Báo cáo hóa học: "EMBEDDING THEOREMS IN BANACH-VALUED B-SPACES AND MAXIMAL B-REGULAR DIFFERENTIAL-OPERATOR EQUATIONS" ppt

SHAKHMUROV Received 28 September 2004; Revised 8 November 2005; Accepted 4 May 2006 The embedding theorems in anisotropic Besov-Lions type spacesB l p,θR n;E0,E are stud-ied; hereE0 and

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(R n;E0,E) are

stud-ied; hereE0 and E are two Banach spaces The most regular spaces E αare found suchthat the mixed differential operators Dαare bounded fromB l p,θ(R n;E0,E) to B s q,θ(R n;E α),

whereE αare interpolation spaces betweenE0andE depending on α =(α1,α2, , α n) and

l =(l1,2, , 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 thequasielliptic partial differential equations and the parabolic Cauchy problems are stud-ied

Copyright © 2006 Veli B Shakhmurov This is an open access article distributed underthe 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 prehensive introduction to the theory of embedding of function spaces and historicalreferences may be also found in [37] In abstract function spaces embedding theoremshave been investigated in [4,5,10,17,21,27,34,40] Lions and Peetre [21] showed thatif

[H0,H] θare interpolation spaces betweenH0andH for 0 ≤ θ ≤1 The similar questionsfor anisotropic Sobolev spacesW l

p(Ω;H0,H),Ω⊂ R nand for corresponding weightedHindawi Publishing Corporation

Journal of Inequalities and Applications

Volume 2006, Article ID 16192, Pages 1 22

DOI 10.1155/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 andspectrum 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 andhistorical references may be found in [15,18,40] In these works, Hilbert-valued functionspaces essentially have been considered The maximalL pregularity and Fredholmness ofpartial 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 similarproblems have been investigated in [13,28–32,36,39,40]

LetE0,E be Banach spaces such that E0is continuously and densely embedded inE.

In the present paper,E-valued Besov spaces B l+s p,θ(R n;E0,E) = B s p,θ(R n;E0)∩ B l+s p,θ(R n;E) are

introduced and called Besov-Lions type spaces The most regular interpolation classE α

betweenE0 and E is found such that the appropriate mixed di fferential operators D α

are bounded fromB l+s(R n;E0,E) to B 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 valued Besov spaces is derived

Banach-The paper is organized as follows.Section 2collects notations and definitions.Section

3presents the embedding theorems in Besov-Lions type spaces

2 Notations and definitions

LetE be a Banach space Let L p(Ω;E) denote the space of all strongly measurable E-valued

functions that are defined onΩ⊂ R nwith the norm



.

(2.1)

The Banach spaceE is said to be a ζ-convex space (see [9,11,12,19]) if there exists

onE × 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, foru ≤1≤ v. (2.2)

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

is bounded inL p( 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, p spaces and theLorentz spacesL pq, p, q ∈(1,∞)

Let C be the set of complex numbers and let

withλ ∈ S ϕ, ϕ ∈(0,π], I is identity operator in E, and L(E) is the space of all bounded

linear operators inE Sometimes A + λI will be written as A + λ and denoted by A λ It is

known [37, Section 1.15.1] that there exist fractional powersA θof the positive operator

A Let E(A θ) denote the spaceD(A θ) with the graphical norm

u E(A θ)=u p+A θ up1/ p

, 1≤ p < ∞,−∞ < θ < ∞. (2.6)LetE0andE be two Banach spaces By (E0,E) σ,p, 0 < σ < 1, 1 ≤ p ≤ ∞we will denotethe interpolation spaces obtained from{E0,E}by theK-method (see, e.g., [37, Section1.3.1] or [10])

LetS(R n;E) denote a Schwartz class, that is, the space of all E-valued rapidly decreasing

smooth functionsϕ on R n.E =C will be denoted byS(R n) LetS(R n;E) denote the space

ofE-valued tempered distributions, that is, the space of continuous linear operators from S(R n) toE.

Letα =(α1,α2, , α n), α iare integers AnE-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 |

holds for allϕ ∈ S(R n)

By using (2.7) the following relations

LetL ∗ θ(E) denote the space of all E-valued function spaces such that

u L ∗ θ(E) =

∞

0

u(t)θ E

dt t

Lets =(s1,s2, , s n) and s k > 0 Let F denote the Fourier transform Fourier-analytic

rep-resentation ofE-valued Besov space on R nis defined as

It should be noted that the norm of Besov space do not depend onκk Sometimes we

will writeu B s

LetE0is continuously and densely embedded intoE W l B s p,θ(R n;E0,E) denotes a space of

all functionsu ∈ B s p,θ(R n;E0)∩ W l B s p,θ(R n;E) with the norm

norm

u B s+l(R n;E0,E) = u B s

p,θ(R n;E0)+u B l+s(R n;E) (2.13)ForE0= E the space B l+s p,θ(R n;E0,E) will be denoted by B l+s p,θ(R n;E).

Letm be a positive integer C( Ω;E) and C m(Ω;E) will denote the spaces of all E-valued

bounded continuous andm-times continuously differentiable functions on Ω, tively We set

The set of all multipliers fromB s

p,θ(R n;E1) toB s

q,θ(R n;E2) will be denoted byM q,θ p,θ(s, E1,

E2).E1= E2= 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, Section2.2.2.], and [4,11,12,14,22], respectively

Let

H k = Ψh∈ M q,θ p,θ

s, E1,E2

,h =h1h2, , h n

q,θ(R n;E1) (2.17)for allh ∈ K and u ∈ S(R n;E1)

Letβ =(β1,β2, , β n) be multiindexes We also define

Definition 2.1 A Banach space E satisfies a B-multiplier condition with respect to p, q,

θ, and s (or with respect to p, θ, and s for the case of p = q) whenΨ∈ C n(R n;L(E)),

1≤ p ≤ q ≤ ∞,β ∈ U n, and ξ ∈ V nif the estimate

ξ1β1+νξ2β2+ν

, ,ξ nβ n+νD β Ψ(ξ)

impliesΨ∈ M q,θ p,θ(s, E).

Remark 2.2. Definition 2.1is a combined restriction toE, p, q, θ, and s This condition

is sufficient for our main aim Nevertheless, it is well known that there are Banach spacessatisfying theB-multiplier condition for isotropic case and p = q, for example, the UMD

spaces (see [4,14])

A Banach spaceE is said to have a local unconditional structure (l.u.st.) if there exists a

constantC < ∞such that for any finite-dimensional subspaceE0ofE there exists a

finite-dimensional spaceF with an unconditional basis such that the natural embedding E0⊂ E

factors asAB with B : E0→ F, A : F → E, and AB ≤ C All Banach lattices (e.g., L p,

L p,q, Orlicz spaces, C[0, 1]) have l.u.st.

The expressionu E1∼u E2means that there exist the positive constantsC1andC2such that

C1 u  E1 ≤  u  E2 ≤ C2 u  E1 (2.20)

for allu ∈ E1∩ E2

Letα1,α2, , α nbe nonnegative and letl1,2, , l nbe positive integers and let

inB s p,θ(R n;E), where a αare complex-valued functions andA(x), A α( x) are possibly

un-bounded operators in a Banach spaceE, here the domain definition D(A) = D(A(x)) of

operatorA(x) does not depend on x For l1= l2=, , = l nwe obtain isotropic equationscontaining the elliptic class of DOE

The function belonging to spaceB s+l p,θ(R n;E(A), E) and satisfying (2.22) a.e onR n issaid to be a solution of (2.22) onR n

Definition 2.3 The problem (2.22) is said to be aB-separable (or B s p,θ(R n;E)-separable) if

the problem (2.22) for all 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)whereL is a realization di fferential operator in B s

p,θ(R n;E) generated by problem (2.22),that is,

Lemma 3.1 Let A be a positive operator in a Banach space E, let b be a positive number,

r =(1,r2, , r n), α =(α1,α2, , α n), and l =(l1,2, , l n), where ϕ ∈(0,π], r k ∈[0,b], l k

are positive and α 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

there exists a constantM0independent onξ, such that

Then the spaces B l+s p,θ(R n;E) and W l B s p,θ(R n;E) are coincided.

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(R n;E) ≤ Cu W l B s

for allu ∈ 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) it is sufficient to prove the followingestimate:

To see this, it is sufficient to show that the function

is Fourier multiplier inL p( R n;E) It is clear to see that for β ∈ U nandξ ∈ V n

It implies the second embedding This completes the prove ofLemma 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,∞ ], and s =(s1,s2, , s n), where s k are positive numbers;

(2)α =(α1,α2, , α n), l =(l1,2, , l n), where α 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, where ϕ ∈(0,π] and 0 < h ≤ h0< ∞.

Then the following embedding

D α B l+s p,θ



R n;E(A), E

⊂ B s q,θ

Proof We have

D α u

B s q,θ(R n;E(A1−κ− μ))=A1−κ− μ D α u

B s q,θ(R n;E) (3.24)for allu such that

D α u

B s q,θ(R n;E(A1−κ− μ))< ∞. (3.25)

On the other hand by using the relation (2.8) we have

A1− α − μ D α u = F −FA1−κ− μ D α u = F −(iξ) α A1−κ− μ Fu. (3.26)Since the operatorA is closure and does not depend on ξ ∈ R nhence denotingFu by u,from the relations (3.24), (3.26) and by definition of the spaceW l B s

p,θ(R n;E0,E) we have

D α u

B s q,θ(R n;E(A1−κ− μ))F −(iξ) α A1−κ− μ u 

B s q,θ(R n;E),

u W l B s p,θ(R n;E0,E)∼Au B s

(3.27)

By virtue ofLemma 3.2and by the above relations it is sufficient to prove that

F −(iξ) α A1−κ− μ u 

B s q,θ(R n;E)

+h −(1− μ)F −u 

B s p,θ(R n;E)

.

(3.28)The inequality (3.23) will be followed if we prove the following inequality

F − 

(iξ) α A1−κ− μ u 

B s p,θ(R n;E) ≤ C μF − 

h μ(A + η)



u

B s p,θ(R n;E) (3.29)for a suitableC μand for allu ∈ B s+l

p,θ(R n;E(A), E), where

Definition 2.1it is clear that the inequality (3.23) will follow immediately from (3.31) if

we can prove that the operator-functionΨ=(iξ) α A1−κ− μ[h μ(A + η)] −1is a multiplier in

M q,θ p,θ(s, E), which is uniform with respect to h Since E satisfies the multiplier condition

with respect top, q, θ, and s, then byDefinition 2.1in order to show thatΨ∈ M q,θ p,θ(s, E),

it suffices to show that there exists a constant Mμ> 0 with

for allξ ∈ R n 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, ,β n) where β k =1 andβ j =0 for

j = k By di fferentiation of the operator-function Ψ(ξ), by virtue of the positivity of A,

and by using (3.34) we have

Result 3.4 Let all conditions ofTheorem 3.3hold Then for allu ∈ B l+s

p,θ(R n;E(A), E) we

have a multiplicative estimate

D α u

B s q,θ(R n;E(A1−κ− μ))≤ C μ u1B − l+s μ(R n;E(A),E) u μ B s

p,θ(R n;E) (3.37)Indeed settingh = u B s

p,θ(R n;E) · u −1

B l+s(R n;E(A),E)in the estimate (3.23) we obtain the aboveestimate

Remark 3.5 It seems from the proof ofTheorem 3.3that the extra condition to space

E (E is UMD space with l.u.st.) and the condition s k /(l k+s k) +s j /(l j+s j)≤1 fork, j =

1, 2, , n are due toLemma 3.2(here the l.u.st condition for the spaceE is required due to

using of Marcinkiewicz-Lizorkin type multiplier theorem [41] inL p(R n;E) space)

There-fore, the proof ofTheorem 3.3implies the following

Result 3.6 Suppose the following conditions hold:

(1)E is a Banach space satisfying the B-multiplier condition with respect to p, q ∈

(1,∞),θ ∈[1,∞] ands =(s1,s2, , s n), where s kare positive numbers;

(2)α = t(α1,α2, , α n),l =(l1,2, , l n), whereα kare nonnegative andl k are positiveintegers such thatκ= |(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< ∞

Then the following embedding

for allu ∈ W l B s p,θ(R n;E(A), E).

Remark 3.7 The condition s k /(l k+s k) + s j /(l j+s j)≤1 fork, j =1, 2, , n inTheorem 3.3

arise due to anisotropic nature of spaceB s p,θ For an isotropic case the above conditionshold without any assumptions

4 Application to vector-valued function spaces

By virtue ofTheorem 3.3we obtain the following

Result 4.1 For A = I we obtain the continuous embedding D α B l+s p,θ(R n;E) ⊂ B s p,θ(R n;E)

and corresponding estimate (3.23) for 0≤ μ ≤1−κin spaceB s+l p,θ(R n;E).

Result 4.2 For E=R m,A= I we obtain the following embedding D α B l+s p,θ(R n;R m)⊂ B s q,θ(R n;

R m) for 0≤ μ ≤1−κ and a corresponding estimate (3.23) For E = R, A = I we get

the embeddingD α B l+s

p,θ(R n)⊂ B s

q,θ(R n) proved in [8, Section 18] for the numerical Besovspaces

Result 4.3 Let l1= l2= ··· = l n = m, s1= s2= ··· = s n = σ, and p = q Then for all E

∈UMD and|α| ≤ m we obtain that the continuous embedding D α B σ+m p,θ (R n;E(A), E) ⊂

(4.1)with the norm

It should not be that the above embedding has not been obtained with a classicalmethod until now.

Consider the following differential-operator equation

p,q(R n;E), where A(x), A α(x) are possible unbounded operators in a Banach space E,

a kare complex-valued functions,l =(l1,2, , l n) and l iare positive integers The imal regularity for DOE was investigated, for example, in [12,14,30] Let us considerDOE with constant coefficients

whereA is a possible unbounded operator in E, A λ = A + λ and b αare complex numbers

Theorem 5.1 Suppose the following conditions hold:

(1)E is UMD space with l.u.st satisfying B-multiplier condition with respect to p ∈

(1,∞ ), q ∈[1,∞ ], and s =(s1,s2, , s n), where s k are positive numbers;

(2)A is a ϕ-positive operator in E with ϕ ∈(0,π] and

Then for all f ∈ B s

p,q(R n;E), for |argλ| ≤ π − ϕ and su fficiently large |λ| > 0 ( 5.2 ) has

a unique solution u(x) that belongs to space B l+s(R n;E(A), E), and the coercive uniform estimate



| α:.l |≤1

|λ|1−| α:.l |D α u

B s p,q+Au B s

p,q ≤ C f  B s

holds with respect to the parameter λ.

Proof By applying the Fourier transform to (5.2) we obtain



K(ξ) + A λ

SinceK(ξ) ∈ S(ϕ) for all ξ ∈ R n, the operatorA + [λ + K(ξ)] is invertible in E So, we

obtain that the solution of (5.5) can be represented in the form

u(x) = F −1 

The function belonging to spaceB s+l p,θ(R n;E(A), E) and satisfying... n) be multiindexes We also define

Definition 2.1 A Banach space E satisfies a B-multiplier condition with respect to p, q,

θ, and s (or with respect to p, θ, and s for... 2.2. Definition 2.1is a combined restriction toE, p, q, θ, and s This condition

is sufficient for our main aim Nevertheless, it is well known that there are Banach spacessatisfying theB-multiplier