fˆ(ξ). (5.5)
SinceK(ξ)∈S(ϕ) for allξ∈Rn, the operatorA+ [λ+K(ξ)] is invertible inE. So, we
obtain that the solution of (5.5) can be represented in the form
By using (5.6) we have
AuBs p,q=
F−1AA+λ+K(ξ)−1ˆ
f
Bs p,q, Dαu
Bs p,q=
F−1iξ
1
α1
·iξ2
α2
,. . .,iξn
αnA+λ+K(ξ)−1fˆ
Bs p,q.
(5.7)
Hence, it is suffices to show that the operator-functions
σ1λ(ξ)=
A+λ+K(ξ)−1,
σ2λ(ξ)= |λ|1−|α:.l|iξ1α1·iξ2α2,. . .,iξnαnA+λ+K(ξ)−1
(5.8)
are multipliers inBs
p,q(Rn;E) uniformly with respect toλ. Firstly, by using the positivity
properties of operatorAwe obtain that the operator functionσλ(ξ) is bounded uniformly
with respect toλ. That is,
σjλ(ξ)
B(E)≤C, j=1, 2. (5.9)
Then by virtue of the same properties of the operatorAwe obtain from (5.9) ξβDβ
ξσjλ(ξ)L(E)≤Mj, β∈Un,ξ∈Vn, j=1, 2. (5.10)
Then in view of (5.10) we obtain that the operator-valued functionsσjλ(ξ) are the
uniform collection of multipliers fromBs
p,q(Rn;E) toBsp,q(Rn;E). So we get that for all f ∈
Bs
p,q(Rn;E) there is a unique solution of (5.2) in the formu(x)=F−1[A+ (λ+K(ξ))]−1fˆ
and the estimate (5.4) holds.
Consider the problem (5.1). Let L0 and L operators in Bsp,q(Rn;E) be generated by
problems (5.2) and (5.1), respectively, that is,
DL0
=D(L)=Bl+s p,q
Rn,E(A),E,
L0u=
|α:.l|=1
aα(x)Dαu+Au,
Lu=L0u+L1u, L1u=
|α:l|<1
Aα(x)Dαu.
(5.11)
Theorem5.2. Suppose condition (1) ofTheorem 5.1holds and let
(1) A(x) be a ϕ positive in E uniformly with respect to x, A(x)A−1(x
0)∈Cb(R;
B(E))∃x0∈(−∞,∞),aα∈Cb(R), whereϕ∈(0,π];
(2)Aα(x)A−(1−|α:l|−μ)∈L∞(Rn;L(E)),0< μ <1− |α:l|;
(3)K(x,ξ)= −|α:.l|=1bα(iξ1)α1·(iξ2)α2,. . .,(iξn)αn∈S(ϕ),|K(x,ξ)|≥Ckn=1|ξk|lk,ξ∈
Then for all f ∈Bs
p,q(Rn;E),|argλ| ≤π−ϕand for sufficiently large|λ| (5.1) has a
unique solutionu(x)that belongs to spaceBl+s
p,q(Rn;E(A),E), and the coercive uniform
esti-mate
α:.l≤1
|λ|1−|α:.l|Dαu Bs
p,q+AuBsp,q≤CfBsp,q (5.12)
holds with respect toλ.
Proof. Letϕj∈C0∞(Rn), j=1, 2,. . .,∞, be a partition of unity such that 0≤ϕj≤1 and
suppϕj⊂Gj,jϕj(x)=1. Letgj∈C∞(Rn) such thatgj(x)≡1 on suppϕj. Then for all
u∈Bl+s
p,q(Rn;E(A),E) we haveu(x)=
juj(x), whereuj(x)=u(x)ϕj(x). From the
equal-ity (5.1) foru∈Bl+s
p,q(Rn;E(A),E) we obtain
(L+λ)uj=
|α:.l|=1
aα(x)Dαuj+Aλ(y)uj(y)= fj(y), (5.13)
where
fj=f ϕj−
|α:.l|<1
bα j(x)Dαu−
|α:.l|<1
Aα(x)Dαuj (5.14)
andbα j(x) are continuous and uniformly bounded functions containing derivatives of
ϕj. Choose a large ballBr0(0) such that|aα(x)−aα(∞)| ≤δ for all|x| ≥r0 andG0=
Rn\B
r0(0). CoverBr0(0) by finitely many ballsGj=Brj(xj) such that|aα(x)−aα(xj)| ≤δ for all|x−xj| ≤rj,j=1, 2,. . .,N. Define coefficients of the local operatorsLjas in [12,
Theorem 5.7], that is,
a0 α(x)=
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
aα(x), x /∈Br0(0),
aα
r2
0
x
|x|2 , x∈Br0(0),
aαj(x)=
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
aα(x), x∈Brj
xj , aα xj+r02
x−xj
x−xj2
, x /∈Brj
xj
(5.15)
for each j=1, 2,. . .,N. Then|aα(x)−aα(xj)| ≤δ for all x∈Rn and j=0, 1, 2,. . .,N.
Freezing the coefficients in (5.13) we obtain that
|α:.l|=1
aα
xj
Dαuj+Aλ
xj
uj(x)=Fj(x), (5.16)
where
Fj= fj+
|α:.l|=1
aα
xj
−aα(x)
Dαu
j+
Axj
By virtue ofTheorem 5.1we obtain that the problem (5.16) has a unique solutionuj,
and for|argλ| ≤π−ϕand sufficiently large|λ|we get
|α:.l|≤1
|λ|1−|α:.l|Dαu jBs
p,q(Gj;E)+AujBsp,q(Gj;E)≤CFjBs p,q
Gj;E
. (5.18)
Whence, using properties of the smoothness of coefficients of (5.14), (5.17) and choos-ing diameters ofGjsufficiently small, we get that
Fj
Bs
p,q(Gj;E)≤εujBsp+,ql(Gj;E(A),E)+C(ε)ujBsp,q(Gj;E), (5.19) whereεis a sufficiently small function andC(δ) is a continuous function. Consequently, from (5.18) and (5.19) we get
|α:.l|≤1
|λ|1−|α:.l|Dαu jBs
p,q(Gj;E)≤CfBsp,q(Gj;E)+δujBs+l
p,q+C(δ)ujBsp,q(Gj;E). (5.20)
Choosingδ <1 from the above inequality we have
|α:.l|≤1
|λ|1−|α:.l|Dαu jBs
p,q(Gj;E)≤C
fGj+ujBs p,q(Gj;E)
. (5.21)
Then by using the equalityu(x)=juj(x) and by virtue of the estimate (5.21) foru∈
Bs+l
p,q(Rn;E(A),E) we have
|α:.l|≤1
|λ|1−|α:.l|Dαu jBs
p,q(Gj;E)≤C
(L+λ)uBs
p,q+uBsp,q
. (5.22)
Letu∈Bs+l
p,q(Rn;E(A),E) be a solution of the problem (5.1). Then for|argλ| ≤π−ϕwe
have
uBs
p,q=(L+λ)u−LuBs p,q≤
1 λ
(L+λ)uBs
p,q+uBsp+,ql
. (5.23)
Then byTheorem 3.3and by virtue of (5.21)–(5.23), for sufficiently large|λ|we have
|α:.l|≤1
|λ|1−|α:.l|Dαu jBs
p,q≤C(L+λ)uBsp,q. (5.24)
The above estimate implies that the problem (5.1) has a unique solution and the op-erator (L+λ) has an invertible operator in its rank space. We need to show that this rank space coincide with the spaceBs
p,q(Rn;E). Let us construct for allj the functionuj, that
is defined on the regionsGjand satisfying the problem (5.1). The problem (5.1) can be
expressed in the form
|α:.l|=1
aα
xj
Dαu
j+Aλ
xj
uj(x)
=
gjf+
Axj
−A(x)uj−
|α:.l|<1
Aα(x)Dαuj
, j=1, 2,. . . .
Consider operatorsOjλinBsp,q(Gj;E) generated by problems (5.25). By virtue of
The-orem5.1for all f ∈Bs
p,q(Gj;E), for|argλ| ≤π−ϕand sufficiently large|λ|we obtain
|α:.l|≤1
|λ|1−|α:.l|DαO−1 jλ fBs
p,q+AO −1 jλ fBs
p,q≤CfBsp,q. (5.26)
Extendingujzero on the outside of suppϕjin equalities (5.25) and passing
substitu-tionsuj=O−jλ1υjwe obtain operator equations with respect toυj:
υj=Kjλυj+gjf, j=1, 2,. . .,N. (5.27)
By virtue ofTheorem 3.3and the estimate (5.26), in view of the smoothness of the coefficients of the expressionKjλfor|argλ| ≤π−ϕand sufficiently large |λ|we have
Kjλ< ε, whereεis sufficiently small. Consequently, (5.27) has a unique solutionυj=
[I−Kjλ]−1gjf and we get
υj
Bs
p,q=I−Kjλ −1
gjfBs
p,q≤ fBsp,q. (5.28) Whence, [I−Kjλ]−1gj are the bounded linear operators fromBsp,q(Rn;E) toBsp,q(Gj;E).
Thus, we obtain that the functionsuj=Ujλf =O−jλ1[I−Kjλ]−1gjf are the solutions of
(5.25). Consider a linear operator (U+λI)=jϕj(y)Ujλf inBsp,q(Rn;E). It is clear from
the constructionsUj and the estimate (5.26) that the operatorsUjλare bounded linear
fromBs
p,q(Rn;E) toBs+lp,q(Rn;E(A),E) and
|α:.l|≤1
|λ|1−|α:.l|DαU−1 jλ fBs
p,q+AU −1 jλ fBs
p,q≤CfBsp,q, (5.29)
for |argλ| ≤π−ϕ and sufficiently large |λ|. Therefore, (U+λI) is a bounded linear operator fromBs
p,q toBsp,q. Then the act of (L+λ) tou=
jϕjUjλf gives (L+λ)u=
f +jΦjλf, whereΦjλ are linear combinations of Ujλ and (d/d y)Ujλ. By virtue of Theorem 3.3, by estimate (5.29), and from the expressionΦjλwe obtain that operators
Φjλare bounded linear fromBsp,q(Rn;E) toBsp,q(Gj;E) andΦjλ< δ. Therefore, there
exists a bounded linear invertible operator
I+
j
Φjλ
−1
. (5.30)
Whence, we obtain that for all f ∈Bs
p,q(Rn;E) the problem (5.1) has a unique solution
u=(U+λI)
I+
j
Φjλ
−1
f, (5.31)
Result 5.3. Theorem 5.2implies that the differential operatorLhas a resolvent operator (L+λ)−1for|argλ| ≤π−ϕ, and for sufficiently large|λ|it has the estimate
|α:.l|≤1
|λ|1−|α:.l|Dα(L+λ)−1 L(Bs
p,q(Rn;E))+A(L+λ) −1
L(Bs
p,q(Rn;E))≤C. (5.32)
Remark 3.5andTheorem 5.2imply the following.
Result 5.4. Suppose the following conditions hold:
(1)Eis a Banach space satisfyingB-multiplier condition with respect to p∈(1,∞) andq∈[1,∞];
(2)Ais aϕ-positive operator inEwithϕ∈(0,π] and
K(ξ)= −
|α:.l|=1
bαiξ1
α1
·iξ2
α2
,. . .,iξnαn∈S(ϕ),
K(x,ξ)≥Cn k=1
ξklk
, ξ∈Rn,x∈Rn;
(5.33)
(3)A(x) is aϕpositive inEuniformly with respect tox,A(x)A−1(x
0)∈Cb(R;B(E)),
x0∈(−∞,∞),aα∈Cb(R), whereϕ∈(0,π];
(4)Aα(x)A−(1−|α:l|−μ)∈L∞(Rn;L(E)), 0< μ <1− |α:l|.
Then for all f ∈Bs
p,q(Rn;E),|argλ| ≤π−ϕand for sufficiently large|λ|(5.1) has a
unique solutionu(x) that belongs to spaceWlBs
p,θ(Rn;E(A),E), and the coercive uniform
estimate
|α:.l|≤1
|λ|1−|α:.l|Dαu Bs
p,q+AuBsp,q≤CfBsp,q (5.34)
holds with respect toλ.
Theorem5.5. Let all conditions ofTheorem 5.2hold forϕ∈(0,π/2). Then the parabolic Cauchy problem (2.24) for|argλ| ≤π−ϕand sufficiently large|λ|is maximalB-regular. Proof. The problem (2.24) can be expressed inBsp,θ(R+;F) in the following form:
du(y)
d y + (L+λ)u(y)= f(t), u(0)=0, y >0, (5.35) whereF=Lp(G;E) andLis the differential operator inBsp,θ(Rn;E) generated by the
prob-lem (5.1). In view ofResult 4.3the operatorLis positive inBs
p,θ(Rn;E) forϕ∈(0,π/2).
Then by virtue of [4, Corollary 8.9] we obtain the assertion.
6. Applications
6.1. Infinite systems of quasielliptic equations. Consider the following infinity systems
of boundary value problem:
(L+λ)um(x)=
|α:.l|=1
aα(x)Dαum(x) +
dm(x) +λ
um(x)
+
|α:.l|<1
∞
k=1
dαkm(x)Dαuk(x)=fm(x), x∈Rn,m=1, 2,. . .,∞.
(6.1)
Let
D(x)=dm(x)
, dm>0, u=
um
, Du=dmum
, m=1, 2,. . .,∞,
lq(D)=
⎧ ⎨
⎩u:u∈lq,ulq(D)= Dulq= ∞
m=1
dmumq
1/q
<∞ ⎫ ⎬ ⎭,
x∈G, 1< q <∞, l=l1,l2,. . .,ln, s=s1,s2,. . .,sn, sk>0, lk∈N.
(6.2)
LetOdenote a differential operator inBs
p,θ(Rn;lq) generated by problem (6.1). Let
B=LBsp,θRn;lq
. (6.3)
Theorem6.1. Letaα∈Cb(Rn),dm∈Cb(Rn),dαkm∈L∞(Rn), andsk,lksuch that
sk
lk+sk
+ sj lj+sj ≤
1, j=1, 2,. . .,n,
∞
k,m=1
dq1 αkmd
−q1(1−|α:l|−μ)
m <∞, 1
q+ 1 q1 =1,
(6.4)
wherep,q∈(1,∞),θ∈[1,∞]. Then
(a)for all f(x)= {fm(x)}∞1 ∈Bsp,θ(Rn;lq),|argλ| ≤π−ϕand for sufficiently large|λ|
the problem (6.1) has a unique solutionu= {um(x)}∞1 that belongs to spaceBs+lp,θ(Rn,lq(D),
lq), and the coercive estimate
|α:l|≤1
Dαu Bs
p,θ(Rn;lq)+duBsp,θ(Rn;lq)≤CfBps,θ(Rn;lq) (6.5)
holds for the solution of the problem (6.1);
(b)for|argλ| ≤π−ϕand for sufficiently large|λ|there exists a resolvent(O+λ)−1of
operatorOand
|α:l|≤1
1 +|λ|1−|α:l|Dα(O+λ)−1
Proof. Really, letE=lq,A(x), andAα(x) be infinite matrices, such that
A=dm(x)δkm
, Aα(x)=
dαkm(x)
, k,m=1, 2,. . .,∞. (6.7)
It is clear to see that operatorAis positive inlq. Therefore, by virtue ofTheorem 5.2we
obtain that the problem (6.1) for allf ∈Bsp,θ(Rn;l
q),|argλ| ≤π−ϕ, and sufficiently large
|λ|has a unique solutionuthat belongs to spaceBs+lp,θ(Rn;l
q(D),lq) and the estimate (6.5)
holds. By virtue of estimate (6.5) we obtain (6.6).
6.2. Cauchy problems for infinite systems of parabolic equations. Consider the
follow-ing infinity systems of parabolic Cauchy problem:
∂um(y,x)
∂y +
|α:.l|=1
aα(x)Dαum(y,x) +
dm(x) +λ
um(y,x) +
|α:.l|<1
∞
k=1
dαkm(x)Dαuk(y,x)
=fm(y,x), um(0,x)=0,m=1, 2,. . .,∞,y∈R+,x∈Rn.
(6.8)
Theorem6.2. Let all conditions ofTheorem 6.1hold. Then the parabolic systems (6.8) for |argλ| ≤π−ϕand for sufficiently large|λ|are maximalB-regular.
Proof. Really, letE=lq,A, andAk(x) be the infinite matrices, such that
A=dm(x)δkm
, Aα(x)=
dαkm(x)
, k,m=1, 2,. . .,∞. (6.9) Then the problem (6.8) can be expressed as the problem (2.24), where
A=dm(x)δkm, Aα(x)=dαkm(x), k,m=1, 2,. . .,∞. (6.10)
Then by virtue of Theorems5.2and5.5we obtain the assertion.
Acknowledgments
This work is supported by Project UDP-748 11.05.2006 of Istanbul University. The author would like to express a deep gratitude to O. V. Besov for his useful advices.
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Veli B. Shakhmurov: Department of Electrical & Electronics Engineering, Engineering Faculty, Istanbul University, Avcilar, 34320 Istanbul, Turkey
