EMBEDDING OPERATORS AND MAXIMAL REGULAR DIFFERENTIAL-OPERATOR EQUATIONS IN BANACH-VALUED FUNCTION SPACES

DIFFERENTIAL-OPERATOR EQUATIONS IN BANACH-VALUED FUNCTION SPACES

VELI B. SHAKHMUROV

Received 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 D^α belong with values; the boundedness and compactness of differential operatorsD^αfrom 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 spacesW_p^l(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 A comprehensive introduction to the differential-operator equa- tions and historical references may be found in [16,32].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 spacesW_p^l(Ω;E0,E), wherel=(l1,l2,. . .,l_n) 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 operatorD^αis continuous from these Banach-valued

Copyright©2005 Hindawi Publishing Corporation

Journal of Inequalities and Applications 2005:4 (2005) 329–345 DOI:10.1155/JIA.2005.329

Sobolev spaces toE_α-valuedL_pspaces, whereE_αare interpolation spaces betweenE0and Edepending on the order of differentiationsD^α. By applying these results, the maximal Lp-regularity of certain class of anisotropic partial differential-operator equations are de- rived.

Letα1,α2,. . .,α_nbe nonnegative integer numbers and D^α=D^α1¹D2^α2···D^α_nⁿ= ∂^α

∂x^α1¹∂x^α2²···∂x^αnⁿ. (1.1) Under certain assumptions to be stated later, we prove that the operatorsu→D^αuare bounded from spaceW_p^l(Ω;E(A),E) to spaceL_q(Ω;E(A^1−κ)), that is, embedding

D^αW_p^lΩ;E(A),E⊂L_qΩ;EA^1−κ (1.2) is continuous. More precisely for 0< µ≤1−κwe prove the estimate

D^αu_L

p(Ω;E(A^1−κ))≤C_µh^µu_W^l_p(Ω;E(A)E)+h^−(1−µ)uLp(Ω;E)

(1.3) for allu∈W^l_p(Ω;E(A),E) and 0≤h≤h0<∞. The constantC_µin the above equation is independent ofu∈W_p^l(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

a_kt_kD_k^lku+Au

|α:l|<1

n k=1

t_k^αk/lkA_α(x)D^αu= f, (1.4) depend on parameterst=(t1,t2,. . .,tn), whereA is a positive operator on the Banach space E,A_α(x) is an operator such that A_α(x)A^{−(1−|α:l|)} is bounded on E, where α= (α1,α2,. . .,αn),l=(l1,l2,. . .,ln), |α:l| =_n

k=1(αk/lk). In general, this equations possess different derivatives and different parameters with respect to the various variables. Taking l1=l2= ··· =l_n=2lin the above equations we obtain elliptic equation with parameters

n k=1

aktkD^2l_ku+Au+

|α|<2l

n k=1

t^α_k^k/2lAα(x)D^αu=f(x). (1.5) We prove the maximal regularity of this differential-operator equations inL_p(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

LetRbe the set of real numbers,Cbe the set of complex numbers. LetEandE0be Banach spaces andL(E0,E) denotes the spaces of bounded linear operators acting fromE0toE.

ForE0=Ewe denoteL(E,E) byL(E),Idenotes the identity operator in the Banach space E. LetAbe a linear operator inE. We will sometimes useA+ξorA_ξ instead ofA+ξI

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)≤M1 +|ξ|₋1

(2.2) withξ∈Sϕ, whereMis a positive constant [28].

EA^θ=

u,u∈DA^θ,uE(A^θ)=A^θu_E+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 D^l_k^k=∂^lk/∂x^l_k^k, k= 1, 2,. . .,n.

We introduce aE0-valued anisotropic function spaceW^l_p(Ω;E0,E) that consist of func- tionsu∈Lp(Ω;E0) such that have the generalized derivativesD^l_k^ku∈Lp(Ω;E) with the norm

u_Wp^l(Ω;E0,E)= uLp(Ω;E0)+ n k=1

D^l_k^ku_Lp(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 spaceW^l_p(Ω;E0,E) parameterized norm

u_W^l_p,t(Ω;E0,E)= uLp(Ω;E0)+ n k=1

tkD^l_k^ku_Lp(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 spaceL_p(R;E), p∈(1,∞), for those and only those Banach spaces Ewhich possess the property ofξ-convexity. Theξ-convex Banach spaces is often called

UMD spaces. UMD spaces containsL_p,l_pspaces and the Lorentz spacesL_pq,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 fromL_p(Rⁿ;E1) toL_q(Rⁿ;E2) if there exists a constantM >0 such that

F⁻¹Ψ(ξ)Fu_Lq(Rn;E2)≤CuLp(Rⁿ;E1) (2.9) for allu∈L_p(Rⁿ;E1), whereFandF⁻¹are Fourier and inverse Fourier transformations, respectively.

We denote the set of all multipliers fromLp(Rⁿ;E1) toLq(Rⁿ;E2) byM^qp(E1,E2). For E1=E2=Ewe denoteM^qp(E1,E2) byM^qp(E). Let

Hk= Ψh∈M^qp

E1,E2

,h=

h1,h2,. . .,hL

∈Q (2.10)

be a collection of multipliers in M^q,γ_p,γ(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⁻¹ΨhFu_L

q(Rⁿ,E2)≤CuLp(Rⁿ,E1) (2.11) for allh∈Kandu∈Lp(Rⁿ;E1).

The exposition of the theory ofL_p-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{r_j}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)∈R^L 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

, ξ^α=ξ1^α1ξ2^α2···ξ_n^αn, |ξ|^α= |ξ|^α1|ξ|^α2···|ξ|^αn.

(2.14)

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) :ξ^{β+1/ p−1/q}D^β_ξΨ(ξ) :ξ∈Vn,β∈Un

(2.15)

areR-bounded, thenΨ∈M^qp(E).

A Banach spaceEhas a property (α), (see, e.g., [12]) if there exists a constantαsuch that

N i,j=1

α_{i j}ε_iε_jx_{i j}

L2(Ω×Ω;E)

d y≤α N i,j=1

ε_iε_jx_{i 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,. . .,ε_N 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

L_A= 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 onL_q, 1≤q≤ ∞[17],Ahas the bounded imaginary powers with (−A^it)B(E)≤Ce^ν|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∈A_p,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∈Mp,γ^p,γ(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 r_k∈ {0,b}. Lett=(t1,t2,. . .,t_n), wheret_k,k=1, 2,. . .,nare nonnegative parameters,0<

t_k≤t0<∞,α=(α1,α2,. . .,α_n)andl=(l1,l2,. . .,l_n),l_k>0such thatκ= |(α+r) :l| ≤1.

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^(α_k^k+r)/lkξ^r(iξ)^αA^1−κ−µh^−µ

A+ n k=1

t_kδξ_k^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−[ⁿ_k=1t_k(δ(ξ)ξ_k)^lk+h⁻¹]∈S(ϕ) for allϕ∈[0,π) then by virtue of the pos- itiveness ofA, operatorB(ξ)=A+ⁿ_k=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^(α_k^k+r)/lk|ξ|^r+αA^1−κ−µu_E

=(hA)^1−κ−µu_Eh^−(1−µ) ht11/l1

ξ1^α1+r1···

htn1/ln

ξn^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≤C_µh^(1−µ)hAu^1−κ−µu^κ+µ ht1

1/l1

ξ1^α1+r1···

ht_n^1/lnξ_n^αn+rn. (3.4) Now, we apply the Young inequality, which states thatg1g2≤g1^k1/k1+g2^k2/k2for any posi- tive real numbersg1,g2andk1,k2with 1/k1+ 1/k2=1, to the product

hAu^1−κ−µ

u^κ+µ ht1

1/l1

ξ1^α1+r1···

ht_n^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)}···

ht_nξ_n^{(αn+rn)/(κ+µ)}. (3.6) Since

n i=1

αi+ri

(κ+µ)⁼ 1 κ+µ

n i=1

αi+ri

l_i ⁼ κ

κ+µ^≤1 (3.7)

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

1 +

n k=1

ξ_k^lk

(3.8)

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 inC_µ, we obtain

ψ_t(ξ)f≤C_µh^µ

Au+ _n

k=1

t_kδξ_kξ_k^lk+h⁻¹

u

. (3.10)

Substituting the value ofu, we get ψt(ξ)f≤CµAB⁻¹(ξ)f+

_n

k=1

tk δξk

ξklk

+h⁻¹

B⁻¹(ξ)f. (3.11)

SinceAis positive operator in the spaceE, we have

A+ n k=1

tk δξk

ξklk

+h⁻¹ −1

f ≤M

1 +

n k=1k

tk δξk

ξklk

+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,. . .,α_n),l=(l1,l2,. . .,l_n), wherel_kare positive andα_kare 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

D^αW^l_pRⁿ;E(A),E⊂L_qRⁿ;EA^1−κ−µ (3.14) is continuous and there exists a constantC_µ>0, depending only onµ, such that

n k=1

t_k^{(αk+1/ p−1/q)/lk}D^αu_Lp(Rn;E(A^1−κ−µ))≤Cµ

h^µu_Wp,t^l _(Rⁿ_;E(A),E)+h^−(1−µ)uLp(Rⁿ;E)

(3.15) for allu∈W^l_p(Rⁿ;E(A),E),tandh.

Proof. We have

D^αu_Lq(Rn;E(A^1−κ−µ))=

Rⁿ

D^αu^q_E(A1−κ−µ)dx 1/q

Rⁿ

A^1−κ−µD^αu^q_Edx 1/q

A^1−κ−µD^αu_L

q(Rⁿ;E)

(3.16)

for allusuch that

D^αu_Lq(Rn;E(A^1−κ−µ))<∞. (3.17) On the other hand we have

A^1−α−µD^αu=F⁻FA^1−κ−µD^αu=F⁻A^1−κ−µFD^αu

=F⁻A^1−κ−µ(iξ)^αFu=F⁻(iξ)^αA^1−κ−µFu. (3.18) Hence denotingFuby ˆu, we get from relations (3.16) and (3.18)

D^αu_L

q(Rⁿ;E(A^1−κ−µ))F⁻(iξ)^αA^1−κ−µuˆ_L

q(Rⁿ;E). (3.19) Moreover, we have

u_W^l_p,t(Rn;E(A),E)= uLp(Rⁿ;E(A))+ n k=1

t_kD_k^lku_Lp(Rn;E)

=F⁻uˆ_Lp(Rn;E(A))+ n k=1

t_kF⁻iξ_k^lkuˆ

Lp(Rⁿ;E)

F⁻¹Auˆ_L

p(Rⁿ;E)+ n k=1

t_kF⁻iξ_k^lkuˆ

Lp(Rⁿ;E)

(3.20)

for allu∈W^l_p(Rⁿ;E(A),E). Thus proving the inequality (3.15) for some constantsC_µ is equivalent to proving

n k=1

t_k^{(αk+1/ p−1/q)/lk}F⁻(iξ)^αA^1−κ−µuˆ_Lq(Rn,E)

≤Cµ

h^µF⁻Auˆ_Lp(Rn,E)+ n k=1

tkF⁻iξk

lkuˆ

Lp(Rⁿ,E)+h^−(1−µ)F⁻uˆ_Lp(Rn,E)

(3.21) for a suitableC_µ. Now if δ is a multiplier of the form described as inExample 2.4, by virtue of multiplier there is constantsC_k>0 for eachk=1, 2,. . .,nsuch that

F⁻1

iδξ_kiξ_k^lkuˆ

Lp(Rⁿ;E)≤C_kF⁻iξ_k^lkuˆ

Lp(Rⁿ;E) (3.22)

for allξ∈Rⁿ. Thus the inequality (3.15) will follow if we prove the following inequality n

k=1

t_k^{(αk+1/ p−1/q)/lk}F⁻(iξ)^αA^1−κ−µuˆ_Lp(Rn;E)

≤Cµ

F⁻

h^µ

A+

n k=1

tk δξk

ξklk

+h^−(1−µ)

ˆ u

Lp(Rⁿ;E)

(3.23)

for a suitableC_µ>0, and for allu∈W_p^l(Rⁿ;E(A),E).

Let us express the left-hand side of (3.23) as follows n

k=1

t_k^{(αk+1/ p−1/q)/lk}F⁻(iξ)^αA^1−κ−µuˆ_Lq(Rn;E)

= n k=1

t^(α_k^{k+1/ p−1/q)/lk}F⁻(iξ)^αA^1−κ−µQ⁻¹(ξ)Q(ξ)_Lq(Rn;E),

(3.24)

where

Q(ξ)=h^µ

A+ n k=1

t_kδξ_kξ_k^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)^αA^1−κ−µQ⁻¹(ξ) is a multiplier inL_p(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< t_k≤T, 0< h≤h0andξ∈(−∞,∞) we obtain

R Ψt,h,µ(ξ) :ξ_∈V_n≤M,

Rξ^{β+1/ p−1/q}D_ξ^βΨt,h,µ(ξ) :β∈U_n:ξ∈V_n≤M_β. (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.

It is possible to stateTheorem 3.2in a more general setting. For this, we use the concept of extension operator.