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

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 spaces B
l
p,θ
(R
n
;E
0
,E)arestud-
ied; here E
0
and E are two Banach spaces. The most regular spaces E
α
are found such
that the mixed differential operators D
α
are bounded from B
l
p,θ
(R
n
;E
0
,E)toB
s
q,θ
(R

n
;E
α
),
where E
α
are interpolation spaces between E
0
and E depending on α = (α
1
,α
2
, ,α
n
)and
l
= (l
1
,l
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 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prehensive introduction to the theory of embedding of function spaces and historical
references may be also found in [37].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
∈ L
2

0,T;H
0

, u
(m)
∈ L
2
(0,T;H), (1.1)
then
u
(i)
∈ L
2

0,T;

H,H
0

i/m


, i = 1,2, ,m − 1, (1.2)
where H
0
, H are Hilbert spaces, H
0
is continuously and densely embedded in H,where
[H
0
,H]
θ
are interpolation spaces between H
0
and H for 0 ≤ θ ≤ 1. The similar questions
for anisotropic Sobolev spaces W
l
p
(Ω;H
0
,H), Ω ⊂ R
n
and for corresponding weighted
Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2006, Article ID 16192, Pages 1–22
DOI 10.1155/JIA/2006/16192
2 Embedding and B-regular operators
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 maximal L
p
regularity 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].
Let E
0
, E be Banach spaces such that E
0
is continuously and densely embedded in E.
In the present paper, E-valued Besov spaces B
l+s
p,θ
(R
n
;E
0
,E) = B
s
p,θ
(R
n
;E
0
) ∩ B
l+s
p,θ

(R
n
;E)are
introduced and cal led Besov-Lions type spaces. The most regular interpolation class E
α
between E
0
and E is found such that the appropriate mixed differential operators D
α
are bounded from B
l+s
p,q
(R
n
;E
0
,E)toB
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 Banach-
valued Besov spaces is derived.
The paper is organized as follows. Section 2 collects notations and definitions. Section
3 presents the embedding theorems in Besov-Lions type spaces
B
s+l

p,q

R
n
;E
0
,E

. (1.3)
Section 4 contains applications of the underlying embedding theorem to vector-valued
function spaces. Section 5 is devoted to the maximal regularity (in B
s
p,q
(R
n
;E)) of the
certain class of anisotropic DOE with var iable coefficients in principal part. Then by us-
ing these results the maximal B-regularity of the parabolic Cauchy problem is shown. In
Section 6 these 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
Let E be a Banach space. Let L
p
(Ω;E) denote the space of all strongly measurable E-valued
functions that are defined on Ω
⊂ R
n
with the norm
 f 
L

p
(Ω;E)
=




f (x)


p
E
dx

1/p
,1≤ p<∞,
 f 
L
∞
(Ω;E)
= esssup
x∈Ω



f (x)


E


, x =

x
1
,x
2
, ,x
n

.
(2.1)
The Banach space E is said to be a ζ-convex space (see [9, 11, 12, 19]) if there exists
on E
× 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,foru≤1 ≤v. (2.2)
Veli B. Shakhmurov 3
A ζ-convex space E is often called a UMD-space and written as E
∈ UMD. It is shown in
[9] that the Hilbert operator
(Hf)(x)
= lim
ε→0

|x−y|>ε
f (y)
x − y
dy (2.3)
is bounded in L

p
(R;E), p ∈ (1,∞) for those and only those spaces E, which possess the
property of UMD spaces. The UMD spaces include, for example, L
p
, l
p
spaces and the
Lorentz spaces L
pq
, p,q ∈ (1,∞).
Let C be the set of complex numbers and let
S
ϕ
=

λ; λ ∈ C,|argλ − π|≤π − ϕ

∪{
0},0<ϕ≤ π. (2.4)
A linear operator A is said to be a ϕ-positive in a Banach space E, with bound M>0if
D( A)isdenseonE and


(A − λI)
−1


L(E)
≤ M


1+|λ|

−1
(2.5)
with λ
∈ S
ϕ
, ϕ ∈ (0, π], I is identity operator in E,andL(E) is the space of all bounded
linear operators in E.SometimesA + λI will be written as A + λ and denoted by A
λ
.Itis
known [37, Section 1.15.1] that there exist fractional powers A
θ
of the positive operator
A.LetE(A
θ
) denote the space D(A
θ
) with the graphical norm
u
E(A
θ
)
=


u
p
+



A
θ
u


p

1/p
,1≤ p<∞, −∞ <θ<∞. (2.6)
Let E
0
and E be two Banach spaces. By (E
0
,E)
σ,p
,0<σ<1, 1 ≤ p ≤∞we will denote
the interpolation spaces obtained from
{E
0
,E} by the K-method (see, e.g., [37,Section
1.3.1] or [10]).
Let S(R
n
;E) denote a Schwar tz class, that is, the space of all E-valued rapidly decreasing
smooth functions ϕ on R
n
. E = C will be denoted by S(R
n
). Let S


(R
n
;E) denote the space
of E-valued tempered distributions, that is, the space of continuous linear operators from
S(R
n
)toE.
Let α
= (α
1
,α
2
, ,α
n
), α
i
are integers. An E-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|

f ,D
α
ϕ

(2.7)
holds for all ϕ
∈ S(R
n
).
By using (2.7) the following relations
F

D
α
x
f

=

iξ
1

α
1

, ,

iξ
n

α
n

f , D
α
ξ

F( f )

=
F

−
ix
n

α
1
, ,

−
ix
n

α

n
f

(2.8)
are obtained for all f ∈ S

(R
n
;E).
Let L
∗
θ
(E) denote the space of all E-valued function spaces such that
u
L
∗
θ
(E)
=


∞
0


u(t)


θ
E

dt
t

1/θ
< ∞,1≤ θ<∞, u
L
∗
∞
(E)
= sup
0<t<∞


u(t)


E
. (2.9)
4 Embedding and B-regular operators
Let s
= (s
1
,s
2
, ,s
n
)ands
k
> 0. Let F denote the Fourier tr a nsform. Fourier-analytic rep-
resentation of E-valued Besov space on R

n
is defined as
B
s
p,θ

R
n
;E

=

u ∈ S


R
n
;E

, u
B
s
p,θ
(R
n
;E)
=






F
−1
n

k=1
t
κ
k
−s
k

1+


ξ
k


κ
k

e
−t|ξ|
2
Fu






L
∗
θ
(L
p
(R
n
;E))
,
p
∈ (1,∞), θ ∈ [1,∞], κ
k
>s
k

.
(2.10)
It should be noted that the norm of Besov space do not depend on
κ
k
.Sometimeswe
will write
u
B
s
p,θ
in place of u
B

s
p,θ
(R
n
;E)
.
Let l
= (l
1
,l
2
, ,l
n
), s = (s
1
,s
2
, ,s
n
), where l
k
are integers and s
k
are positive numbers.
Let W
l
B
s
p,θ
(R

n
;E) denote an E-valued Sobolev-Besov space of all functions u ∈ B
s
p,θ
(R
n
;E)
such that they have the generalized derivatives D
l
k
k
u=∂
l
k
u/∂x
l
k
k
∈B
s
p,θ
(R
n
;E), k = 1, 2, , n
with the norm
u
W
l
B
s

p,θ
(R
n
;E)
=u
B
s
p,θ
(R
n
;E)
+
n

k=1


D
l
k
k
u


B
s
p,θ
(R
n
;E)

< ∞. (2.11)
Let E
0
is continuously and densely embedded into E. W
l
B
s
p,θ
(R
n
;E
0
,E) denotes a space of
all functions u
∈ B
s
p,θ
(R
n
;E
0
) ∩ W
l
B
s
p,θ
(R
n
;E) with the norm
u

W
l
B
s
p,θ
=u
W
l
B
s
p,θ
(R
n
;E
0
,E)
=u
B
s
p,θ
(R
n
;E
0
)
+
n

k=1




D
l
k
k
u



B
s
p,θ
(R
n
;E)
< ∞. (2.12)
Let l
= (l
1
,l
2
, ,l
n
), s = (s
1
,s
2
, ,s
n

), where s
k
are real numbers and l
k
are positive num-
bers. B
l+s
p,θ
(R
n
;E
0
,E) denotes a space of all functions u ∈ B
s
p,θ
(R
n
;E
0
) ∩ B
l+s
p,θ
(R
n
;E) with the
norm
u
B
s+l
p,θ

(R
n
;E
0
,E)
=u
B
s
p,θ
(R
n
;E
0
)
+ u
B
l+s
p,θ
(R
n
;E)
. (2.13)
For E
0
= E the space B
l+s
p,θ
(R
n
;E

0
,E) will be denoted by B
l+s
p,θ
(R
n
;E).
Let m be a positive integer. C(Ω;E)andC
m
(Ω;E) will denote the spaces of all E-valued
bounded continuous and m-times continuously di fferentiable functions on Ω,respec-
tively. We set
C
b
(Ω;E) =

u ∈ C(Ω;E), lim
|x|→∞
u(x) exists

. (2.14)
Let E
1
and E
2
be two Banach spaces. A function Ψ ∈ C
m
(R
n
;L(E

1
,E
2
)) is called a multi-
plier from B
s
p,θ
(R
n
;E
1
)toB
s
q,θ
(R
n
;E
2
)forp ∈ (1,∞)andq ∈ [1,∞]ifthemapu → Ku=
F
−1
Ψ(ξ)Fu, u ∈ S(R
n
;E
1
), is well defined and extends to a bounded linear operator
K : B
s
p,θ


R
n
;E
1

−→
B
s
q,θ

R
n
;E
2

. (2.15)
Veli B. Shakhmurov 5
The set of all multipliers from B
s
p,θ
(R
n
;E
1
)toB
s
q,θ
(R
n
;E

2
) will be denoted by M
q,θ
p,θ
(s,E
1
,
E
2
). E
1
= E
2
= 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,Section
2.2.2.], and [4, 11, 12, 14, 22], respectively.
Let
H
k
=

Ψ
h
∈ M
q,θ
p,θ


s,E
1
,E
2

, h =

h
1
h
2
, ,h
n

∈
K

(2.16)
be a collection of multipliers in M
q,θ
p,θ
(s,E
1
,E
2
). We say that H
k
is a uniform collection of
multipliers if there exists a constant M
0

> 0, independent on h ∈ K,suchthat


F
−1
Ψ
h
Fu


B
s
p,θ
(R
n
;E
2
)
≤ M
0
u
B
s
q,θ
(R
n
;E
1
)
(2.17)

for all h
∈ K and u ∈ S(R
n
;E
1
).
Let β
= (β
1
,β
2
, ,β
n
) be multiindexes. We also define
V
n
=

ξ =

ξ
1
,ξ
2
, ,ξ
n

∈
R
n

, ξ
i
= 0, i = 1,2, ,n

,
U
n
=

β : |β|≤n

, ξ
β
= ξ
β
1
1
ξ
β
2
2
, ,ξ
β
n
n
, ν =
1
p
−
1

q
.
(2.18)
Definit ion 2.1. ABanachspaceE satisfies a B-multiplier condition with respect to p, q,
θ,ands (or with respect to p, θ,ands for the case of p
= q)whenΨ ∈ C
n
(R
n
;L(E)),
1
≤ p ≤ q ≤∞, β ∈ U
n
,andξ ∈ V
n
if the estimate


ξ
1


β
1
+ν


ξ
2



β
2
+ν
, ,


ξ
n


β
n
+ν


D
β
Ψ(ξ)


L(E)
≤ C (2.19)
implies Ψ
∈ M
q,θ
p,θ
(s,E).
Remark 2.2. Definition 2.1 is a combined restriction to E, p, q, θ,ands. This condition
is sufficient for our main aim. Nevertheless, it is well known that there are Banach spaces

satisfying the B-multiplier condition for isotropic case and p
= q, for example, the UMD
spaces (see [4, 14]).
ABanachspaceE is said to have a local unconditional str ucture (l.u.st.) if there exists a
constant C<
∞ such that for any finite-dimensional subspace E
0
of E there exists a finite-
dimensional space F with an unconditional basis such that the natural embedding E
0
⊂ E
factors as AB with B : E
0
→ F, A : F → E,andAB≤C. All Banach lattices (e.g., L
p
,
L
p,q
, Orlicz spaces, C[0,1]) have l.u.st.
The expression
u
E
1
∼ u
E
2
means that there exist the positive constants C
1
and C
2

such that
C
1
u
E
1
≤u
E
2
≤ C
2
u
E
1
(2.20)
for all u
∈ E
1
∩ E
2
.
6 Embedding and B-regular operators
Let α
1
,α
2
, ,α
n
be nonnegative and let l
1

,l
2
, ,l
n
be positive integers and let
1
≤ p ≤ q ≤∞,1≤ θ ≤∞, |α: .l|=
n

k=1
α
k
l
k
, κ =
n

k=1
α
k
+1/p− 1/q
l
k
,
D
α
= D
α
1
1

D
α
2
2
, ,D
α
n
n
=
∂
|α|
∂x
α
1
1
∂x
α
2
2
, ,∂x
α
n
n
, |α|=
n

k=!
α
k
.

(2.21)
Consider in general, the anisotropic differential-operator equation
(L + λ)u
=

|α:.l|=1
a
α
(x) D
α
u + A
λ
(x) u +

|α:.l|<1
A
α
(x) D
α
u = f (2.22)
in B
s
p,θ
(R
n
;E), where a
α
are complex-valued functions and A(x), A
α
(x) are possibly un-

bounded operators in a B anach space E, here the domain definition D(A)
= D(A(x)) of
operator A(x) does not depend on x.Forl
1
= l
2
=, ,= l
n
we obtain isotropic equations
containing the elliptic class of DOE.
The function belonging to space B
s+l
p,θ
(R
n
;E(A),E) and satisfying (2.22)a.e.onR
n
is
said to be a solution of (2.22)onR
n
.
Definit ion 2.3. The problem (2.22)issaidtobeaB-separable (or B
s
p,θ
(R
n
;E)-separable) if
the problem (2.22)forall 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)
where L is a realization differential operator in B
s
p,θ
(R
n
;E) generated by problem (2.22),
that is,
D( L)
= B

s+l
p,θ

R
n
;E(A),E

, Lu =

|α:.l|=1
a
α
(x) D
α
u + A(x)u +

|α:.l|<1
A
α
(x) D
α
u. (2.25)
We say that the parabolic Cauchy problem (2.24)issaidtobeamaximalB-regular,
if for all f
∈ B
s
p,θ
(R
n+1
+

;E) there exists a unique solution u satisfying (2.24) almost every-
where on R
n+1
+
and there exists a positive constant C independent on f ,suchthatithas
the estimate




∂u(y,x)
∂y




B
s
p,θ
(R
n+1
+
;E)
+ Lu
B
s
p,θ
(R
n+1
+

;E)
≤ C f 
B
s
p,θ
(R
n+1
+
;E)
. (2.26)
3. Embedding theorems
In this section we prove the boundedness of the mixed differential operators D
α
in the
Besov-Lions type spaces.
Veli B. Shakhmurov 7
Lemma 3.1. Let A be a positive operator in a Banach space E,letb be a positive number,
r
= (r
1
,r
2
, ,r
n
), α = (α
1
,α
2
, ,α
n

),andl = (l
1
,l
2
, ,l
n
),whereϕ ∈ (0,π], r
k
∈ [0,b], l
k
are positive and α
k
, k = 1,2, ,n, are nonnegative integers such that κ =|(α + r):l|≤1.
For 0 <h
≤ h
0
< ∞ and 0 ≤ μ ≤ 1 − κ the operator-function
Ψ(ξ)
= Ψ
h,μ
(ξ) =


ξ
1


r
1



ξ
2


r
2
, ,


ξ
n


r
n
(iξ)
α
A
1−κ−μ
h
−μ

A + η(ξ)

−1
(3.1)
is a bounded operator in E uniformly with respect to ξ and h, that is, there is a constant C
μ
such that



Ψ
h,μ
(ξ)


L(E)
≤ C
μ
(3.2)
for all ξ
∈ R
n
,where
η
= η(ξ) =
n

k=1


ξ
k


l
k
+ h
−1

. (3.3)
Proof. Since
−η(ξ) ∈ S(ϕ), for all ϕ ∈ (0,π]andA is a ϕ-positive in E, then the operator
A + η(ξ)isinvertiableinE.Let
u
= h
−μ

A + η(ξ)

−1
f. (3.4)
Then


Ψ(ξ) f


E
=


(hA)
1−κ−μ
u


E
h
−(1−μ)



h
1/l
1
ξ
1


α
1
+r
1
, ,


h
1/l
n
ξ
n


α
n
+r
n
. (3.5)
Using the moment inequality for powers of positive operators, we get a constant C
μ

de-
pending only on μ such that


Ψ(ξ) f


E
≤ C
μ
h
−(1−μ)
hAu
1−κ−μ
u
κ+μ


h
1/l
1
ξ
1


α
1
+r
1
, ,



h
1/l
n
ξ
n


α
n
+r
n
. (3.6)
Now, we apply the Young inequality, which states that ab
≤ a
k
1
/k
1
+ b
k
2
/k
2
for any positive
real numbers a, b and k
1
, k
2

with 1/k
1
+1/k
2
= 1totheproduct
hAu
1−κ−μ


u
κ+μ


h
1/l
1
ξ
1


α
1
+r
1
, ,


h
1/l
n

ξ
n


α
n
+r
n

(3.7)
with k
1
= 1/(1 − κ − μ), k
2
= 1/(κ + μ)toget


Ψ(ξ) f


E
≤C
μ
h
−(1−μ)

(1 − κ − μ)


hAu



+(κ +μ)

h
1/l
1


ξ
1



(α
1
+r
1
)/(κ+μ)
, ,

h
1/l
n


ξ
n




(α
n
+r
n
)/(κ+μ)
u

.
(3.8)
Since
n

i=1
α
i
+ r
i
(κ + μ)
=
1
κ + μ
n

i=1
α
i
+ r
i
l

i
=
κ
κ + μ
≤ 1, (3.9)
8 Embedding and B-regular operators
there exists a constant M
0
independent on ξ,suchthat


ξ
1


(α
1
+r
1
)/(κ+μ)
, ,


ξ
n


(α
n
+r

n
)/(κ+μ)
≤ M
0

1+
n

k=1


ξ
k


l
k

(3.10)
for all ξ
∈ R
n
. Substituting this on the inequality (3.8) and absorbing the constant coeffi-
cients in C
μ
,weobtain


ψ(ξ) f



≤
C
μ

h
μ


Au +
n

k=1


ξ
k


l
k
u

+ h
−(1−μ)
u

. (3.11)
Substituting the value of u we get



ψ(ξ) f



≤
C
μ
h
μ



A

A + η(ξ)

−1
f


+
n

k=1


ξ
k



l
k



A + η(ξ)

−1
f



+ h
−(1−μ)




A + η(ξ)

−1
f



.
(3.12)
By using the properties of the positive operator A for all f
∈ E we obtain from (3.12)



Ψ(ξ) f


E
≤ C
μ
 f 
E
. (3.13)

Lemma 3.2. Let E be a UMD space with l.u.st., p ∈ (1,∞), θ ∈ [1,∞] and let for all k, j ∈
(1,n)
s
k
l
k
+ s
k
+
s
j
l
j
+ s
j
≤ 1. (3.14)
Then the spaces B
l+s

p,θ
(R
n
;E) and W
l
B
s
p,θ
(R
n
;E) are coinc ided.
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
p,θ
(R

n
;E)
≤ Cu
W
l
B
s
p,θ
(R
n
;E)
(3.15)
for all u
∈ 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)itissufficient to prove the following
estimate:






F
−1
n

k=1
t
κ
k
−l
k
−s
k

1+


ξ
k


κ
k

e
−t|ξ|

2
Fu





L
θ
p
≤ C





F
−1
n

k=1
t
κ
k
−s
k

1+



ξ
k


κ
k

e
−t|ξ|
2
Fυ





L
θ
p
,
(3.16)
where
L
θp
= L
∗
θ

L
p


R
n
;E

, υ = F
−1

1+
n

k=1
ξ
l
k
k

Fu. (3.17)
Veli B. Shakhmurov 9
To see this, it is sufficient to show that the function
φ(ξ)
=
n

k=1

1+


ξ

k


l
k
+s
k
+δ


n

k=1

1+


ξ
k


s
k
+δ


−1

1+
n


k=1


ξ
k


l
k

−1
, δ>0 (3.18)
is Fourier multiplier in L
p
(R
n
;E). It is clear to see that for β ∈ U
n
and ξ ∈ V
n


ξ
1


β
1



ξ
2


β
2
, ,


ξ
n


β
n


D
β
φ(ξ)


L(E)
≤ C. (3.19)
Then in view of [41, Proposition 3] we obtain that the function φ is Fourier multiplier in
L
p
(R
n

;E).
In the second step we prove that the embedding B
l+s
p,θ
(R
n
;E) ⊂ W
l
B
s
p,θ
(R
n
;E)iscontin-
uous. In a similar way as in the first step we show that for s
k
/(l
k
+ s
k
)+s
j
/(l
j
+ s
j
) ≤ 1the
function
ψ(ξ)
=


n

k=1

1+


ξ
k


s
k
+δ


1+
n

k=1


ξ
k


l
k


n

k=1

1+


ξ
k


l
k
+s
k
+δ


−1
(3.20)
is Fourier multiplier in L
p
(R
n
;E). So, we obtain for all u ∈ B
l+s
p,θ
(R
n
;E)theestimate






F
−1
n

k=1
t
κ
k
−s
k

1+


ξ
k


κ
k


1+
n


k=1
ξ
l
k
k

e
−t|ξ|
2
Fu





L
θ
p
≤ C





F
−1
n

k=1
t

κ
k
−l
k
−s
k

1+


ξ
k


κ
k

e
−t|ξ|
2
Fu





L
θ
p
.

(3.21)
It implies the second embedding. This completes the prove of Lemma 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,∞],ands = (s
1
,s
2
, ,s
n
),wheres
k
are positive numbers;
(2) α
= (α
1
,α
2
, ,α
n
), l = (l
1
,l
2
, ,l
n
),whereα
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,whereϕ
∈ (0,π] and 0 <h≤ h
0
< ∞.
Then the following embedding
D
α
B
l+s
p,θ


R
n
;E(A),E

⊂
B
s
q,θ

R
n
;E

A
1−κ−μ

(3.22)
is continuous and there exists a positive constant C
μ
depending only on μ, such that


D
α
u


B
s

q,θ
(R
n
;E(A
1−κ−μ
))
≤ C
μ

h
μ
u
B
l+s
p,θ
(R
n
;E(A),E)
+ h
−(1−μ)
u
B
s
p,θ
(R
n
;E)

(3.23)
for all u

∈ B
l+s
p,θ
(R
n
;E(A),E).
10 Embedding and B-regular operators
Proof. We have


D
α
u


B
s
q,θ
(R
n
;E(A
1−κ−μ
))
=


A
1−κ−μ
D
α

u


B
s
q,θ
(R
n
;E)
(3.24)
for all u such that


D
α
u


B
s
q,θ
(R
n
;E(A
1−κ−μ
))
< ∞. (3.25)
On the other hand by using the relation (2.8)wehave
A
1−α−μ

D
α
u = F
−
FA
1−κ−μ
D
α
u = F
−
(iξ)
α
A
1−κ−μ
Fu. (3.26)
Since the operator A is closure and does not depend on ξ
∈ R
n
hence denoting Fu by u,
from the relations (3.24), (3.26) and by definition of the space W
l
B
s
p,θ
(R
n
;E
0
,E)wehave



D
α
u


B
s
q,θ
(R
n
;E(A
1−κ−μ
))



F
−
(iξ)
α
A
1−κ−μ
u


B
s
q,θ
(R

n
;E)
,
u
W
l
B
s
p,θ
(R
n
;E
0
,E)
∼ Au
B
s
p,θ
(R
n
;E)
+
n

k=1


F
−1
ξ

l
k
k
u


B
s
p,θ
(R
n
;E)
.
(3.27)
By virtue of Lemma 3.2 and by the above relations it is sufficienttoprovethat


F
−
(iξ)
α
A
1−κ−μ
u


B
s
q,θ
(R

n
;E)
≤ C
μ

h
μ



F
−
Au


B
s
p,θ
(R
n
;E)
+
n

k=1


F
−


ξ
l
k
k
u



B
s
p,θ
(R
n
;E)

+ h
−(1−μ)


F
−
u


B
s
p,θ
(R
n
;E)


.
(3.28)
The inequality (3.23) will b e followed if we prove the following inequality


F
−

(iξ)
α
A
1−κ−μ
u



B
s
p,θ
(R
n
;E)
≤ C
μ


F
−


h
μ
(A + η)


u


B
s
p,θ
(R
n
;E)
(3.29)
for a suitable C
μ
and for all u ∈ B
s+l
p,θ
(R
n
;E(A),E), where
η
= η(ξ) =
n

k=1



ξ
k


l
k
+ h
−1
. (3.30)
Let us express the left-hand side of (3.29)asfollows:


F
−

(iξ)
α
A
1−κ−μ
u



B
s
q,θ
(R
n
;E)
(3.31)

=


F
−
(iξ)
α
A
1−κ−μ

h
μ
(A + η)

−1

h
μ
(A + η)


u


B
s
q,θ
(R
n
;E)

. (3.32)
(Since A is the positive operator in E and
−η(ξ) ∈ S(ϕ) so it is possible). By virtue of
Definition 2.1 it is clear that the inequality (3.23) will follow immediately from (3.31)if
we can prove that the operator-function Ψ
= (iξ)
α
A
1−κ−μ
[h
μ
(A + η)]
−1
is a multiplier in
Veli B. Shakhmurov 11
M
q,θ
p,θ
(s,E), which is uniform with respect to h.SinceE satisfies the multiplier condition
with respect to p, q, θ,ands,thenbyDefinition 2.1 in order to show that Ψ
∈ M
q,θ
p,θ
(s,E),
it suffices to show that there exists a constant M
μ
> 0with


ξ

1


β
1
+ν


ξ
2


β
2
+ν
, ,


ξ
n


β
n
+ν


D
β
ξ

Ψ(ξ)


L(E)
≤ M
μ
(3.33)
for all β
∈ U
n
, ξ ∈ V
n
,and0<h≤ h
0
< ∞. To see this, we apply Lemma 3.1 and get a
constant M
μ
> 0 depending only on μ such that


ξ
1


ν


ξ
2



ν
, ,


ξ
n


ν


Ψ(ξ)


L(E)
≤ M
μ
(3.34)
for all ξ
∈ R
n
and ν = 1/p − 1/q. This shows that the inequality (3.33)issatisfiedfor
β
= (0, ,0). We next consider (3.33)forβ = (β
1
, ,β
n
)whereβ
k

= 1andβ
j
= 0for
j
= k.Bydifferentiation of the operator-function Ψ(ξ), by virtue of the positivity of A,
and by using (3.34)wehave




∂
∂ξ
k
Ψ(ξ)




L(E)
≤ M
μ


ξ
k


−(1+ν)
, k = 1,2 ,n. (3.35)
Repeating the above process we obtain the estimate (3.33). Thus the operator-function

Ψ
h,μ
(ξ) is a uniform multiplier with respect to h, that is,
Ψ
h,μ
∈ H
K
⊂ M
q,θ
p,θ
(s,E), K =
R
+
. (3.36)
This completes the proof of Theorem 3.3.

Result 3.4. Let all conditions of Theorem 3.3 hold. Then for all u ∈ 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(A
1−κ−μ
))
≤ C
μ
u
1−μ
B
l+s
p,θ
(R
n
;E(A),E)
u
μ
B
s
p,θ
(R
n
;E)
. (3.37)
Indeed setting h
=u
B

s
p,θ
(R
n
;E)
·u
−1
B
l+s
p,θ
(R
n
;E(A),E)
in the estimate (3.23) we obtain the above
estimate.
Remark 3.5. It seems from the proof of Theorem 3.3 that 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
) ≤ 1fork, j =
1,2, ,n are due to Lemma 3.2 (here the l.u.st. condition for the space E is required due to

using of Marcinkiewicz-Lizorkin type multiplier theorem [41]inL
p
(R
n
;E)space).There-
fore, the proof of Theorem 3.3 implies 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 = (s
1
,s
2
, ,s
n
), where s
k
are positive numbers;
(2) α
= t(α
1
,α
2
, ,α
n
), l = (l
1
,l
2
, ,l

n
), where α
k
are nonnegative and l
k
are positive
integers such that
κ =|(α +1/p− 1/q):l|≤1andlet0≤ μ ≤ 1− κ;
(3) A is a ϕ-positive operator in E,whereϕ
∈ (0,π]and0<h≤ h
0
< ∞.
12 Embedding and B-regular operators
Then the following embedding
D
α
W
l
B
s
p,θ

R
n
;E(A),E

⊂
B
s
q,θ


R
n
;E

A
1−κ−μ

(3.38)
is continuous and there exists a positive constant C
μ
depending only on μ such that


D
α
u


B
s
q,θ
(R
n
;E(A
1−κ−μ
))
≤ C
μ


h
μ
u
W
l
B
s
p,θ
(R
n
;E(A),E)
+ h
−(1−μ)
u
B
s
p,θ
(R
n
;E)

(3.39)
for all u
∈ 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
) ≤ 1fork, j = 1,2, ,n in Theorem 3.3
arise due to anisotropic nature of space B
s
p,θ
. For an isotropic case the above conditions
hold without any assumptions.
4. Application to vector-valued function spaces
By virtue of Theorem 3.3 we 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)for0≤ μ ≤ 1 − κ in space B
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
)for0≤ μ ≤ 1 − κ and a corresponding estimate (3.23). For E = R, A = I we get
the embedding D
α
B
l+s
p,θ
(R
n
) ⊂ B
s
q,θ
(R
n
)provedin[8, Section 18] for the numerical Besov
spaces.
Result 4.3. Let l
1
= l
2
= ···= l
n
= m, s
1
= s
2
= ···= s
n
= σ,andp = q.ThenforallE
∈ UMD and |α|≤m we obtain that the continuous embedding D

α
B
σ+m
p,θ
(R
n
;E(A),E) ⊂
B
σ
p,θ
(R
n
;E(A
1−|α|/m
)) and a corresponding estimate (3.23) for the isotropic Besov-Lions
spaces B
σ+m
p,θ
(R
n
;E(A),E).
Result 4.4. Let σ be a positive number. Consider the following space [37, Section 1.18.2]:
l
σ
q
=

u; u =

u

i

, i = 1,2, ,∞, u
i
∈ C

(4.1)
with the norm
u
l
σ
q
=

∞

i=1
2
iqσ


u
i


q

1/q
< ∞. (4.2)
Note that l

0
q
= l
q
.LetA be an infinite matrix defined in l
q
such that
D( A)
= l
σ
q
, A =

δ
ij
2
si

, (4.3)
where δ
ij
= 0, when i = j, δ
ij
= 1, when i = j, i, j = 1,2, ,∞. It is clear to see that
this operator A is positive in l
q
.ThenfromTheorem 3.3 for s
k
/(l
k

+ s
k
)+s
j
/(l
j
+ s
j
) ≤ 1,
k, j
= 1,2, ,n and 0 ≤ μ ≤ 1 − κ, κ =

n
k
=1
(α
k
+1/p
1
− 1/p
2
)/l
k
we obtain the contin-
uous embedding D
α
B
l+s
p
1

,θ
(Ω;l
σ
q
,l
q
) ⊂ B
s
p
2
,θ
(Ω;l
σ(1−κ −μ)
q
) and the corresponding estimate
(3.23).
Veli B. Shakhmurov 13
It should not be that the above embedding has not been obtained with a classical
method until now.
5. Maximal B-regular DOE in R
n
Consider the following differential-operator equation
(L + λ)u
=

|α:.l|=1
a
α
(x) D
α

u + A
λ
(x) u +

|α:.l|<1
A
α
(x) D
α
u = f (5.1)
in B
s
p,q
(R
n
;E), where A(x), A
α
(x) are possible unbounded operators in a Banach space E,
a
k
are complex-valued functions, l = (l
1
,l
2
, ,l
n
)andl
i
are positive integers. The max-
imal regularity for DOE was investigated, for example, in [12, 14, 30]. Let us consider

DOE with constant coefficients

L
0
+ λ

u =

|α:.l|=1
b
α
D
α
u + A
λ
u = f , (5.2)
where A is a possible unbounded operator in E, A
λ
= A+ λ and b
α
are 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, ∞],ands = (s
1
,s
2
, ,s
n

),wheres
k
are positive numbers;
(2) A is a ϕ-positive operator in E with ϕ
∈ (0,π] and
K(ξ)
=−

|α:.l|=1
b
α
(iξ
1
)
α
1
·

iξ
2

α
2
, ,

iξ
n

α
n

∈ S(ϕ),


K(ξ)


≥
C
n

k=1


ξ
k


l
k
, ξ ∈ R
n
;
(5.3)
(3) s
k
/(l
k
+ s
k
)+s

j
/(l
j
+ s
j
) ≤ 1 for k, j = 1,2, ,n.
Then for all f
∈ B
s
p,q
(R
n
;E),for|argλ|≤π − ϕ and sufficiently large |λ| > 0 (5.2)has
auniquesolutionu(x) that belongs to space B
l+s
p,q
(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
p,q
(5.4)
holds with respect to the parameter λ.
Proof. By applying the Fourier transform to (5.2)weobtain

K(ξ)+A
λ

ˆ
u(ξ)
=
ˆ
f (ξ). (5.5)
Since K(ξ)
∈ S(ϕ)forallξ ∈ R
n
,theoperatorA +[λ + K(ξ)] is invertible in E.So,we
obtain that the solution of (5.5)canberepresentedintheform
u(x)

= F
−1

A + λ+ K(ξ)

−1
ˆ
f. (5.6)
14 Embedding and B-regular operators
By using (5.6)wehave
Au
B
s
p,q
=



F
−1
A

A +

λ + K(ξ)

−1
ˆ
f




B
s
p,q
,


D
α
u


B
s
p,q
=



F
−1

iξ
1

α
1
·


iξ
2

α
2
, ,

iξ
n

α
n

A +

λ + K(ξ)

−1
ˆ
f



B
s
p,q
.
(5.7)
Hence, it is suffices to show that the operator-functions
σ

1λ
(ξ) =

A +

λ + K(ξ)

−1
,
σ
2λ
(ξ) =|λ|
1−|α:.l|

iξ
1

α
1
·

iξ
2

α
2
, ,

iξ
n


α
n

A +

λ + K(ξ)

−1
(5.8)
are multipliers in B
s
p,q
(R
n
;E) uniformly with respect to λ. Firstly, by using the positivity
properties of operator A we obtain that the operator function σ
λ
(ξ) is bounded uniformly
with respect to λ. That is,


σ
jλ
(ξ)


B(E)
≤ C, j = 1,2. (5.9)
Then by virtue of the same properties of the operator A we obtain from (5.9)



ξ
β
D
β
ξ
σ
jλ
(ξ)


L(E)
≤ M
j
, β ∈ U
n
, ξ ∈ V
n
, j = 1,2. (5.10)
Then in view of (5.10) we obtain that the operator-valued functions σ
jλ
(ξ)arethe
uniform collection of multipliers from B
s
p,q
(R
n
;E)toB
s

p,q
(R
n
;E). So we get that for all f ∈
B
s
p,q
(R
n
;E) there is a unique solution of (5.2)intheformu(x) = F
−1
[A +(λ + K(ξ))]
−1
ˆ
f
and the estimate (5.4)holds.
Consider the problem (5.1). Let L
0
and L operators in B
s
p,q
(R
n
;E) be generated by
problems (5.2)and(5.1), respectively, that is,
D

L
0


=
D( L) = B
l+s
p,q

R
n
,E(A),E

,
L
0
u =

|α:.l|=1
a
α
(x) D
α
u + Au,
Lu
= L
0
u + L
1
u, L
1
u =

|α:l|<1

A
α
(x) D
α
u.
(5.11)

Theorem 5.2. Suppose condition (1) of Theorem 5.1 holds and let
(1) A(x) be a ϕ positive in E uniformly with respect to x, A(x)A
−1
(x
0
) ∈ C
b
(R;
B(E))
∃x
0
∈ (−∞,∞), a
α
∈ C
b
(R),whereϕ ∈ (0,π];
(2) A
α
(x) A
−(1−|α:l|−μ)
∈ L
∞
(R

n
;L(E)), 0 <μ<1 −|α : l|;
(3) K(x,ξ)
=−

|α:.l|=1
b
α
(iξ
1
)
α
1
· (iξ
2
)
α
2
, ,(iξ
n
)
α
n
∈S(ϕ), |K(x, ξ)|≥C

n
k
=1
|ξ
k

|
l
k
, ξ ∈
R
n
, x ∈ R
n
.
Veli B. Shakhmurov 15
Then for all f
∈ B
s
p,q
(R
n
;E), | arg λ|≤π − ϕ and for sufficiently large |λ| (5.1)hasa
unique solution u(x) that belongs to space B
l+s
p,q
(R
n
;E(A),E), and the coercive uniform esti-
mate

α:.l≤1
|λ|
1−|α:.l|



D
α
u


B
s
p,q
+ Au
B
s
p,q
≤ C f 
B
s
p,q
(5.12)
holds with respect to λ.
Proof. Let ϕ
j
∈ C
∞
0
(R
n
), j = 1,2, ,∞, be a partition of unity such that 0 ≤ ϕ
j
≤ 1and
suppϕ
j

⊂ G
j
,

j
ϕ
j
(x) = 1. Let g
j
∈ C
∞
(R
n
)suchthatg
j
(x) ≡ 1onsuppϕ
j
.Thenforall
u
∈ B
l+s
p,q
(R
n
;E(A),E)wehaveu(x) =

j
u
j
(x), where u

j
(x) = u(x)ϕ
j
(x). From the equal-
ity (5.1)foru
∈ B
l+s
p,q
(R
n
;E(A),E)weobtain
(L + λ)u
j
=

|α:.l|=1
a
α
(x) D
α
u
j
+ A
λ
(y)u
j
(y) = f
j
(y), (5.13)
where

f
j
= fϕ
j
−

|α:.l|<1
b
αj
(x) D
α
u −

|α:.l|<1
A
α
(x) D
α
u
j
(5.14)
and b
αj
(x) are continuous and uniformly bounded functions containing derivatives of
ϕ
j
.ChoosealargeballB
r
0
(0) such that |a

α
(x) − a
α
(∞)|≤δ for all |x|≥r
0
and G
0
=
R
n
\ B
r
0
(0). Cover B
r
0
(0) by finitely many balls G
j
= B
r
j
(x
j
)suchthat|a
α
(x) − a
α
(x
j
)|≤δ

for all
|x − x
j
|≤r
j
, j = 1,2, ,N.Definecoefficients of the local operators L
j
as in [12,
Theorem 5.7], that is,
a
0
α
(x) =
⎧
⎪
⎪
⎨
⎪
⎪
⎩
a
α
(x), x/∈ B
r
0
(0),
a
α

r

2
0
x
|x|
2

, x ∈ B
r
0
(0),
a
j
α
(x) =
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
a
α
(x), x ∈ B
r
j

x

j

,
a
α

x
j
+ r
2
0
x − x
j


x − x
j


2

, x/∈ B
r
j

x
j

(5.15)
for each j

= 1, 2, , N.Then|a
α
(x) − a
α
(x
j
)|≤δ for all x ∈ R
n
and j = 0,1,2, ,N.
Freezing the coefficients in (5.13)weobtainthat

|α:.l|=1
a
α

x
j

D
α
u
j
+ A
λ

x
j

u
j

(x) = F
j
(x), (5.16)
where
F
j
= f
j
+

|α:.l|=1

a
α

x
j

−
a
α
(x)

D
α
u
j
+

A


x
j

−
A(x)

u
j
. (5.17)
16 Embedding and B-regular operators
By virtue of Theorem 5.1 we obtain that the problem (5.16) has a unique solution u
j
,
and for
|argλ|≤π − ϕ and sufficiently large |λ| we get

|α:.l|≤1
|λ|
1−|α:.l|


D
α
u
j


B
s

p,q
(G
j
;E)
+


Au
j


B
s
p,q
(G
j
;E)
≤ C


F
j


B
s
p,q

G
j

;E

. (5.18)
Whence, using properties of the smoothness of coefficients of (5.14), (5.17)andchoos-
ing diameters of G
j
sufficiently small, we get that


F
j


B
s
p,q
(G
j
;E)
≤ ε


u
j


B
s+l
p,q
(G

j
;E(A),E)
+ C(ε)


u
j


B
s
p,q
(G
j
;E)
, (5.19)
where ε is a sufficiently small function and C(δ) is a continuous function. Consequently,
from (5.18)and(5.19)weget

|α:.l|≤1
|λ|
1−|α:.l|


D
α
u
j



B
s
p,q
(G
j
;E)
≤ C f 
B
s
p,q
(G
j
;E)
+ δ


u
j


B
s+l
p,q
+ C(δ)


u
j



B
s
p,q
(G
j
;E)
. (5.20)
Choosing δ<1 from the above inequality we have

|α:.l|≤1
|λ|
1−|α:.l|


D
α
u
j


B
s
p,q
(G
j
;E)
≤ C


f 

G
j
+


u
j


B
s
p,q
(G
j
;E)

. (5.21)
Then by using the equality u(x)
=

j
u
j
(x) and by virtue of the estimate (5.21)foru ∈
B
s+l
p,q
(R
n
;E(A),E)wehave


|α:.l|≤1
|λ|
1−|α:.l|


D
α
u
j


B
s
p,q
(G
j
;E)
≤ C



(L + λ)u


B
s
p,q
+ u
B

s
p,q

. (5.22)
Let u
∈ B
s+l
p,q
(R
n
;E(A),E) be a solution of the problem (5.1). Then for |argλ|≤π − ϕ we
have
u
B
s
p,q
=


(L + λ)u − Lu


B
s
p,q
≤
1
λ




(L + λ)u


B
s
p,q
+ u
B
s+l
p,q

. (5.23)
Then by Theorem 3.3 and by virtue of (5.21)–(5.23), for sufficiently large
|λ| we have

|α:.l|≤1
|λ|
1−|α:.l|


D
α
u
j


B
s
p,q

≤ C


(L + λ)u


B
s
p,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 space B
s
p,q
(R
n
;E). Let us construct for all j the function u
j
,that
is defined on the regions G
j
and satisfying the problem (5.1). The problem (5.1)canbe
expressed in the form

|α:.l|=1
a
α

x

j

D
α
u
j
+ A
λ

x
j

u
j
(x)
=

g
j
f +

A

x
j

−
A(x)

u

j
−

|α:.l|<1
A
α
(x) D
α
u
j

, j = 1,2,
(5.25)
Veli B. Shakhmurov 17
Consider operators O
jλ
in B
s
p,q
(G
j
;E) generated by problems (5.25). By virtue of The-
orem 5.1 for all f
∈ B
s
p,q
(G
j
;E), for | argλ|≤π − ϕ and sufficiently large |λ| we obtain


|α:.l|≤1
|λ|
1−|α:.l|


D
α
O
−1
jλ
f


B
s
p,q
+


AO
−1
jλ
f


B
s
p,q
≤ C f 
B

s
p,q
. (5.26)
Extending u
j
zeroontheoutsideofsuppϕ
j
in equalities (5.25) and passing substitu-
tions u
j
= O
−1
jλ
υ
j
we obtain operator equations with respect to υ
j
:
υ
j
= K
jλ
υ
j
+ g
j
f , j = 1,2, ,N. (5.27)
By virtue of Theorem 3.3 and the estimate (5.26), in view of the smoothness of the
coefficients of the expression K
jλ

for |argλ|≤π − ϕ and sufficiently large |λ| we have
K
jλ
 <ε,whereε is sufficiently small. Consequently, (5.27) has a unique solution υ
j
=
[I − K
jλ
]
−1
g
j
f and we get


υ
j


B
s
p,q
=



I − K
jλ

−1

g
j
f


B
s
p,q
≤f 
B
s
p,q
. (5.28)
Whence, [I − K
jλ
]
−1
g
j
are the bounded linear operators from B
s
p,q
(R
n
;E)toB
s
p,q
(G
j
;E).

Thus, we obtain that the functions u
j
= U
jλ
f = O
−1
jλ
[I − K
jλ
]
−1
g
j
f are the solutions of
(5.25). Consider a linear operator (U + λI)
=

j
ϕ
j
(y)U
jλ
f in B
s
p,q
(R
n
;E). It is clear from
the constructions U
j

and the estimate (5.26) that the operators U
jλ
are bounded linear
from B
s
p,q
(R
n
;E)toB
s+l
p,q
(R
n
;E(A),E)and

|α:.l|≤1
|λ|
1−|α:.l|


D
α
U
−1
jλ
f


B
s

p,q
+


AU
−1
jλ
f


B
s
p,q
≤ C f 
B
s
p,q
, (5.29)
for
|argλ|≤π − ϕ and sufficiently large |λ|. Therefore, (U + λI)isaboundedlinear
operator from B
s
p,q
to B
s
p,q
. Then the act of (L + λ)tou =

j
ϕ

j
U
jλ
f gives (L + λ)u =
f +

j
Φ
jλ
f ,whereΦ
jλ
are linear combinations of U
jλ
and (d/dy)U
jλ
.Byvirtueof
Theorem 3.3, by estimate (5.29), and from the expression Φ
jλ
we obtain that operators
Φ
jλ
are bounded linear from B
s
p,q
(R
n
;E)toB
s
p,q
(G

j
;E)andΦ
jλ
 <δ. Therefore, there
exists a bounded linear invertible operator

I +

j
Φ
jλ

−1
. (5.30)
Whence, we obtain that for all f
∈ B
s
p,q
(R
n
;E)theproblem(5.1) has a unique solution
u
= (U + λI)

I +

j
Φ
jλ


−1
f , (5.31)
that is, we obtain the assertion of Theorem 5.2.

18 Embedding and B-regular operators
Result 5.3. Theorem 5.2 implies that the differential operator L has a resolvent oper ator
(L + λ)
−1
for |argλ|≤π − ϕ,andforsufficiently large |λ| it has the estimate

|α:.l|≤1
|λ|
1−|α:.l|


D
α
(L+λ)
−1


L(B
s
p,q
(R
n
;E))
+



A(L + λ)
−1


L(B
s
p,q
(R
n
;E))
≤ C. (5.32)
Remark 3.5 and Theorem 5.2 imply the following.
Result 5.4. Suppose the following conditions hold:
(1) E is a Banach space satisfying B-multiplier condition with respect to p
∈ (1,∞)
and q
∈ [1,∞];
(2) A is a ϕ-positive operator in E with ϕ
∈ (0,π]and
K(ξ)
=−

|α:.l|=1
b
α

iξ
1

α

1
·

iξ
2

α
2
, ,

iξ
n

α
n
∈ S(ϕ),


K(x,ξ)


≥
C
n

k=1


ξ
k



l
k
, ξ ∈ R
n
, x ∈ R
n
;
(5.33)
(3) A(x)isaϕ positive in E uniformly with respect to x, A(x)A
−1
(x
0
) ∈ C
b
(R;B(E)),
x
0
∈ (−∞,∞), a
α
∈ C
b
(R), where ϕ ∈ (0,π];
(4) A
α
(x) A
−(1−|α:l|−μ)
∈ L
∞

(R
n
;L(E)), 0 <μ<1 −|α : l|.
Then for all f
∈ B
s
p,q
(R
n
;E), |argλ|≤π − ϕ and for sufficiently large |λ| (5.1)hasa
unique solution u(x) that belongs to space W
l
B
s
p,θ
(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
p,q
(5.34)
holds with respect to λ.
Theorem 5.5. Let all conditions of Theorem 5.2 hold for ϕ
∈ (0,π/2). Then the parabolic
Cauchy problem (2.24)for
|argλ|≤π − ϕ and sufficiently large |λ| is maximal B-regular.
Proof. The problem (2.24) can be expressed in B
s
p,θ
(R
+
;F) in the following form:
du(y)
dy
+(L + λ)u(y)
= f (t), u(0) = 0, y>0, (5.35)
where F
= L
p

(G;E)andL is the differential operator in B
s
p,θ
(R
n
;E) generated by the prob-
lem (5.1). In view of Result 4.3 the operator L is positive in B
s
p,θ
(R
n
;E)forϕ ∈ (0,π/2).
Then by virtue of [4, Corollary 8.9] we obtain the assertion.

Remark 5.6. There are lots of positive operators in concrete Banach spaces. Therefore,
putting concrete Banach spaces instead of E and concrete positive differential, pseudo
differential operators, or finite, infinite matrices, and so forth, instead of operator A on
DOE (5.1), by virtue of Theorem 5.2 we can obtain the maximal regularity of different
class of BVP’s for partial differential equations or system of equations. Here we give some
of its applications.
Veli B. Shakhmurov 19
6. Applications
6.1. Infinite systems of quasielliptic equations. Consider the following infinity systems
of boundary value problem:
(L + λ)u
m
(x) =

|α:.l|=1
a

α
(x) D
α
u
m
(x)+

d
m
(x)+λ

u
m
(x)
+

|α:.l|<1
∞

k=1
d
αkm
(x) D
α
u
k
(x) = f
m
(x), x ∈ R
n

, m = 1,2, ,∞.
(6.1)
Let
D( x)
=

d
m
(x)

, d
m
> 0, u =

u
m

, Du =

d
m
u
m

, m = 1,2, ,∞,
l
q
(D) =
⎧
⎨

⎩
u : u ∈ l
q
, u
l
q
(D)
=Du
l
q
=

∞

m=1


d
m
u
m


q

1/q
< ∞
⎫
⎬
⎭

,
x
∈ G,1<q<∞, l =

l
1
,l
2
, ,l
n

, s =

s
1
,s
2
, ,s
n

, s
k
> 0, l
k
∈ N.
(6.2)
Let O denote a differ ential operator in B
s
p,θ
(R

n
;l
q
) generated by problem (6.1). Let
B
= L

B
s
p,θ

R
n
;l
q

. (6.3)
Theorem 6.1. Let a
α
∈ C
b
(R
n
), d
m
∈ C
b
(R
n
), d

αkm
∈ L
∞
(R
n
),ands
k
, l
k
such that
s
k
l
k
+ s
k
+
s
j
l
j
+ s
j
≤ 1, j = 1,2, ,n,
∞

k,m=1
d
q
1

αkm
d
−q
1
(1−|α:l|−μ)
m
< ∞,
1
q
+
1
q
1
= 1,
(6.4)
where p,q
∈ (1,∞), θ ∈ [1,∞].
Then
(a) for a ll f (x)
={f
m
(x)}
∞
1
∈ B
s
p,θ
(R
n
;l

q
), |argλ|≤π − ϕ and for sufficiently large |λ|
the problem (6.1)hasauniquesolutionu ={u
m
(x)}
∞
1
that belongs to space B
s+l
p,θ
(R
n
,l
q
(D),
l
q
), and the coercive estimate

|α:l|≤1


D
α
u


B
s
p,θ

(R
n
;l
q
)
+ du
B
s
p,θ
(R
n
;l
q
)
≤ C f 
B
s
p,θ
(R
n
;l
q
)
(6.5)
holds for the solution of the problem (6.1);
(b) for
|argλ|≤π − ϕ and for sufficiently large |λ| there exists a resolvent (O + λ)
−1
of
operator O and


|α:l|≤1

1+|λ|

1−|α:l|


D
α
(O + λ)
−1


B
+


d(O + λ)
−1


B
≤ M. (6.6)
20 Embedding and B-regular operators
Proof. Really , let E
= l
q
, A(x), and A
α

(x) be infinite matrices, such that
A
=

d
m
(x) δ
km

, A
α
(x) =

d
αkm
(x)

, k,m = 1,2, ,∞. (6.7)
It is clear to see that operator A is positive in l
q
. Therefore, by virtue of Theorem 5.2 we
obtain that the problem (6.1)forall f
∈ B
s
p,θ
(R
n
;l
q
), | argλ|≤π − ϕ,andsufficiently large

|λ| has a unique solution u that belongs to space B
s+l
p,θ
(R
n
;l
q
(D),l
q
) and the estimate (6.5)
holds. By virtue of estimate (6.5)weobtain(6.6).

6.2. Cauchy problems for infinite systems of parabolic equations. Consider the follow-
ing infinity systems of parabolic Cauchy problem:
∂u
m
(y,x)
∂y
+

|α:.l|=1
a
α
(x) D
α
u
m
(y,x)+

d

m
(x)+λ

u
m
(y,x)+

|α:.l|<1
∞

k=1
d
αkm
(x) D
α
u
k
(y,x)
= f
m
(y,x), u
m
(0,x) = 0, m = 1,2, ,∞, y ∈ R
+
, x ∈ R
n
.
(6.8)
Theorem 6.2. Let all conditions of Theorem 6.1 hold. Then the parabolic systems (6.8)for
|argλ|≤π − ϕ and for sufficiently large |λ| are maximal B-regular.

Proof. Really , let E
= l
q
, A,andA
k
(x) be the infinite matrices, such that
A
=

d
m
(x) δ
km

, A
α
(x) =

d
αkm
(x)

, k,m = 1,2, ,∞. (6.9)
Then the problem (6.8) can be expressed as the problem (2.24), where
A
=

d
m
(x) δ

km

, A
α
(x) =

d
αkm
(x)

, k,m = 1,2, ,∞. (6.10)
Then by virtue of Theorems 5.2 and 5.5 we obtain the assertion.

Acknowledgments
This work is supported by Project UDP-748 11.05.2006 of Istanbul Un iversity. The author
would like to express a deep gratitude to O. V. Besov for his useful advi ces.
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Ve li B. Shakhmurov: Department of Electrical & Electronics Engineering, Engineering Faculty,
Istanbul University, Avcilar, 34320 Istanbul, Turkey
E-mail address: