DIFFERENTIAL-OPERATOR EQUATIONS
IN BANACH-VALUED FUNCTION SPACES
VELI B. SHAKHMUROVReceived 11 November 2003 and in revised form 27 December 2004
This study focuses on anisotropic Sobolev type spaces associated with Banach spacesE0, E. Several conditions are found that ensure the continuity and compactness of embedding operators that are optimal regular in these spaces in terms of interpolations ofE0andE. In particular, the most regular class of interpolation spacesEαbetweenE0,E, depending of αand order of spaces are found that mixed derivatives Dα belong with values; the
boundedness and compactness of differential operatorsDαfrom this space toE
α-valued
Lpspaces are proved. These results are applied to partial differential-operator equations
with parameters to obtain conditions that guarantee the maximalLpregularity uniformly
with respect to these parameters.
1. Introduction
Embedding theorems in function spaces have been elaborated in detail by [5,28]. A comprehensive introduction to the theory of embedding of function spaces and histor-ical references may be also found in [28]. In abstract function spaces embedding the-orems have been studied by [3,18,22,23,24,25,26]. Lions-Peetre [18] showed that, ifu∈L2(0,T;H0), u(m)∈L2(0,T;H) then u(i)∈L2(0,T; [H,H0]i/m), i=1, 2,. . .,m−1,
whereH0,H are Hilbert spaces,H0 is continuously and densely embedded in H and [H0,H]θare interpolation spaces betweenH0,Hfor 0≤θ≤1. In [22,23,24,25,26] the similar questions were investigated for anisotropic Sobolev spacesWl
p(Ω;H0,H),Ω⊂Rn. Moreover, boundary value problems for differential-operator equations have been stud-ied in detail by [16,27,30,32]. The solvability and the spectrum of boundary value prob-lems for elliptic differential-operator equations have also been refined by [1,2,4,8,10,11, 13,22,23,24,25,26]. A comprehensive introduction to the differential-operator equa-tions and historical references may be found in [16,32]. In these works Hilbert-valued function spaces essentially have been considered. In the present paper, are to be intro-duced a Banach-valued function spacesWl
p(Ω;E0,E), wherel=(l1,l2,. . .,ln) andE0,Eare Banach spaces such thatE0is continuously and densely embedded inE. The properties of continuity and compactness of embedding operators in these spaces are obtained. We prove that the generalized derivative operatorDαis continuous from these Banach-valued
Copyright©2005 Hindawi Publishing Corporation
Sobolev spaces toEα-valuedLpspaces, whereEαare interpolation spaces betweenE0and Edepending on the order of differentiationsDα. By applying these results, the maximal
Lp-regularity of certain class of anisotropic partial differential-operator equations are
de-rived.
Letα1,α2,. . .,αnbe nonnegative integer numbers and
Dα=Dα1
1 D2α2···Dαnn=
∂α
∂xα1
1 ∂xα22···∂xαnn. (1.1)
Under certain assumptions to be stated later, we prove that the operatorsu→Dαuare
bounded from spaceWl
p(Ω;E(A),E) to spaceLq(Ω;E(A1−κ)), that is, embedding
DαWl p
Ω;E(A),E⊂Lq
Ω;EA1−κ (1.2)
is continuous. More precisely for 0< µ≤1−κwe prove the estimate Dαu
Lp(Ω;E(A1−κ))≤Cµ
hµu
Wl
p(Ω;E(A)E)+h
−(1−µ)u
Lp(Ω;E)
(1.3)
for allu∈Wl
p(Ω;E(A),E) and 0≤h≤h0<∞. The constantCµin the above equation is
independent ofu∈Wl
p(Ω;E(A),E) and of the choice ofh. Further, we prove
compact-ness of this embedding operator. Furthermore, we consider certain applications of these theorems. This kind of embedding theorems arise in the investigation of boundary value problems for anisotropic partial differential-operator equations
n
k=1
aktkDklku+Au
|α:l|<1
n
k=1 tαk/lk
k Aα(x)Dαu= f, (1.4)
depend on parameterst=(t1,t2,. . .,tn), whereA is a positive operator on the Banach
space E,Aα(x) is an operator such that Aα(x)A−(1−|α:l|) is bounded on E, where α=
(α1,α2,. . .,αn),l=(l1,l2,. . .,ln), |α:l| =nk=1(αk/lk). In general, this equations possess
different derivatives and different parameters with respect to the various variables. Taking l1=l2= ··· =ln=2lin the above equations we obtain elliptic equation with parameters
n
k=1
aktkD2klu+Au+
|α|<2l n
k=1 tαk/2l
k Aα(x)Dαu=f(x). (1.5)
We prove the maximal regularity of this differential-operator equations inLp(Rn;E)
uniformly with respect to parametert. In this direction we should mention the works [10,22,23,24,25,26,31].
2. Notations and definitions
for a scalarξand (A−ξI)−1denotes the inverse of the operatorA−ξIor the resolvent of operatorA.
Let
Sϕ= ξ,ξ∈C,|argξ−π| ≤π−ϕ
∪ {0}, 0< ϕ≤π. (2.1)
A linear operatorAis said to be positive in a Banach spaceE, ifD(A) is dense onEand (A−ξI)−1
L(E)≤M
1 +|ξ|−1 (2.2)
withξ∈Sϕ, whereMis a positive constant [28].
EAθ=u,u∈DAθ,uE(Aθ)=AθuE+u<∞,−∞< θ <∞
. (2.3)
We denote byLp(Ω;E) the space of strongly measurableE-valued functions onΩ⊂Rn
with the norm
uLp= uLp(Ω;E)=
Ω
u(x)p Edx
1/ p
, 1≤p <∞. (2.4)
Let l=(l1,l2,. . .,ln), where li,i=1, 2,. . .,n positive integers and Dlkk=∂lk/∂xlkk, k=
1, 2,. . .,n.
We introduce aE0-valued anisotropic function spaceWlp(Ω;E0,E) that consist of func-tionsu∈Lp(Ω;E0) such that have the generalized derivativesDlkku∈Lp(Ω;E) with the
norm
uWl
p(Ω;E0,E)= uLp(Ω;E0)+
n
k=1 Dlk
kuLp(Ω;E)<∞, 1≤p <∞. (2.5)
Let be t=(t1,t2,. . .,tn), wheretk,k=1, 2,. . .,n are nonnegative parameters. Let us
define in the spaceWl
p(Ω;E0,E) parameterized norm
uWl
p,t(Ω;E0,E)= uLp(Ω;E0)+
n
k=1 tkDlk
kuLp(Ω;E). (2.6)
The Banach spaceEis said to beξ-convex [7] if there exists on E×Ea symmetric functionξ(u,v) which is convex with respect to every one of the variables and satisfies the condition
ξ(0, 0)>0, ξ(u,ν)≤ u+v foruE= vE=1. (2.7)
It is shown in [7] that a Hilbert operator
(H f)(x)=lim
ε→0
|x−y|>ε
f(y)
x−yd y (2.8)
is bounded in the spaceLp(R;E), p∈(1,∞), for those and only those Banach spaces
UMD spaces. UMD spaces containsLp,lpspaces and the Lorentz spacesLpq,p,q∈(1,∞)
for instance.
C(l)(Ω;E) denotes the space ofE-valued continuously differentiable functions oflth order. LetE1andE2be Banach spaces. A functionΨ∈C(l)(Rn;L(E1,E2)) is called a mul-tiplier fromLp(Rn;E1) toLq(Rn;E2) if there exists a constantM >0 such that
F−1Ψ(ξ)Fu
Lq(Rn;E2)≤CuLp(Rn;E1) (2.9)
for allu∈Lp(Rn;E1), whereFandF−1are Fourier and inverse Fourier transformations, respectively.
We denote the set of all multipliers fromLp(Rn;E1) toLq(Rn;E2) byMqp(E1,E2). For E1=E2=Ewe denoteMpq(E1,E2) byMqp(E). Let
Hk= Ψh∈Mqp
E1,E2
,h=h1,h2,. . .,hL
∈Q (2.10)
be a collection of multipliers in Mqp,,γγ(E1,E2). We say that Ψh=Ψh(ξ) is a uniformly
bounded multipliers with respect tohif there exists a constantC >0, independent of h∈B(h), such that
F−1Ψ
hFuLq(Rn,E2)≤CuLp(Rn,E1) (2.11)
for allh∈Kandu∈Lp(Rn;E1).
The exposition of the theory ofLp-multipliers of the Fourier transformation, and
some related references, can be found in [28, Sections 2.2.1, 2.2.2, 2.2.3, and 2.2.4]. On the other hand, in vector-valued function spaces, Fourier multipliers have been studied, for example, by [3,6,12,15,20,21,29].
A setK⊂B(E1,E2) is calledR-bounded [6,29] if there is a constantCsuch that for all T1,T2,. . .,Tm∈Kandu1,u2,. . .,um∈E1,m∈N.
1
0
m
j=1
rj(y)Tjuj
E2 d y≤C
1
0
m
j=1 rj(y)uj
E1
d y, (2.12)
where{rj}is a sequence of independent symmetric [−1, 1]-valued random variables on
[0, 1].
A set K(h)⊂B(E1,E2) depending on parameters h=(h1,h2,. . .,hL)∈B(h)∈RL is
called uniformlyR-bounded with respect tohif there is a constantC such that for all T1(h),T2(h),. . .,Tm(h)∈Kandu1,u2,. . .,um∈E1,m∈N.
1
0
m
j=1
rj(y)Tj(h)uj
E2 d y≤C
1
0
m
j=1 rj(y)uj
E1
d y, (2.13)
where a positive constantCis independent of parametersh. Let
Un= β=β1,β2,. . .,βn,βi∈(0, 1),i=1, 2,. . .,n,
Vn= ξ=ξ1,ξ2,. . .,ξn∈Rn,ξi=0,i=1, 2,. . .,n,
α=α1,α2,. . .,αn
, ξα=ξα1
1 ξ2α2···ξnαn, |ξ|α= |ξ|α1|ξ|α2···|ξ|αn.
Definition 2.1. The Banach spaceEis said to be a space satisfying a multiplier condition with respect top,q∈(1,∞),p≤qwhen forΨ∈C(n)(Rn;B(E)) if the set
Ψ(ξ) :ξβ+1/ p−1/qDξβΨ(ξ) :ξ∈Vn,β∈Un
(2.15)
areR-bounded, thenΨ∈Mqp(E).
A Banach spaceEhas a property (α), (see, e.g., [12]) if there exists a constantαsuch that
N
i,j=1
αi jεiεjxi j
L2(Ω×Ω;E) d y≤α
N
i,j=1 εiεjxi j
L2(Ω×Ω;E)
(2.16)
for allN∈N,xi,j∈E,αi j∈ {0, 1},i,j=1, 2,. . .,N, and all choices of independent,
sym-metric,{−1, 1}-valued random variablesε1,ε2,. . .,εN,ε1,ε2,. . .,εN on probability spaces Ω,Ω. For example, the spacesLp(Ω), 1≤p <∞has the property (α).
Remark 2.2. IfEis UMD space with property (α) then these spaces are satisfy the multi-plier condition with respect top∈(1,∞) (see [12]).
Definition 2.3. Theϕ-positive operatorAis said to be aR-positive in the Banach spaceE if there existsϕ∈(0,π] such that the set
LA= 1 +|ξ|
(A−ξI)−1:ξ∈S
ϕ
(2.17)
isR-bounded.
Note that in the Hilbert spaces every norm bounded set isR-bounded. Therefore, in the Hilbert spaces all positive operators areR-positive. IfAis a generator of a con-traction semigroup onLq, 1≤q≤ ∞[17],Ahas the bounded imaginary powers with
(−Ait)
B(E)≤Ceν|t|,ν< π/2 inE∈UMD [8,9] then those operators areR-positive. It is well known (see, e.g., [19]) that any Hilbert space satisfies the multiplier condition. By virtue of [21] Mikhlin conditions are not sufficient for operator-valued multiplier theorem. There are however, Banach spaces which are not Hilbert spaces but satisfy the multiplier condition, for example, UMD spaces (see, e.g., [29]).
Byσ∞(E) will be denoted a space of compact operators acting inE.
Example 2.4. Ifγ∈Ap,δ∈C∞(R) with δ(y)≥0 for all y≥0, δ(y)=0 for |y| ≤1/2
andδ(−y)= −δ(y) for all y, thenδ∈Mpp,,γγ(R). Really it clear to see thatδ(y) satisfies
multiplier conditions [28, Section 2.3.3].
3. Embedding theorems
Lemma3.1. LetAbe a positive operator on a Banach spaceEandr=(r1,r2,. . .,rn)where
rk∈ {0,b}. Lett=(t1,t2,. . .,tn), wheretk,k=1, 2,. . .,nare nonnegative parameters,0<
Letδ be a multiplier of the form described inExample 2.4. Then for0≤h≤h0<∞and 0≤µ≤1−κthe operator-function
Ψt(ξ)=Ψt,r,h,µ(ξ)
=
n
k=1 t(αk+r)/lk
k ξr(iξ)αA1−κ−µh−µ
A+
n
k=1 tk
δξk
lk
+h−1 −1
(3.1)
is bounded operator inEuniformly with respect toξ∈Rn,handt, that is, there is a constant
Cµsuch that
Ψt,h,µ(ξ)
L(E)≤Cµ (3.2)
for allξ,tandh.
Proof. Since−[nk=1tk(δ(ξ)ξk)lk+h−1]∈S(ϕ) for allϕ∈[0,π) then by virtue of the
pos-itiveness ofA, operatorB(ξ)=A+nk=1tk(δ(ξk)ξk)lk+h−1is invertible in the spaceE. Let
u=h−µB−1(ξ)f. Then Ψt(ξ)f
E= n
k=1 t(αk+r)/lk
k |ξ|r+αA1−κ−µuE
=(hA)1−κ−µuEh−(1−µ)ht11/l1ξ1
α1+r1
···htn1/lnξn αn+rn
.
(3.3)
Using the moment inequality for powers of a positive operators, we get a constantCµ
depending only onµsuch that Ψt(ξ)
E≤Cµh(1−µ)hAu1−κ−µuκ+µ
ht1
1/l1 ξ1
α1+r1
···htn
1/lnξ
n αn+rn
. (3.4) Now, we apply the Young inequality, which states thatg1g2≤g1k1/k1+g2k2/k2for any posi-tive real numbersg1,g2andk1,k2with 1/k1+ 1/k2=1, to the product
hAu1−κ−µuκ+µht
1 1/l1
ξ1
α1+r1
···htn
1/lnξ
n αn+rn
(3.5)
withk1=1/(1−κ−µ),k2=1/(κ+µ) to get Ψt(ξ)f
E≤Cµh−(1−µ)(1−κ−µ)hAu
+ (κ+µ)ht1ξ1
(α1+r1)/(κ+µ)
···htnξn
(αn+rn)/(κ+µ). (3.6)
Since
n
i=1 αi+ri
(κ+µ)= 1
κ+µ
n
i=1 αi+ri
li = κ
κ+µ≤1 (3.7)
there exists a positive constantM0independent ofξ, such that ξ1(α1+r1)/(κ+µ)
···ξn
(αn+rn)/(κ+µ)≤M
0
1 +
n
k=1 ξklk
for allξ∈Rn. It is clear that|y|l≤(δ(y)y)lfor all|y|>1/2. Therefore
ξ1(α1+r1)/(κ+µ)
···ξn(αn+rn)/(κ+µ)≤M1
1 +
n
k=1
δξk
ξk
lk
(3.9)
for a suitableM1>0 and allξ∈Rn. Substituting this on the inequality (3.6) and absorb-ing the constant coefficients inCµ, we obtain
ψt(ξ)f≤Cµhµ
Au+
n
k=1 tk
δξk
ξk
lk
+h−1
u
. (3.10)
Substituting the value ofu, we get ψt(ξ)f≤CµAB−1(ξ)f+
n
k=1
tkδξkξklk+h−1
B−1(ξ)f. (3.11)
SinceAis positive operator in the spaceE, we have
A+
n
k=1
tkδξkξklk+h−1
−1 f
≤M
1 +
n
k=1k
tkδξkξklk+h−1
−1
f (3.12)
for all f ∈E. Combining those with the inequality (3.11) we obtain Ψt(ξ)f
E≤CµfE (3.13)
for all f ∈E,handt. The inequality (3.13) implies the estimate (3.2). Theorem3.2. Suppose the following conditions hold:
(1)Eis a Banach space satisfying the multiplier condition with respect topandq, where 1< p≤q <∞;
(2)t=(t1,t2,. . .,tn), wheretk,k=1, 2,. . .,nare nonnegative parameters0< tk≤t0<∞ and0≤h≤h0<∞;
(3)α=(α1,α2,. . .,αn),l=(l1,l2,. . .,ln), wherelkare positive andαkare nonnegative real
numbers such thatκ= |(α+ 1/ p−1/q) :l| ≤1, and let0≤µ≤1−κ; (4)Ais aR-positive operator onE.
Then an embedding
DαWl p
Rn;E(A),E⊂L q
Rn;EA1−κ−µ (3.14)
is continuous and there exists a constantCµ>0, depending only onµ, such that n
k=1
t(αk+1/ p−1/q)/lk
k DαuLp(Rn;E(A1−κ−µ))≤Cµ
hµuWl
p,t(Rn;E(A),E)+h
−(1−µ)u
Lp(Rn;E)
(3.15)
for allu∈Wl
Proof. We have
Dαu
Lq(Rn;E(A1−κ−µ))=
Rn
Dαuq
E(A1−κ−µ)dx
1/q
Rn
A1−κ−µDαuq Edx
1/q
A1−κ−µDαu Lq(Rn;E)
(3.16)
for allusuch that
Dαu
Lq(Rn;E(A1−κ−µ))<∞. (3.17)
On the other hand we have
A1−α−µDαu=F−FA1−κ−µDαu=F−A1−κ−µFDαu
=F−A1−κ−µ(iξ)αFu=F−(iξ)αA1−κ−µFu. (3.18)
Hence denotingFuby ˆu, we get from relations (3.16) and (3.18) Dαu
Lq(Rn;E(A1−κ−µ))
F−(iξ)αA1−κ−µuˆ
Lq(Rn;E). (3.19)
Moreover, we have
uWl
p,t(Rn;E(A),E)= uLp(Rn;E(A))+
n
k=1 tkDlk
kuLp(Rn;E)
=F−uˆLp(Rn;E(A))+
n
k=1 tkF−
iξk
lkuˆ
Lp(Rn;E)
F−1AuˆL
p(Rn;E)+
n
k=1 tkF−
iξk
lkuˆ
Lp(Rn;E)
(3.20)
for allu∈Wl
p(Rn;E(A),E). Thus proving the inequality (3.15) for some constantsCµ is
equivalent to proving
n
k=1
t(αk+1/ p−1/q)/lk
k F−(iξ)αA1−κ−µuˆLq(Rn,E)
≤Cµ
hµF−AuˆLp(Rn,E)+
n
k=1 tkF−
iξk
lkˆ
u
Lp(Rn,E)+h
−(1−µ)F−uˆ
Lp(Rn,E)
(3.21)
for a suitableCµ. Now if δ is a multiplier of the form described as inExample 2.4, by
virtue of multiplier there is constantsCk>0 for eachk=1, 2,. . .,nsuch that
F−1
iδ
ξk
iξk
lkuˆ
Lp(Rn;E)
≤CkF−
iξk
lkuˆ
for allξ∈Rn. Thus the inequality (3.15) will follow if we prove the following inequality n
k=1
t(αk+1/ p−1/q)/lk
k F−
(iξ)αA1−κ−µuˆ Lp(Rn;E)
≤Cµ
F−
hµ
A+
n
k=1
tkδξkξklk
+h−(1−µ)
ˆ u
Lp(Rn;E)
(3.23)
for a suitableCµ>0, and for allu∈Wpl(Rn;E(A),E).
Let us express the left-hand side of (3.23) as follows
n
k=1
t(αk+1/ p−1/q)/lk
k F−
(iξ)αA1−κ−µuˆLq(Rn;E)
=
n
k=1
t(αk+1/ p−1/q)/lk
k F−(iξ)αA1−κ−µQ−1(ξ)Q(ξ)Lq(Rn;E),
(3.24)
where
Q(ξ)=hµ
A+
n
k=1 tk
δξk
ξk
lk
+h−(1−µ). (3.25)
(SinceAis the positive operator inEso it is possible.) By virtue of definition of mul-tiplier it is clear that the inequality (3.23) will follow immediately if we can prove that the operator-functionΨt,h,µ=(iξ)αA1−κ−µQ−1(ξ) is a multiplier inLp(Rn;E), which is
uniform with respect to parameterstandh.
Firstly by usingLemma 3.1we obtain that the operator functionΨt,h,µ(ξ) is bounded
uniformly with respect tohandt. That is, Ψt,h,µ(ξ)
B(E)≤C. (3.26)
By virtue of theR-positivity of operatorAand by virtue of the homogenous properties ofR-bounds with respect to product by scalar and the triangle inequality (see, e.g., [8, Proposition 3.4]) by using (3.26) for 0< tk≤T, 0< h≤h0andξ∈(−∞,∞) we obtain
R Ψt,h,µ(ξ) :ξ∈Vn
≤M,
Rξβ+1/ p−1/qDβ
ξΨt,h,µ(ξ) :β∈Un:ξ∈Vn
≤Mβ.
(3.27)
By virtue of (3.27) we obtain that the operator-valued functionsΨt,h,µ(ξ) are uniformly
R-bounded multipliers with respect to t,hand R-bounds are independent oftand h. Then in view ofDefinition 2.1it follows that the operator-valued functionΨt,h,µ(ξ) are
uniformly bounded Fourier multipliers fromLp(Rn;E) toLq(Rn;E). This completes the
proof ofTheorem 3.2.
Condition 3.3. Let be the regionΩ⊂Rnsuch that there exists a bounded linear extension
operatorPacting fromLq(Ω;E) toLq(Rn;E) also fromWlp(Ω;E(A),E) toWlp(Rn;E(A),E),
for 1< p≤q <∞.
Remark 3.4. IfΩ⊂Rn is a region satisfying the strongl-horn condition (see [5, page
117]) andl=(l1,. . .,ln),li,i=1, 2,. . .,nare nonnegative integers numbers, E=R,A=
I, then there exists a bounded linear extension operator fromWl
p(Ω)=Wpl(Ω;R,R) to
Wl
p(Rn)=Wlp(Rn;R,R).
Theorem3.5. Suppose all conditions ofTheorem 3.2andCondition 3.3are hold. Then an embedding
DαWl p
Ω;E(A),E⊂Lq
Ω;EA1−κ−µ (3.28)
is continuous and there exists a constantCµdepending only onµsuch that
n
k=1
t(αk+1/ p−1/q)/lk
k DαuLp(Ω;E(A1−κ−µ))≤Cµ
hµu
Wl
p,t(Ω;E(A),E)+h
−(1−µ)u
Lp(Ω;E)
(3.29)
for allu∈Wl
p(Ω;E(A),E),tandh.
Proof. It is suffices to prove the estimate (3.29). LetPis a bounded linear extension oper-ator fromLp(Ω;E) toLp(Rn;E) and fromWpl(Ω;E(A),E) toWpl(Rn;E(A),E), and letPΩ
be the restriction operator fromRntoΩ. Then for anyu∈Wl
p(Ω;E(A),E) we have n
k=1
t(αk+1/ p−1/q)/lk
k DαuLq(Ω;E(A1−κ−µ))
=
n
k=1
t(αk+1/ p−1/q)/lk
k DαPΩPuLq(Ω;E(A1−κ−µ))
≤C
n
k=1
t(αk+1/ p−1/q)/lk
k DαPuLq(Rn;E(A1−κ−µ))
≤Cµ
hµPu
Wl
p,t(Rn;E(A),E)+h
−(1−µ)Pu
Lp(Rn;E)
≤Cµhµu Wl
p,t(Ω;E(A),E)+h
−(1−µ)u
Lp(Ω;E)
.
(3.30)
Result 3.6. Let all conditions ofTheorem 3.5holds. Then for allu∈Wl
p(Ω;E(A),E) we
have multiplicative estimate Dαu
Lq(Ω;E(A1−κ−µ))≤Cµu
1−µ Wl
p(Ω;E(A),E)u
µ
Lp(Ω;E). (3.31)
Indeed settingh= uLp(Ω;E)· u−
1
Wl
p(Ω;E(A),E)in estimate (3.29) we obtain (3.31).
Theorem3.7. Assume that all conditions ofTheorem 3.5are satisfied. LetΩbe a bounded region onRnandA−1be compact operator in the spaceE. Then for0< µ <1−κan embed-dingDαWl
Proof. By virtue of [22, Theorem 1] an embeddingWl
p(Ω;E(A),E)⊂Lq(Ω;E) is compact.
Then by the estimate (3.31) we obtain the assertion ofTheorem 3.7. Theorem3.8. Suppose all conditions of theoremA2are satisfied.
Then for0< µ <1−κan embedding
DαWlpΩ;E(A),E⊂LpΩ;E(A),Eκ (3.32)
is continuous and there exists a positive constantCµsuch that
n
k=1
t(αk+1/ p−1/q)/lk
k DαuLp(Ω;(E(A),E)κ+µ,p)
≤Cµ
hµ
AuLp(Ω;E)+
n
k=1 tkDlk
kuLp(Ω;E)
+h−(1−µ)uLp(Ω;E)
(3.33)
for allu∈Wl
p(Ω;E(A),E)and0< h≤h0<∞.
Proof. Let at first to show the theorem for the caseΩ=Rn. Then it is sufficient to prove
the estimate
n
k=1
t(αk+1/ p−1/q)/lk
k DαuLq(Rn;(E(A),E)κ+µ,p)≤Cµ
hµu
Wl
p(Rn;E(A),E)+h
−(1−µ)u
Lp(Rn;E)
(3.34)
for allu∈Wl
p(Rn;E(A),E),tandh. By the definition of interpolation spaces (E(A),E)κ+µ
(see [28, Section 1.14.5]) the estimate (3.34) is equivalent to the inequality
n
k=1
t(αk+1/ p−1/q)/lk
k F−1y1−κ−µ−1/ p
Aχ+µ(A+y)−1ξαuˆ
Lq(Rn;Lp(R÷;E))
≤Cµ
F−
hµ
A+
n
k=1 tk
δξk
ξk
lk
+h−(1−µ)
ˆ u
Lp(Rn;E)
.
(3.35)
By virtue of the definition of the multiplier it is clear that the inequality (3.34) will follow immediately from (3.35) if we can prove that the operator-function
Ψt,h,µ=(iξ)α n
k=1
t(αk+1/ p−1/q)/lk
k y1−κ−µ−1/ p
Aχ+µ(A+y)−1
×
hµ
A+
n
k=1 tk
δξk
ξk
lk
+h−(1−µ)
−1 (3.36)
is the multiplier fromLp(Rn;E) toLq(Rn;Lp(R+;E)). This fact is proved by the same man-ner as inTheorem 3.2.
Therefore, we get the estimate (3.35). Then by using the extension operator we obtain
Result 3.9. Let all conditions ofTheorem 3.5holds. Then for all
u∈Wpl
Ω;E(A),E (3.37)
we have a multiplicative estimate Dαu
Lq(Ω;(E(A),E)κ+µ,p)≤Cµu
1−µ Wl
p(Ω;E(A),E)u
µ
Lp(Ω;E). (3.38)
Indeed setting
h= uLp(Ω;E)· u−1Wl
p(Ω;E(A),E) (3.39)
in the estimate (3.33) we obtain (3.38).
Theorem 3.10. Assume that all the conditions of Theorem 3.8are satisfied. Let Ωbe a bounded region inRnandA−1is a compact operator in the spaceE. Then for0< µ <1−κ, 1< p <∞an embedding
DαWplΩ;E(A),E⊂Lq
Ω;E(A),Eκ+µ,p (3.40)
is compact.
Proof. By virtue of [22] an embedding
Wl p
Ω;E(A),E⊂Lp(Ω;E) (3.41)
is compact. Then by the estimate (3.38) we obtain the assertion ofTheorem 3.10. Result 3.11. Ifl1=l2= ··· =ln=lthen we obtain continuity of embedding operators in
isotropic class
Wl p
Ω,E(A)E. (3.42)
Remark 3.12. IfE=Handp=q=2,Ω=(0,T),l1=l2= ··· =ln=l,A=A×≥cIthen
we obtain the result of Lions-Peetre [18] and even in the one dimensional case the result of Lions-Peetre are improving for in general, non self adjoint positive operatorsA.
IfE=R,A=Ithen we obtain embedding theorems
DαWlp(Ω)⊂Lq(Ω) (3.43)
proved in [5] for numerical Sobolev spacesWl p(Ω).
4. Applications
4.1. Embedding in vector-valued spaces. Lets∈R,s >0. Let us consider the space [28, Section 1.18.2]
lσs=
u;u= ui,i=1, 2,. . .,∞,ui∈C,
∞
i=1
2ipsuiσ
1/σ
<∞
with the norm
uls
σ=
∞
i=1
2ipsuiσ
1/σ
<∞, 1< σ <∞. (4.2)
Note thatl0
σ=lσ. Let Ais infinite matrix defined in the space lσ such thatD(A)=lsσ,
A=[δi j2si], whereδi j=0, wheni=j,δi j=1, wheni=j,i,j=1, 2,. . .,∞. It is clear to
see that, this operatorAis positive inlp. Then byTheorem 3.5we obtain that for 0≤µ≤
1−κ,κ=nk=1(αk+ 1/ p−1/q)/lk the embeddingDαWlp(Ω,lσs,lσ)⊂Lq(Ω,ls
(1−κ−µ)
σ ) is
continuous and also an estimate of type (3.29) is satisfied.
It should be noted that the above embedding has not been obtained with classical method until now.
4.2. Maximal regular differential-operator equations. Let us consider diff erential-operator equations
Lu=
n
k=1 (−1)lkt
kDk2lku+Aλu+
|α:2l|<1
n
k=1 tαk/2lk
k Aα(x)Dαu=f (4.3)
in the spaceLp(Rn;E), where,Aλ=A−λI,AandAα(x) are in general, unbounded
oper-ators in Banach spaceE,tk,k=1, 2,. . .,nparameters,l=(l1,l2,. . .,ln),li-positive integers.
Theorem4.1. Suppose the following conditions hold: (1) 0< tk≤t0<∞,k=1, 2,. . .,n,0< ϕ≤π;
(2)Eis a Banach space satisfying multiplier condition with respect top,1< p <∞; (3)Ais aR-positive operator inEand
Aα(x)A−(1−|α:2l|−µ)∈L∞Rn,L(E), 0< µ <1− |α: 2l|. (4.4)
Then for all f ∈Lp(Rn;E)and for sufficiently large|λ|>0,λ∈S(ϕ)(4.3) has a unique
solutionu(x)that belongs to spaceW2l
p(Rn;E(A),E)and the estimate hold n
k=1
tkD2klkuLp(Rn;E)+AuLp(Rn;E)≤CfLp(Rn;E). (4.5)
Proof. At first we will consider principal part of (4.3), that is, differential-operator equa-tion
L0u=
n
k=1 (−1)lkt
kD2klku+Aλu=f . (4.6)
Then we apply Fourier transform to (4.6) with respect tox=(x1,. . .,xn) and obtain
n
k=1
In view of (1) condition, nk=1tkξk2lk ≥0 for all ξ=(ξ1,. . .,ξn)∈Rn. Therefore, λ−
n
k=1tkξk2lk∈S(π) for allξ∈Rn. That is, an operatorA−[λ−
n
k=1tkξk2lk]I is invertible
inE. Hence from (4.7) we obtain that the solution of (4.6) can be represented in the form
u(x)=F−1
A−
λ−
n
k=1 tkξk2lk
I
−1
fˆ. (4.8)
It is clear to see that operator-functionϕλ,t(ξ)=[A−(λ−kn=1tkξk2lk)I]−1is multiplier
inLp(Rn;E) uniformly toλandt. Actually, by virtue ofϕ-positiveness of operatorAfor
allξ∈Rnandλ∈S(ϕ) we get
ϕλ(ξ)
L(E)≤M
1 +λ−
n
k=1 tkξk2lk
−1
≤M0. (4.9)
Moreover, sinceDkϕλ,t(ξ)=[A−(λ−nk=1tkξk2lk)]−22lktkξk2lk−1then
ξkDkϕλ,t
L(E)≤2lktkξk2lk
A−
λ−
n
k=1 tkξk2lk
−2
≤2lktkξk2lk
1 +
λ−
n
k=1 tkξk2lk
−2
≤M.
(4.10)
Using the estimate (4.10) and theR-positiveness of operatorA for operator-functions ϕkλ,t(ξ)=ξk2lkϕλ,t,k=1, 2,. . .,nandϕ0λ,t=Aϕλ,twe have
Rξβϕ
k,λ,t(ξ), β∈Un:ξ∈Vn
≤Cβ,
Rξβϕ
0,λ,t(ξ), β∈Un:ξ∈Vn
≤Mβ.
(4.11)
Then by virtue of condition (2) and estimates (4.11) we obtain that operator-functions ϕλ,t,ϕkλ,t,ϕ0λ,t are multiplier in the spaceLp(Rn;E). By using the representation of (4.8)
we have
D2lk
k uLp=
F−1iξ
k
2lkϕ
λ,t(ξ)fˆL
p, AuLp=F−1AuˆLp=F
−1Aϕ
λ,t(ξ)fˆLp.
(4.12)
By the definition of multiplier we obtain that for all f ∈Lp(Rn;E) there is unique solution
of (4.6) in the form (4.8) and holds estimate
n
k=1
tkD2klkuLp+AuLp≤CfLp. (4.13)
In the spaceLp(Rn;E), we consider the differential operatorL0−λthat is generated by the problem (4.6), that is
DL0−λ
=W2l p
Rn,E(A),E, L
0−λ
u=
n
k=1 (−1)lkt
The estimate (4.13) implies that the operatorL0−λhas a bounded inverse acting from Lp(Rn;E) intoW2pl(Rn;E(A),E). We denote byL−λthe differential operator inLp(Rn;E)
that is generated by the problem (4.3). Namely,
D(L−λ)=W2l p
Rn,E(A),E, (L−λ)u=L0−λu+L1u, (4.15)
where
L1u=
|α:2l|<1
n
k=1 tαk/2lk
k Aα(x)Dαu. (4.16)
In view of condition (3) and by virtue ofTheorem 3.5foru∈W2l
p(Rn;E(A),E) we
have
L1u
Lp≤
|α:2l|<1
n
k=1 tαk/2lk
k Aα(x)DαuLp
≤
|α:2l|<1
n
k=1 tαk/2lk
k A1−|α:2l|−µDαuLp
≤C
hµ
n
k=1
tkDk2lkuLp+AuLp
+h−(1−µ)u
Lp
.
(4.17)
Then from estimates (4.13) and (4.17) foru∈W2l
p(Rn;E(A),E) we obtain
L1u
Lp≤C
hµL
0−λ
uL
p+h
−(1−µ)u
Lp
. (4.18)
SinceuLp=(1/λ)(L0−λ)u+L0uLpforu∈W
2l
p(Rn;E(A),E) we get
uLp≤
1
|λ|L0−λ
uLp+L0uLp
≤|1λ|L0−λ
uLp+ 1
|λ|
n
k=1
tkDk2lkuLp+AuLp
.
(4.19)
From estimates (4.13) and (4.17), (4.18), and (4.19) foru∈W2l
p(Rn;E(A),E) obtain
L1u
Lp≤Ch
µL
0−λ
u+C1|λ|−1h−(1−µ)L0−λ
u. (4.20)
Then choosinghandλsuch thatChµ<1,C
1|λ|−1h−(1−µ)<1 from (4.20) obtain
L1
L0−λ −1
L(E)<1. (4.21)
Using relation (4.15), estimates (4.13) and (4.21) and perturbation theory of linear op-erators [14], we establish that the differential operatorL−λis invertible fromLp(Rn;E)
intoW2l
Remark 4.2. There are a lot of positive operators in the different concrete Banach spaces. Therefore, putting instead ofE, concrete Banach spaces and instead of operatorA, con-creteR-positive differential, pseudo differential operators, or finite, infinite matrices on the differential-operator equations (4.3), by virtue ofTheorem 4.1we can obtain the dif-ferent class of maximal regular partial differential equations or system of equations.
Acknowledgment
This work was supported by project of UDP-227/18022004 of Istanbul University.
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Veli B. Shakhmurov: Department of Electric-Electronics Engineering, Faculty of Engineering, Istanbul University, Avcilar, 34850 Istanbul, Turkey
